IUTAM Symposium on Modelling Nanomaterials and Nanosystems: Proceedings of the IUTAM Symposium held in Aalborg, Denmark, 19-22 May, 2008 (IUTAM Bookseries)

IUTAM Symposium on Modelling Nanomaterials and Nanosystems

IUTAM BOOKSERIES Volume 13 Series Editors G.M.L. Gladwell, University of Waterloo, Waterloo, Ontario, Canada R. Moreau, INPG, Grenoble, France Editorial Board J. Engelbrecht, Institute of Cybernetics, Tallinn, Estonia L.B. Freund, Brown University, Providence, USA A. Kluwick, Technische Universität, Vienna, Austria H.K. Moffatt, University of Cambridge, Cambridge, UK N. Olhoff Aalborg University, Aalborg, Denmark K. Tsutomu, IIDS, Tokyo, Japan D. van Campen, Technical University Eindhoven, Eindhoven, The Netherlands Z. Zheng, Chinese Academy of Sciences, Beijing, China

Aims and Scope of the Series The IUTAM Bookseries publishes the proceedings of IUTAM symposia under the auspices of the IUTAM Board.

For other titles published in this series, go to www.springer.com/series/7695

R. Pyrz • J.C. Rauhe Editors

IUTAM Symposium on Modelling Nanomaterials and Nanosystems Proceedings of the IUTAM Symposium held in Aalborg, Denmark, 19–22 May 2008

R. Pyrz Department of Mechanical Engineering Aalborg University Aalborg, Denmark

ISBN: 978-1-4020-9556-6

J.C. Rauhe Department of Mechanical Engineering Aalborg University Aalborg, Denmark

e-ISBN: 978-1-4020-9557-3

Library of Congress Control Number: 2008941672 © Springer Science+Business Media, B.V. 2009 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 987654321 springer.com

Table of Contents

Preface

ix

Multiscale Failure Modeling: From Atomic Bonds to Hyperelasticity with Softening K.Y. Volokh

1

Crack Initiation, Kinking and Nanoscale Damage in Silica Glass: Multimillion-Atom Molecular Dynamics Simulation Y.C. Chen, K. Nomura, Z. Lu, R. Kalia, A. Nakano and P. Vashishta

13

Multiscale Modelling of Layered-Silicate/PET Nanocomposites during Solid-State Processing Ł. Figiel, F.P.E. Dunne and C.P. Buckley

19

Modelling Transient Heat Conduction at Multiple Length and Time Scales: A Coupled Non-Equilibrium Molecular Dynamics/Continuum Approach K. Jolley and S.P.A. Gill

27

Multiscale Modeling of Amorphous Materials with Adaptivity V.B.C. Tan, M. Deng, T.E. Tay and K.M. Lim

37

Thermodynamically-Consistent Multiscale Constitutive Modeling of Glassy Polymer Materials P.K. Valavala and G.M. Odegard

43

Effective Wall Thickness of Single-Walled Carbon Nanotubes for Multiscale Analysis: The Problem and a Possible Solution L.C. Zhang and C.Y. Wang

53

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Table of Contents

Discrete-Continuum Transition in Modelling Nanomaterials R. Pyrz and B. Bochenek Looking beyond Limitations of Diffraction Methods of Structural Analysis of Nanocrystalline Materials B. Palosz, E. Grzanka, S. Gierlotka, M. Wojdyr, W. Palosz, T. Proffen, R. Rich and S. Stelmakh Multiscale Modelling of Mechanical Anisotropy of Metals G. Winther Micromechanical Modeling of the Elastic Behavior of Multilayer Thin Films; Comparison with In Situ Data from X-Ray Diffraction G. Geandier, L. Gélébart, O. Castelnau, E. Le Bourhis, P.-O. Renault, Ph. Goudeau and D. Thiaudière

63

75

89

99

Two Minimisation Approximations for Joining Carbon Nanostructures D. Baowan, B.J. Cox, N. Thamwattana and J.M. Hill

109

On the Eigenfrequencies of an Ordered System of Nanoobjects V.A. Eremeyev and H. Altenbach

123

Monitoring of Molecule Adsorption and Stress Evolutions by In-Situ Microcantilever Systems H.L. Duan, Y. Wang and X. Yi

133

Using Thermal Gradients for Actuation in the Nanoscale E.R. Hernández, R. Rurali, A. Barreiro, A. Bachtold, T. Takahashi, T. Yamamoto and K. Watanabe

141

Systematic Design of Metamaterials by Topology Optimization O. Sigmund

151

Modeling of Indentation Damage in Single and Multilayer Coatings J. Chen and S.J. Bull

161

Reverse Hall–Petch Effect in Ultra Nanocrystalline Diamond I.N. Remediakis, G. Kopidakis and P.C. Kelires

171

Elastic Fields in Quantum Dot Structures with Arbitrary Shapes and Interface Effects H.J. Chu, H.L. Duan, J. Wang and B.L. Karihaloo

181

Table of Contents

vii

Numerical Modelling of Nano Inclusions in Small and Large Deformations Using a Level-Set/Extended Finite Element Method J. Yvonnet, E. Monteiro, H. Le Quang and Q.-C. He

191

Thermo-Elastic Size-Dependent Properties of Nano-Composites with Imperfect Interfaces H.L. Duan, B.L. Karihaloo and J. Wang

201

Modeling the Stress Transfer between Carbon Nanotubes and a Polymer Matrix during Cyclic Deformation C.C. Kao and R.J. Young

211

Atomistic Studies of the Elastic Properties of Metallic BCC Nanowires and Films P.A.T. Olsson and S. Melin

221

Advances Continuum-Atomistic Model of Materials Based on Coupled Boundary Element and Molecular Approaches T. Burczy´nski, W. Ku´s, A. Mrozek, R. Górski and G. Dziatkiewicz

231

Finite Element Modelling Clay Nanocomposites and Interface Effects on Mechanical Properties J.Y.H. Chia

241

Small Scale and/or High Resolution Elasticity I. Goldhirsch and C. Goldenberg

249

Multiscale Molecular Modelling of Dispersion of Nanoparticles in Polymer Systems of Industrial Interest M. Fermeglia and S. Pricl

261

Structural-Scaling Transitions in Mesodefect Ensembles and Properties of Bulk Nanostructural Materials. Modeling and Experimental Study O.B. Naimark and O.A. Plekhov

271

Modeling Electrospinning of Nanofibers T.A. Kowalewski, S. Barral and T. Kowalczyk

279

Use of Reptation Dynamics in Modelling Molecular Interphase in Polymer Nano-Composite J. Jancar

293

Appendix 1: Presentations without Paper

303

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Table of Contents

Appendix 2: Scientific Program

305

Appendix 3: List of Participants

309

Colour Section

317

Author Index

339

Preface

The IUTAM Symposium on Modelling Nanomaterials and Nanosystems was held in Aalborg, Denmark, from 19th to 22nd May 2008. Participation in the Symposium was reserved for invited participants, suggested by the members of the Scientific Committee. The Symposium brought together 53 researchers from 19 countries representing a broad range of backgrounds relevant to the topic of the meeting. Modelling and simulation on various length and time scales have become a major field of Materials Science and Engineering in academia as well as in industrial research and development. Multiscale materials modelling encompasses all the tools which physicists, chemists, mechanical engineers and materials scientists have been developing to describe materials and their behaviour. The goal of the Symposium was namely to bring together scientists from different disciplines who are striving for an improvement and enhancement of virtual design and development of nanomaterials, and of virtual development and testing of components. A special emphasis has been placed on transitional frameworks that are essential to link and to complement continuum and atomistic methods. Different upscaling methodologies using results from a lower-scale calculation to obtain parameters for a higher-scale have been presented. Extensions of continuum approaches that incorporate atomistic/molecular phenomena not observable in classical continuum settings were discussed as well. Several contributions presented appropriate validation experiments which are crucial to verify that the models predict the correct behaviour at each length scale, ensuring that the linkages are directly enforced. All lectures were followed by lively and interesting discussions. Many participants expressed the opinion that the Symposium was very informative and challenging since problem formulations, methods of solution and applications contained a high degree of multidisciplinarity comprising physics, chemistry, materials science and solid mechanics aspects of nanomaterials and nanosystems. The volume contains 30 papers published in an order as it appears in the scientific programme of the Symposium. Some of the lectures are not represented, mainly due to a prior commitments to publish elsewhere. The list of these lectures, the scientific programme and the list of participants appears as an Appendix.

ix

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Preface

The International Scientific Committee comprised following: Professor A. Argon (USA) Professor L.B. Freund (USA – IUTAM representative) Professor M. Griebel (Germany) Professor E. Hernández (Spain) Professor R. Pyrz (Denmark), Chairman Professor J. Wang (PRChina) Professor B. Yakobson (USA) The financial support from the International Union of Theoretical and Applied Mechanics, the Obels Family Foundation, Spar Nord Foundation and NEMT Laboratory, Aalborg University is gratefully acknowledged. The smooth running of the Symposium owes much to the efforts of Dr. Jens C. Rauhe, Dr. Lars R. Jensen, Dr. Jan Schjødt-Thomsen and the Symposium Secretary Lisbeth H. Kolmorgen. To all of them many thanks for their help which cannot be overestimated. R. Pyrz Aalborg, September 2008

Multiscale Failure Modeling: From Atomic Bonds to Hyperelasticity with Softening K.Y. Volokh

Abstract Separation of two particles is characterized by a magnitude of the bond energy that limits the accumulated energy of the particle interaction. In the case of a solid comprised of many particles there exist a magnitude of the average bond energy that limits the energy that can be accumulated in a small material volume. The average bond energy can be calculated if the statistical distribution of the bond density is known for a particular material. Alternatively, the average bond energy can be determined in macroscopic experiments if the energy limiter is introduced in a material constitutive model. Traditional continuum models of materials do not have energy limiters and, consequently, allow for the unlimited accumulation of the strain energy. The latter is unphysical, of course, because no material can sustain large enough strains without failure. The average bond energy limits the strain energy and controls material softening, which indicates failure. Thus, by limiting the strain energy we include a description of material failure in the constitutive model. Generally, elasticity including energy limiters can be called softening hyperelasticity because it can describe material failure via softening. We illustrate the capability of softening hyperelasticity in examples of brittle fracture and arterial failure. First, we analyze the overall strength of arteries under the blood pressure. For this purpose we enhance various arterial models with the energy limiters. The models vary from the phenomenological Fung-type theory to the microstructural theories regarding the arterial wall as a bi-layer fiber-reinforced composite. Based on the simulation results we find, firstly, that residual stresses accumulated during artery growth can significantly delay the onset of arterial rupture like the pre-existing compression in the pre-stressed concrete delays the crack opening. Secondly, we find that the media layer is the main load-bearing layer of the artery. And, thirdly, we find that the strength of the collagen fibers dominates the media strength. Second, we numerically simulate tension of a thin plate with a preexisting central crack within a softening hyperelasticity framework and we find that the critical load essentially depends on the crack sharpness: the sharper is the crack the lower is the K.Y. Volokh Technion – Israel Institute of Technology, Haifa 32000, Israel; e-mail: [email protected]

R. Pyrz and J.C. Rauhe (eds.), IUTAM Symposium on Modelling Nanomaterials and Nanosystems, 1–12. © Springer Science+Business Media B.V. 2009

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critical load. The latter also means that the fracture toughness of brittle materials cannot be calibrated in experiments uniquely. Such a conclusion qualitatively corresponds to the results of the experimental tests on the calibration of the fracture toughness of ceramics, for example. The practical implication of our results is a recommendation to calibrate toughness in experiments where the size of the notch tip is comparable with a characteristic length of the material microstructure, e.g. grain size, atomic distance etc. In other words, toughness can be calibrated only under conditions where the hypothesis of continuum fails.

1 Introduction The problem of the modeling of the failure nucleation and propagation in solids is still largely open despite the enormous progress in computational materials science and engineering during past decades. The existing methods are too restrictive and computationally involved to be finally accepted as an optimal approach to modeling failure. For example, molecular dynamic simulations are restricted by length and time scales to such a degree that no real macroscopic materials and structures can be practically analyzed. On the other hand, continuum models that can handle macroscopic length and time scales are phenomenological and their experimental calibration is far from trivial, moreover, they are sophisticated mathematically and computationally. The existing continuum models of material failure can be divided in two groups: surface and bulk models. The surface models, pioneered by Barenblatt [1], appear by name of Cohesive Zone Models (CZM) in the modern literature. They present material surfaces – cohesive zones – where displacement discontinuities occur. The discontinuities are enhanced with constitutive laws relating normal and tangential displacement jumps with the corresponding tractions. There are a number of proposals of constitutive equations for the cohesive zones: for example, Dugdale [6], Rice and Wang [21], Tvergaard and Hutchinson [24], Xu and Needleman [31], and Camacho and Ortiz [3]. All CZM are constructed qualitatively as follows: tractions increase, reach a maximum, and then approach zero with increasing separation. Such a scenario is in harmony with our intuitive understanding of the rupture process. Since the work by Needleman [19] CZM are used increasingly in finite element simulations of crack tip plasticity and creep; crazing in polymers; adhesively bonded joints; interface cracks in bimaterials; delamination in composites and multilayers; fast crack propagation in polymers and etc. Cohesive zones can be inside finite elements or along their boundaries [2,5,31]. Crack nucleation, propagation, branching, kinking, and arrest are natural outcomes of the computations where the discontinuity surfaces are spread over the bulk material. This is in contrast to the traditional approach of fracture mechanics where stress analysis is separated from a description of the actual process of material failure. The CZM approach is natural for simulation of fracture at the internal material interfaces in polycrystals, composites, and multilayers. It is less natural for modeling fracture of the bulk because it leads to the simultaneous use of two material models for the same real material: one

Multiscale Failure Modeling

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model describes the bulk while the other model describes CZM imbedded in the bulk. Such two-model approach is rather artificial physically. It seems preferable to incorporate a material failure law directly in the constitutive description of the bulk. Remarkably, the first models of bulk failure – damage mechanics – proposed by Kachanov [14] and Rabotnov [20] for analysis of the gradual failure accumulation and propagation in creep and fatigue appeared almost simultaneously with the cohesive zone approach. The need to describe the failure accumulation, i.e. evolution of the material microstructure, explains why damage mechanics is very similar to plasticity theories including (a) the internal damage variable (inelastic strain), (b) the critical threshold condition (yield surface), and (c) the damage evolution equation (flow rule). The subsequent development of the formalism of damage mechanics [15, 17, 18, 22] left its physical origin well behind the mathematical and computational techniques and, eventually, led to the use of damage mechanics for the description of any bulk failure. Theoretically, the approach of damage mechanics is very flexible and allows reflecting physical processes triggering macroscopic damage at small length scales. Practically, the experimental calibration of damage theories is not easy because it is difficult to measure the damage parameter directly. The experimental calibration should be implicit and include both the damage evolution equation and criticality condition. A physically-motivated multiscale alternative to damage mechanics in the cases of failure related with the bond rupture has been considered by Gao and Klein [10]; Klein and Gao [16] who showed how to mix the atomic/molecular and continuum descriptions in order to simulate material failure. They applied the Cauchy–Born rule linking micro- and macro-scales to empirical potentials, which include a possibility of the full atomic separation. The continuum-atomistic link led to the formulation of the macroscopic strain energy potentials allowing for the stress/strain softening and strain localization. The continuum-atomistic method is very effective at small length scales where purely atomistic analysis becomes computationally intensive. Unfortunately, a direct use of the continuum-atomistic method in macroscopic failure problems is not very feasible because its computer implementation includes a numerically involved procedure of the averaging of the interatomic potentials over a representative volume. In order to bypass the computational intensity of the continuum-atomistic method while preserving its sound physical basis the softening hyperelasticity approach was proposed by Volokh [25, 26]. The basic idea of the approach was to formulate an expression of the stored macroscopic energy, which would include the energy limiter – the average bond energy. Such a limiter automatically induces strain softening, that is a material failure description, in the constitutive law. The softening hyperelasticity approach is computationally simple yet physically appealing and we describe and use it below.

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Fig. 1 Lennard–Jones potential (left) and Cauchy–Born rule (right).

2 Softening Hyperelasticity Let us start with the interaction of two particles (atoms, molecules, etc.) and let us choose, to be specific, the Lennard–Jones potential, ϕ, for the description of the particle interaction ϕ(l) = 4ε((σ/ l)12 − (σ/ l)6 ) (1) where l is the distance between particles ε and σ are the bond energy and length constants accordingly (see Figure 1). Let L designate the distance between particles in a reference state and F is the one-dimensional deformation gradient. In the latter case we have l = FL

(2)

ϕ(F ) = 4ε((σ/F L)12 − (σ/F L)6 )

(3)

Substituting (2) in (1) we have

Assuming that deformation increases to infinity we have ϕ(F → ∞) = 0

(4)

On the other hand, we have at the reference state ϕ0 = ϕ(F = 1) = 4ε((σ/L)12 − (σ/L)6 )

(5)

In the absence of external loads the energy of the interaction tends to minimum and it√is natural to choose the minimum energy state – equilibrium – at distance L = 6 2 σ where no forces are acting between the particles. In the latter case we have ϕ0 = −ε (6) We notice that energy is negative in the equilibrium state according to the classical Lennard–Jones potential. The latter is inconvenient in solid mechanics and we

Multiscale Failure Modeling

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modify the classical LJ potential by shifting its reference energy to zero (Figure 1) ψ =ϕ+ε

(7)

We further formalize the described energy shift as follows ψ(F ) = ϕ(F ) − ϕ0

(8)

ϕ0 = min ϕ(F = 1)

(9)

L

Equations (8) and (9) are essential in the subsequent consideration of assemblies of many particles. It is important to emphasize that we cannot increase energy unlimitedly by increasing deformation. The energy increase is limited ψ(F → ∞) = −ϕ0 =  = constant

(10)

Now we extend all considerations for a pair of particles given above to large assemblies of particles comprising solid bodies. Consider particles placed at ri in the 3D space. Generally, the volumetric density of the total potential energy, i.e. the strain energy, can be written with account of two-particle interactions as follows: 1
ϕ(rij ) 2V

(11)

i,j

where rij = |rij | = |ri − rj | and V is the volume occupied by the system. According to the Cauchy–Born rule [23, 30], originally applied to the crystal elasticity, the current rij and initial (reference) Rij = Ri − Rj relative positions of the same two particles can be related by the deformation gradient: rij = FRij

(12)

where F = ∂x/∂X is the deformation gradient of a generic material macro-particle of body  occupying position X at the reference state and position x(X) at the current state of deformation (see Figure 1). Substituting (12) in (11) yields 1
1
ϕ(rij ) = ϕ(rij (C)) 2V 2V i,j

(13)

i,j

where C = FT F is the right Cauchy–Green deformation tensor. Direct application of (13) to analysis of material behavior can be difficult because of the large amount of particles. Gao and Klein [10] and Klein and Gao [16] considered the following statistical averaging procedure:  1 ϕ(l) = ϕ(l)DV dV (14) V0 V0∗

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 l = rij = L ξ · Cξ = L|Fξ |

(15)

where L = Rij = |Ri − Rj | is the reference bond length; ξ = (Ri − Rj )/L is the reference bond direction; V0 is the reference representative volume; ϕ(l) is the bond potential (Lennard–Jones); DV is the volumetric bond density function; and V0∗ is the integration volume defined by the range of influence of ϕ. Now the average strain energy takes form  1 ϕ(C) = 4ε((σ/LC)12 − (σ/LC)6 )DV dV (16) V0 V0∗ where C =



ξ · Cξ

(17)

Analogously to the case of the pair interaction considered in the previous subsection – Equations (8) and (9) – we define the shifted strain energy, which is zero at the equilibrium reference state, ψ(C) = ϕ(C) − ϕ0

(18)

ϕ0 = minϕ(C = 1)

(19)

L

Analogously to (10), we can define the average bond energy by setting the unlimited increase of deformation  = ψ(C → ∞) = −ϕ0 = constant

(20)

Thus, the average bond energy sets a limit for the energy accumulation. This conclusion generally does not depend on the choice of the particle potential and it is valid for any interaction that includes a possible particle separation – the bond energy. Contrary to the conclusion above traditional hyperelastic models of materials do not include the energy limiter. The stored energy of hyperelastic materials is defined as ψ =W (21) Here W is used for the strain energy of the intact material, which can be characterized as follows: C → ∞ ⇒ ψ = W → ∞ (22) where . . . is a tensorial norm. In other words, the increasing strain increases the accumulated energy unlimitedly. Evidently, the consideration of only intact materials is restrictive and unphysical. The energy increase of a real material should be limited as it was shown above, C → ∞ ⇒ ψ →  = constant

(23)

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where the average bond energy,  = constant, can be called the material failure energy. Equation (23) presents the fundamental idea of introducing a limiter of the stored energy in the elasticity theory. Such a limiter induces material softening, indicating material failure, automatically. The choice of the limited stored energy expression should generally be material-specific. Nonetheless, a somewhat universal formula [26] can be introduced to enrich the already existing models of intact materials with the failure description ψ(W ) =  −  exp(−W/)

(24)

where ψ(W = 0) = 0 and ψ(W = ∞) = . Formula (24) obeys condition C → ∞ ⇒ ψ(W (C)) →  and, in the case of the intact material behavior, W , we have ψ(W ) ≈ W preserving the features of the intact material.

3 Arterial Failure In this section we apply the softening hyperelasticity method to analyze arterial failure. We consider artery as an infinite axisymmetric pipe under internal (blood) pressure [27]. We start by using the classical Fung’s function of the stored energy density [4, 8, 9]: W =

c 2 2 2 exp(c1 ERR + c2 E + c3 EZZ + 2c4 ERR E 2  + 2c5 EZZ E + 2c6 ERR EZZ ) − 1

(25)

with c the only dimensional elastic parameter and ci dimensionless. EAA are the components of the Green–Lagrange strain tensor, E = (FT F − 1)/2, in cylindrical coordinates. Pressure-radius curves for the following set of material parameters for anisotropic artery [4]: c1 = 0.0089, c2 = 0.9925, c3 = 0.4180, c4 = 0.0193, c5 = 0.0749, c6 = 0.0295; are shown in Figure 2 where also the distribution of the Cauchy stresses is shown across the artery wall. Arteries accumulate residual stress during growth. To account for such prestresses it is usually assumed that when an arterial ring is cut radially it opens up in the form of a circular sector with the opening angle ω. Introducing this angle in analysis [8] it is possible to estimate the influence of the residual stresses. Particularly, the results for the unprestressed state with ω = 0◦ and the internal and external reference radii A = 0.71 mm and B = 1.10 mm accordingly are shown on the top of Figure 2. We demonstrate stresses for dimensionless pressure g¯ = 0.5, which corresponds to pressure g = 13.47 KPa for the shear modulus

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Fig. 2 Pressure-radius (left) and true stresses (right) curves for artery without (top) and with (bottom) pre-stress. Notation: g¯ = g/c; σ¯ ij = σij /c; a¯ = a/A; r¯ = r/A.

c = 26.95 KPa. The results for the prestressed state with ω = 160◦ and the internal and external reference radii A = 1.43 mm and B = 1.82 mm accordingly are shown on the bottom of Figure 2. We notice that the pressure increase always corresponds to the radius increase, i.e. the artery deformation is always stable and no failure is observed. The latter is unphysical and we enhance the classical formulation of the stored energy density with the average bond energy, , as in (24). The pressure-radius curves for the same ¯ = /c = 1. set of material parameters are shown in Figure 3 for  We observe that failure appears on the pressure-radius curve as a limit point where static instability occurs. Though the decreasing branch of the curve is shown for the sake of consistency, it should be clearly realized that it is not statically stable and the dynamic failure propagation should be monitored after the limit point. Remarkably, the residual stresses delay the onset of failure (and the delay increases with the increasing average bond energy,  [27]). It is also interesting that the prestress makes the distribution of the hoop stresses more uniform optimizing the stress distribution in a loaded artery. It is worth emphasizing that the considered model is purely phenomenological though it includes the anisotropy description in the theoretical setting. A more microscopically-informed model can be developed which includes the layered structure of the arterial wall including media and adventitia and a separate description of the collagen fibers and non-collagen matrix [28]. Being more sophisticated the micro-structural model, however, shows results similar to the discussed above concerning the influence of pre-stressing on the overall arterial failure. Besides,

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Fig. 3 Pressure-radius (left) and true stresses (right) curves for artery without (top) and with (bot¯ = /c = 1. tom) pre-stress for 

the micro-structural model allows bringing new insights into the role of adventitia/media and matrix/collagen components in the overall arterial strength. Our numerical simulations led to the following three findings. Firstly, it was found that the fiber strength dominates the overall arterial strength. Such a conclusion has immediate experimental implication: it is necessary to calibrate the mechanical models of individual fibers in order to predict the global arterial strength. Of course, the role of the fiber binding energy may also be important. The latter is the reason why the experiments with the fiber bundles are of great interest too. Secondly, it was also found that the media dominates the overall arterial strength and plays the crucial role in the load-bearing capacity of arteries. Such a conclusion is in a good qualitative agreement with the fact that the rupturing saccular aneurysms lack the media layer. Thirdly, it was found that residual stresses can increase the overall arterial strength significantly. The pre-existing compression in arteries delays the onset of rupture like the pre-existing compression in the pre-stressed concrete delays the crack opening.

4 Crack Sharpness and Fracture Toughness Now we apply the softening hyperelasticity approach to fracture mechanics. We are motivated by a controversy between the observations of the influence of the crack

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sharpness on the fracture toughness in experiments and the ignorance of the crack sharpness in the classical theories of brittle fracture [7, 29]. To gain new insight into the controversy we numerically simulated the onset of the crack propagation in thin plates under the hydrostatic tension [29]. The critical tension, when fracture starts, occurs when material fails at the tip of the crack. The failure is driven by the strain softening induced in the material constitutive model with the help of the energy limiter – the average bond energy. The material – Hookean – model enhanced with the failure description was plugged in ABAQUS and crack simulations were performed on very fine meshes to examine the influence of the crack sharpness on the onset of fracture. Small elliptic and straight cracks were considered with constant length and varying width, i.e. the tip curvature or the crack sharpness. It was observed that sharper cracks led to lower magnitudes of the critical tension. The latter, in turn, led to the lower magnitudes of the critical stress intensity factors – material toughness – in harmony with the experimental observations. Our observations are in agreement with the well-known Inglis [13] finding that the stress at the tip of an elliptic crack strongly depends on its sharpness. Assuming that the stress at the tip controls material strength it is possible to expect that the crack sharpness affects the onset of material failure. Such a scenario was considered by Inglis using linear elasticity. Comparing the approach of Inglis with the softening hyperelasticity approach used in the present work we should emphasize the difference between them. Inglis uses local – strength of materials – criteria of failure which are separated from the constitutive description of material. No global experiment on the calibration of the fracture toughness can be reproduced within the simplistic framework of strength of materials. The softening hyperelasticity approach is different. It allows tracking the global failure/instability of the structure with cracks due to the inclusion of the strain softening in the constitutive description of material. Thus, softening hyperelasticity allows reproducing the real physical experiments where the global instability/failure is observed. Why are the Griffith theory and LEFM ignorant of the crack sharpness? Such ignorance can be explained by the notion that the classical theories of brittle fracture are based on the energy balance considerations, which are integral and because of that they ‘smear’ the real stress/strain concentration at the tip of a real crack. The latter is explicit in the Griffith work where the energy balance is the basis of the theory. The energy nature of LEFM appears in disguise. Indeed, the critical stress intensity factors (SIF) that indicate the onset of fracture are the coefficients in the local asymptotic expansions of stress fields. At first glance, they are not formally related to any energy consideration. However, the stress intensity factors are ‘truly esoteric quantities’ [12] unless they are physically interpreted within the energetic framework of Griffith and the link is established between the critical SIF and the critical energy release rate. Thus, the fracture criteria of LEFM are essentially energetic though they appear in a form related to the local stress. It is remarkable that though the classical theories of brittle fracture ignore the crack sharpness they are capable of describing the influence of the crack length on the critical load very well in the case where the crack sharpness is constant. Our simulations of the straight

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cracks show that the critical tension depends inversely on the square root of the crack length in full harmony with the Griffith finding [11]. Unfortunately, that is true only for the equivalent cracks, i.e. cracks with the same tips. The main practical implication of our results is a conclusion that generally material toughness cannot be uniquely calibrated in experimental tests because its numerical magnitude significantly depends on the sharpness of the crack/notch used for the calibration. The crack sharpness controls the stress/strain concentration, which in turn controls the onset of fracture. It is possible, however, to decrease the radius of the tip of the crack/notch to a magnitude where our conclusion based on the classical continuum considerations is not applicable. Such a magnitude should be related with a characteristic length of the material microstructure, e.g. grain size, atomic distance, etc.

References 1. Barenblatt, G.I., 1959. The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially-symmetric cracks. J. Appl. Math. Mech. 23, 622–636. 2. Belytschko, T., Moes, N., Usiu, S., and Parimi, C., 2001. Arbitrary discontinuities in finite elements. Int. J. Num. Meth. Engng. 50, 993–1013. 3. Camacho, G.T. and Ortiz, M., 1996. Computational modeling of impact damage in brittle materials. Int. J. Solids Struct. 33, 2899–2938. 4. Chuong, C.J. and Fung, Y.C., 1983. Three-dimensional stress distribution in arteries. J. Biomech. Engng. 105, 268–274. 5. De Borst, R., 2001. Some recent issues in computational failure mechanics. Int. J. Numer. Meth. Engng. 52, 63–95. 6. Dugdale, D.S., 1960. Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100– 104. 7. Emmerich, F.G., 2007. Tensile strength and fracture toughness of brittle materials. J. Appl. Phys. 102, 073504. 8. Fung, Y.C., 1993. Biomechanics: Mechanical Properties of Living Tissues, 2nd edn., SpringerVerlag, New York. 9. Fung, Y.C., Fronek, K., and Patitucci, P., 1979. Pseudoelasticity of arteries and the choice of its mathematical expression. Amer. J. Physiol. 237, H620–H631. 10. Gao, H. and Klein, P., 1998. Numerical simulation of crack growth in an isotropic solid with randomized internal cohesive bonds. J. Mech. Phys. Solids 46, 187–218. 11. Griffith, A.A., 1921. The phenomena of rupture and flow in solids. Phil. Trans. Roy. Soc. London A221, 163–198. 12. Hutchinson, J.W., 2002. Life as a Mechanician: 1956– . Timoshenko Medal Acceptance Speech, http://imechanica.org/node/195. 13. Inglis, C.E., 1913. Stresses in a plate due to presence of cracks and sharp corners. Proc. Inst. Naval Architects 55, 219–241. 14. Kachanov, L.M., 1958. Time of the rupture process under creep conditions. Izv. Akad. Nauk SSSR, Otdelenie Teckhnicheskikh Nauk 8, 26–31. 15. Kachanov, L.M., 1986. Introduction to Continuum Damage Mechanics, Martinus Nijhoff, Dordrecht, the Netherlands. 16. Klein, P. and Gao, H., 1998. Crack nucleation and growth as strain localization in a virtualbond continuum. Engng. Fract. Mech. 61, 21–48. 17. Krajcinovic, D., 1996. Damage Mechanics, North Holland Series in Applied Mathematics and Mechanics, Elsevier.

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18. Lemaitre, J. and Desmorat, R., 2005. Engineering Damage Mechanics: Ductile, Creep, fatigue and Brittle Failures, Springer, Berlin. 19. Needleman, A., 1987. A continuum model for void nucleation by inclusion debonding. J. Appl. Mech. 54, 525–531. 20. Rabotnov, Y.N., 1963. On the equations of state for creep. In: Progress in Applied Mechanics (Prager Anniversary Volume), MacMillan, New York. 21. Rice, J.R. and Wang, J.S., 1989. Embrittlement of interfaces by solute segregation. Mat. Sci. Engng. A 107, 23–40. 22. Skrzypek, J. and Ganczarski, A., 1999. Modeling of Material Damage and failure of Structures, Springer, Berlin. 23. Tadmor, E.B., Ortiz, M., and Phillips, R., 1996. Quasicontinuum analysis of defects in solids. Phil. Mag. 73, 1529–1563. 24. Tvergaard, V. and Hutchinson, J.W., 1992. The relation between crack growth resistance and fracture process parameters in elastic-plastic solids. J. Mech. Phys. Solids 40, 1377–1397. 25. Volokh, K.Y., 2004. Nonlinear elasticity for modeling fracture of isotropic brittle solids. J. Appl. Mech. 71, 141–143. 26. Volokh, K.Y., 2007. Hyperelasticity with softening for modeling materials failure. J. Mech. Phys. Solids 55, 2237–2264. 27. Volokh, K.Y., 2008a. Fung’s arterial model enhanced with a failure description. Mol. Cell. Biomech. 5, 207–216. 28. Volokh, K.Y., 2008b. Prediction of arterial failure based on a microstructural bi-layer fibermatrix model with softening. J. Biomech. 41, 447–453. 29. Volokh, K.Y. and Trapper, P., 2008. Fracture toughness from the standpoint of softening hyperelasticity. J. Mech. Phys. Solids 56, 2459–2472. 30. Weiner, J.H., 1983. Statistical Mechanics of Elasticity, Wiley, New York. 31. Xu, X.P. and Needleman, A., 1994. Numerical simulations of fast crack growth in brittle solids. J. Mech. Phys. Solids 42, 1397–1434.

Crack Initiation, Kinking and Nanoscale Damage in Silica Glass: Multimillion-Atom Molecular Dynamics Simulations Y.C. Chen, K. Nomura, Z. Lu, R. Kalia, A. Nakano and P. Vashishta

Abstract We present molecular dynamics (MD) simulations of crack kinking, growth, propagation and healing at the nanometer scale in a confined, pre-cracked silica glass subjected to dynamic mode II loading. The pre-crack tip kinks towards maximum mode I tension and nanometer scale cavities are nucleated in the tensile region ahead of the kink. Nanocavities coalesce with kinks to form wing cracks. Dynamics of wing cracks are dominated by coalescence with damage cavities, and the growth and healing of crack and cavities are controlled by confinement.

1 Introduction Glassy materials are ubiquitous in the modern society. Goldschmidt [1] first proposed the network hypothesis of glass structure in 1926 and Zachariasen [2] later pointed out that the coordination number in glass must be approximately the same as in the crystal. For example, in crystalline SiO2 , the silicon atom is surrounded by four oxygen atoms, which form SiO4 tetrahedra. The silica glass also consists of SiO4 tetrahedra, which are connected in a corner-sharing configuration via the oxygen atoms. Numerous studies have been made to understand mechanisms underlying crack initiation and propagation in glasses [3, 4]. For decades, it was believed that under the influence of tensile stress, glass acts as a brittle solid with no observable plasticity. Thus, the brittleness of a glass was described in terms of its strength. The theoretical strength σth of a glass is determined from
Eγ σth = , a0 Y.C. Chen, K. Nomura, Z. Lu, R. Kalia, A. Nakano and P. Vashishta Collaboratory for Advanced Computing and Simulations, University of Southern California, Los Angeles, CA 90089-0242, USA

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where γ is the surface energy, E is the Young’s modulus and a0 is the atomic spacing. For a typical silicate glass, a value of 35 GPa is obtained. However, most glasses have flaws, which make the practical strength several orders of magnitude smaller than the theoretical strength. Despite the efforts to understand brittleness of glasses, fundamental mechanisms of fracture in glasses are still not well understood.

2 Molecular Dynamics Simulations Molecular Dynamics (MD) simulations reported here involve 15-million atoms, and these simulations were performed on a parallel PC cluster. The dimensions of the amorphous silica system are 120 nm × 120 nm × 15 nm in the x, y and z directions, respectively. Amorphous silica is obtained by melting the crystobalite phase and then by gradually quenching the melt. The amorphous system is pre-cracked and confined by repulsive walls in the x and y directions while periodic boundary condition is used along z. The inset in Figure 1 schematically shows the MD simulation setup, which includes the silica glass (light blue) with a pre-existing crack (dark blue) and a rigid indenter (white). The latter moves at a constant velocity in the x direction and exerts a repulsive force on atoms above the pre-crack surface. The indenter speeds, vimpact , were chosen to be 5% and 12.5% of the Rayleigh wave speed (vR ≈ 3,000 m/s) in silica glass.

3 Results The indenter pushes in the top half of the system, generating a compression wave that propagates from right to left in the inset of Figure 1. We calculated stresses around the pre-crack tip immediately after the load was applied, but before the compression wave reached the crack tip. For distances greater than 10 nm from the tip, the radial and angular dependences of atomistic-level stresses are in good agreement with mode II stress distributions in Linear Elastic Fracture Mechanics (LEFM) [5]. However, within a few nanometers from the pre-crack tip, the discrete nature of the material becomes relevant, and the stresses in MD simulation differ significantly from LEFM results. Yet, surprisingly, we find agreement between the MD and LEFM results for the direction in which the pre-crack kinks under compression. In MD simulation, nanometer scale pores open up just below the pre-crack tip shortly after the compression wave arrives there. The crack tip bends 700 counterclockwise around the z-axis in the direction of maximum mode I tension, as predicted by LEFM calculations [6] and observed in experiments at the macroscopic scale [7–10]. Nanocavities grow in this tensile stress region and coalesce with kinks to form nanocolumns. Figure 1

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Fig. 1 Nucleation of nanocavities and crack nanocolumns in a tensile-stress region around the pre-crack tip. The inset shows the setup of the simulation. The white rectangular plate is a rigid indenter, the light blue parallelepiped is the silica glass of dimensions 120 × 120 × 15 nm3 , and the dark blue region denotes a pre-crack of length 40 nm and width 15 nm. Dotted line indicates the direction along which the nanocavities nucleate. Damage nanocavities (red, orange, yellow, green) and nanocolumns (blue) in the tensile stress region. The impact loading speed is 0.05vR , where vR is the Rayleigh wave speed.

Fig. 2 Formation of a wing crack via growth and coalescence of nano-columns and nanocavities at an impact loading speed of 0.05vR . (a) A snapshot taken after 19 ps shows cavities (red, orange, yellow, green and dark blue) around nanocolumns (blue). (b) In the next couple of picoseconds, nanocolumns merge and coalesce with nanocavities to form a wing crack.

is a snapshot of these damage nanocavities (red, orange, yellow, green) and nanocolumns (blue).

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Fig. 3 Second healing of the wing crack at the loading speed of 0.05vR . (Right) A snapshot of the wing and primary cracks (blue) just after healing begins. The wing-crack tip is split up into two nanocolumns and there are a few damage cavities (green and red) near the tip. (Middle) In 4 ps the wing crack has receded considerably and left several cavities (red, yellow, green and blue). (Left) A snapshot of the wing crack and cavities after the crack stops healing. The residual length of the wing crack is slightly less than half of the maximum length.

Figure 2 provides an atomistic view of the wing-crack (blue) formation and growth and of nanocavitation (red, yellow, orange, green and dark blue are cavities) in the damage zone around the crack. Figure 2a is a snapshot taken 1.3 ps after Figure 1 at the same constant impact loading speed of 0.05vR . In this duration, the nanocolumns have propagated and new nanocavities have nucleated in the tensile region. Figure 2b shows that in a couple of pico seconds the nanocolumns merge to form a wing crack. The average velocity of the wing crack (1,000 m/s) is onethird of the speed of Rayleigh waves. Figure 2b shows that most of the cavities have coalesced with the advancing wing crack. Damage cavities continue to nucleate (and are also annihilated) in the tensile region ahead of the moving crack front. The growth of the wing crack ceases when it encounters a compression wave reflected from the left end of the system (see the inset in Figure 1). In the next 7 ps, the wing crack heals completely and the average healing speed is 1,300 m/s. After the healing of the primary crack stops, the wing crack reemerges and propagates at a higher average speed (1,500 m/s) through the damage zone of nanoscale cavities. Figure 3 displays three noteworthy events during the second retreat of the wing crack (blue) after it reaches a maximum length of 27 nm. The right snapshot, taken just after the wing crack encounters a compression wave (71 ps), shows two large damage cavities (green and red) at the wing-crack tip and a few cavities ahead of the primary crack. The middle snapshot (76 ps) shows two large (blue and green) and a few small nanocavities (red and yellow) left behind by the receding wing crack. The compression wave continues to heal the wing crack and damage cavities. The average speed of the receding wing crack is about 800 m/s. The left snapshot in Figure 3 shows the wing crack and damage cavities after the

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second healing stops (80 ps). The residual length of the wing crack is 13 nm. During this entire simulation, the wing crack propagates and retreats repeatedly without rupturing the glass. In conclusion, our simulations have shown that LEFM predictions are accurate down to a few nanometers. In particular, when a crack is subjected to external shear, a wing crack forms in the direction of maximum mode I tension. Damage induced by disorder intrinsically linked to the amorphous structure is shown to be the main mechanism for the nucleation and growth of the wing crack, and confinement is shown to control healing [11].

References 1. V.M. Goldschmidt, Geochemische Verteilungsgesetze der Elemente 8, 137 (1926). 2. W.H. Zachariasen, Journal of American Chemical Society 54, 3841 (1932). 3. R.C. Bradt, and A.G. Evans, Fracture Mechanics of Ceramics, Plenum Press, New York, London (1974). 4. C.R. Kurkjian, Strength of Inorganic Glass, Plenum, New York (1985). 5. T.L. Anderson, Fracture Mechanics Fundamentals and Applications, CRC Press, Florida (1995). 6. B. Cotterell and J.R. Rice, International Journal of Fracture 16, 155 (1980). 7. S. Lee and G. Ravichandran, Optics and Lasers in Engineering 40, 341 (2003). 8. W.N. Chen and G. Ravichandran, Journal of the Mechanics and Physics of Solids 45, 1303 (1997). 9. M.F. Ashby and S.D. Hallam, Acta Metallurgica 34, 497 (1986). 10. M.F. Ashby and C.G. Sammis, Pure and Applied Geophysics 133, 489 (1990). 11. Z. Lu et al., Physical Review Letters 95, 133501 (2005).

Multiscale Modelling of Layered-Silicate/PET Nanocomposites during Solid-State Processing Łukasz Figiel, Fionn P.E. Dunne and C. Paul Buckley

Abstract This work aims to develop a continuum, multi-scale, physically-based model of the forming process for layered-silicate nanocomposites based on poly(ethylene terephthalate) (PET) matrices, as might be used for packaging. This challenge is tackled using: (1) a physically-based model of PET implemented into the FEM-based code ABAQUS, (2) RVEs with prescribed morphologies reflecting TEM images, and (3) nonlinear computational homogenisation. As a result, 2-D two-scale FEM-based simulations under biaxial deformations (constant width) enabled the extraction of macroscopic stress-strain curves at different silicate contents and processing temperatures. In particular, interesting features in terms of morphology changes and its impact on the macroscopic stress response were captured: (a) morphology change by particle re-orientation and pronounced bending, (b) macroscopic strain hardening due to platelet re-orientation and local strain stiffening, (c) facilitated platelet alignment, and platelet delamination in tactoids through sheardominated deformations and increasing temperature.

1 Introduction The main unresolved issue related to layered-silicate polymer nanocomposites is insufficient delamination (exfoliation) of silicates. Quasi-solid state processing offers attractive means for modifying nanocomposite morphology, and hence increasing the level of exfoliation and varying particle orientation. Advanced modelling of this process appears to be of great assistance to understand better the relationship between processing conditions (strain, strain rate, temperature), and morphology and properties (mechanical and barrier) of the nanocomposite. This issue has not yet been addressed in the literature from the modelling perspective. Łukasz Figiel, Fionn P.E. Dunne and C. Paul Buckley Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK; e-mail: [email protected]

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The present work forms part of a wider multidisciplinary project, aiming to enhance physical properties of layered-silicate polymer nanocomposites via quasisolid state processing. For that purpose, various experimental (TEM, XRD, AFM), image analysis and modelling (molecular- and continuum-based) techniques are exploited. The ultimate goal of this specific work is to develop a continuum, multi-scale, physically-based computational model in order to simulate the large deformation behaviour of a layered-silicate/poly(ethylene terephthalate) (PET) nanocomposite around Tg , as experienced during shaping processes. Three main challenges related to this problem are tackled and described in this paper: (1) physically-based constitutive modelling of PET and its implementation into the Finite Element Method (FEM) framework, (2) reconstruction of the nanocomposite morphology using the RVE concept and the Monte-Carlo algorithm, (3) scale transition and computational homogenisation to determine the macroscopic response of the nanocomposite. The next section describes the modelling methodology adopted in this work. Then, some details about the polymer constitutive model for PET and its FEMbased implementation are presented. Further, RVE generation and computational homogenization are briefly described. Finally, results of simulations are presented and discussed.

2 Modelling Methodology TEM images of PET nanocomposites samples obtained from the extrusion process suggest that main features of nanocomposite morphology can be captured through two descriptors, namely particle aspect ratio and orientation. Moreover, they point out that those descriptors can be captured at the length scales between few micrometres and several hundreds of nanometres. Additionally, the time scale at which a typical forming process takes place is much longer than typical conformations of polymer chains at the forming temperature. Hence, we believe that this experimental evidence and practical reasons allow us here to take advantage of the continuum approach and avoid time consuming molecular simulations. Thus, a length scale at which particle geometry and orientation can be captured is used to define a Representative Volume Element (RVE). Due to the lack of experimental evidence about any interphase region around particles, each RVE consists of two phases: (1) silicate platelets in different configurations and (2) the polymer (PET) matrix assumed to be unaffected by the presence of particles. Further, each particle is treated as continuum and isotropic, and hence described by two elasticity constants. Description of the polymer behaviour is challenged through a constitutive model scaling main physical phenomena occurring in PET at the molecular level onto the continuum length scale of the RVE. Hence, description of PET is challenged in terms of physically meaningful parameters. A scale transition from the RVE level into the macroscopic level is carried out using a computational homogenisation concept assuming local periodicity of the nanocomposite morphology at the RVE level.

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3 Modelling of the PET Behaviour In order to capture complex both temperature and strain-rate dependent behaviour of PET, an extension of the single-mode constitutive model originally proposed by Adams et al. [1], has been recently devised in our lab and it is used herein. In particular, the total Cauchy stress tensor is assumed to be composed of two parts, namely the one dominating below Tg (called B-stress), and the one resulting from the behaviour above Tg (C-stress). The isochoric part of the B-stress, σˆ , is described through a differential equation ◦

σˆ B +

1 ˆB σˆ B = 2GB D τs

and τs = τs (aT , as , aσ ),

(1)

◦

where σˆ B is the Jaumann rate, GB is the shear modulus representing stiffness of ˆ B stands for the true rate of deformation process, while the reatomic bonds, D laxation time τs accounts for the viscous molecular processes and is a function of temperature aT , polymer structure as and applied stress aσ . The rate of the viscoplastic process and the softening behaviour are described through the Eyring equation [1] and the fictive temperature concept, Tf , T˙f =

T − Tf + κ ε˙ˆ V P , τs

(2)

the fictive temperature evolves as a function of the invariant of the viscoplastic strain rate ε˙ˆ VP , and κ is a material constant. The isochoric part of the C-stress, σˆ C , aims at describing a rubbery-viscous behaviour of an entangled molecular network above Tg . The rubbery response is captured through the strain energy function AC = AC (λˆ i(E) , Ns , T , η, α) for entangled polymers [1], and depends on principal elastic stretches λˆ i(E) , number of entanglements per unit volume Ns , temperature T , entanglement mobility η and chain inextensibility between entanglements α. Differentiation of AC with respect to elastic stretches, λˆ i(E) , results in isochoric Cauchy stresses σˆ C

3 1
= λˆ i(E) J i=1



1
∂AC ∂AC − 3 ∂ λˆ i(E) ∂ λˆ k(E) 3

 (i)

(i)

υE ⊗ υE ,

(3)

k=1

and υ (i) E transforms stresses from principal to fixed axes. The viscous behaviour is described by the non-Newtonian viscosity that accounts for the shear thinning effect. A very important aspect of the onset of crystallization is described here through an empirical, temperature- and stress-dependent, criterion. Finally, the hydrostatic part of the PET response is assumed to be elastic and described in terms of the bulk modulus. For the purpose of nanocomposite simulations the PET model is implemented into a FEM-based package ABAQUS using a UMAT subroutine. This requires an

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objective incremental formulation based on the nonlinear continuum mechanics framework. The main assumption in our formulation is an additive decomposition ˆ E and inelastic D ˆ V parts. A robust of the rate of deformation tensor into elastic D (stable for large time increments) numerical scheme for integration of B-stresses is used here, along with the Runge–Kutta method for integration of the fictive temperature. A specific structure of the PET model related to the calculation of principal C-stresses from the principal stretches, requires some special handling of the elastic ˆ E . This is currently resolved by employing a copart of the deformation rate, D rotational rate of the elastic part of the left Cauchy–Green tensor Bˆ E , following the idea of Leonov [4]: ◦

ˆ E Bˆ E + Bˆ E D ˆ E, Bˆ E = D

ˆE = D ˆ −D ˆV. where D

(4)

This enables an objective tackling of large strains and rotations occurring in the presence of silicate particles. As a result, the material time derivative of Bˆ E is calculated using the Jaumann rate. A constitutive tangent matrix consistent with the underlying strain energy function AC is derived analytically and implemented into the UMAT to ensure rapid convergence of the solution process.

4 RVE Generation, Macro-to-RVE and RVE-to-Macro Transitions Appropriate definition of the nanocomposite at the RVE level is crucial for reliable prediction of the macroscopic behaviour of nanocomposites. Therefore, an algorithm for the generation of RVEs was coded, and its current version enables for a flexible, automatic generation of different nanocomposite morphologies under assumptions of local periodicity. The TEM images quantified by experimentalists provide an input for RVE generation in terms of particle distribution, aspect ratio and orientation. The main challenge for this algorithm is generation of high aspect ratio particles, such that they are prevented from overlapping each other. It is realised through a rejection algorithm connected with the Monte-Carlo based sample generation. A Fortran programme for RVE generation is combined with ABAQUS/CAE through a Python script that generates FE models. Finally, periodic boundary conditions are imposed through an external, user-written Fortran programme. The macro-to-RVE transition was necessary to impose macroscopic boundary conditions at the RVE level. Assumption of local periodicity, allowed for the decomposition of the displacement vector uRVE (x) at the location x within an RVE to be expressed by macroscopic umacro and periodic uper parts uRVE (x) = umacro + uper

and umacro = εmacro x,

(5)

where macroscopic strain field εmacro was assumed to cause a uniform deformation of an RVE. Hence, it was sufficient to consider a single RVE of volume VRVE to

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Fig. 1 Simulated stress-strain curves – effect of silicate loading; T = 95◦ C.

predict the macroscopic behaviour. Thus, virtually a single integration point was used to represent the nanocomposite macromodel and was related to a single RVE. Displacement-controlled boundary conditions were applied to the RVE corners and complemented by the periodicity conditions imposed on the RVE edges. The RVE-to-macro transition was performed to calculate macroscopic stresses from average stresses within an RVE. It was carried out following approach of Kouznetsova et al. [3] by integrating surface tractions t distributed over RVE boundaries, which arose from applied displacement and periodicity conditions. That integration was carried out over the RVE boundary RVE , and as a result of it, an average stress tensor for the RVE σ¯ RVE was extracted  1 t ⊗ x d RVE ⇒ σ¯ RVE = σ macro . (6) σ¯ RVE = VRVE RVE The ultimate RVE average stress tensor was then associated with the macroscopic stress tensor σ macro , due to the assumption of uniform macroscopic deformation within each RVE, and used to generate macroscopic stress-strain curves.

5 Results and Discussion Several plane strain simulations of PET nanocomposites were carried out using developed numerical tools. The plane strain assumption corresponds to biaxial (constant width) conditions, frequently experienced during bottle forming process. Stress-strain curves were simulated at different particle volume fractions, and tem-

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Fig. 2 Simulated deformation and contour plots representing onset of crystallization at applied strains: (A) 0.5, (B) 1; temperature: 95◦ C; applied strain rate: 1 s−1 ; legend: 1 – lock-up of viscous flow due to crystallization at all integration points of a finite element, 0 – no lock-up.

peratures above Tg . Random orientation of particles, constant particle aspect ratio equal to 100, nominal strain rate around 1s−1 were used in all simulations. Effects of particle volume fraction on the macroscopic nanocomposite response are shown in Figure 1. As expected, the initial modulus (initial slope of the stress-strain curve) increases with increasing particle volume fraction. In particular, normalised (by matrix modulus) initial nanocomposite stiffness is ∼1.96 (5%), ∼1.74 (2.5%), ∼1.27 (1%), ∼1.16 (0.5%) suggesting a relatively small effect of particles, compared to the neat PET. However, one can notice a gap in the initial stiffness between volume fractions 2.5% and 1%. This might suggest that despite particle agglomeration in tactoids with increasing volume fraction (hence reduction in an effective particle aspect ratio), particle interaction enhances the initial modulus. Significant strain stiffening is observed as applied strains increase, as shown in Figure 1. This is in qualitative agreement with initial experiments. This increasing strain stiffening results partially from reorientation of particles – particles align with the applied load, which allows for more efficient stress transfer. We believe though, that a precipitated onset of stress-induced crystallization near particles is also responsible for the macroscopic strain stiffening. In particular, particles act as stress concentrators, hence allow for crystallization to occur earlier for reinforced than unfilled PET. This is shown in Figure 2 with contour plots representing the onset of crystallization. The onset of crystallization occurs in some nanocomposite regions already at strains around 0.2, and progresses with applied strain. In the case of unfilled PET this happens at strains around 1.6 (simulated under the same conditions). The macroscopic strain stiffening effect observed in Figure 1 can be quantified through a strain amplification factor concept, originally proposed for fillerreinforced rubber by Mullins and Tobin [5]. Here, it is calculated by relating macroscopic strains of the nanocomposite with those for the unfilled PET, at the specified

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Fig. 3 Strain amplification factor at different volume fractions; T = 95◦ C; applied strain rate: 1 s−1 .

Fig. 4 Effect of processing temperature on the nanocomposite morphology; (A) T = 100◦ C, (B) T = 110◦ C; strain rate: 1 s−1 .

stress level. As an example, strain amplification factors were calculated at different stress levels and volume fractions at temperature 95◦C, and the results are shown in Figure 3. We believe that a generalisation of the strain amplification factor to general deformations can help to match the simulated nanocomposite response solely with an enhanced constitutive model for PET, similarly to Bergström and Boyce [2]. As expected a softening of macroscopic stress-strain curves was found as a result of increasing processing temperature, for all volume fractions. However, it was observed that increasing processing temperature can facilitate particle re-orientation (as shown in Figure 4) and enhances separation of particles, initially locked in tact-

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oids. This confirms that quasi-solid state processing can be an effective way in improving morphology and hence nanocomposite properties.

6 Conclusions In this work, a physically-based, multiscale modelling of layered-silicate PET nanocomposites was challenged in the quasi-solid state regime. Combination of a physically-based constitutive model for PET, modelling tools for generation of the nanocomposite morphology, and a nonlinear computational homogenisation enabled prediction of the macroscopic nanocomposite response. Analysis of macroscopic stress-strain curves and morphology changes at the RVE level resulted in the following conclusions: (1) combined effect of volume fraction, particle reorientation and stress-induced onset of crystallization results in the macroscopic strain stiffening, (2) quasi-solid state processing leads to changes in morphology and enhancement of nanocomposite properties. Acknowledgements The work was supported by Research Grant EP/C006984/1 from the Engineering and Physical Sciences Research Council.

References 1. Adams AM, Buckley CP, Jones DP (2000) Biaxial hot drawing of poly(ethylene terephthalate): Measurements and modelling of strain-stiffening. Polymer 41:771–786. 2. Bergström JS, Boyce MC (2000) Large strain time-dependent behaviour of filled elastomers. Mech. Mater. 32(11):627–644. 3. Kouznetsova V, Brekelmans WAM, Baaijens FPT (2001) An approach to micro-macro modeling of heterogeneous solids. Comp. Mech. 27:37–48. 4. Leonov AI (1976) Nonequilibrium thermodynamics and rheology of viscoelastic polymer media. Rheol. Acta 15(2):85–98. 5. Mullins L, Tobin NR (1965) Stress softening in rubber vulcanizates. Part I. Use of a strain amplification factor to describe the elastic behavior of filler-reinforced vulcanized rubber. J. Appl. Polymer Sci. 9:2993–3009.

Modelling Transient Heat Conduction at Multiple Length and Time Scales: A Coupled Non-Equilibrium Molecular Dynamics/Continuum Approach Kenny Jolley and Simon P.A. Gill

Abstract A method for controlling the thermal boundary conditions of nonequilibrium molecular dynamics simulations by concurrent coupling with a continuum far field region is presented. The method is simple to implement into a conventional molecular dynamics code and independent of the atomistic model employed. It regulates the temperature in a thermostatted boundary region by feedback control to achieve the desired temperature at the edge of an inner region where the true atomistic dynamics are retained. This is necessary to avoid intrinsic boundary effects in non-equilibrium molecular dynamics simulations. A stadium damping thermostat is employed to avoid the adverse reflection of phonons that occurs at an MD interface. The effectiveness of the algorithm is demonstrated for the example of transient heat flow down a three-dimensional atomistic composite rod.

1 Imposing a Steady State Temperature Gradient on a Molecular Dynamics Simulation The boundary conditions for molecular dynamics (MD) simulations in the condensed phase are a compromise between correct representation of the far field and minimization of the system size due to computational constraints. In recent years, concurrent multiscale methods have been developed for crystalline solids in which the complex response of the far field is represented by a coarse-grained continuum region constructed from finite elements [1–11]. These multiscale modelling methodologies have mainly focused on the far-field representation of the elastic field at zero or constant temperature, although a few authors have looked at the thermal far-field [7, 10, 11]. This paper describes a recently developed algorithm which allows for (a) the temperature of an MD simulation to be precisely controlled away Kenny Jolley and Simon P.A. Gill Department of Engineering, University of Leicester, University Road, Leicester LE1 7RH, UK; e-mail: [email protected]

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Fig. 1 Steady state temperature profile along a one-dimensional 100 atom Lennard–Jones chain with fixed ends. The temperature difference is imposed by thermostatting the two end atoms to 50 K and 40 K respectively using a stochastic Langevin thermostat. The expected steady state temperature distribution is shown as a dashed line. The large deviation from the expected result arises from the discontinuity in the temperature profile at each end due to phonon scattering.

from thermal equilibrium; and (b) MD simulations to be concurrently coupled with a continuum representation of the thermal far-field. Most molecular dynamics simulations are sampled from the micro-canonical (constant energy) or canonical (constant temperature) ensembles, although there have been a number of studies where a steady state temperature gradient has been imposed on such an atomistic simulation [12–21]. The technique is simply to use conventional thermostatting techniques to enforce different temperatures on opposite ends of the sample. These non-equilibrium molecular dynamics (NEMD) simulations can then be used to determine the effective thermal conductivity of the medium, k, from Fourier’s law for macroscopic heat flow q = −k∇T ,

(1)

where q is the heat flux (averaged over time and space) in the unthermostatted region between the thermostats, and ∇T is the “measured” steady state temperature gradient. Importantly the measured temperature gradient is not the same as the desired temperature gradient imposed by the thermostats (as we will see in Figure 1). The situation described above is therefore not as straightforward as it may appear. In this paper we restrict our interest to ballistic heat transport in insulators via phonon interaction (i.e. conduction by electrons is neglected). This is an inherently non-linear phenomenon as phonons do not interact in the harmonic limit. The temporal evolution of a thermostatted particle of mass mi at a position xi is described by the usual equations of motion

Modelling Transient Heat Conduction at Multiple Length and Time Scales

mi x¨ i = −

∂V − γ mi x˙ i + Rf, ∂xi

29

(2)

where γ is a damping √ coefficient, −1 ≤ R ≤ 1 is a uniformly distributed random variable and fn = 6γ mi Tc /t is the magnitude (n = x, y or z) of the stochastic force f for a target temperature Tc and a time step t. The stochastic Langevin thermostat is active in regions where γ is non-zero. In regions where γ is zero the true (unthermostatted) dynamics are retained. The local Langevin thermostat is advantageous for NEMD simulations as it allows for the (average) temperature of each atom to be specified at the boundaries. It is also very easy to implement in a simulation code. One drawback is that there is no feedback between the actual temperature and the target temperature for the Langevin thermostat. This is reasonable for equilibrium thermostatting, for which it was designed, but far from equilibrium there is no guarantee that the target temperature will be achieved or maintained. It will prove desirable to introduce some feedback control for this thermostatting technique later in the paper. Also, local thermostats are beneficial for rapidly changing transient boundary conditions as they respond quickly to local changes in temperature. We now investigate the ability of the Langevin thermostat to impose a steady state temperature gradient along an atomistic rod. The Lennard–Jones interatomic potential for solid argon is used. Initially we consider a 1D atomistic chain in which thermostats are applied to the penultimate atoms at each end (i = 1 and i = N) where N = 100 is the number of atoms in the chain. We take the fixed target temperatures for the thermostats to be TL = 50K and TR = 40K at the left and right ends respectively. The resulting steady state temperature distribution along the length of the chain is shown in Figure 1. Fourier’s 1st law (1) predicts that the temperature profile will vary linearly between the target temperatures at each end for a constant thermal conductivity (as illustrated by the dashed line in Figure 1). It is clear that the simulation results do not conform to this expectation. There is a large temperature discontinuity at the ends, such that the temperature gradient observed in the simulation is not the applied temperature gradient. This effect has been widely observed and is generally attributed to phonon mismatch at the interface with the thermostats [16]. For real physical interfaces this is known as the Kapitza effect, where it is observed that the thermal conductivity, like most physical properties, deviates from the bulk value near an interface. Even if the interface/boundary is artificial, as is the case here, it is difficult to avoid. It exists in one, two and three-dimensions, although the most extreme effects are seen in one-dimensional chains. The effective conductivity at a point must be proportional to the inverse of the temperature gradient such that k ∝ 1/∇T . The large temperature gradient at each end implies that the conductivity is very small at the thermostat interface. To couple the MD simulation with a continuum far field it is necessary to gain complete control over the temperature at the boundary. This has clearly not been achieved in Figure 1. To do this we control the thermostat using feedback control. In addition, we adopt a variant of Langevin called stadium damping which has been shown to be an effective means of phonon absorption [7, 22] and importantly to produce the expected canonical ensemble [2]. In this case, the damping coefficient

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Fig. 2 Precise imposition of a steady state temperature gradient along a three-dimensional (8×8× 100 atom rod) NEMD simulation by feedback control of the thermal boundary conditions using stadium damping thermostats. The time-averaged temperature profile along the rod is shown. The target temperatures in the left and right thermostatted regions (TRs) are regulated at TL and TR by (3) such that the prescribed temperatures at the edges of the true dynamics region (TDR) (at j = 0 and j = 50) are maintained at T0 = 50 K and TM = 40 K. A buffer region (BR) of 10 atomic slices is introduced between the TRs and the central TDR to avoid Kapitza effects at the TR/BR interfaces.

is simply a function of position. As shown in Figure 2, γ is linearly ramped from a maximum value of γ = γ0 at the rod ends down to zero at the edge of the thermostatted region (TR). This forms a diffuse interface which allows phonons to move into the TR and be slowly absorbed as they move through it. This avoids many of the problems associated with phonon reflection at a sharp interface [10]. The Kapitza boundary effect is still observed for stadium damping although much reduced. The unthermostatted region between the thermostatted ends is the true dynamics region (TDR) except for a small (unthermostatted) buffer region (BR) of 10 atoms between the TRs and the TDR to accommodate the Kapitza effect, as shown in Figure 2. Now the temperature at the edges of the TDR (not the TR) is stipulated to be T0 = 50K and TM = 40K at the left and right ends respectively, where M is the number of atoms in the TDR. The target temperatures in the two TRs, TL and TR , are varied dynamically during the simulation to achieve this using a simple feedback loop [T0 − T0 ] T˙L = QT

[TM − TM ] , T˙R = QT

(3)

where T0  and TM  are the time ensemble averaged temperatures of the atoms at the left and right edges of the TDR respectively, and the constant QT determines the responsiveness of the thermostat. One benefit of this algorithm is that it provides

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the required feedback for the Langevin thermostat to ensure that the desired nonequilibrium temperature is achieved. The resulting steady state temperature profile for TRs of 15 atoms, BRs of 10 atoms and a TDR of 50 atoms is shown in Figure 2 for a three-dimensional atomic rod totalling 100 atoms in length and 8 × 8 atoms in cross-section. The feedback thermostatic control correctly imposes the prescribed temperature gradient on the TDR. The temperature distribution in the TR is fairly linear in accordance with the linear variation in the damping parameter in these regions. Note that a large difference between the thermostat target temperatures, TL − TR , is required for the stochastic thermostats to achieve the prescribed temperature difference, T0 − TM , due to the Kapitza effect.

2 A Coupled Atomistic/Continuum Model for Transient Heat Flow The feedback system developed in Section 1 for controlling the temperature at the boundaries of a MD simulation is now extended to allow the MD region to be coupled to a compatible continuum model. A continuum finite difference (FD) model is employed at the two ends of the rod. These continuum regions (CRs) overlap the TRs and the BRs in the MD simulation. These latter atomistic regions are now of no interest and only exist as a means for controlling the temperature at the edge of the atomistic TDR. For this reason, the temperature in the TRs and BRs is no longer shown. A one-dimensional FD grid is chosen to match the initial regular positions of the atoms. The nodal temperatures on the finite difference grid are denoted as T˜j and are at a fixed position x˜j , where j denotes the corresponding 8 × 8 slice of atoms in the MD region. These continuum temperatures can evolve by the usual FD algorithm cT˙˜ j = q˜j − q˜j −1 , (4)


where q˜j = k

T˜j + T˜j +1 2



(T˜j +1 − T˜j ) (x˜j +1 − x˜j )

(5)

is the continuum heat flux between nodal points j and j + 1 and c is the (constant) heat capacity of a slice. The thermal conductivity k(T ) is assumed to be a linear function of temperature √ such that k(T ) = k0 +∇kT . The √ constants were determined to be k0 = 1.037/ m W/(m.K) and ∇k = −0.021/ m W/(m.K) where m is the (dimensionless) atomic mass. The continuum and atomistic models are coupled through the imposition of two sets of boundary conditions. Firstly, for transient analysis, formula (3) is abandoned in favour of ensuring conservation of heat flux across the boundary from the CR to the TDR. The instantaneous atomistic heat flux is rapidly changing so conservation of heat is enforced on average over time such that

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Fig. 3 The steady state temperature profile for a coupled atomistic/continuum (MD/FD) model of a composite rod. The temperatures are fixed at the edges of the continuum regions (CRs) to be 40 K and 20 K respectively. The temperature in the atomistic TRs and BRs is not shown. A region in which atoms have a large mass (m=10) is located in the centre of the atomistic rod. There is a large drop in the temperature at the interface between the regions of different masses due to the Kapitza effect. Results for the pure FD model do not reflect this unless a reduced interface thermal conductivity k is used.

1 T˙L = Qq



t o

L (q˜CR − qTLDR ) dt

1 T˙R = Qq



t 0

R (q˜CR − qTRDR ) dt,

(6)

L is the continuum heat flux from the left-hand CR into the TDR, q L where q˜CR T DR is the atomistic heat flux entering the TDR from the left-hand BR, etc., and determines the response rate of the system to disparities in the heat flux. The integral ensures that no heat is lost over time. The second boundary condition connects the temperatures in the atomistic and continuum regions. This is achieved by defining the continuum temperature at the CR/TDR interface to be the time-averaged value at the same point in the atomistic simulation such that

T˜0 = T0 

T˜M = TM .

(7)

The thermal response of a composite atomistic rod to rapid changes in boundary temperature is now investigated to demonstrate the ability of the model to deal with demanding applications. The composite rod is modelled within an atomistic TDR of 80 atoms. It consists of two different atomic species. The interatomic potentials are the same, only their masses differ. The majority of the atoms have a mass of m = 1 but a thin central layer (30 atoms in length and 8 × 8 in cross-section) contains atoms with a significantly higher mass of m = 10. Firstly, the steady state response of the assembly is determined. The temperature profile is shown in Figure 3 for fixed remote temperatures at the far continuum FD boundaries of 40 K and 20 K.

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The coupled FD-MD model result shows a continuous temperature and temperature gradient across the CR/TDR interface as required. There is a large jump in the temperature at the interface between the m = 1 and m = 10 regions. As before, this is due to the Kapitza effect (although for a real interface this time, not an artificial one). This indicates the conductivity at the interface between the two mass regions is greatly reduced due to phonon scattering. The results of pure FD simulations are also shown in Figure 3. If the reduced interface conductivity k is not accounted for the steady state response is not accurate. To model this, a much lower conductivity is used in the interface element between the two mass regions with k0 = 0.0001 and k = 0.001. This revised FD model, using the reduced interface conductivity, is shown in Figure 3 and agrees well with the coupled FD-MD results. The transient response of the system is now investigated. Figure 4 shows the dynamic response of the composite rod and the coupling of the MD-FD system to an instantaneous change in the temperatures of one of the continuum boundaries. Figure 4a demonstrates the response to heating. Initially, at time t = 0, the assembly is at a uniform temperature of 20 K. The left-hand CR boundary is then changed to 40 K. The FD and MD regions respond dramatically, although at all times the temperature and thermal gradient at the continuum/atomistic interface are continuous. The central m = 10 region heats up slowly due to the slow conduction of heat across the interface between the two mass regions. Eventually the system achieves the steady state profile shown in Figure 3 as expected. The results of a pure FD simulation (with reduced interfacial k) are also shown. A small difference with the atomistic simulation becomes apparent at later times, probably due to the intrinsic non-linear dynamic response of the MD region. Figure 4b shows a similar situation for cooling. The simulation starts with the steady state profile of Figure 3 at t = 0. The left-hand continuum boundary temperature is then dropped from 40 K to 20 K. Residual heat is seen to remain for sometime in the m = 10 region due to Kapitza effect. These two transient simulations demonstrate that the coupling algorithm copes equally well with heat flowing from the continuum into the atomistic region and vice versa. Again there are some small disparities between the pure FD and the coupled MD-FD model, but overall they are reasonably similar. In summary, a method for controlling the thermal boundary conditions of nonequilibrium molecular dynamics (NEMD) simulations has been presented in Section 1. The method is simple to implement into a conventional molecular dynamics code and independent of the atomistic model employed. The body is thermostatted at the boundaries to control the temperature at the edges of the true dynamics region (TDR). The target temperatures for the thermostats to impose a given temperature gradient is not known in advance and is determined dynamically using a simple feedback control loop. The method for controlling the boundary conditions of NEMD simulations has been extended in Section 2 to allow atomistic/continuum (MD-FD) models to be thermally coupled concurrently for the analysis of steady state and transient heat conduction problems. The effectiveness of this algorithm has been demonstrated through the study of an atomistic composite rod. The results are found to be compatible with finite difference (FD) predictions using Fourier’s law if the reduced conductivity of interfaces is accounted for. The true benefit of

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Fig. 4 Two examples demonstrating the control of transient boundary conditions for a coupled atomistic/continuum simulation (solid line). (a) Initially at a uniform temperature of 20 K, the far left continuum boundary temperature is instantly changed to 40 K at t = 0 to heat the system. (b) Initially at 40 K, the far left boundary is dropped to 20 K at t = 0 to cool the system. The dashed line is a pure FD solution using the reduced interfacial conductivity k.

this technique will be in the investigation of non-linear atomistic phenomena such as defects (e.g. grain boundaries) or nanofeatures (e.g. voids, composites) [23, 24].

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Acknowledgements S. Gill and K. Jolley gratefully acknowledge the support of a Royal Academy of Engineering-Leverhulme Senior Research Fellowship and EPSRC grant no. EP/P501547/1 respectively.

References 1. Broughton, J.Q., Abrahams, F.F., Bernstein, N., Kaxiras, E.: Concurrent coupling of length scales: Methodology and application, Phys. Rev. B60, 2391 (1999). 2. Curtin, W.A., Miller, R.E: Atomistic/continuum coupling in computational materials science, Modelling Simul. Mater. Sci. Eng. 11 R33–R68 (2003). 3. Gill, S.P.A., Jia, Z., Leimkuhler, B., Cocks, A.C.F.: Rapid thermal equilibration in coarsegrained molecular dynamics, Phys. Rev. B 73 184304 (2006). 4. Li, X., Weinan, E.: Multiscale modelling of the dynamics of solids at finite temperature, J. Mech. Phys. Solids 53 1650–1685 (2005). 5. Liu, W.K., Park, H.S., Qian, D., Karpov, E.G., Kadowaki, H., Wagner, G.J.: Bridging scale methods for nanomechanics and materials, Comput. Methods Appl. Mech. Engrg. 195 1407– 1421 (2006). 6. Park, H.S., Karpov, E.G., Liu, W.K.: A temperature equation for coupled atomistic/continuum simulations, Comput. Methods Appl. Mech. Engrg. 193 1713–1732 (2004). 7. Qu, S., Shastry, V., Curtin, W.A., Miller, R.E.: A finite temperature dynamic coupled atomistic/discrete dislocation method, Modelling Simul. Mater. Sci. 13 1101 (2005). 8. Rudd, R.E., Broughton, J.Q.: Concurrent coupling of length scales in solid state systems, Phys. Stat. Sol. B217, 5893 (2000). 9. Shilkrot, L.E., Miller, R.E., Curtin, W.E.: Multiscale plasticity modelling: coupled atomistic and discrete dislocation mechanics, J. Mech. Phys. Solids 52 755–787 (2004). 10. Xiao, S.P., Belytschko, T.: A bridging domain method for coupling continua with molecular dynamics, Comput. Methods Appl. Mech. Engrg. 193 1645–1669 (2004). 11. Dupuy, L.M., Tadmor, E.B., Miller, R.E., Phillips, R.: Finite-temperature quasicontinuum: Molecular dynamics without all the atoms, Phys. Rev. Lett. 95 060202 (2005). 12. Bhowmick, S., Shenoy, V.B.: Effect of strain on the thermal conductivity of solids, J. Chem. Phys. 125 164513 (2006). 13. Heino, P.: Thermal conductivity and temperature in solid argon by nonequilibrium molecular dynamics simulations, Phys. Rev. B 71 144302 (2005). 14. Huang, Z., Tang, Z.: Evaluation of momentum conservation influence in non-equilibrium molecular dynamics methods to compute thermal conductivity, Physica B 373 291–296 (2006). 15. Lepri, S., Livi, R., Politi, A,: Energy transport in anharmonic lattices close to and far from equilibrium, Physica D 119 140–147 (1998). 16. Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices, Physics Rep. 377(1) 1–80 (2003). 17. Prasher, R.: Diffraction limited phonon thermal conductance of nanoconstrictions, Appl. Phys. Lett. 91 143119 (2007). 18. Schelling, P.K., Phillpot, S.R., Keblinski, P.: Comparison of atomic-level simulation methods for computing thermal conductivity, Phys. Rev. B 65 144306 (2002). 19. Segal, D., Nitzan, A.: Thermal conductance through molecular wires, J. Chem. Phys. 119 13 6840 (2003). 20. Watanabe, T., Ni, B., Phillpot, S.R.: Thermal conductance across grain boundaries in diamond from molecular dynamics simulation, J. Appl. Phys. 102 063503 (2007). 21. Terao, T., Muller–Plathe, F.: A nonequilibrium molecular dynamics method for thermal conductivities based on thermal noise, J. Chem. Phys. 122 081103 (2005). 22. Holian, B.L., Ravelo, R.: Fracture simulations using large-scale molecular-dynamics, Phys. Rev. B 51 11275–11288 (1995).

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23. Huang, W., Huang, G., Wang, L., Huang, B.: Phonon-cavity-enhanced low temperature thermal conductance of a semiconductor nanowire with narrow constrictions, Phys. Rev. B 75 233415 (2007). 24. Tian, W., Yang, R.: Effect of interface scattering on phonon thermal conductivity percolation in random nanowire composites, Appl. Phys. Lett. 90 263105 (2007).

Multiscale Modeling of Amorphous Materials with Adaptivity V.B.C. Tan, M. Deng, T.E. Tay and K.M. Lim

Abstract We present a method to reduce the degrees freedom (DOF) in molecular mechanics simulation. Although it is formulated particularly for amorphous materials, it is also equally applicable to crystalline materials. Concurrent multiscale simulation is carried out by reducing the DOF in regions where displacement gradients are small while simultaneously using classical molecular mechanics (MM) for regions undergoing large deformation. The accuracy and computational efficiency of the approach is demonstrated through the simulation of a domain of polymerlike macromolecular chains stretched to fracture. The region around an initial slit in the polymer is modelled by classical molecular mechanics while the region further away has the degrees of freedom (DOFs) reduced by about 50 times. The simulations are adaptive in that regions of reduced DOFs are automatically reverted back to classical MM in-situ when deformation gradients become high.

1 Introduction In the study of the mechanical behaviour of materials at the sub-micron length scale, atomistic simulations such as molecular mechanics (MM) and molecular dynamics (MD) simulations are normally the tools of choice when the number of atoms are in excess of hundreds of atoms. Smaller systems may be studied by more the computationally expensive ab initio quantum mechanics. In many engineering problems, such as those involving damage propagation and material fracture, molecular reconfiguration at a local region is highly dependent on far field loading conditions. In these situations, events at the molecular level are strongly linked to events at a length scale where atomistic simulations cannot be performed in a reasonable amount of time on computing platforms of today and even the near future. Multiscale simulation techniques to bridge molecular and continuum levels are needed to accurately V.B.C. Tan, M. Deng, T.E. Tay and K.M. Lim Department of Mechanical Engineering, National University of Singapore, Singapore 117576

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describe these multi length scale problems. A multiscale simulation technique, formulated to accommodate the amorphous nature of polymer systems [1], is used to simulate the fracture of a polymer-like system in this paper.

2 Multiscale Methods Of the various multiscale approaches that have been proposed, those that have seen significant applications can be broadly classified into Coarsed Grained Molecular Dynamics, handshake models and Quasicontinuum models. These three methods represent the common strategies to performing multiscale simulations. Coarsed Grained Molecular Dynamics (CGMD) is widely adopted for the simulation of polymeric systems. Rather than computing the displacements of all molecules, clusters of molecules which can be identified a priori as reasonably rigid are grouped together to form a bead or a grain. Each bead is then treated as a large molecule in what is then essentially classical Molecular Dynamics simulation [2–4]. The main idea in handshaking models is to extend the atomistic domain and the continuum domain slightly into each other to create an overlapping region known as the ‘handshake’ region. The computation proceeds independently for each domain, e.g., MD can be used for the atomistic domain while finite element method can be used for the continuum domain. Coupling of the two domains is achieved by assuming that a field variable (normally the potential energy) takes on an arbitrarily weighted combination of the magnitudes of the same variable in the continuum and atomistic domains [5, 6]. In the Quasicontinuum (QC) method, a tessellation of cells is overlaid onto the molecules in the domain of interest. By using the Cauchy–Born rule, i.e., the deformation gradient is assumed to be uniform within each cell. The energy associated with each cell can then be computed simply by looking at the deformation at one point in the cell. This is carried out by having the vertices of the cells coincide with some pre-selected representative molecules. The degrees of freedom are the position (or displacement) of the representative molecules. The energy of each cell is computed by assuming the position of each molecule in a cell to be a function of the representative molecules of the cell and the average energy of a cell can be approximated from a group of molecules within the cell [7–10]. The QC method seeks the configuration of representative molecules that gives the minimum potential energy of the computational domain.

3 The Pseudo Amorphous Cell All the three approached presented have their own advantages and disadvantages. The ‘handshaking’ method is easy to implement but the coupling between scales is not rigorously established. Coarse graining lacks the capability to scale up to a

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continuum level while the Cauchy–Born rule in the QC method is only applicable to materials with regular or simple lattice structures. Hence, to perform the simulations presented here which involve polymers, we adopt the pseudo amorphous cell (PAC) method [1]. The mechanical modeling of polymers even at solely atomistic scale is by itself an area that has not been firmly established. Different methods of modeling polymers are still being proposed. Modeling polymers is complex because they are generally amorphous and molecular interaction comprises pair-wise bonds, long range non-bond interaction, bond bending and bond torsion. Moreover polymer chains can comprise more than 10,000 monomers per chain. Unlike lattices of metal atoms, the equilibration of polymer chains in bulk normally does not end up with the same equilibrated configurations. One way of determining the mechanical properties of polymers in bulk is through the use of the amorphous cell [11]. In this method, the bulk polymer model comprises a periodic assembly of repeating parent polymer chains. The repeat intervals in the three spatial directions are conveniently represented by a cuboid or cell. The parent chains are obtained by equilibrating chains of specified number of monomers with the constraints that the chains replicate themselves according to the dimensions of the cell. The PAC method for multiscale simulation makes use of the amorphous cell as building blocks of polymer domains to be investigated. Rather than solving for the position of all the molecules of the constituent amorphous cell, displacement of the vertices of the cells (a.k.a. nodes) are the unknowns. The molecular displacements are then recovered from the nodal displacement through a mapping function, T. The PAC method allows regions where atomistic details are important to be modelled by classical molecular mechanics (MM). The connection between the PAC cell regions and the MM region is seamlessly effected by augmenting the T transformation with unit matrices where the molecular displacements are to be solved directly from MM. The computational effort to solve the coupled system of equations can be shown to be dependent on the size of the MM region if the PAC cell regions do not undergo large deformation. This is in correspondence with the way the polymer domain is expected to be partitioned – PAC cell and continuum methods at regions of small strain gradients with computationally costly MM domains at regions of high deformation. In the PAC method, the T matrix is derived to extract the molecular displacements according to the modes of deformation of the cells (i.e. stretch and shear) which turn out to be very different from linear interpolation.

4 Results and Discussion To demonstrate the PAC cell method, simulations of tensile loading of a polymerlike domain were performed. The polymer domain is 11 cells across by 26 cells deep. The PAC cell method is applied to the top and bottom eight rows of cell of the domain while molecular mechanics was used for the centre 10 rows. A horizontal ‘pre-crack’ of one cell length was introduced by cutting several polymer chains lying

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Fig. 1 Strain contours from multiscale simulation just before crack propagation.

across the crack. The parent chain of the amorphous cell consists of 50 monomers. Hence, a 50:1 reduction in degrees of freedom is achieved for the PAC cell domain. The tensile process was also simulated using a fully molecular mechanics simulation for the entire domain for comparison. The interatomic potential of atoms which are bonded to each other is given by
−0.5kR02 ln[1 − (r/R0 )2 ], r < R0 , φbond(r) = (1) ∞, r ≥ R0 and the Lennard–Jones pairwise potential for nonbonded atoms is given by    σ 12  σ 6 φLJ (r) = 4ε − r r

(2)

with a cutoff of r = 2.5. The parameters of the Lennard–Jones potential are set to unity while the parameters for the bond potential are k = 30 and R0 = 1.5. Figures 1 and 2 are the strain contours of the polymer obtained from the multiscale simulations just prior to the propagation of the crack and at the end of the simulation respectively. Although the displacements of atoms within PAC cells are not, and need not, be solved during the simulation, individual atom displacement can be easily obtained through the mapping function T at any stage of the simulation. This implies that whenever and wherever higher resolution in displacement is required, PAC cells can selectively be reverted to atomistic domains. The lines running left to right in Figures 1 and 2 show how the PAC cells have receded during the simulation to

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Fig. 2 Strain contours at the end of multiscale simulation.

Fig. 3 Comparison of tensile force vs displacement graphs from multiscale and molecular mechanics simulations.

demonstrate adaptive simulations. For the adaptive simulations, the PAC cells are reverted to atomistic domains when they experience a strain (εy ) of 2%. The tensile force versus displacement curves from the fully molecular mechanics and multiscale simulations are shown in Figure 3. It shows that the tensile force predicted from both simulations is almost identical. Acknowledgements The research reported herein was carried out while the authors were supported by the Asian Office of Aerospace Research and Development (AOARD).

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References 1. Tan V.B.C., Zeng X.S., Deng M., Lim K.M. and Tay T.E., Multiscale modeling of polymers – The Pseudo Amorphous Cell, Computer Methods in Applied Mechanics and Engineering 197, 536–554, 2008. 2. Muller-Plathe F., Coarse-graining in polymer simulation: From the atomistic to the mesoscopic scale and back, ChemPhysChem 3(9), 754–769, 2002. 3. Tsige M. and Stevens M.J., Effect of cross-linker functionality on the adhesion of higly crosslinked polymer networks: A molecular dynamics study of epoxies, Macromolecules 37(2), 630–637, 2004. 4. Baschnagel J., Binder K., Doruker P., Gusev A.A., Hahn O., Kremer K., Mattice W.L., MullerPlathe F., Murat M., Paul W., Santos S., Suter U.W. and Tries V. Bridging the gap between atomistic and coarse-grained models of polymers: Status and perspectives. In Viscoelasticity, Atomistic Models, Statistical Chemistry, Advances in Polymer Science, Vol. 152, Springer Verlag, pp. 41–156, 2000. 5. Abraham F.F., Broughton J.Q., Bernsten N. and Kaxiras E., Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture, Europhysics Letters 44(6), 783– 787, 1998. 6. Broughton J.Q., Abraham F.F., Bernsten N. and Kaxiras E., Concurrent coupling of length scales: methodology and application, Physical Review B 60(4), 2391–2403, 1999. 7. Tadmor E.B., Ortiz M. and Phillips R., Quasicontinuum analysis of defects in solids, Philosophical Magazine A 73, 1529–1563, 1996. 8. Tadmor E.B., Phillips R. and Ortiz M., Mixed atomistic and continuum models of deformation in solids, Langmuir 12(19), 4529–4534, 1996. 9. Shenoy V.B., Miller R., Tadmor E.B., Phillips R. and Ortiz M., Quasicontinuum models of interfacial structure and deformation, Physical Review Letters 80(4), 742–745, 1998. 10. Rodney D. and Phillips R., Structure and strength of dislocation junctions: An atomic level analysis, Physical Review Letters 82(8), 1704–1707, 1999. 11. Theodorou D.N. and Suter U.W., Detailed molecular structure of a vinyl polymer glass, Macromolecules 18(7), 1467–1478, 1985.

Thermodynamically-Consistent Multiscale Constitutive Modeling of Glassy Polymer Materials Pavan K. Valavala and Gregory M. Odegard

Abstract The multiscale modeling of polymer-based materials has many challenges that not yet been fully addressed in the literature. These challenges are summarized and discussed. A particular challenge is the modeling of the large distribution of microstates that exist in many bulk engineering polymer materials. A multiscale modeling approach is proposed to predict the bulk Young’s modulus of polymers using a series of molecular models of individual polymer microstates and a statistics-based micromechanical modeling method. The method is applied to polyimide and polycarbonate material systems.

1 Introduction Polymer-based composite and nanocomposite materials have the potential to provide significant increases in specific stiffness and specific strength relative to current materials used for many engineering structural applications. To facilitate the design and development of polymer nanocomposite materials, structure-property relationships must be established that predict the bulk mechanical response of these materials as a function of the molecular- and micro-structure. Many multiscale modeling techniques have been developed to predict the mechanical properties of thermoplastic polymers based on the molecular structure and behavior [1–34]. However, all of these techniques are limited in terms of their treatment of time-dependent deformations, molecular behavior detail, applicability to large deformations, and/or spatial distributions of conformations of polymer chains (henceforth referred to as microstates). The proper incorporation of these issues into a multiscale framework may provide efficient and accurate tools for establishing structure-property relationships of polymer-based materials. Pavan K. Valavala and Gregory M. Odegard Department of Mechanical Engineering – Engineering Mechanics, Michigan Technological University, Houghton, MI 49931, USA; e-mail: [email protected]

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The objective of this paper is to establish an approach for incorporating the above-listed issues for multiscale modeling of thermoplastic polymer materials. The modeling of amorphous, time-dependent materials subjected to large deformations will be discussed first, followed by the development of a micromechanics approach for incorporating the statistics-based distribution of microstates. Results from this modeling are briefly discussed.

2 Multiscale Modeling of Polymers Many multiscale modeling approaches have been developed for crystalline nanostructured materials [35]. These approaches often rely on establishing equivalent continuum models whose kinematic responses match identically with the representative volume element (RVE) of the corresponding molecular model. The motion of each individual atom is the same as the motion of the corresponding continuum material points in the equivalent continuum model. Although this approach works very well for materials with a high degree of molecular structural repeatability (e.g. crystals and carbon nanotubes), it becomes very difficult to use in amorphous materials that contain a many different bonding types (e.g. directional bonding) and molecular geometries. Therefore, for polymer materials, alternative multiscale modeling approaches are necessary. Another method for establishing an equivalent-continuum model is to match energies of deformation for identical deformations applied to the boundaries of RVEs of molecular and equivalent-continuum models [1–5]. With this approach, there is no exact correspondence between the kinematics of equivalent-continuum points and individual atoms, however, the overall response of the RVE is modeled. This approach also assumes the bulk-level material symmetry of the equivalent-continuum material, with a corresponding strain-energy function of an assumed form based on the material symmetry. This assumption, in effect, establishes a very simple scale-up from the properties predicted from the molecular RVE to the bulk-level properties. This approach provides an accurate and efficient technique to predict bulk responses of polymer-based materials. Alternative to energy equivalence, the stress-strain relationships of the equivalent-continuum and molecular models can be matched for establishing the properties of an equivalent continuum model [6-8]. Because of the one-to-one correspondence between the stress components and the strain energy in an equivalentcontinuum elastic material, the results from matching energies and stresses should ideally result in the same equivalent-continuum properties. However, several issues may arise with this approach. First, if the material symmetry in not taken into consideration, more computational simulations are required to characterize the six independent stress components (assuming a symmetric stress tensor is used) in Hooke’s law with a fourth-order elasticity tensor. For hyperelastic energy equivalence, the material symmetry is embedded in the formulation, thus reducing the number of required deformations to the minimum value. Second, a form of the virial stress func-

Multiscale Constitutive Modeling of Glassy Polymer Materials

45

tion must be assumed in the stress equivalence approach. This is an open subject of debate in the literature [36] based on the kinematic terms of the virial stress function. With the energy equivalence approach, no assumptions need to be made regarding the form of the stress because it is dictated by the hyperelastic strain energy function and the assumed deformation measure. Third, the energy equivalence approach allows for an easy extension to a hyperelastic framework, which is necessary when modeling large deformations of polymer materials, especially elastomers. It is important to note that the stress-equivalence and energy-equivalence approaches have been used [8] in a manner to determine the individual elastic constants for a system with bulk-level isotropic symmetry without taking advantage of the expected material symmetry. As a result, non-symmetric or non-isotropic stiffness tensors can be established that violate the bulk-level material symmetry. Therefore, this approach predicts bulk-level properties based completely on a single, finite-sized, molecular RVE that are not consistent with the continuum-mechanic framework for which the elastic constants are based. In the energy-equivalence approach, the material symmetry is inherent in the assumed hyperelastic strain energy function so that the predicted mechanical properties obey the expected material symmetry operations. Whether the energy- or stress-equivalence approaches are used to establish equivalent-continuum properties, three basic options exist for applying the deformation. Some multiscale approaches use fluctuation methods [9–15]. These methods simulate relatively small deformations in both tension and compression with a linear-elastic response. However, many polymers, particularly lightly-crosslinked elastomers, are used in engineering applications that require hyperelastic deformations. Furthermore, non-linear elastic responses are often observed. Therefore, a generalized multiscale modeling approach needs to be easily adaptable to large deformations and non-linear material responses, which the fluctuation methods are not. Other multiscale modeling approaches include static methods using molecular mechanics (MM) [1, 3, 5, 8, 16–29], and dynamic approaches using molecular dynamics (MD) [5, 25, 30–34]. With dynamic approaches, the properties are determined based on the material behavior over a very short time interval (on the order of picoseconds), whereas with static approaches the properties are estimated based on the material behavior over an ambiguous time interval associated with the molecular equilibration. The time-based ambiguity with static simulations is due to the energy equilibration of the molecular structure irrespective of a simulated time. Time-dependent (viscoelastic) responses in thermoplastic polymers are beyond the reach of time scales that are achievable with current time-stepping schemes using MD approaches (1 × 1015 simulated time steps for a fully atomistic model). Therefore, dynamic approaches simulate deformation rates that are unrealistically high (for engineering purposes), while static approaches efficiently simulate deformations at relatively large, yet quantitatively undefined time intervals that are more realistic with engineering experience. Because thermoplastic polymers exhibit timedependent behavior, it is expected that dynamic simulations will predict mechanical properties that are unrealistically high, while static approaches should predict mechanical behavior that is more agreeable with experiment [5].

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Therefore, it follows that the prediction of mechanical properties of polymerbased materials using multiscale modeling techniques is best-served with staticbased molecular modeling approaches that rely on an energy-equivalence approach to establishing an equivalent-continuum. This is the approach used in the current study.

3 Thermodynamically-Consistent Modeling of Polymers Many different conformations of multiple polymer chains are possible for a particular polymer RVE in an equilibrated or non-equilibrated state [37–40]. As a result, the molecular structure, and thus density, of a polymer material varies substantially on the nanometer length-scale [40–47]. Each combination of chain conformations in a RVE has an associated potential energy which can be interpreted as an energy landscape in the conformational space of the polymer network. The conformational space does not necessarily have a one-to-one correspondence to the energy landscape. Therefore, the energy landscape generally consists of multiple local minima. As a result, for a RVE consisting of a finite number of polymer chains, multiple locally-equilibrated states can exist. A majority of high performance polymer-based materials operate at temperatures much below their glass transition temperatures. An amorphous polymeric material in a glassy state can be envisioned as a super-cooled liquid that is ‘frozen’ in a local potential energy equilibrium state, which is not necessarily a globally-minimized potential energy state. The different microstates that are not at the global-minimum energy state are essentially ‘metastable’ states with exceptionally long relaxation times, as the energy barriers to cross over to the global minimum energy state are generally very high. Therefore, the bulk material behavior can be imagined to be an average response from all the available conformational microstates. In order to accurately predict the bulk-level behavior of polymer-based systems based on molecular structure, a range of conformational microstates of a polymeric network must be included in multiscale constitutive modeling approaches. Molecular simulations were carried out on both polyimide and polycarbonate systems. Multiple RVEs representing samples from the conformational space were obtained for the two systems. Nine thermally-equilibrated structures were obtained for the polycarbonate, each consisting of 5958 atoms with 9 polymer chains and each chain comprised of 20 repeat units, and seven for the polyimide, each consisting of 6622 atoms with 11 chains and 10 repeat units per chain. Each of the structures represented a single microstate. Representative RVEs of the two polymer systems are shown in Figure 1. Once each initial molecular model RVE was established with the expected bulk density (∼1 g/cc) using the methods outlined elsewhere [5], a series of MD simulations were used to establish thermally-equilibrated solid structures for each microstate in the following order at 300 K: (1) A 50 ps simulation with the NVT ensemble to prepare the structure for further equilibration, (2) a 100 ps simulation

Multiscale Constitutive Modeling of Glassy Polymer Materials

47

Table 1 Polyimide microstate properties. Microstate

Er (GPa)


r (kcal/mole)

pr

1 2 3 4 5 6 7

0.02 0.24 4.71 11.10 3.82 10.20 16.70

42,788.42 52,780.01 79,859.22 84,264.01 84,682.77 85,469.25 97,465.05

0.233 0.189 0.125 0.118 0.118 0.116 0.102

Table 2 Polycarbonate microstate properties. Microstate

Er (GPa)


r (kcal/mole)

pr

1 2 3 4 5 6 7 8 9

2.29 5.14 2.01 12.00 20.90 0.09 5.69 7.08 1.65

13,706.69 13,722.27 15,905.73 18,093.05 21,674.48 23,372.80 38,850.61 50,629.76 55,728.69

0.175 0.175 0.151 0.133 0.111 0.103 0.062 0.047 0.043

with the NPT ensemble at 100 atm to evolve the system to higher densities as the structure was prepared from a low-density structure, (3) a 100 ps NPT simulation at 1 atm to reduce the effects of high-pressure simulations and to let the system evolve to a state of minimal residual stresses, and (4) a 100 ps NVT simulation to allow the system to equilibrate at the simulated temperature and density for a specific microstate. The equivalent continuum Young’s modulus was determined for each microstate of the two polymer systems using the static approach described above (Tables 1 and 2). The details of these simulations and calculations are given elsewhere [5]. Because a given RVE has a unique combination of chain conformations, it is expected that the above-discussed approach to predicting the elastic properties of a polymer system will generally yield different predicted properties for different nanometer-scale RVEs. It is also expected that the bulk material response will be dependent on the mechanical response of all such microstates that are possible for a given polymer system. Because of the computational difficulty of establishing every possible microstate for a molecular RVE of a polymer system, the modeling procedure described herein is restricted to the finite number of microstate RVEs obtained as described above. A micromechanics technique was used to determine the equivalent bulk-level response of the two polymer systems based on the mechanical response of the microstates. Because the geometry of the individual microstates is generally unknown, it was assumed that the bulk Young’s modulus of the polymers could be described

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P.K. Valavala and G.M. Odegard

Fig. 1 Molecular RVEs of two polymer systems.

by a modified form of the Halpin–Tsai equation E = Eave

1+ 1−


N

r=2 ηr pr


N

r=2 ηr pr

(1)

where Eave is the average of the microstate Young’s moduli, N is the total number of microstates, pr is the probability of existence of microstate r, and ηr is defined as Er − Eave ηr = (2) Er + Eave The traditional Halpin–Tsai approach is to use the Young’s modulus of the composite matrix as the reference modulus instead of Eave , to have only two phases (matrix and one reinforcement), and to use the volume fractions of both phases instead of the probabilities. However, the modified form shown above was adapted to provide an efficient and accurate estimate of the bulk properties of a polymer with multiple microstates of unknown geometry. The distribution of microstates in a polymer is generally unknown, so an exact form of pr cannot be established. However, a physically-intuitive option is to base the distribution of microstates, and thus the probability of existence of a particular microstate pr , on equilibrium statistical mechanics. Due to the statistical nature of the growth of polymer networks during the synthesis of addition polymers, the networks to not crystallize and the chain dynamics are significantly hindered due to the

Multiscale Constitutive Modeling of Glassy Polymer Materials

49

formation of elaborate entanglements. The entanglements resist the free movement of the polymer chains and therefore hinder the network evolution to a globallyminimized potential energy state. Therefore, it is expected that the lower-energy microstates are more common, with a finite number of higher-energy microstates with smaller probabilities. It is postulated that the probability of a given microstate has a skewed distribution based on energy given by
−1 pr =
N r −1 s=1
s

(3)

where
r is the potential energy of microstate r calculated with a chosen force field. From Equation (3) the summation of pr over all of the microstates is unity. Tables 1 and 2 show the potential energies corresponding to each microstate of the two polymer systems. Using Equations (1)–(3) and the data in Tables 1 and 2, the predicted bulk Young’s moduli for the polyimide and polycarbonate materials are 3.1 and 4.4 GPa, respectively. Experimentally-obtained values of Young’s modulus for these materials are 3.9 and 2.2 GPa, respectively [48, 49]. Therefore, the predicted and experimentally-obtained Young’s moduli for the polyimide are reasonably close. The predicted Young’s modulus for the polycarbonate is twice as high as the experimentally-obtained value. There are two possible reasons for the discrepancies between the measured and predicted Young’s moduli. First, the molecular systems modeled in the current study represent polymer structures without any chain length distribution and unreacted monomer. Second, only a finite number of microstates were modeled. Increasing the number of microstates could potentially result in more accurate predicted mechanical properties. Acknowledgements This research was jointly sponsored by NASA Langley Research Center under grant NNL04AA85G and the National Science Foundation under grant DMI-0403876.

References 1. Odegard GM, Frankland SJV, Gates TS (2005) Effect of Nanotube Functionalization on the Elastic Properties of Polyethylene Nanotube Composites. AIAA Journal 43:1828–1835. 2. Odegard GM, Gates TS, Nicholson LM, Wise KE (2002) Equivalent-Continuum Modeling of Nano-Structured Materials. Composites Science and Technology 62:1869–1880. 3. Odegard GM, Gates TS, Wise KE, Park C, Siochi E (2003) Constitutive Modeling of Nanotube-Reinforced Polymer Composites. Composites Science and Technology 63:1671–1687. 4. Odegard GM, Pipes RB, Hubert P (2004) Comparison of Two Models of SWCN Polymer Composites. Composites Science and Technology 64:1011–1020. 5. Valavala PK, Clancy TC, Odegard GM, Gates TS (2007) Nonlinear Multiscale Modeling of Polymer Materials. International Journal of Solids and Structures 44:1161–1179. 6. Anand L, Su C (2005) A Theory for Amorphous Viscoplastic Materials Undergoing Finite Deformations, with Application to Metallic Glasses. Journal of the Mechanics and Physics of Solids 53:1362–1396.

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31. Frankland SJV, Caglar A, Brenner DW, Griebel M (2002) Molecular Simulation of the Influence of Chemical Cross-Links on the Shear Strength of Carbon Nanotube-Polymer Interfaces. Journal of Physical Chemistry B 106:3046–3048. 32. Griebel M, Hamaekers J (2004) Molecular Dynamics Simulations of the Elastic Moduli of Polymer-Carbon Nanotube Composites. Computer Methods in Applied Mechanics and Engineering 193:1773–1788. 33. Qi D, Hinkley J, He G (2005) Molecular Dynamics Simulation of Thermal and Mechanical Properties of Polyimide-Carbon Nanotube Composites. Modeling and Simulation in Materials Science and Engineering 13:493–507. 34. Sheng N, Boyce MC, Parks DM, Rutledge GC, Abes JI, Cohen RE (2004) Multiscale Micromechanical Modeling of Polymer/Clay Nanocomposites and the Effective Clay Particle. Polymer 45:487–506. 35. Liu WK, Karpov EG, Park HS (2006) Nano Mechanics and Materials: Theory, Multiscale Methods and Applications. John Wiley & Sons, Ltd., Hoboken, NJ. 36. Zhou M (2003) A New Look at the Atomic Level Virial Stress: On Continuum-Molecular System Equivalence. Proceedings of the Royal Society of London Series A: Mathematical Physical and Engineering Sciences 459:2347–2392. 37. Capaldi FM, Boyce MC, Rutledge GC (2004) Molecular Response of a Glassy Polymer to Active Deformation. Polymer 45:1391–1399. 38. Laot CM, Marand E, Schmittmann B, Zia RKP (2003) Effects of Cooling Rate and Physical Aging on the Gas Transport Properties in Polycarbonate. Macromolecules 36:8673–8684. 39. Lyulin AV, Michels MAJ (2002) Molecular Dynamics Simulation of Bulk Atactic Polystyrene in the Vicinity of T-g. Macromolecules 35:1463–1472. 40. Stachurski ZH (2003) Strength and Deformation of Rigid Polymers: Structure and Topology in Amorphous Polymers. Polymer 44:6059–6066. 41. Cohen MH, Turnbull D (1970) On the Free-volume Model of Liquid-Glass Transition. Journal of Chemical Physics 52:3038. 42. Dlubek G, Pointeck J, Shaikh MQ, Hassan EM, Krause-Rehberg R (2007) Free Volume of an Oligomeric Epoxy Resin and Its Relation to Structural Relaxation: Evidence from Positron Life-time and Pressure-volume-temperature Experiments. Physical Review E 75. 43. Hinkley JA, Eftekhari A, Crook RA, Jensen BJ, Singh JJ (1992) Free-Volume in Glassy Poly(arylene ether ketone)s. Journal of Polymer Science Part B: Polymer Physics 30:1195– 1198. 44. Roe RJ, Curro JJ (1983) Small-Angle X-Ray-Scattering Study of Density Fluctuation in Polystyrene Annealed below the Glass-Transition Temperature. Macromolecules 16:428–434. 45. Roe RJ, Rigby D (1990) Free-Volume Distribution and Local Chain Motion in Polymer Liquid and Glass Studied by MD-Simulation. Abstracts of Papers of the American Chemical Society 199:361-POLY. 46. Rutledge GC, Suter UW (1991) Calculation of Mechanical-Properties of Poly(P-phenylene terephthalamide) by Atomistic Modeling. Polymer 32:2179–2189. 47. Wilks BR, Chung WJ, Ludovice PJ, Rezac ME, Meakin P, Hill AJ (2006) Structural and Freevolume Analysis for Alkyl-substituted Palladium-catalyzed Poly(norbornene): A Combined Experimental and Monte Carlo Investigation. Journal of Polymer Science Part B: Polymer Physics 44:215-233. 48. Hergenrother PM, Watson KA, Smith JG, Connell JW, Yokota R (2002) Polyimides from 2,3,3’,4’-Biphenyltetracarboxylic Dianhydride and Aromatic Diamines. Polymer 43:5077– 5093. 49. Christopher WF, Fox DW (1962) Polycarbonates. Reinhold Publishing Corporation, New York.

Effective Wall Thickness of Single-Walled Carbon Nanotubes for Multi-Scale Analysis: The Problem and a Possible Solution L.C. Zhang and C.Y. Wang

Abstract Continuum mechanics models have been used to characterize the mechanical behaviour of carbon nanotubes, but their validity down to the nanometer scale has not been fully verified. A typical example is the effective wall thickness of single-walled carbon nanotubes (SWCNTs), which has not been well defined after years of effort. This paper proposes a sufficient condition for determining the effective wall thickness h and Young’s modulus E of an SWCNT using in-plane stiffness, Kin−plane , torsion stiffness, Dtorsion, bending stiffness, Dbending, and off-plane torsion stiffness, Ktorsion, as the independent elastic constants of a continuum model. The paper concludes that when the Vodenitcharova–Zhang’s necessary condition and Dbending/Kin−plane = Dtorsion/Ktorsion are satisfied, the intersect of Dbending and Kin−plane curves in the E–h plane will determine a unique h, and in turn, leads to a defined E. For SWCNT (10, 10), h ≈ 0.1 nm and E ≈ 3.5 TPa.

1 The Problem It has been reported that under external loads a carbon nanotube (CNT) behaves like a continuum structure and has both membrane and bending capacities. Nevertheless, continuum mechanics models so far have been associated with strong assumptions, leaving some fundamentals to clarify. To characterize the mechanical properties of a CNT, one often uses directly the mechanics quantities defined by continuum mechanics, such as Young’s modulus. However, a CNT has a discrete molecular structure and its ‘wall’, comprising of only a number of atoms, does not L.C. Zhang School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia; e-mail: [email protected] C.Y. Wang Civil and Computational Engineering Centre, School of Engineering, Swansea University, Swansea SA2 8PP, UK; e-mail: [email protected]

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L.C. Zhang and C.Y. Wang

have a continuous material distribution spatially, whereas to calculate the Young’s modulus under the framework of continuum mechanics one needs to know the wall thickness. In the literature, mechanics modelling has been based on various assumptions of h, treating an SWCNT, e.g., as a truss member [1], a solid beam [2], a solid cylinder [3], or simply letting h be the inter-planar spacing of two graphite layers (= 3.4 Å) [4–11]. As a result, the Young’s modulus of SWCNTs calculated with different h values varies in a wide range. Vodenitcharova and Zhang [12] introduced a concept of an effective wall thickness under the umbrella of continuum mechanics. Based on the consideration of force equilibrium, they proposed a necessary condition that the effective wall thickness of an SWCNT must be smaller than the theoretical diameter of a carbon atom, about 0.142 nm [13]. Their argument is that a cross-section of a nanotube contains only a number of atoms and the forces in the tube are transmitted through these atoms; but in a continuum mechanics model the same forces are transmitted through a continuous wall area. As such, the effective wall thickness must be smaller than the theoretical diameter of a carbon atom; as otherwise, the tube equilibrium cannot be maintained. According to this condition, continuum models using a wall thickness greater than or equal to the diameter of a carbon atom are unreasonable, while the others below that are possible but their validity needs to be confirmed. In an attempt to address the continuum-atomic modelling, Zhang et al. [14] and Zhang et al. [15] established a direct link between the atomic structure of an SWCNT with a continuum constitutive model, by equating the strain energy stored in the equivalent constitutive model to that in the atomic bonds described by the Tersoff–Brenner potential [16, 17]. This method seems to be reasonable because the Tersoff–Brenner potential has been shown to be appropriate for analysing SWCNTs [18]. An isotropic constitutive model was therefore derived for SWCNTs subjected to in-plane deformation with two elastic constants, the in-plane stiffness, Kin−plane, and the in-plane shear stiffness, Ktorsion. Then the following relationship was obtained by a comparison with a 3D thin shell of thickness h: Kin−plane = Eh/(1 − ν 2 )

and Ktorsion = Gh

(1)

where G = E/2(1+ν) is the shear modulus and ν is the Poisson ratio. However, the effective thickness of SWCNTs cannot be determined in this way because the model involves only the in-plane deformation. Huang et al. [19] considered both the inplane and off-plane deformation of SWCNTs. By using the Tersoff–Brenner potential, V , and the modified Cauchy–Born rule [14,15], they obtained two-dimensional isotropic constitutive relations, where the bending stiffness, Dbending , and off-plane torsion stiffness, Dtorsion, as well as Kin−plane and Ktorsion of SWCNTs were calculated as √
 3 ∂V Dbending = , Dtorsion = 0, 2 ∂ cos θij k

Effective Wall Thickness of Single-Walled Carbon Nanotubes

1 Kin−plane = √ 2 3



∂ 2V ∂rij2

 + 0

55



A , 8

Ktorsion =

A √ 16 3

(2)

where θij k (k = i, j ) is the angle between bonds i–j and i–k, rij is the i–j bond length, and A is a function of the first and second-order derivatives ofV with respect to rij , θij k and θij l (l = i, j , k). In their derivation, the bending of SWCNTs was considered as a result of the σ -bond angle change while the off-plane torsion is independent of the deformation of the σ -bonds. Now, if we use a 3D continuum thin shell theory, where bending and off-plane torsion are due to the deformation across the wall thickness, we can see that Dbending = Eh3 /12(1 − v 2 )

and Dtorsion = Gh3 /12.

(3)

Combining Eqs. (1) and (3) leads to the following condition: Dtorsion Ktorsion 1−ν = = Dbending Kin−plane 2

(4)

The above discussion indicates that to clarify the problem of the effective wall thickness and Young’s modulus of SWCNTs, it is necessary to first establish a mechanics model without using E and h directly, and then determine them by integrating the model’s prediction with the results obtained independently from atomistic analyses or experimental measurements.

2 A Shell Model 2.1 Formulation To avoid using the debatable variables E and h, let us consider an SWCNT as a 2D isotropic shell. Unlike a classical 3D shell [20], the Dbending, Dtorsion, Kin−plane and Ktorsion in the 2D shell are treated as independent parameters which are not necessarily related to the thickness via the classical formulae, such as Eqs. (1) and (3) above, and thus do not have to satisfy Eq. (4). Then the solution to the free vibration equations of an SWCNT of radius R and mass density per unit area ρ can be written as u = U cos kx x · cos nθ · eiωt , w = W sin kx x · cos nθ · eiωt ,

v = V sin kx x · sin nθ · eiωt , (5)

where U , V and W are the vibration amplitudes, kx is the wave vector in x-direction, n is the circumferential wave number and ω is angular frequency related to frequency f by ω = 2π · f . The condition of non-zero solution of (U , V , W ) gives the vibration frequencies and their associated modes for SWCNTs.

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2.2 Determination of Elastic Constants Now let us see if the above shell solution can describe the free vibration behaviour of an SWCNT. To reach a conclusion, there are two issues to clarify. First, we need to understand the sensitivity of the vibration modes to the material parameters Dbending, Dtorsion, Kin−plane , and Ktorsion; and secondly, to determine them. A parametric analysis of SWCNT (10, 10) demonstrates that the axisymmetric radial (R), longitudinal (L) and torsional (T ) modes (n = 0), and beam-like bending mode (n = 1) of the SWCNT are primarily governed by Kin−plane and Ktorsion but are almost independent of Dtorsion and Dbending. Thus, these vibration modes are ideal for determining the values of Kin−plane and Ktorsion. For the vibration modes with n = 2−5, when kx = 0, Dbending determines the frequencies, but when kx > 0, both Dbending and Dtorsion contribute. Hence, the vibration modes with n = 2 − 5 can be used to determine the values of Dtorsion and Dbending. To determine parameters Dbending, Dtorsion, Kin−plane and Ktorsion in our model, we use the results independent of our derivation. Since a direct experimental measurement is unavailable, to our knowledge, the simplest way is to see whether solution (5) can give the same predictions by the published atomistic analyses. We found that when taking Kin−plane = 360 J/m2 and Ktorsion = 144 J/m2 , our model’s predictions of the R, L and T modes with n = 0 are in excellent agreement with the results from the force constant model [21], the lattice dynamic models [22–25], and the continuum model [26]. This justifies that the values of Kin−plane and Ktorsion used are reasonable. With the Kin−plane and Ktorsion thus determined, the values of Dtorsion and Dbending can be found by the best fit of the present model prediction to the results of [21–26] with n = 2–5. This leads to Dbending ≈ 2 eV and Dtorsion ≈ 0.8 eV. It is important to point out that with the Dbending, Dtorsion, Kin−plane and Ktorsion values determined above, Eq. (4) is satisfied with a constant effective wall thickness of the SWCNT.

3 Analysis In the elastic shell theory of continuum mechanics, the bending stiffness Dbending and in-plane stiffness Kin−plane have definite relationships with Young’s modulus E, wall thickness h and Poisson’s ratio ν, as defined in Eqs. (1) and (3). Hence we can plot Dbending and Kin−plane on a single E–h coordinate plane. On the other hand, when using an atomistic theory, the bending stiffness can be calculated by 1 ∂ 2 W (κ) Dbending = · Vα ∂κ 2 [27–31], where W (κ) represents the energy required to roll up flat graphite sheet to a cylindrical surface and κ denotes the curvature change in the process. Similarly,

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the in-plane stiffness Kin−plane can be calculated by Kin−plane =

1 ∂ 2 W (εaxial ) · 2 Vα ∂εaxial

[6, 8, 27–30, 32–38], or alternatively by the force method, i.e., Kin−plane =

Faxial /εaxial 2πR

[30, 36, 39–41], where Vα is the volume of a carbon atom, εaxial is the axial strain of an SWCNT under a uniaxial force Faxial and W (εaxial ) is the strain energy shared by each carbon atom in the SWCNT. Hence, the Dbending and Kin−plane from an atomistic study can also be plotted on the E–h plane using the E or h value reported by that study. As shown in Figure 1, the majority of the results (dots 1 to 24) in the literature collapse to the curve of Kin−plane = 360 J/m2 , with the lowest at Kin−plane ≈ 300 J/m2 (dot 6) and the highest at Kin−plane ≈ 420 J/m2 (dots 23 and 24). This shows that different modelling techniques give similar in-plane stiffness, indicating that the in-plane deformation of SWCNTs can be adequately described by almost all aforementioned atomistic theories. Experimentally ‘measured’ Young’s modulus of SWCNTs is also presented in Figure 1, where the equivalent in-plane stiffness Kin−plane is obtained by fitting the Euler beam model to the data measured in thermal vibration (a dashed line labelled 25) [10], or by three-point bending experiments (the three dotted lines labelled 26 from three samples [42]). We can see that these Kin−plane values are very consistent with the theoretical results and therefore verifies that the theoretically predicted Kin−plane , around 360 J/m2 , is correct. There are a few exceptions (dots 1, 2 and 12), but as has been pointed out by Wang and Zhang [45], Chen and Cao’s arguments in defending their results [43] are questionable according to the molecular dynamics analysis by Mylvaganam and Zhang [18] while the validity of Wang’s model [44] needs to be further examined. The Dbending values of SWCNTs reported in the literature, on the other hand, scatter in a wide range as shown in Figure 1, i.e., Dbending = 0.69 ∼ 0.85 eV (dots 3, 5, 6), 1 ∼ 1.3 eV (dots 1, 4, 7, 8, 12), 1.49 ∼ 2 eV (dots 2, 9–11, 13), and 3.28 eV or larger (dots 14–15). Such diversity can be due to the discrepancy between atomistic models in dealing with the ‘equivalent macroscopic’ bending of SWCNTs. For example, the LDA (dot 8) shows that both the σ -bond change and π electron resonance contribute to the bending resistance of an SWCNT. Nevertheless, the MM model (dots 1 and 12) assumes that Dbending is merely determined by the change of the π-orbital electron density. Similarly, different potentials used in the same simulations also lead to significantly different Dbending values, e.g., dots 13 and 14. This shows that Dbending is much more sensitive to the models used than Kin−plane , and must be properly addressed in determining the effective wall thickness of an SWCNT. As shown in Figure 1, the variation of Dbending from 0.85 to 2 eV substantially changes the h and E values when Kin−plane is fixed at 360 J/m2 .

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Fig. 1 Comparison of Kin−plane and Dbending in the E–h plane. The sources of the dots in this figure are: 1: [44]; 2: [43]; 3: [27]; 4: [29]; 5: [31]; 6: [12]; 7: [35]; 8: [33]; 9: [30]; 10: [28]; 11: [48]; 12: [44]; 13 & 14: [36]; 15: [38]; 16: [41]; 17: [37]; 18: [8]; 19: [40]; 20: [32]; 21: [49]; 22: [39]; 23 & 25: [10]; 24: [34]; 26: [42].

In addition, the E and h values associated with dots 2, 6, 13, 14 and 15 in Figure 1 are extracted directly by fitting continuum models to atomistic simulations. We can see that when a shell model [36, 43] is used to calculate E and h, the corresponding Dbending values (dots 2 and 13) obtained based on Eq. (3) are close to those given by TB model (dot 4), LDA (dot 8), and ab initio calculation (dots 5, 9 and 10). On the other hand, the Eand h values calculated by a ring [12], or a structure frame model [31] lead to a significantly smaller or lager Dbending (dots 6 or 15) compared with those (dots 4–5 and 9–10) given by the atomistic simulations. These suggest that the E and h values obtained for a continuum model other than shell could be different from those of an equivalent shell model for SWCNTs and naturally, the resulting Dbending value based on the shell formula Eq. (3) will deviate from the atomistic simulations. For example, the effective thickness of 0.147 nm [38] given by dot 15 is in fact the diameter of the micro-beams used to describe interatomic interaction between two adjacent atoms, which is obviously larger than that of a continuum thin shell according to Vodenitcharova and Zhang [12] and, thus, gives an unreasonably high Dbending in Figure 1. From the above analysis we see that in determining the effective thickness h and Young’s modulus E of SWCNTs there exists large uncertainty in atomistic theories and unclear definition of effective concepts of SWCNTs, which leads to a large scatter of the h and E values.

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4 A Possible Solution Figure 1 shows that the effective thickness of SWCNTs in the literature fall into two groups: those smaller than the atomic diameter of carbon atoms 0.142 nm and those greater. As pointed out in [45], only those smaller than the atomic diameter are reasonable because they satisfy the Vodenitcharova–Zhang’s necessary condition [12]. The question is: among the many reasonable values, which is right? This can be answered if a sufficient condition of determining h can be established. As discussed above, if an SWCNT is modelled as a 3D isotropic shell, the elastic constants of the shell must satisfy Dbending/Kin−plane = Dtorsion/Ktorsion, which reflects the bending mechanism of a continuum shell and ensures the existence of an effective thickness. Nevertheless, the deformation of SWCNTs in atomistic models is caused by the changes in chemical bonds between neighbouring atoms. Thus, if the Dbending, Dtorsion, Kin−plane and Ktorsion given by an atomistic theory cannot satisfy the above condition, then this atomistic model cannot give rise to a consistent effective thickness and an effective Young’s modulus. On the other hand, our discussion in the previous sections has shown that if condition Dbending/Kin−plane = Dtorsion/Ktorsion and the Vodenitcharova–Zhang’s necessary condition are satisfied at the same time, the intersect of the Dbending and Kin−plane curves in the E–h plane determines a unique h and a corresponding E. It is therefore reasonable to propose the following sufficient condition for determining the effective thickness h of an SWCNT: The h and E values can be determined by the intersect of the Dbending and Kin−plane curves in the E–h plane by a continuum or an atomistic model, when condition Dbending/Kin−plane = Dtorsion/Ktorsion and the Vodenitcharova–Zhang’s necessary condition are satisfied at the same time. As a specific example, our analysis shown in Figure 1 (dot 11) has obtained that the correct effective thickness of SWCNT (10, 10) is h ≈ 0.1 nm and its effective Young’s modulus is E ≈ 3.5 TPa, which is in good agreement with the most recent simulation result of h ≈ 0.1 nm and E = 3.2–3.4 TPa [50].

5 Conclusions Based on an elastic shell theory and a thorough comparison with numerous results in the literature, this paper has proposed a possible sufficient condition for determining the effective thickness of an SWCNT. It concludes that when the Vodenitcharova– Zhang’s necessary condition and Dbending/Kin−plane = Dtorsion/Ktorsion are satisfied, the intersect of Dbending and Kin−plane curves in the E–h plane will determine a unique h, and in turn, leads to a defined E. For the SWCNT (10, 10) studied in the present paper, we have found that h ≈ 0.1 nm and E ≈ 3.5 TPa.

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Discrete-Continuum Transition in Modelling Nanomaterials Ryszard Pyrz and Bogdan Bochenek

Abstract In the present investigation we elaborate on the development of a secondorder elastic deformation gradient in discrete/atomistic system. Whereas kinematics are typically characterized by the Cauchy–Born rule that enforces homogeneous deformation, the second-order deformation gradient allows to capture highly nonhomogeneous deformations. This is particularly important in disordered molecular systems where nonaffine deformations are responsible for the mechanical behaviour of nanomaterials. The local inhomogeneity measure has been defined to determine variability of the deformation field of nanostructures under loading. Several application examples have been worked out comprising fullerene structures, diamond plates and nanowires.

1 Introduction Nanoscale systems are intrinsically of a discrete nature and the applicability of macroscopic continuum theories at that scale is not always obvious. The only cases for which microscopically based derivation of elasticity are documented are uniformly strained lattices. A continuum theory breaks down also for disordered, amorphous systems below a certain length scale. Classical continuum mechanics is size independent which is in contradiction to the physical observations that at the size scale of a few nanometres, deformations and elastic state are size dependent, and a departure from classical mechanics can be expected [2,13,16,29,33]. The key point here is that the information involved in these two approaches is biased; i.e., conRyszard Pyrz Department of Mechanical Engineering, Aalborg University, Pontoppidanstræde 101, 9220 Aalborg East, Denmark; e-mail: [email protected] Bogdan Bochenek Institute of Applied Mechanics, Cracow University of Technology, Al. Jana Pawla II 37, 31-864 Cracow, Poland

R. Pyrz and J.C. Rauhe (eds.), IUTAM Symposium on Modelling Nanomaterials and Nanosystems, 63–74. © Springer Science+Business Media B.V. 2009

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tinuum approach is concerned with classical field quantities such as Cauchy stress tensor and small deformation strain tensor whereas these quantities do not have the same format in discrete systems. Merging these approaches can be done either by using more sophisticated continuum models [1, 3, 20, 28, 32] or elaborating new discrete quantities which are “dual” and have their appropriate counterpart within the continuum model. The latter is not as simple as it seems due to the ambiguity in stress calculations at atomic level. The concept of Cauchy stress tensor is essentially macroscopic and cannot be used directly to the set of atoms/molecules which constitute a discrete system. The most frequently used form for the stress at atomic level is based upon the Clausius virial theorem, which determines the stress field applied to the surface of a fixed volume containing interacting particles (atoms). It has been shown that the virial stress cannot be directly related to the classical Cauchy stress and several modifications have been proposed [4, 5, 9, 14, 15, 17, 22, 34, 35]. It is essential to recognize that the stress at the location of an atom depends on the details of the interatomic interactions and the positions of interacting neighbours. Hence, the atomic stress is a non-local function of the state of the matter at all points in some vicinity of the reference atom, in contrast to the local stress field used in classical continuum theories. Furthermore, atoms in bonded polymeric chains are subject to bending and torsion moments, which are not included in the definition of virial stress. It seems that a concept of common deformation measure between atomic scale simulations and the continuum framework is not as ambiguous as the concept of atomic stress. Although different strain measures can be formulated all of them rely on the coordinates of atoms. A locally constant atomic strain tensor has been calculated in [7] based on the relative motion of a reference atom and its nearest neighbours. The atomic displacements have been interpolated using a Voronoi tessellation [19]. The local atomic deformation gradient is defined as a weighted average of the deformation gradients of adjacent tetrahedrons and strain tensors are computed directly from their definitions in terms of the deformation gradient. A similar approach has been used in [25, 26], where the local deformation gradient was first computed by minimizing a least square error between the transformed reference configuration and actual configuration, and subsequently it was modified by the addition of correction factors arising from the optimization procedure. In the present work we aim at deriving deformation measures that are based on the definition of a discrete equivalent to the continuum second-order deformation gradient that accounts for non-homogeneous deformations. The Cauchy–Born rule that assumes an affine mapping of atomic distance vectors from the reference configuration to the current configuration is no longer valid if the deformation is not sufficiently homogeneous. This is particularly true for regular, crystalline systems and the Cauchy–Born rule fails completely in amorphous systems due to necessary relaxation movement of atoms/molecules towards an equilibrium state. Furthermore, inclusion of second-order gradients in deformation characteristics of discrete, atomic systems allows making a natural connection to higher-order continua such as the strain gradient and the couple stress theories which might be related to the underlying molecular origin [12, 21].

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Fig. 1 Deformation mapping from the reference to the current configuration.

2 Atomic Second-Order Deformation Gradient We introduce the deformation mapping
that relates reference coordinates x to current coordinates X. In the continuum settings, x is a continuous variable whereas it assumes discrete values at atomic positions in discrete systems. Thus the mapping
in the discrete system can be seen as taking discrete values from such continuous displacement field (Figure 1). The quadratic approximation of the deformation mapping is assumed in the following form:
(x) = X → ⎧ 2 2 2 ⎪ ⎨ a0 + a1 x + a2 y + a3 z + a4 x + a5 y + a6 z + a7 xy + a8 xz + a9 yz = X b0 + b1 x + b2 y + b3 z + b4 x 2 + b5 y 2 + b6 z2 + b7 xy + b8 xz + b9 yz = Y (1) ⎪ ⎩ c0 + c1 x + c2 y + c3 z + c4 x 2 + c5 y 2 + c6 z2 + c7 xy + c8 xz + c9 yz = Z The values of 30 coefficients are determined based on information of the reference (x, y, z) and current (X, Y, Z) positions of N neighbouring atoms including the central atom. The number of neighboring atoms (including central atom) is taken as N > 10, in order to get number of equations greater than the number of unknowns. The best approximation of current atomic coordinates in the sense of least square differences is obtained if CT CU = CT R where C(N × 10) is the matrix of polynomial components, U(10 × 3) is the matrix of unknown polynomial coefficients and the matrix R(N × 3) represents coordinate components in the current configuration. The Gauss elimination technique can be applied in order to solve this equation. If the Delaunay tessellation is performed the number of equations Ncan be taken as number of nearest neighbours found that way provided that it is not smaller than 10.

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Otherwise a selected value of N can be chosen the same for each atom and equal to average number of near neighbours. This procedure provides a set of approximated atomic coordinates in the current configuration. In order to obtain deformation gradients of atomic bonds the deformation mapping
˘amust be applied to inter-atomic distance vectors d(ij ) , Figure 1. The quadratic approximation to distance vectors mapping is written as
(d) = D → ⎧ a1 dx + a2 dy + a3 dz + a4 dx2 + a5 dy2 + a6 dz2 + a7 dx dy + a8 dx dz + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + a9 dy dz = Dx ⎪ ⎪ ⎨ b d + b d + b d + b d2 + b d2 + b d2 + b d d + b d d + 1 x 2 y 3 z 4 x 5 y 6 z 7 x y 8 x z ⎪ + b9 dy dz = Dy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ c1 dx + c2 dy + c3 dz + c4 dx2 + c5 dy2 + c6 dz2 + c7 dx dy + c8 dx dz + ⎪ ⎪ ⎩ + c9 dy dz = Dz

(2)

where 27 polynomial coefficients are determined based on information of distances between central atom and its N neighbours in the reference (dx , dy , dz ) and current (Dx , Dy , Dz ) configuration. The method of solution to this problem is the same as for the previous case. Having an explicit form of the deformation mapping the transformation of the distance vector from an atom (i) to its neighbour (j ) can thus be expressed as 1 D(ij ) = Fhom d(ij ) + G : [d(ij ) ⊗ d(ij ) ] 2

(3)

where Fhom is the first-order derivative of the mapping
and corresponds to the linear deformation gradient whereas G is the second-order derivative of the mapping and represents the second-order deformation gradient.

3 Calculation Examples The stable C60 and diamond sheet structures have been calculated using semiempirical PM3 method which is based on the neglect of diatomic differential overlap (NDDO) approximation. PM3 is primary used for organic species, but is also parameterized for many main group elements. The stable ZnO nanowire structure and its electronic properties has been calculated using semi-empirical MNDO/d method which is a Modified Neglect of Diatomic Overlap method based on the neglect of diatomic differential overlap approximation. It follows from ab initio calculations that inclusion of d-orbitals significantly improves the predictions for compounds involving second-row elements and zinc groups. The major problem in including d-orbitals in the neglect of diatomic differential overlap approximation is the significant increase in distinct two-electron integrals which ultimately must be assigned

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Fig. 2 Comparison of deformed atomic positions from the molecular optimization of the C60 structure with solutions based on the least square and the second-order deformation gradients.

suitable values based on the experimental data. A workable MNDO/d model has been constructed in [31], and it is implemented in commercially available package HyperChem
. Above mentioned semi-empirical methods were used within the Polak–Ribiere conjugate gradient minimization method in all geometry optimization runs. The approximation to quasi-static uniaxial loading is achieved in two consecutive steps. First, the boundary atoms of the structure are moved to the position that corresponds to a predetermined displacement step. This is followed by the equilibration of the structure with boundary atoms fixed at their current positions. Figure 2 shows initial and deformed configurations of a C60 fullerene ball. The deformed state has been obtained by displacing aromatic rings at the top and bottom poles towards each other. The configuration of the deformed state i.e. the coordinates of all atoms has been calculated using the least square method [25,26] and the present second-order gradient method. Atomic coordinates from these two methods are compared with the coordinates that result from the molecular optimization procedure. A significant discrepancy exists between deformed atomic positions and those calculated from the least square method, particularly this is true for the atoms that underwent relatively large displacements. On the other hand, the second-order deformation gradient method provides very close approximation to the deformed atomic positions. A similar situation is observed when the diamond sheet is subject to three increasing shear loading (Figure 3). The solution from the second-order deformation gradient method follows accurately deformed atomic positions whereas the results

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Fig. 3 Comparison of deformed atomic positions from the molecular optimization of the diamond sheet with solutions based on the least square and the second-order deformation gradients.

Fig. 4 Non-homogeneity measure for three shear deformation levels of the diamond sheet.

from the least square method exhibit significant discrepancies with growing deformation. Characteristics of a non-homogeneous deformation are contained in the secondorder gradient G. An absolute non-homogeneity measure can be constructed from norms of the corresponding components of G [28], where each component is eval-

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Fig. 5 Non-homogeneity measure for the real structure and the structure deformed in an affine manner (a); colour code slip vector module that indicates displacement difference between nonaffine (real) and affine displacement field (b).

uated at the site of each atom from all neighbours used to calculate the mapping
and the maximum value is selected. A colour coded non-homogeneity measure is illustrated in Figure 4. An overall deformation of the diamond sheet seems to be homogeneous for all three loadings; nevertheless there exist pronounced local variations of the non-homogeneity measure particularly in the case of largest shear deformation. In the classical theory of elasticity the diamond sheet is viewed as a spatially homogeneous medium subject to uniform stresses at its boundaries and it would deform in an affine manner i.e. the deformation would be homogeneous with a constant strain. This notion of affine deformation is only valid at length scales large compared to any locally variable inhomogeneities. The non-affine deformation plays a significant role in understanding atomic/molecular response to external mechanical stimulus [6, 10, 11, 23]). We create an artificial diamond structure in which all atoms are moved in the affine manner in such a way that the overall affine shear deformation corresponds to the largest deformation of the real structure, Figure 5a. The non-homogeneity measure for the affine deformation is constant throughout the structure and due to numerical rounding errors is not exactly null but several orders of magnitude smaller than values of the scale bar. The deviation from affinity can be measure by a slip vector si given by si = Xi − xi − Xj − xj 

(4)

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Fig. 6 Colour code atomic shear strain components for three shear deformation levels.

Fig. 7 Energy gap in the nanowire electronic structure during tensile loading (a) and absorptivity spectra for different strain levels (b).

where the average   runs over j neighbours of atom i taken in the calculation of the second-order deformation gradient. Taking the module of the slip vector, it follows from Figure 5b that the smallest deviation from affinity is exhibited by central atoms where surface effects are suppressed. The Lagrangian atomic strain tensor can be calculated from components of the deformation gradient F appearing in Eq. (3) and its shear components are shown in Figure 6. Values of the shear strain indicated in this figure correspond to the average of the atomic shear strain. Interestingly, the overall value of the shear strain applied to the diamond sheet is very close to it and reads 0.024, 0.065 and 0.214 for selected deformation level, respectively. Thus the local atomic strain variations are averaged providing the overall strain imposed on the structure. The electronic properties of nanostructures can be dramatically altered by mechanical deformation [8, 18, 30]. Being one of the most important functional semiconductor materials, ZnO has been widely applied in the optoelectronic industry because of its excellent optical (high fluorescence yield in the UV and green), elec-

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Fig. 8 Non-homogeneity measure at different strain levels.

Fig. 9 Colour code atomic tensile strain components at different deformation levels.

trical and piezoelectrical properties. Figure 7 shows a stable structure of smallest ZnO nanowire subject to tensile loading. The electronic orbitals and the electronic spectrum have been extracted from configuration interaction calculations at the optimized geometry [24]. A large HOMO-LUMO jump of energy gap appears at strain 0.07, Figure 7a. The strain induces changes in the electronic structure of the nanowire and it can be expected that it influences absorbance of electromagnetic radiation in the visible and ultraviolet (UV-visible) region. Absorption bands observed in the UV-visible region are associated with changes of orbital occupations. Figure 7b shows absorption spectra at selected points. There is a clear indication that the visible spectrum, which starts approximately at 400 nm upwards, changes its form as deformation increases. In the visible range, there are two absorbance peaks until the deformation level reaches the strain value equal to 0.07. Subsequent loading steps show no absorbance in the visible range and what is left is the ultraviolet absorbance below 250 nm. Intensities and positions of peaks within the visible spectrum change as deformation progresses. These changes can be related to the strain level at which they

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were recorded, thus providing the deformation sensing by measuring the absorbance spectra. It appears that the coefficient that is equal to the ratio of the first and the second peak wavelength is proportional to the strain level, which is similar to the calibration curve in Raman spectroscopic test where the Raman peak shift is related to the strain level [24]. The non-homogeneity measure at selected strain levels is illustrated in Figure 8. The non-homogeneity measure at atomic sites is uniformly distributed if the overall strain is below the limit where the large band gap jump appears. A sudden change of the non-homogeneity measure distribution occurs for strains that are above the limiting value. Thus, the non-homogeneity measure can be used as a mechanical indicator of electronic changes that are induced by the mechanical stimulus. The atomic tensile strain components are shown in Figure 9. There is a clear indication that atoms at the left side of the wire move more than the rest which corresponds very well with the appearance of two zones in the non-homogeneity distribution at limiting value of the overall strain. Similarly to the diamond sheet case, values of the average atomic tensile strain are very close to corresponding overall values.

4 Conclusions A new approach for computing the deformation measures in discrete atomic systems has been presented. This work demonstrated that the second-order deformation gradient and related atomic strain tensor provide means for a detailed characterization of a deformation pattern that exists in nanostructures under mechanical loading. The non-homogeneity measure derived from the second-order deformation gradient appears to be a good indicator of electronic changes resulting from mechanical loading. However, the true test of the atomic strain concept is how well it approximates total strain of the simulation cell by summing local atomic strains over all atoms present in the system. In the best case this sum should be equal or very close to the total strain calculated from boundary conditions of the simulation cell as it has been documented in the present work. Once the second-order deformation gradient is computed the deformed configuration of structures can be recovered with very good accuracy even for large deformations. Another benefit of the present approach comes from the fact that the second-order deformation gradient is calculated from the deformation mapping which possesses an explicit form. Thus it allows calculating other parent quantities such as deviatoric stretch gradient and rotational deformation gradient. Averaging these quantities over all atoms of the structure provides a direct bridge to continuum theories such as a strain gradient elasticity and couple stress theory and from an atomic scale it allows determining constitutive constants that appear in these theories. This aspect will be pursued in a future work.

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References 1. Arndt, M., Griebel, M.: Derivation of higher order gradient continuum models from atomistic models for crystalline solids. Multiscale. Model. Simul. 4, 531–562 (2005). 2. Chen, Y., Lee, J.D., Eskandarian, A.: Examining the physical foundation of continuum theories from the viewpoint of phonon dispersion relation. Int. J. Engng. Sci. 41, 61–83 (2003). 3. Chen, Y., lee, J.D., Eskandarian, A.: Atomistic counterpart of micromorphic theory. Acta Mech. 161, 81–102 (2003). 4. Cormier, J., Rickman, J.M., Delph, T.J.: Stress calculation in atomistic simulations of perfect and imperfect solids. J. Appl. Phys. 89, 99–104 (2001). 5. Delph, T.J.: Conservation laws for multibody interatomic potentials. Modelling Simul. Mater. Sci. Eng. 13, 585–594 (2005). 6. DiDonna, B.A., Lubensky, T.C.: Nonaffine correlations in random elastic media. Phys. Rev. E 72, 066619 (2005). 7. Falk, M.L., Langer, J.S.: Dynamics of viscoplastic deformation in amorphous solids. Phys. Rev. E. 57, 7192–7205 (1998). 8. Guo, W., Tang, C., Guo, Y.: Nanointelligent materials and systems. Int. J. Nanosci. 5, 671–676 (2006). 9. Hardy, R.J.: Formulas for determining local properties in molecular dynamics simulations: Shock waves. J. Chem. Phys. 76, 622–628 (1982). 10. Hatami-Marbini, H., Picu, R.C.: Scaling of nonaffine deformation in random semiflexible fiber networks. Phys. Rev. E 77, 062103 (2008). 11. Head, D.A., Levine, A.J., MacKontosh, F.C.: Distinct regimes of elastic response and deformation modes of cross-linked cytoskeletal and semiflexible polymer networks. Phys. Rev. E 68, 061907 (2003). 12. Lam, D.C.C., Chong, A.C.M.: Effect of cross-link density on strain gradient plasticity in epoxy. Mat. Sci. Engng. A281, 156–161 (2000). 13. Leonforte, F., Boissière, R., Tanguy, A., Wittmer, J.P., Barrat, J.-L.: Continuum limit of amorphous elastic bodies. III. Three-dimensional systems. Phys. Rev. B 72, 224206 (2005). 14. Lutsko, J.F.: Stress and elastic constants in anisotropic solids: Molecular dynamics techniques. J. Appl. Phys. 64, 1152–1154 (1988). 15. Machová, A.: Stress calculations on the atomistic level. Modelling Simul. Mater. Sci. Eng. 9, 327–337 (2001). 16. Maranganti, R., Sharma, P.: Length scales at which classical elasticity breaks down for various materials. Phys. Rev. Lett. 98, 195504 (2007). 17. Maranganti, R., Sharma, P., Wheeler, L.: Quantum notion of stress. J. Aerospace Engng. 20, 22–37 (2007). 18. Minot, E.D., Yaish, Y., Sazonova, V., Park, J-Y., Brink, M., McEuen, P.L.: Tuning carbon nanotube band gaps with strain. Phys. Rev. Lett. 90, 156401 (2003). 19. Mott, P.H., Argon, A.S., Suter, U.W.: The atomic strain tensor. J. Comput. Phys. 101, 140–150 (1992). 20. Nakane, M., Shizawa, K., Takahashi, K.: Microscopic discussions of macroscopic balance equations for solids based on atomic configurations. Archiv Appl. Mech. 70, 533–549 (2000). 21. Nikolov, S., Han, C.-S., Raabe, D.: On the origin of size effects in small-strain elasticity of solid polymers. Int. J. Solids Struct. 44, 1582–1592 (2007). 22. Nishioka, K., Takai, T., Hata, K.: Interpretation of the atomic formulae for stress and stiffness coefficients. Philosophical Mag. A 65, 227–244 (1992). 23. Onck, P.R., Korman, T., van Dillen, T., van der Giessen, E.: Alternative explanation of stiffening in cross-linked semiflexible networks. Phys. Rev. Lett. 95, 178102 (2005). 24. Pyrz, R.: Properties of ZnO nanowires and functional nanocomposites. Int. J. Nanosci. 7, 29– 35 (2008). 25. Pyrz, R., Bochenek, B.: Atomic/continuum transition at interfaces of nanocomposite materials. Key Engng. Mat. 334–335, 657–660 (2007).

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26. Pyrz, R., Bochenek, B.: Discrete-continuum transition at interfaces of nanocomposites. Bull. Pol. Ac.: Tech. 55, 1–10 (2007). 27. Shen, H.: Tensile properties and electronic structures of C240 nanotube and 4C60 fullerene polymers. Int. J. Nanosci. 5, 99–107 (2006). 28. Sunyk, R., Steinmann, P.: On higher gradients in continuum-atomistic modeling. Int. J. Solids Struct. 40, 6877–6896 (2003). 29. Tanguy, A., Wittmer, J.P., Leonforte, F., Barrat, J-L.: Continuum limit of amorphous elastic bodies: A finite-size study of low-frequency harmonic vibrations. Phys. Rev. B 66, 174205 (2002). 30. Thean, A., Leburton, J.P.: Strain effects in large silicon nanocrystal quantum dots. Appl. Phys. Lett. 79, 1030–1032 (2001). 31. Thiel, W., Voityul, A.A.: Extension of MNDO to d orbitals: Parameters and results for the second-row elements and for the zinc group. J. Phys. Chem. 100, 616–626 (1996). 32. Triantafyllidis, N., Bardenhagen, S.: On higher order gradient continuum theories in 1-D nonlonear elasticity. Derivation from and comparison to the corresponding discrete models. J. Elast. 33, 259–293 (1993). 33. Zhang, X., Sharma, P.: Size dependency of strain in arbitrary shaped anisotropic embedded quantum dots due to nonlocal dispersive effects. Phys. Rev. B 72, 195345 (2005). 34. Zhou, M.: A new look at the atomic level virial stress: on continuum-molecular system equivalence. Proc. R. Soc. Lond. A 459, 2347–2392 (2003). 35. Zimmerman, J.A., Webb III, E.B., Hoyt, J.J., Jones, R.E., Klein, P.A., Bammann, D.J.: Calculation of stress in atomistic simulation. Modelling Simul. Mater. Sci. Eng. 12, S319–S332 (2004).

Looking beyond Limitations of Diffraction Methods of Structural Analysis of Nanocrystalline Materials Bogdan Palosz, Ewa Grzanka, Stanisław Gierlotka, Marcin Wojdyr, Witold Palosz, Thomas Proffen, Ryan Rich and Svitlana Stelmakh

Abstract In this work we discuss how to learn about the real atomic structure of nanocrystalline materials without misinterpreting the results of powder diffraction experiments. We discuss implications of nano-size on powder diffractograms based on some theoretical models of nanograins. Examples of experimental studies on nanocrystalline diamond and SiC are demonstrated.

1 Introduction It is safe to consider a nanocrystal as a unique “piece of matter” with its individual atomic structure and properties which are dependent on its characteristic dimensions. Development of “nanoscience” is driven by a common belief that there is a variety of new (still undiscoved) properties of nanocrystals which could be useful in new material technologies. Indeed, there are clear indications that there are significant differences in the properties between nano- and larger crystals [1–5]. Although we are often able to make use of the special properties of nanomaterials, the origin of specific “nano-properties” is usually not well recognized and not really understood. One of the key problems in understanding the behavior of nanomaterials is in Bogdan Palosz, Ewa Grzanka, Stanisław Gierlotka and Marcin Wojdyr Institute of High Pressure Physics, Polish Academy of Sciences, Warsaw, Poland Witold Palosz Brimrose Corporation, Baltimore, MD, USA Thomas Proffen Los Alamos National Laboratory, Los Alamos, NM, USA Ryan Rich TCU, Department of Physics and Astronomy, Fort Worth, TX, USA Svitlana Stelmakh Institute of High Pressure Physics, Polish Academy of Sciences, Warsaw, Poland; and TCU, Department of Physics and Astronomy, Fort Worth, TX, USA

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Fig. 1 Tentative models of a nanocrystal.

designing the right experiments and in applying the right tools for the experimental data elaboration that is to determine the specific size-dependent physical properties of the materials. The basic and essentially the only well established experimental technique for determination of the structure of matter with atomic resolution is diffraction. However, due to a complex structure of nanomaterials, standard methods of structural analysis are of limited use and non-standard and new approaches and tools for structural analysis of nanocrystals are necessary [6]. There is a variety of specific problems that need to be addressed: (i) what is the nanocrystal basic crystallographic structure; (ii) what are the differences beween similar crystal phases present in nanoand micro-polycrystalline samples (if those differences really exist); (iii) are there any lattice defects specific for nanograins, and others. To start with, one needs a reference model to elaborate the experimental diffraction data. In crystals of dimensions measured in micrometers, only a very small fraction of all atoms resides at the surface, therefore their effect on the diffraction patterns is insignificant: the expected signal from the surface is outside the sensitivity limit of even the best radiation sources and instruments available. The situation is very different for nanocrystals with sizes on the order of a few nm, where the total number of atoms located at and near the surface becomes comparable to the number of atoms in the grain interior [1]. Since interatomic distances between similar atoms may vary depending on the positions of the atoms in the grain (bulk vs. surface positions), a single lattice parameter might be insufficient for characterization of nanostructure [6]. The relation between interatomic potentials and the atomic structure of nanocrystals is therefore much more complex than in larger crystals. Thus the goals of structural studies on nano-crystals have to be formulated differently than those for poly-crystalline materials with relatively large, micron-size grains. Although the goals are different, the same experimental technique, powder diffraction, has to be used to study both kinds of materials [7].

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There is variety of models of a nanocrystal offered in the literature, starting from the assumption that nano-crystal is basically a small single crystal (Figure 1a). Usually, however, it is assumed that nanocrystals have a core-shell structure, where interatomic distances at the surface (in the surface shell with a given thickness) are different than those in the bulk (Figure 1b). One might also consider complex models assuming a modulation of the lattice density, what is equivalent to changes of interatomic distances within the particle volume. Such a structure might look like a sequence of tensile followed by compressive strains between the surface and the grain interior (Figure 1c). The question that crystallographer faces before elaborating a diffraction experiment is, which model best describes the results of the diffraction data analysis. Obviously the question concerns both qualitative and quantitative description of the sample. There are two basic methods of analysis of powder diffraction: 1. reciprocal space analysis, which refers to characteristic Bragg scattering [7, 8], and 2. real space analysis called atomic Pair Distribution Function analysis (PDF) [9–13]. Both techniques give information on the atomic structure which is, however, a volumetric average of the sample. That is the major concern of structural analysts of poly-crystalline materials in general, and nanocrystals in particular. None of those techniques is capable of providing a complete, unique description of a nano-crystal [14], but each has a unique capabilities with respect to information about the structure. An analysis made in “reciprocal space”, which refers to a unit cell and is based on examination of characteristic Bragg-type scattering, is sensitive directly to the long range-atomic order. The weakness of this approach is, that it is basically applicable only to crystalline materials without providing information on structural phases that do not show a long range order and do not contribute to Bragg scattering. In a routine procedure of structural analysis of conventional polycrystalline materials practically the only questions which one has to answer is to confirm that the examined sample is a single or a mixture of several crystallographic phases. If needed, structural refinement, e.g. with application of the Rietveld refinement software [8], is being done. In this case the diffuse scattering is ignored. Analysis made in “real space”, which provides information on the length and abundance of interatomic distances between pairs of atoms, is based on examination of the total scattering (this includes both characteristic Bragg reflections and diffuse scattering underneath the Bragg peaks) and thus, in principle, provides information on every single atom present in the sample [10]. The problem is that there are no simple (straightforward) methods which would allow to link specific interatomic distances to different structural components of the sample. In our case, it is impossible to find out in which part of the sample volume, the core or the surface, given interatomic distances occur. Neither real- nor reciprocal space analysis alone is well suited for nanocrystallography. The best approach encompasses a combination of both techniques, with extraction of consistent information derived from both methods. That may be

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a very effective searching tool for a unique model of nanocrystals [14, 20]. In this paper we discuss various models of nanograins and compare the effects that specific structures have on Bragg scattering and on interatomic distances (determined with PDF analysis).

2 Methodology of Diffraction Analysis of Nanocrystals: Theoretical Calculations Searching for a model which can decribe experimental diffraction data we refer to theoretical calculations of diffraction patterns performed for atomic models which we build based on our expertise, constructive creativity, etc. Interpretation of the experimental data (evaluation of the parameters describing the atomic structure) is based on comparison and best match of the experimental patterns with those computed theoretically. Theoretical patterns are calculated using the Debye equation [7–9]. One has to remember that information which we are able to extract from a diffraction experiment is always a volumetric average of the sample. These are the structural parameters of the real samples which should be referred to as average parameters describing the model. Since the reference material for any kind of crystalline sample examined by diffraction is a perfect crystal lattice, it is reasonable to ask about deviations of the structure of the examined samples from the perfect lattice. Lattice parameters as determined from Bragg reflections describe average dimensions of the unit cell representing the whole sample volume. Similarly, interatomic distances determined from PDF analysis (ri ) are values averaged over the whole sample volume. For any structural model and real samples we derive relative changes of interatomic distances

ri /ri0  (=
ri  − ri0 )/ri0 , where
ri  is the average value of a given interatomic distance and ri0 is the corresponding distance in the perfect crystal lattice) and the ratio of alp/a0 , where alp is the lattice parameter calculated from a given Bragg reflection and a0 is the lattice parameter of the perfect reference crystal. A standard examination of powder diffraction data of a crystalline material starts with determination of the lattice parameter, which can be calculated from the position of a single Bragg reflection. It appears that for nanocrystals with dimensions up to several tens of nanometers and having a perfect crystal lattice, the positions of Bragg peaks do not coincide with those corresponding to a perfect crystal lattice, and the value of individually calculated “lattice parameters” change from reflection to reflection [6]. This is due to small dimensions of the crystallites size which is smaller than the coherence length of the scattered beam. This effect is demonstrated in Figure 2 which shows, that although interatomic distances in the model are identical with those in the bulk single crystal (Figure 2a), the values of the lattice parameters calculated from individual reflections change with diffraction vector Q (= 4π sin /λ) (Figure 2b). That means, that for nanocrystals the term “the lattice parameter” as calculated from the Bragg equation no longer has a meaning of a

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Fig. 2 Diamond nanocrystal of 5 nm in diameter with a perfect lattice. (a) Average interatomic distances in the grain; (b) values of the lattice parameters calculated from individual Bragg reflections.

Fig. 3 Nanocrystal of a diamond of 4.6 nm in average diameter with a core-shell structure. (a) Average interatomic distances calculated for the grain; (b) values of the lattice parameters calculated from individual Bragg reflections (alp-Q dependence).

constant but, instead, constitutes a variable. Thus we need to replace the term “the lattice parameter” with the term “the apparent lattice parameter”, alp, and measure its value for individual Bragg reflections. The alp values plotted versus diffraction vector Q might be then used as the basis for evaluation of the real structure of nanograins.

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Fig. 4 Model of a nanocrystal with a double-shell structure. (a) Average interatomic distances calculated for the grain; (b) alp-Q dependence calculated from individual Bragg reflections.

3 Average Interatomic Distances and alp-Q Dependences in the Core-Shell Model of a Nanocrystal A widely accepted tentative model of a nanocrystal is a core-shell model where the core has a well defined crystal structure while the shell has a similar structure with either shorter or longer, relative to the core, inter-atomic distances (the surface shell is either shrinked or expanded with respect to the grain interior). The actual physical dimension of a nanocrystal, which is commonly used as its “characteristic dimension”, is in fact the diameter of the grain core plus twice the surface shell thickness. From this point of view, considering a core-shell model of a nanocrystal one needs to determine two “characteristic dimensions”: its physical dimension and the thickness of the surface shell. The parameters which fully characterize this simple model are: 2R is the grain diameter, s0 is the thickness of the surface shell, (
r)shell is the change of the shortest interatomic distance in the shell relative to the core. The effect of a decrease in the shortest interatomic distance in the shell by 5% on all average interatomic distances and on the alp-Q plot is demonstrated in Figures 3a and 3b, respectively. Shortening of the interatomic distances in the shell has the strongest effect on the shortest and longest (close to the grain diameter) average r values, and a relatively small effect on the distances closer to the grain radius. Because changes in the average values of interatomic distances are directly related to changes in the average values of inter-planar spacings d (each Brag reflection corresponds to a given d value), it is obvious that there is a correlation between changes in the average r values and changes in the apparent lattice parameters corresponding to specific d values. As seen in Figure 3b, the alp value tends to reach maximum, but they will never reach the alp value corresponding to the lattice parameter of the grain core (similarly the average r value corresponding to the grain radius will never reach the real value of interatomic distances in the core).

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Fig. 5 Model of a nanocrystal with a five-shell structure. (a) Average interatomic distances calculated for the grain; (b) values of the lattice parameters calculated from individual Bragg reflections.

Figure 4 shows a model of a nanograin with a double-shell – core structure, where the outer shell with compressed lattice is followed by an expanded shell. The total volumetric compression of the outer shell is equal to the total expansion of the inner shell and, as a result, the physical dimension of the grain remains unchanged compared to the grain with a uniform lattice equal to that in the grain core. The plot in Figure 4a shows changes in the average interatomic distances for the model. The values of the shortest and the longest distances, r1 and r30 , are equal to those in the grain core. Other average inter-atomic distances first increase at the extreme distances and then decrease to some local minimum occurring at half-diameter distance, cf. Figure 3. Figure 4b shows that in the whole range of the diffraction vector Q, the alp values are larger than in the grain core, and that alp tends to reach minimum which is still larger than the lattice parameter of the grain core by a factor of about 1.0025, the same factor by which the average r value of the length equal to the half-diameter is enlarged relative to the real value of r0 . The last model which we examine here is a multi-shell structure with a modulation of the lattice density, where tensile and compressive strains occur alternately and extend from the grain surface to the grain interior. In terms of structural parameters this situation would correspond to elongation and shortening of interatomic distances along the particle radius (Figure 5). The plot showing changes of the average values of interatomic distances ri is very complex and shows minima and maxima which reflect the modulation of the interatomic distances implemented in the model. Correlation between average r values, Figure 5a, and the minimum of alp-Q dependence, Figure 5b, is similar to that observed for simpler models, Figures 3 and 4.

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Fig. 6 Experimental results on SiC crystals of 11 and 40 nm in diameter. (a) Distribution of changes of the average interatomic distances; (b) corresponding alp-Q plots.

Comparison of Figures 3 to 5 shows that alp-Q plots of various core-shell models have similar character, which means that in case of such complex structures a fully unique interpretation of the apparent lattice parameters might be questionable. On the other hand, the plots of
ri /ri for different models show distinctive differences with the relation symmetrical about the distance corresponding to the grain radius. This shows that the results of structural analysis performed with application of real and reciprocal space methods which we demonstrated above give consistent results, and that for a unique interpretation of the diffraction data real space analysis is necessary.

4 Experimental Results: SiC and Diamond Nanocrystals The first condition for a meaningful diffraction experiment which could provide a valuable information on nanocrystals with dimensions on the order of a few nm is an access to a large diffraction vector range. A conventional laboratory X-ray sources using Cu and Mo radiation are far insufficient to collect diffraction data which would serve the purpose: the maximum accessible Q ranges for those sources are only about 6 and 12 Å−1 , respectively. Q range required for examination of alpQ dependence is about 20–25 Å−1 [6], while PDF analysis with sufficient resolution requires Q much in excess of 25 Å−1 [9]. That means that structural studies on nanocrystals can nowadays be performed only with application of hard synchrotron radiation or “hot” neutrons. Figure 6 presents the results of examination of SiC powders with grains 11 and 40 nm in average sizes. Figure 7 shows our results for diamond nanocrystals of 4.6 nm in diameter. The results are based on neutron diffraction data collected at NPDF Station, LANSCE at LANL, Los Alamos, USA. The data were elaborated by PDF analysis using PDF_getN program.

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Fig. 7 Experimental results for diamond nanocrystals of 4.6 nm in average diameter. (a) Distribution of changes of the average interatomic distances in samples which underwent differrent chemical and temperature processing; (b) corresponding alp-Q plots. I: nanodiamond obtained by detonation technique after purification with ozone; II: the same nanodiamond purified with a mixture of sulfuric acid/chromic anhydrite mixture [16]; III: UD96 (commercial powder from Microdiamant AG); IV: UD96 powder annealed at 1200◦ C under vacuum and containing about 15% of non-diamond carbon [15].

Figure 6a shows changes of the average r values with r, Figure 6b shows the corresponding alp-Q plots. Due to a limited resolution of the g(r) function derived from PDF analysis [9] and a strong overlap of individual g(r) maxima (which correspond to specific interatomic distances), only r values up to about 20 Å were identified. This covers only a part of all interatomic distances in the grain, thus a complete shape of these relations (parts (a) of Figures 3 to 5) cannot be deduced from our data. What is evident from these plots, however, is that nanocrystalline SiC is definitely not a small single crystal (in which case there would be no change in the r values, Figure 2), and that the simple core-shell model is insufficient for an accurate description of the structure of nano-SiC, cf. Figure 3. The analysis of the alp-Q plots of these samples (Figure 6b), based on application of the core-shell model with a single shell suggests, that the SiC grain has about 0.5 nm thick shell with interatomic distances smaller by about 3% relative to those in the grain interior. Within this approximation, the interatomic distances in the interior of the grains are larger than in a single crystal SiC at least by about 0.1% for 11 nm grain, and slightly larger than that for 40 nm SiC powder. Figure 7 shows experimentally measured changes of interatomic distances r and corresponding alp-Q plots of nanodiamond powders obtained from the same synthetic raw material but purified by different chemical treatment procedures, samples I-III. Sample IV was obtained from III by annealing at 1200◦C under vacuum, what resulted in a partial graphitization of its surface.

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The relation of relative changes in the average ri ’s versus ri in Figure 7a are similar to those calculated for the 5-shell model presented in Figure 5. That suggests, that nanodiamond grains show a non uniform structure where the length of interatomic distances along the grain radius changes within about 3%. The outmost shell is expanded compared to the reference diamond and this tensile strain is then relaxed stepwise towards the grain center. Although the shapes of plots I to III in Figure 7a are similar, the differences between the maxima and minima decrease (the curves flatten) from sample I thru III. This suggests that the strains (which we define as a deviation of the average r values from the reference values of the large crystal) present in sample I are the strongest and they get weaker in samples II and III. It is worth noticing that this behavior is accompanied by a shift of all
r’s to smaller values, which means that the physical dimensions of the grains decrease. This conclusion is consistent with the alp-Q plots presented in Figure 7b, which show a shift to smaller values when going from sample I thru III. The r values of plot II are roughly those of plot I multiplied by the factor of 0.9995, those of sample III by a factor of about 0.9992. The analysis of alp-Q plots performed with application of a simple core-shell model showed, that those plots may be interpreted assuming that the surface shell has the thickness of about 0.3 nm and the interatomic distances in the shell are longer by about 3% than those in the grain core [15]. (Examination of complex models with multi-shell structure is presently under way and will be published at a later time.) A presence of graphite sp2 bonds (the length of which is about 92% of that of sp3 bonds in diamond) in sample IV leads to a very pronounced shortening of all r values, particularly the shortest ones, relative to samples I–III. However, a presence of carbon atoms with sp2 bonds at the surface of nanodiamond has no effect on Bragg scattering (and, thus, on the alp-Q plots which are dependent on the diamond structure alone). From the alp-Q plot of sample IV it can be seen that the entire nano-diamond grain is shrunk relative to samples I–III. It was suggested in the literature that a presence of sp2 carbon on the surface of nanodiamond leads to generation of a strong compressive stress on the diamond grain [17]. The present study supports this suggestion, but it also shows that similar effects occur when the surface is modified by chemical or thermal treatment.

5 Discussion and Conclusions In this work we discussed tentative models of a nanograin. We are aware of the fact that there is still a variety of alternate reasonable models which can be constructed. Obviously, a single technique is insufficient to prove that a given model matches the real structure best. When it comes to diffraction techniques, a satisfactory verification of a given model may be assumed if it describes properly the effects that are measured experimentally and allows for extraction of parameters which provide a quantitative decription of the model. From this point of view we are quite satisfied with the multi-shell model which explains rather well changes of experimental in-

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teratomic distances and alp-Q dependencies measured for SiC and diamond. The agreement between the experimental data and theoretical calculations obtained for diamond nanocrystals is quite convincing, although this kind of experimental data is available only for the smallest, a few nm in size grains. Obtaining similar data for larger grains is beyond limitations imposed by powder diffraction technique due to experimentally limited range of diffraction vector Q. Also the resolutions of diffraction pattern itself (measurement of the positions of individual Bragg reflections) and of g(r) function (obtained by PDF analysis) are limited. Therefore, although powder diffraction technique is the main method for examination of the smallest nanocrystals, there are technical limitations of using it as a standard/routine technique for examination of nanomaterials of all sizes [6]. Undoubtedly, the ideal method to determine the structure of a nanograin that one could think of is a structural analysis performed on a single nano-grain. This kind of studies are already under development but they require much better and much stronger radiation sources than presently available [18, 19]. It is certain that studies on nanocrystals with use of coherent scattering will become an alternative to powder diffraction in predictable future, and it will come along with development of X-ray laser sources and the advantage of their great brightness and excellent coherence. It is quite natural that working in the field of experimental powder diffraction we are well aware of the shortcomings imposed by diffraction methods, both by their technical limitations and by the available methods of elaboration of the experimental data. Those limitations affect our ability to set novel tasks and envision new research horizons. Therefore it is worthwhile to go beyond those limitations and to embrace a different point of view, where the “impossible is possible”. Such opportunity is brought about by numerical simulations, i.e. by a virtual experiment having no technical limitations of experimental reality. It is the area of a great potential which has essentially not been applied to studies on nanomaterials yet. It could be a tremendous inspiration for experimenters, crystallographers, and a great opportunity for the modelers to verify their theoretical predictions. There are only a very few studies on simulations performed with the goal of determining the atomic structure of the entire volume of a nanocrystal. A good example is a combination of simulation methods, powder diffraction, and EXAFS (XANES) performed for 3 nm ZnS [19]. The study showed undoubtedly, that Molecular Dynamics simulations give an oversimplified image of nanoparticles, and that there exists a strong disorder between the interior and surface of the grains [20]. There are numerous MD studies performed on specific structures of the surface (grain boundaries) of large and small objects [21], and large scale modelings to explain the unique mechanical properties of nanocrystalline materials [22–26]. Since those properties are related to the outmost surface atomic layers (which form grain boundaries), implementation of short range interatomic potentials in simulations might lead to satisfactory results. For such purposes an analysis of the atomic structure underneath the surface might not be necessary. A good example is SiC for which extended and quite successful studies were performed on the properties of sintered materials [27, 28]. However, the lattice properties calculated with application of similar interatomic potentials do not reproduce well the specific structural properties of

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this material observed experimentally [29] (a presence of stacking faults, different lattice densities of the cubic and trigonal modifications of SiC, changes of the c/a ratio with a change in the lattice symmetry [29, 30]). It is obvious that structural properties are not reproduced because interatomic potentials used for the simulations are insufficient (too much simplified) to reproduce symmetry-dependent relationships. From the point of view of a crystallographer, a good criterion of strength of simulation methods would be its sensitivity to lattice symmetry and lattice parameters, in particular their changes with the grain diameter and particle environment. In the present work we showed that real- and reciprocal-space analysis perfomed using powder diffraction may provide valuable information on the atomic structure of nanocrystals with the size on the order of several nanometers. Although we are confident that the models which we propose reproduce well the real structure of nanograins, there is a number of unanswered questions concerning the origin of (multi)shell-core structures and interactions responsible for such structures. To answer those questions, a combination of diffraction with other techniques could be very effective, particularly if different methods lead to consistent conclusions. At present it is well established that a nano-crystal is not a single crystal surrounded by a surface shell, but a complex structure with strains (disordering) extending towards its interior. This conclusion follows both from the analysis of Bragg scattering and from that of total scattering (PDF) [31]. Finding a geommetrical model which would match the diffraction results, and getting similar model starting from the physical rules and interactions is very challenging. To perform a diffraction study on nanopowders one needs a reference model. On the other hand those that use simulations and create models, need its verification, which can be done by application of experimental diffraction. Thus a close cooperation between modelers and experimentalist can be very fruitful. Acknowledgements This work has benefited from the use of NPDF station at the Lujan Center at Los Alamos Neutron Science Center. The support of the US Department of Energy/LANLLANSCE (projects #20062110, 20062129 and 20071133) is greatfully acknowledged. This study was supported by NSF grant DMR 0502136 and by the Polish Ministry of Education and Science grant 3T08A 02029. The authors are grateful to Olga Shenderova, International Technology Center, Raleigh, NC, for providing nanodiamond powders.

References 1. C. Suryanarayana (Ed.), Non-equilibrium Processing of Materials, Pergamon, 1999. 2. H. Gleiter, Nanostructured materials: Basic concepts and microstructure, Acta Mater. 48 (2000) 1–29. 3. A.P. Alivisatos, Nanocrystals: Building blocks for modern materials design, Endeavour 21 (1997) 56–60. 4. Z.L. Wang, Characterization of Nanophase Materials, Wiley-VCH, 2000. 5. J.T. Lue, A review of characterization and physical property studies of metallic nanoparticles, J. Phys. Chem. Solids 62 (2001) 1599–1612.

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6. B. Palosz, S. Stelmakh, E. Grzanka, S. Gierlotka, and W. Palosz, Application of the apparent lattice parameter to determination of the core-shell structure of nanocrystals, Nanocrystallography, Z. Krist. 222 (2007) 580–594. 7. D.L. Bish and J.E. Post, Modern powder diffraction, Rev. Mineral. Geochem. 20 (1989). 8. R.A. Young, The Rietveld Method, International Union of Crystallography, Oxford University Press, 1993. 9. T. Egami and S.J.L. Billinge, Underneath the Bragg Peaks: Structural Analysis of Complex Materials, Pergamon, 2003. 10. S.J.L. Billinge and M.F. Thorpe, Local Structure from Diffraction, Plenum Press, New York, 1998. 11. Th. Proffen and S.J.L. Billinge, PDTFIT, a program for full profile structural refinement of the Atomic Pair Distribution function, J. Appl. Cryst. 32 (1999) 572–575. 12. Th. Proffen, S.J.L. Billinge, T. Egami, and D. Louca, Structural analysis of complex materials using the atomic pair distribution function – A practical guide, Z. Krist. 218 (2003) 132–143. 13. V. Petkov, S.J.L. Billinge, P. Larson, S.D. Mahanti, T. Vogt, K.K. Rangan, and M.G. Kanatzidis, Structure of nanocrystalline materials using atomic pair distribution function analysis: Study of LiMoS2 , Phys. Rev. B65 (2002) 092105. 14. B. Palosz, E. Grzanka, S. Gierlotka, S. Stelmakh, R. Pielaszek, U. Bismayer, J. Neuefeind, H.-P. Weber, Th. Proffen, R. Von Dreele, and W. Palosz, Analysis of short and long range atomic order in nanocrystalline diamonds with application of powder diffractometry, Z. Krist. 217 (2002) 497–509. 15. B. Palosz, C. Pantea, E. Grzanka, S. Stelmakh, Th. Proffen, T.W. Zerda, and W. Palosz, Investigation of relaxation of nano-diamond surface in real and reciprocal spaces, Diamond and Related Materials 15 (2006) 1813–1817. 16. I. Petrov, O. Shenderova, V. Grishko, V. Grichko, T. Tyler, G. Cunningham, and G. McGuire, Detonation nanodiamonds simultaneously purified and modified by gas treatment, Diamond and Related Materials 16 (2007) 2098–2103. 17. L. Sun and F. Banhart, Graphitic onions as reaction cells on the nanoscale, Appl. Phys. Lett. 88 (2006) 193121. 18. I.K. Robinson and I.A. Vartanyants, Use of coherent X-ray diffraction to map strain fields in nanocrystals, Appl. Surface Sci. 182 (2001) 186–191. 19. W.J. Huang, R. Sun, J. Tao, L.D. Menard, R.G. Nuzzo, and J.M. Zuo, Coordination-dependent surface atomic contraction in nanocrystals revealed by coherent diffraction, Nature Mater. 7 (2008) 308–313. 20. B. Gilbert, H. Zhang, F. Huang, J.F. Banfield, Y. Ren, D. Haskel, J.C. Lang, G. Srajer, A. Jürgensen, and G. Waychunas, Analysis and simulation of structure of nanoparticles that undergo a surface-driven structural transformation, J. Chem. Phys. 120 (2004) 11785–11795. 21. M. Kohyama, Computational studies of grain boundaries in covalent materials, Modeling Simul. Mater. Sci. Eng. 10 (2002) R31–R59. 22. H. Svygenhoven, D. Farkas, and A. Caro, Grain-boundary structure in polycrystalline metals at the nanoscale, Phys. Rev. B62 (2000) 831–838. 23. S.H. Svygenhoven, P.M. Derlet, and A. Hasnaoui, Atomistic modeling of strength of nanocrystalline metals, Adv. Engrg. Mater. 5 (2003) 345–350. 24. J. Schitz, F.D. DiTolla, and K.W. Jacobsen, Softening of nanocrystalline metals at very small grain sizes, Nature 391 (1998) 561–563. 25. J. Schitz, T. Vegge, F.D. DiTolla, and K.W. Jacobsen, Atomic-scale simulations of the mechanical deformation of nanocrystalline materials, Phys. Rev. B60 (1999) 11971–11983. 26. S.R. Phillpot, D. Wolf, and H. Gleiter, Molecular-dynamic study of the synthesis and characterization of a fully dense, three-dimensional nanocrystalline material, J. Appl. Phys. 78 (1995) 847–861. 27. I. Szlufarska, A. Nakano, and P. Vashishta, A crossover in the mechanical response of nanocrystalline ceramics, Science 309 (2005) 911. 28. I. Szlufarska, R.K. Kalia, A. Nakano A., et al., Atomistic mechanisms of amorphization during nanoindentation of SiC: A molecular dynamics study, Phys. Rev. B71 (2005) 174113.

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29. B. Palosz, S. Stelmakh, E. Grzanka, S. Gierlotka, S. Nauyoks, T.W. Zerda, and W. Palosz, Origin of macro- and micro-strains in diamond-SiC nanocomposites based on the core-shell model, J. Appl. Phys. 102 (2007) 074303. 30. M.C. Righi, C.A. Pignedoli, G. Borghi, R.Di Felice, and C.M.Bertoni, Surface-induced stacking transition at SiC(0001), Phys. Rev. B66 (2002) 045320 31. A.S. Masadeh, E.S. Bozin, C.L. Farrow, G. Paglia, P. Juhas, S.J.L. Billinge, A. Karkamkar, and M.G. Kanatzidis, Quantitative size-dependent structure and strain determination of CdSe nanoparticles using atomic pair distribution function analysis, Phys. Rev. B76 (2007) 115413.

Multiscale Modelling of Mechanical Anisotropy of Metals Grethe Winther

Abstract The mechanical anisotropy of a rolled metal sheet depends both on its texture and dislocation boundary characteristics (boundary plane, misorientation angle and boundary spacing), thereby linking dislocation interactions at the nanometre scale to bulk properties through phenomena involving individual grains of dimensions of the order of 100 micrometres. The focus of the modelling is on the boundary plane. Atomistic and dislocation dynamics simulations have not yet been able to produce sufficiently realistic dislocation structures to provide the planes needed. Instead the boundary planes can be predicted at the grain scale based on the slip systems operating during rolling, which generate the dislocations available for inclusion in the boundaries. The predicted boundary planes are verified by transmission electron microscopy and the predicted anisotropy by mechanical testing.

1 Introduction Rolling of metal sheets and subsequent forming of the sheet, for example by deep drawing, is a standard process in the metallurgical industry. The beverage can is a well-known example. The mechanical anisotropy of the rolled sheets influences the formability of the sheet significantly. The anisotropy is the source of ‘earing’ and also leads to an uneven wall thickness. The anisotropy is typically investigated by tensile tests along different directions in the sheet, and the properties in focus are the flow stress and the ratio of the two strain components perpendicular to the elongation strain (also known as the contraction ratio or the Lankford coefficient). The flow stress may be measured easily (taken as the value at 0.2% elongation)

Grethe Winther Center for Fundamental Research: Metal Structures in Four Dimensions, Materials Research Division, Risø National Laboratory for Sustainable Energy, Technical University of Denmark, DK-4000 Roskilde, Denmark; e-mail: [email protected]

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while the contraction ratio is more difficult to measure precisely as the changes in the sample dimensions involved are relatively small. While the crystallographic texture is an important source of mechanical anisotropy, it is not always enough to account for the experimentally observed anisotropy in rolled sheets. In aluminium, for example, the flow stress perpendicular to the rolling direction systematically exceeds that along the rolling direction [1], even after correction for texture effects, and the difference increases with the rolling strain. This is due to alignment of the deformation induced dislocation structure as confirmed by experiments showing that this additional anisotropy can be suppressed by annealing of the sheet [2, 3], addition of solutes [2] and processing of the sheet by cross-rolling instead of conventional monotonic rolling [4], which destroys or alters the dislocation structure. The dislocation structure is formed by interaction between individual dislocations in the nanometre regime to assemble into dislocation boundaries. The boundary spacing is typically a few micrometres. At the scale of the grain (∼100 micrometres) deformation by slip must propagate across these boundaries and the mechanical anisotropy is a bulk property measured at the scale of the sample. Modelling these phenomena therefore becomes a multiscale problem.

2 Dislocation Boundaries As seen in the transmission electron micrograph in Figure 1 dislocation boundaries may be classified as either short, randomly oriented boundaries (also termed incidental dislocation boundaries (IDBs)) or extended planar boundaries (geometrically necessary boundaries (GNBs)). The latter are revealed as long straight lines in the micrograph, and are emphasized in the tracing below. The boundaries may be further characterised by a number of quantitative parameters, which are all important when modelling the resulting anisotropy: • Misorientation angle: The crystallographic lattices on the two sides of a boundary are misoriented with an angle, which is inversely proportional to the dislocation spacing in the boundary. • Boundary spacing: The spacing between the boundaries is of the order of a few micrometres at low strain. • Boundary plane: The GNBs are planar and parallel boundaries. The plane with which they align may be characterised either in the sample coordinate system, typically in relation to the deformation axes, or in the crystallographic lattice.

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Fig. 1 Transmission electron micrograph of the typical dislocation structure in metals like aluminium and steel deformed to low strains. The dislocation boundaries are marked in the tracing below together with the crystallographic misorientation across the boundaries. From [5].

2.1 Plane of the GNBs In the sample coordinate system the GNBs exhibit a general alignment with the most stressed sample planes which for rolling are the planes inclined 45◦ to the rolling direction and parallel to the transverse direction of the sheet. The mean inclination angle to the rolling direction is, however, often smaller and the width of the distribution of inclination angles around 30◦ . In the crystallographic lattice the GNB plane depends on the crystallographic grain orientation. For tension the grain orientation dependence of the GNB plane has been fully characterised for the entire relevant orientation space [6]. For rolling, however, the orientation space is eight times larger, making a full characterisation very time consuming and this has not been done to date. Sufficient data are, however, available to demonstrate that also in rolling the crystallographic GNB planes depend on the grain orientation [5, 6].

3 Multiscale Modelling Scheme Polycrystal plasticity models for the evolution of deformation textures as well as the associated mechanical anisotropy are well-established and typically proceed by an

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Fig. 2 Sketch of the typical polycrystal plasticity model.

iterative procedure as sketched in Figure 2. For each small deformation step and for each grain the slip systems are calculated from the initial grain orientation (arrow 1). Subsequently the lattice rotation is calculated (arrow 2), leading to an updated grain orientation, which is then the basis for the next deformation step (arrow 3). The flow stress may be derived from the slip system activity (arrow 4), often involving a hardening law, which may also be updated. The evolution of the dislocation structure is more difficult to model. Neither atomistic simulations nor dislocation dynamics have so far been able to produce GNBs with the experimentally observed characteristics. Dislocation boundaries are often believed to be low energy dislocation structures (LEDS) [7], meaning that the dislocations in the boundaries screen each others’ stress fields so that the boundaries are free of long range stresses. Detailed studies of individual GNBs have revealed that a large fraction of the dislocations in the boundary corresponds to those generated by the expected active slip systems [8, 9]. These observations are in agreement with calculations assuming that the GNBs are low energy dislocation structures [8– 10]. However, the LEDS principle has not been capable of actually predicting the observed dislocation structure characteristics. The evolution of angles and spacings with the strain follows some common principles: as the strain increases, the boundary spacing decreases while the misorientation angle increases. In the present paper typical values measured experimentally are employed. The grain orientation dependence of the crystallographic GNB plane is believed to originate from an underlying dependence of the active slip systems. Recently, universal relations between slip systems and crystallographic GNB planes have been formulated [11]. These relations are independent of deformation mode. They have been derived based on data from tension and rolling for grains where the slip systems are unambiguously determined by the Taylor model and a Schmid factor analysis. These two methods represent extremes in only considering the strain and stress

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Fig. 3 Modification of the standard modelling scheme for prediction of lattice rotations (see Figure 2) to model the coupling of lattice rotations and dislocation structures through the slip systems.

conditions, respectively, of the grain. The predictive capacity of the relations has, however, only been tested on a few examples from torsion deformation and only on symmetric grain orientations. In view of the existing modelling schemes in Figure 2 for texture effects based on the slip systems it would be ideal to also model the GNB planes by their relation to the slip systems as sketched in Figure 3 where the traditional scheme is extended by a new branch (arrow 5). Such a procedure has previously been proposed [12] but due to the lack of a general predictive model for the dislocation structure evolution it has not been fully implemented. This paper exploits the recent understanding of the relations between slip systems and GNB planes to make progress in the modelling of the coupled effects of texture and dislocation structures on the flow stress anisotropy of rolled sheets. The first step is to develop the predictive model for the grain orientation dependence of the GNB planes in rolling and the second is to incorporate it into a previously developed flow stress model [13–15].

4 Prediction of GNB Planes in Rolling Table 1 summarises the relations between slip systems and GNB planes identifed in [11], where more details may be found. Slip systems which either operate in the same slip plane or in the same direction and combinations of these give rise to different GNB planes, and may also lead to a structure consisting only of randomly oriented cells, i.e. no planar GNBs. In the following the slip systems are predicted by the Taylor model. The fraction of the total slip in the grain which is accounted for by the slip system combinations

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G. Winther Table 1 Summary of relations between slip systems and GNB planes.

Slip system characteristics

GNB plane

Single or coplanar slip Codirectional slip

{111} {xyy}, closer to the more active slip plane and parallel to the slip direction no GNBs but only cells {351}, {441} or {115}, depending on the exact combination

2× easily cross slipping codirectional Combinations of 3 systems, of which one is coplanar and codirectional, respectively, to the others

Fig. 4 Predicted GNB planes based on two models (see text) and experimental data for grain the orientation space of rolling.

in Table 1 is the parameter considered for each grain.1 The slip system combination accounting for the highest slip fraction is assumed to be the one, which determines the GNB plane. The result is presented in Figure 4a, where different shades of grey represent different predicted GNB planes. Experimental data points for aluminium [5, 6] and copper [6] are superimposed. Experimentally, the GNB planes have only been classified as aligned with {111} or not-aligned with {111}. Some grains form one of each. It is known from studies of grains close to a stable rolling texture fibre that the dominant set of GNBs not-aligned with {111} in these grains lie on {xyy} planes. For the present comparison the majority of the grains observed to have 1

Only considering systems which account for more than 10% if the total slip.

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GNBs on both {111} and another plane should therefore be taken as having GNBs on {xyy} planes, which is in good agreement with the prediction in Figure 4a. The prediction of grain orientations with only cells is also good. Grains observed to have GNBs aligned with {111} are also predicted to have these GNB planes. However, a substantial number of grains observed to have GNBs, which are not aligned with {111}, are seen to be predicted to have GNBs on {111}. This discrepancy is ascribed to inaccuracies in the slip system prediction using the Taylor model. A Schmid factor analysis of these grain orientations predicts slip systems of the type which give GNBs on {351}, {441} or {115} planes, i.e. which agree with the observations of GNBs not aligned with {111}. To obtain the best presently available prediction of the grain orientation dependent GNB planes in rolled sheets a threshold value is employed so that grains where the fraction of slip giving rise to GNBs on {111} exceeds 45% are predicted to actually form such GNBs. This parameter is evaluated first and the procedure used for Figure 4a is then applied to the remaining grains. Such a threshold has previously been succesful in predicting GNBs on {111} [16], where the GNB planes in the remaining grains were not considered. The new prediction is shown in Figure 4b and is seen to be in even better agreement with experiment than Figure 4a. The prediction in Figure 4b is consequently used in the flow stress calculations.

5 Modelling Flow Stress Anisotropy of Rolled Sheets 5.1 Anisotropic Critical Resolved Shear Stresses During deformation the gliding dislocations interact with the dislocation boundaries. This interaction is assumed to be similar to the interaction with ordinary grain boundaries and consequently a Hall–Petch type relation is employed [17]. The spacing between the GNBs as experienced by the individual slip system therefore becomes the important parameter changing the critical resolved shear stress from system to system. This spacing is proportional to the perpendicular GNB spacing, K, but must be multiplied by a geometric factor, di , which takes into account the orientation of the GNB plane relative to each slip system. The factor di is here taken as the distance measured along the slip direction. The critical resolved shear stress of slip system i, then becomes: −1/2

τcrss,i = τ0 + x · K −1/2 · di

,

(1)

where τ0 is the isotropic part common to all slip systems and x the Hall–Petch slope. Considering various mechanistic models, this parameter has been seen to be proportional to the misorientation angle across the GNBs [18] and therefore depends on the spacing of the dislocations in the GNB. The isotropic part of the critical resolved shear stress, τ0 , is the sum of contributions from the friction stress of the undeformed metal and from the randomly

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G. Winther Table 2 Parameters for the critical resolved shear stress for two rolling strains. From [13]. τp

τ0 (MPa)

× (MPa·µm1/2 )

K (µm)

0.11 0.2

17.7 20.1

26.8 35.0

4.3 3.5∗

∗ For

the largest grain size the GNB spacing K is 5 µm [1].

oriented part of the dislocation structure, namely the IDBs and loose dislocations. The latter contributions are given by the dislocation densities, normally relating the dislocation density for the IDBs to their misorientation angle and spacing [19].

5.2 Importance of Grain Orientation Dependent GNB Planes The flow stress anisotropy is predicted for three commercially pure aluminium samples (AA1050) with different grain sizes. The anisotropy of these materials is investigated experimentally in [1], where it is also established that all the materials have weak rolling textures. The parameters τ0 and x of Equation (1) used in the following predictions are those calculated in [13] based on typical microstructural data for aluminium (see table 1). The three materials are assumed to have identical parameters except for the GNB spacing, K, which for the coarse grained material is measured to be almost twice that of the two other materials [1]. Two different flow stress predictions are included in Figure 5, the first assuming that all GNBs align with the macroscopically most stressed planes and the second incorporating the predicted grain orientation dependent GNB planes.

5.2.1 Macroscopically Aligned GNBs It is assumed that half of the grains of each orientation has one set of GNBs inclined +45◦ to the rolling direction while the other half has GNBs inclined −45◦ . In Figure 5 the flow stress along different directions in the rolled sheet (represented by the angle β between the original rolling direction and the tensile direction) is shown as the dashed curve. The predicted flow stress is seen to overestimate the absolute value of the flow stress and in particular for the medium grain size to underestimate the anisotropy. While fine tuning (with an element of fitting) of the parameters in Equation (1) can bring the absolute flow stress level in agreement with experiment the degree of anisotropy for the medium grain size cannot be modelled quantitatively assuming only macroscopically aligned GNBs.

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Fig. 5 Flow stress anisotropy for three aluminium samples with different grain sizes and rolled to different prestrains (εp ) before tensile testing along different directions in the sheet given by the angle β, with β = 0◦ along the rolling direction and β = 90◦ along the transverse direction. Experimental data (symbols) from [1]. The anisotropy is calculated using (i) macroscopically aligned GNBs (dashed curve), (ii) grain orientation dependent GNB planes (solid curve).

5.2.2 Grain Orientation Dependent GNB Planes The flow stress prediction including the grain orientation dependence, i.e. no GNBs, GNBs aligned with {111} or {xyy} planes and macroscopically aligned GNBs as shown in Figure 4b, is the solid curve in Figure 5. As it is known that the GNBs always deviate a little from the ideal {111} plane they are given a deviation of 3◦ in the calculations and this deviation is also used for the GNBs aligned with {xyy} planes. For the grain orientations with no GNBs x is set equal to 0 in Equation (1). It is seen in Figure 5 that the flow stress level is lowered and the anisotropy increased. The prediction is now in much better agreement with experiment both with respect to the level and the degree of anisotropy. The origin of this effect is that some of the slip directions in grains with GNBs aligned with {111} and {xyy} planes are practically parallel to the GNB plane, resulting in a relatively low critical resolved shear stress for these systems. It is clear that introduction of the grain orientation dependent GNB planes leads to a dramatic improvement of the flow stress anisotropy prediction. One might, therefore, suspect that complete understanding of also the exact plane of the GNBs taken as macroscopically aligned in the calculations would have a large impact. It is, however, expected that many of them align with {351}, {441} or {115} planes. These planes do not align with any of the slip systems and they will therefore not induce as large a difference in the critical resolved shear stresses of a grain as GNBs

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aligned with {111} or {xyy}. The effect of knowing the exact GNB plane in these orientations is therefore comparable to knowing whether a macroscopically aligned GNB is inclined positively or negatively to the rolling direction. The effect of the latter has been demonstrated to be small in calculations where all GNBs are assumed to be macroscopically aligned [14], leading to the conclusion that further improvements of the GNB plane prediction will not in practice affect the flow stress anisotropy results.

6 Conclusions The assembly of dislocations at the nanometre scale into boundaries with micrometre spacing within grains with dimensions of the order of 100 micrometres needs to be included in the modelling of mechanical properties of metals. In particular the parallelism of planar dislocation boundaries is important for the mechanical anisotropy, as exemplified for rolled metal sheets. The exact plane of the boundaries is an important parameter, which must be included in the modelling to obtain good predictions of the mechanical properties. These cannot at present be modelled in the form of dislocation interactions but they can be modelled at the grain scale by their relation to the slip systems, which also generate the dislocations during deformation.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Juul Jensen D, Hansen N, Acta Metall Mater 38, 1990, 1369. Eardley ES, Coulet A, Court SA, Humphreys F, Bate P, Mat Sci Forum 426–432, 2003, 363. Eardley ES, Humphreys F, Court SA, Bate P, Mat Sci Forum 396–402, 2002, 1085. Juul Jensen D, Hansen N, In: Brandon DG, Chaim R, Rosen A (Eds), Proceedings of ICSMA 9, Haifa, 1991, p 179. Liu Q, Juul Jensen D, Hansen N, Acta Mater 46, 1998, 5819. Huang X, Winther G, Phil Mag A 87, 2007, 5189. Kühlmann-Wilsdorf D, Mat Sci Eng A 113, 1989, 1. Hughes DA, Khan S, Godfrey A, Zbib H, Mat Sci Eng. A 309, 2001, 220. McCabe RJ, Misra A, Mitchell TE, Acta Mater 52, 2004, 705. Wert JA, Liu Q, Hansen N, Acta Metall Mater 43, 1995, 4153. Winther G, Huang X, Phil Mag A 87, 2007, 5215. Winther G, In: Szpunar (Ed), Proceedings of ICOTOM 12, Montreal, 1999, p 387. Li ZJ, Winther G, Hansen N, Acta Mater 54, 2006, 401. Winther G, Scripta Mater 52, 2005, 995. Winther G, Juul Jensen D, Hansen N, Acta Mater 45, 1997, 2455. Winther G, Juul Jensen D, Hansen N, Acta Mater 45, 1997, 5059. Hansen N, Juul Jensen D, Acta Metall Mater 40, 1992, 3265. Hansen N, Mat Sci Eng A 409, 2005, 39. Hansen N, Huang X, Hughes DA, Mat Sci Eng A 317, 2001, 3.

Micromechanical Modeling of the Elastic Behavior of Multilayer Thin Films; Comparison with In Situ Data from X-Ray Diffraction G. Geandier, L. Gélébart, O. Castelnau, E. Le Bourhis, P.-O. Renault, Ph. Goudeau and D. Thiaudière

Abstract Our goal is to address the elastic stiffness of polycrystalline materials as the grain size decreases down to the nanometric scale. Tensile tests on W/Cu multilayers exhibiting various (nanometric) thicknesses have been carried out under synchrotron radiation. Analyses of X-ray diffraction data provide the average axial elastic strain in the two types of layers and along different crystallographic orientation. To interpret those results, we use a mean-field homogenization approach based on two scale transitions. The accuracy of this micromechanical model is compared to reference results obtained with a full-field approach in which a simple periodic microstructure is considered as for the grain distribution within each layer. Experimental data well match model predictions.

1 Introduction For a few years, there has been an increasing interest in the elastic properties of thin films [2, 22, 23, 26]. Literature data seem to show that the elastic behavior of metallic thin films can differ significantly from their bulk counterparts because of their specific microstructure (texture, defects, high density of interface, interface mixing) and of size effects [9, 21, 24, 27]. Studying and tailoring the size effect on elastic G. Geandier and O. Castelnau LPMTM – CNRS, Institut Galilée, Université Paris 13, 93430 Villetaneuse, France; e-mail: [email protected] L. Gélébart CEA-Saclay DEN/DMN/SRMA, 91190 Gif sur Yvette, France E. Le Bourhis, P.-O. Renault and Ph. Goudeau PHYMAT, CNRS, Université Poitiers, SP2MI, 86962 Futuroscope Chasseneuil, France D. Thiaudière Synchrotron SOLEIL, 91192 Gif sur Yvette, France

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Fig. 1 STEM image of a W/Cu multilayer specimen, deposited on the polyimide substrate. Respective layer thicknesses are 6 nm and 18 nm.

constants of polycrystalline thin films requires controlling the nanostructure (grain size, residual stresses, texture). One way to control grain size along one direction at nanometric scales is to prepare multilayer structures. Elastic behavior of supported thin films can be determined by combining tensile [5, 6, 18] or four-point bending [14, 15, 19] tests with X-ray diffraction (XRD). X-ray diffraction is phase selective and hence it is a unique technique to determine both the size-sensitive mechanical behavior and the microstructural state of the diffracting phases. The interpretation of experimental data requires a micromechanical model accounting for the actual specimen microstructure and grain behavior, and aiming at the estimation of the effective behavior of the whole specimen, as well as stress and strain localizations in individual grains. The aim of this paper is to show how the elastic behavior of multilayers can be studied from the in-situ loading of the specimen during XRD experiments. We report a tensile test study of a W/Cu multilayer deposited on a polyimide (Kapton
) substrate. Then, we propose a model for W/Cu multilayers taking into account the actual thin-film microstructure (crystallographic texture and multilayered specimen), based on two scale transitions (from the grains to the layer, and from the layers to the specimen), and making use of mean-field homogenization approaches. This approach is further compared to “exact” (or at least “reference”) results obtained by a full-field (Finite Element) method.

2 Experimental Procedure and Results We provide here only a short outline of the experimental method, more details being given in [4]. Basically, W/Cu multilayers thin films were deposited at room temperature by physical vapor deposition on a 127.5 µm thick Kapton substrate. Several thicknesses and thickness ratio for the W and Cu layers were investigated [8], but we report here only results for a specimen comprising 20 periods of 24 nm each,

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composed of 6 nm W and 18 nm Cu (Figure 1). W presents the advantage of exhibiting a high X-ray scattering factor. Cu was chosen because of its immiscibility with W under thermodynamic equilibrium conditions. Besides, W and Cu exhibiting very different elastic behaviors (W is about 3 times stiffer than Cu), the obtained multilayer can be considered as a composite material with high mechanical contrast between the constituents, as shown later. The structure of the specimen (layer thickness, periodicity, chemical compositions, crystallographic textures, etc.) was carefully analyzed by several experimental techniques, including XRD, profilometry, X-ray reflectometry, and energy dispersive X-ray analyses. It was in particular observed that grains exhibit columnar shapes with a lateral extend of the same order than the film thickness. Moreover, pronounced {111} (resp. {110}) crystallographic texture were evidenced for Cu (resp. W) layers. The elastic response of the dog-bone shape specimen was characterized at the synchrotron beamline 11.3.1 of the ALS (Berkeley, USA). A tensile machine was installed on the beamline, and the displacement of several Bragg diffraction peaks was recorded in situ for different load levels applied to the specimen. Diffraction peaks corresponding to selected {hkl} diffracting planes (according to Bragg’s law) of the crystal lattice and exhibiting different orientations with respect to the applied stress axis were recorded on a two-dimensional detector. The position of Bragg peaks (given by their center of gravity) provides a direct measurement of the average axial elastic strain (so-called “lattice strain”) in the diffracting volume and in a direction parallel to the diffraction vector. This is a general feature that is independent on the actual strain distribution within the diffracting volume, i.e. also valid for nanomaterials with high volume fraction of grain boundaries exhibiting significant lattice disorder. Note also that the diffracting volume (denoted
) comprises only grains fulfilling Bragg’s conditions, and therefore the technique is orientation selective. Besides, grain size being here much smaller than the beam cross section (which is typically 100 µm), and X-ray attenuation length being much larger than the specimen thickness, measurements are statistically relevant and representative of the whole specimen volume. There are two contributions to the local elastic strain in the material. The first one is due to the localization of the applied macroscopic stress in the different grains, associated to the purely elastic response of the specimen, which is the quantity of interest here. The second contribution is associated to the residual stress generated during the elaboration process, which is not known. The advantage of performing in situ experimental tests is that once this second contribution has been characterized in the unloaded state, the purely elastic response of the specimen can be investigated, independently on the residual stress level and distribution. In practice, this is achieved by measuring the shift of each Bragg peak with respect to its position in the unloaded configuration, which is a valid procedure as far as elasticity is linear [3, 13]. Figure 2 shows the X-ray strain measurements as a function of the total applied stress for both W and Cu sublayers. The measured strain is either negative or positive depending on the orientation of the diffracting planes with respect to the specimen surface and the applied load direction. Strain is found to be proportional to the ap-

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Fig. 2 X-ray strain measurements (points) and elastic strain estimations based on the micromechanical SC model (dotted lines) for the sublayers versus the uniaxial stress applied to the multilayer/kapton composite. (left) W sublayers, (right) Cu sublayers.

plied stress, indicating that deformations remain elastic, whereas the slight waviness of the data is attributed to experimental uncertainties. However, to interpret these experimental results, the use of a micromechanical model is unavoidable owing to the complex microstructure of the material.

3 Micromechanical Modeling Based on Mean-Field Homogenization Our goal is to construct a simple micromechanical model that captures the main features of the material microstructure and that provides accurate and statistically relevant results. Thus, use of mean-field homogenization approaches is straightforward. We will assume that grains exhibit a uniform elastic behavior throughout their volume, thus neglecting the probable different stiffness at the interfaces (grain boundaries) which may become important at very small sizes. Here, three characteristic scales can be distinguished: the scale of the grain (nm), the scale of the Representative Volume Element (µm) of each layer, comprising a large number of grains, and the macroscopic scale of the specimen (mm) with its laminate structure. The two last scales are of course related to in-plane dimensions of the film. Since the three scales differ by several orders of magnitude, the mechanical problem can be split in a first approximation into two easier scale transition problems. The first scale transition consists in evaluating the effective behavior of each layer according to its microstructure and to the elastic behavior of grains. Indeed, the anisotropy of the local elastic stiffness C leads to the building of a mechanical interaction between the grains upon macroscopic loading, resulting in a heterogeneous distribution of stresses (and associated elastic strains) inside and between

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grains. The stress distribution can be expressed by means of an elastic localization tensor B(x) relating the local stress σ (x) at a spatial position x to the macroscopic stress σ¯ applied on a single layer, σ (x) = B(x) : σ¯ . The estimation of B(x) is in general a complex problem since it depends on the actual grain arrangement. Its ˜ of each layer. The knowledge allows the determination of the effective behavior C lattice strain reads −1 {ε}hkl : B
: σ¯ ) (1) , = n ⊗ n : (C with n the unit vector (defined by angles  and ) normal to the {hkl} diffracting planes, and ·
denoting the volume average over the diffracting volume. As for the W component, the problem simplifies considerably owing to the elastic isotropy of W grains so that, from the mechanical point of view, W layers are made of a uniform isotropic material exhibiting homogeneous stress and strain, independently of the actual microstructure. The stress localization tensor thus exactly reduces to identity. Concerning the Cu component, because of the strong elastic anisotropy of Cu grains, the average of B(x) over each crystallographic orientation has to be estimated. Several simple models have been proposed in the literature for that purpose [16, 25]. Here, we apply the Self-Consistent (SC) scheme [11] since, unlike those cited previously, this model provides the exact response for at least some particular classes of bulk polycrystals with general local anisotropy. The microstructure of such polycrystals consists in a 3-D random arrangement of grains and an infinite graduation of grain size. Note that an excellent match with polycrystal exhibiting realistic microstructure has also been obtained [12]. Here however, the SC model can only provide an estimation of the effective behavior of Cu sublayers since the microstructure of this component, with basically only one grain in the layer thickness, does not really match the one accounted for by this scheme. The second scale transition consists of approximating the actual specimen by a laminate structure. Indeed, owing to the small thickness of each layer and small grain size compared to the lateral extend of layers, it is reasonable to consider the specimen as an infinite multilayer with homogeneous stiffness for each layer. In ˜˜ of the whole specimen can be calculated exactly doing so, the effective stiffness C for general anisotropy of the layers (see [17] and references therein)
 ˜˜ −1 −  −1 = [σ (σ I − C) ˜ −1 − ]−1 σ0 (σ0 I − C) 0 0

(2)

with σ0 an arbitrary scalar, I the identity tensor, and  a geometrical tensor. The ˜˜ enables the calculation of the (average) stresses and strains in each knowledge of C layers, since these quantities are exactly uniform within each layer under the assumptions specified just before. It is worth noting that the model accounts for the actual microstructure (at least the crystallographic texture) of each sublayer, through ˜ Eventually, stress relocalization and associated elastic strain in inthe estimate of C. dividual crystal direction in each layer can be estimated, and therefore results can be compared to experimental data.

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Model calculations have been performed using elastic constants relevant for large (i.e. micrometric) grains. For W, Young’s modulus and Poisson ratio have been set to 409 GPa and 0.28 respectively, whereas for Cu, we have used C11 = 166.1 GPa, C12 = 119.9 GPa, and C44 = 75.6 GPa (standard Voigt notation). Young’s modulus and Poisson ratio have been set to 5 GPa and 0.40 for the Kapton substrate. Moreover, the crystallographic texture of the Cu component has been approximated by a {111} fiber with 15◦ dispersion at Full Width at Half Maximum. Accordingly, it is found that application of a uniaxial macroscopic stress at the whole specimen gives rise to a biaxial stress state in both W and Cu layers, owing to the different elastic response of those layers. However, the transverse stress component remains small compared to the axial component (about one tenth of it). Model results are shown in Figure 2. Globally a good agreement is found with the experiments, with however slightly superior results for W sublayers than for Cu. This shows that (i) the average stress level predicted for each sublayer is accurate, and (ii) within each sublayer, the average localization of the stress in grains with different crystallographic orientations is well estimated by this simple model. To this point, more complete results will be published elsewhere. It is worth mentioning that much worse results (not shown here) are obtained when applying the Reuss bound instead of the SC scheme, as also shown below.

4 Comparison with Exact Solutions To go deeper in the interpretation of these results, and although comparison to experimental data was satisfactory, a validation of the model is necessary. For doing this, we have generated several exact solutions corresponding to the actual material structure, and to which the above described mean-field model can be compared. We therefore concentrate now on the W/Cu laminate structure and get rid of the polyimide substrate. The elastic behavior of the laminate material is obtained from a large number (here 100) of simulations performed on random domains, each of these domains (or realization) being sufficiently small in size to allow fast numerical simulations, but large enough to account for grain to grain interactions, thus assuming ergodicity [1, 10, 20]. The goal of performing this ensemble average of several realizations is to increase the statistical relevance of the results which otherwise could be attained only with numerical power exceeding by far standard computer capacities. The investigated multi-layer polycrystalline material consists of two layers of columnar grains (grains boundaries are parallel to the layer normal), the Cu layer being 3 times thicker than the W one as for the real specimen investigated above. The in-plane microstructure is statistically isotropic and obtained by standard periodic Voronoi tessellation, with an average of 100 grains in each layer for a single realization (Figure 3). Unlike in previous section, and for the sake of simplicity, grains are assumed to exhibit random crystallographic orientation (isotropic crystallographic texture), so that each layer exhibits an isotropic effective behavior.

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Fig. 3 Typical bilayer periodic microstructure and Finite Element mesh used for generating fullfield reference solutions. The top and thick layer is soft and made of anisotropic grains (“Cu-like” behaviour). The thin bottom layer is made of isotropic W grains.

As for the local behavior, we associated to the grains of the thin layer the isotropic and stiff elastic behavior of W. The behavior of the thicker layer (referenced as “Cu-like” hereafter) is assumed to exhibit a cubic symmetry as Cu, but here we took advantage of the possibility of investigating the specimen response for different anisotropy parameter a = 2C44 /(C11 − C12 ) of those grains (a = 3.27 for copper). The elastic constants C11 , C12 , and C44 were then chosen so that the Voigt (uniform strain) estimate of the Cu-like layer is independent on a, which can be achieved when considering C11 = k˜ +

20 µ˜ V , 3(2 + 3a)

C12 = k˜ −

10 µ˜ V , 3(2 + 3a)

C44 =

5a µ˜ V , (3) 2 + 3a

where the expressions k˜ = (C11 + 2C12 )/3 and µ˜ V = (C11 − C12 + 3C44 )/5 are evaluated with the numerical values given in the previous section. The mechanical problem is solved by the Finite Element Method (CAST3M, [28]) using a regular mesh of (8 nodes-8 Gauss points) cubic elements, with 8 elements in the specimen thickness. Periodic boundary conditions were imposed on all specimen surfaces as described in [7], and a uniaxial effective stress of 100 MPa was prescribed macroscopically. The distribution of equivalent stress in each layer is illustrated in Figure 4, for a = 100. A very large heterogeneity in the Cu-like layer is observed, the stress ranging from almost zero to about 250 MPa. This comes from the large grain anisotropy that generates strong intergranular interactions since each grain reacts very differently to the applied stress. The situation is somehow different for the W layer. From the mechanical point of view, this layer is homogeneous owing to the isotropy of W. The observed stress heterogeneity, which is of less extend than that for the Cu-like layer, comes only from the interaction with Cu grains that deform heterogeneously. In Figure 5, the average axial stress in each layer, estimated by the mean-field model, is compared to those reference results. For a = 1, since both layers exhibit

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Fig. 4 Distribution of equivalent elastic strain in (top) the soft “Cu-like” layer, and (bottom) the stiff W layer. The applied macroscopic axial stress is 100 MPa. Results generated for a = 100. Note the different color scales for both figures.

locally an isotropic behavior, both predictions match exactly and correspond to a real laminate structure, described by Equation (2). As a increases, load is transferred to the stiff W layer. This feature is well reproduced by the mean-field model for relatively low a values, say a < 10, showing that the mean-field model remains accurate for the vast majority of elastic materials. This result is consistent with the good match to experimental data shown above. This numerical study allows the investigation of larger a values, which are of high interest for (i) a more general validation of the model for extremely anisotropic (even unrealistic) elastic materials, but also (ii) in view of an eventual application of the model to the plastic regime in which the mechanical contrast between grains is known to be particularly large. In that case, it is observed that the mean-field approach departs from about 10% from the reference solution. It is also observed that replacing the SC estimate by simpler Reuss or Voigt bounds leads to worse results.

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Fig. 5 Average axial stress in the (set of top curves) stiff W and (set of bottom curves) soft Cu-like layers as obtained by the mean-field approach associated to Reuss, Voigt, and SC estimates, and compared to the reference results provided by the full-field approach, for various anisotropy factor a for the soft layer.

5 Conclusion We have shown that the X-ray diffraction technique, combined with synchrotron radiation and in-situ mechanical tests is powerful for the investigation of the constitutive behavior of laminate polycrystalline structures with nanometric grains. We propose a simple micromechanical model based on mean-field homogenization techniques and accounting for the main features of the specimen microstructure. It has been shown that the model provides a good match of experimental data. However, comparison of the model with reference results generated by a full-field approach shows some discrepancies when the local elastic behavior of the Cu-like layer becomes very large, which is probably a limitation of the model for future extension in the plastic regime. This work will be pursued in the following directions. From the experimental point of view, complex deformation paths will be investigated, in particular using a biaxial tension device actually under development and soon available at the new French synchrotron SOLEIL. From the theoretical point of view, there is room to improve the proposed model. First of all, one has to account more carefully for the grain structure in each layer, with one grain in the layer thickness. The proposition of [6] using pancake shape two-point statistics will be checked with regard to our new exact solutions. Eventually, to deal with very small grain size, it is clear that the specific behavior of the interphases will have to be considered. Acknowledgements This project was partially funded by the French “Agence Nationale de la Recherche” (project Cmonano).

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References 1. Akiniwa, Y., Machiya, S., Serizawa, K., Tanaka, K., Proc. 2004 Int. Symp. on Micro-Nano Mechatronics, Nagoya, Japan, pp. 75–80 (2004). 2. Badawi, K.F., Villain, P., Goudeau, P., Renault, P. O., Appl. Phys. Lett., 80, 4705 (2002). 3. Bretheau, T., Castelnau, O., In: Rayons X et Matière, RX2006, P. Goudeau, P., Guinebretière, R. (Eds.), p. 123, Hermes Science (2006). 4. Castelnau, O., Geandier, G., Renault, P.-O., Goudeau, Ph., Le Bourhis, E., Thin Solid Films, 516(2–4), 320–324 (2007). 5. Faurie, D., Renault, P.-O., Le Bourhis, E., Villain, P., Goudeau, Ph., Badawi, K.F., Thin Solid Films, 469–470, 201 (2004). 6. Faurie, D., Castelnau, O., Brenner, R., Renault, P.O., Le Bourhis, E., Goudeau, Ph., Patriarche, G., Appl. Phys. Lett., 89, 061911 (2006). 7. Gélébart, L., Colin, C., J. Nucl. Mater., in press. 8. Girault, B., Villain, P., Le Bourhis, E., Goudeau, P., Renault, P.-O. Surface and Coatings Technol., 201, 4372–4376 (2006). 9. Jaouen, M., Pacaud, J., Jaouen, C., Phys. Rev. B, 64, 144106 (2001). 10. Kanit T., Forest S., Galliet I., Mounoury V., Jeulin D., Int. J. Solids Struct., 40, 3647–3679, (2003). 11. Kröner, E., J. Phys. F: Met. Phys., 8, 2261 (1978). 12. Lebensohn, R.A., Castelnau, O., Brenner, R., Gilormini, P., Int. J. Solids Struct., 42, 5441 (2005). 13. Letouzé, N., Brenner, R., Castelnau, O., Bechade, J.L., Scripta Mater., 47, 595 (2002). 14. Lu, Y.H., Lai M.O., Lu L., Zheng G.Y., Surface and Coatings Techn., 200, 4006 (2006). 15. Martinschitz, K.J., Eiper, E., Massl, S., Köstenbauer, H., Daniel, R., Fontalvo, G., Mitterer, C., Keckes, J., J. Appl. Cryst., 39, 777 (2006). 16. Matthies, S., Priesmeyer, H.G., Daymond, M.R., J. Appl. Cryst., 34, 585 (2001). 17. Milton G.W., The Theory of Composites. Cambridge University Press (2002). 18. Noyan, I.C., Sheikh, G., Mat. Res. Soc. Symp. Proc., 308, 3 (1993). 19. Pina, J., Dias, A., Lebrun, J.L., Mat. Sci. Eng. A, 267, 130 (1999). 20. Siska, F., Forest, S., Gumbsch, R., Weygand, D., Model. Simul. Mater. Sci. Eng., 15, 217–238, (2007). 21. Schiøtz, J., Vegge, T., Di Tolle, F.D., Jacobsen, K.W., Phys. Rev. B, 60, 11971 (1999). 22. Shenoy, V.B., Phys. Rev. B, 71, 094104, (2005). 23. Van Workum, K., de Pablo, J.J., Phys. Rev. E, 67, 031601 (2003). 24. Villain, P., Goudeau, Ph., Renaul,t P.-O., Badawi, K.F., Appl. Phys. Lett., 81, 4365 (2002). 25. Welzel, U., Leoni, M., Mittemeijer, E.J., Phil. Mag., 83, 603 (2003). 26. Yu, D.Y.W., Spaepen, F., J. Appl. Phys., 95, 2991 (2004). 27. Zhou, L.G., Huang, H., Appl. Phys. Lett., 86, 1, (2004). 28. http://www-cast3m.cea.fr

Two Minimisation Approximations for Joining Carbon Nanostructures Duangkamon Baowan, Barry J. Cox, Ngamta Thamwattana and James M. Hill

Abstract Two simple least squares approaches for connecting two carbon nanostructures are determined here. We speculate that the basis of joining carbon nanostructures is an underlying requirement that each inter-atomic distance be as close as possible to the ideal carbon-carbon bond length, or that the bond angle be as close as possible to the ideal bond angle. Both least squares approaches to bond lengths and to bond angles are applied for three systems, including nanotori formed from two and three distinct carbon nanotube sections, the joining between a carbon nanotube and a flat graphene sheet and nanobuds, which comprise a carbon nanotube joined to a fullerene. Moreover, Euler’s theorem is utilised to verify that the correct polygons occur at the connection sites. We comment that these purely geometrical approaches can be formally related to certain numerical energy minimisation methods used by a number of authors.

1 Introduction Although classical applied mathematical modelling has been widely used for solving many problems in areas such as economics and medical science, to date it has received less usage in the field of nanotechnology which is dominated by experiments and molecular dynamics simulations. The aim of this paper is to understand the forming mechanisms for connecting two carbon nanostructures. Two least squares approaches, which are the minimisation of the total variation in the bond length and the minimisation of the total variation in the bond angle, are employed to determine nanotori, composed of either two or three distinct carbon nanotubes, the joining between a carbon nanotube and a flat graphene sheet and the joining of a carbon nanotube and a fullerene to form the so-called nanobud. In Duangkamon Baowan, Barry J. Cox, Ngamta Thamwattana and James M. Hill Nanomechanics Group, School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia; e-mail: [email protected]

R. Pyrz and J.C. Rauhe (eds.), IUTAM Symposium on Modelling Nanomaterials and Nanosystems, 109–121. © Springer Science+Business Media B.V. 2009

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terms of the variation in bond length, we speculate that the principle underlying carbon nanostructures is that each inter-atomic distance be as close as possible to the ideal carbon-carbon bond length, which is taken to be σ = 1.42 Å. For the bond angle approach, we suggest that bond angles in the hexagonal network of the nanostructure need to be as close as possible to 120◦ which is the ideal bond angle. Both of these purely geometrical approaches can be directly related to certain bonded potential energy numerical minimisation methods [4, 14, 15, 18] who consider the requirement of energy minimisation. In the following section, the two least squares minimisation approaches are detailed. Elbow structures, which are the basic components of certain nanotori and comprise the joining of two distinct carbon nanotubes, are considered in Section 3. We examine two different nanotori molecules which are formed from both two and three distinct carbon nanotubes. In Section 4, the joining between a carbon nanotube and a graphene sheet is determined which forms a nanostructure considered to be the basic unit necessary to transmit signals in nano-electrical devices. The novel carbon nanostructure formed from a C60 fullerene and a carbon nanotube, namely a nanobud, is also constructed by the two simple mathematical approaches and is presented in Section 5. Results and discussion are presented in Section 6, and finally, a summary is given in Section 7.

2 Two Minimisation Methods Following the methodology proposed by the authors [1–3, 5] two least squares approaches, which are the variation in bond length and the variation in bond angle, are employed to determine the combined structures obtained by joining two carbon nanostructures. Throughout this study, the carbon-carbon bond length is taken to be σ = 1.42 Å. All the atom positions and bond angles for the carbon nanotubes are calculated using the new geometrical model of carbon nanotubes proposed by Cox and Hill [6], the graphenes are assumed to be perfect with a bond angle 120◦ and all 60 atom coordinates for a C60 fullerene are determined utilising “TruncatedIcosahedron" command in MAPLE with a radius of 3.55 Å.

2.1 Variation in Bond Length For this method, we seek to minimise the variation in distance between two atoms at the junction from the ideal carbon-carbon bond length σ . We start by defining the ith terminal atom at a join location by the position vectors ai = (axi , ayi , azi ) and bi = (bxi , byi , bzi ) for the two carbon nanostructures. Therefore, the Euclidean distance between the atoms is given by |ai − bi | = [(axi − bxi )2 + (ayi − byi )2 + (azi − bzi )2 ]1/2 .

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Fig. 1 Position vectors for the variation in bond angle approach.

In addition, the two structures are allowed to translate in three dimensional space by amounts X, Y ,  and to rotate by an angle ϕ in a certain direction depending on the system. Given the distances between joining atoms, our procedure is to attempt to determine the parameters X, Y ,  and ϕ by minimising the least squares variation of these distances from the ideal carbon-carbon bond length σ = 1.42 Å. In other words, we wish to minimise the following objective function:
f (X, Y, , ϕ) = (|ai − bi | − σ )2 . (1) i

2.2 Variation in Bond Angle In this subsection we examine the variation in bond angle approach. We attempt to minimise the variation of the bond angle at each atom-atom junction from the maximum normal physical value for both carbon nanostructures. For the graphene sheet, the ideal bond angle is assumed to be 120◦, while the ideal bond angles for the nanotubes are taken from the new model of carbon nanotubes given by Cox and Hill [6] which properly incorporates curvature. Since the atomic networks for carbon nanostructures are formed from hexagonal rings, a general procedure to determine the position vectors of all atoms at the junction is given by 1. 2. 3. 4.

Find the point M which is the mid-point of A1 and A2 , Find the vector U = MA3 , Find the unit vector V = A1 A2 /|A1 A2 | which is perpendicular to U, Determine the vector W which is perpendicular to both U and V and has the same magnitude as U; namely W = U × V, 5. The atom position is then given by M + U cos θ + W sin θ ,

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where A1 , A2 and A3 are the atoms position as shown in Figure 1. The atomic position A3 which is the joining position is assumed to move around a circular path and whose position is determined by an angle θ . Moreover, each bond length which joins between the atoms is restricted to be 1.42 Å. We refer the reader to [1] for the full details of this method. We comment that the complication of this method depends on the symmetries of the system and the number of atoms at the junction.

3 Nanotori Dunlap [7] first proposed the torus as a stable form of graphitic carbon. He constructs toroidal molecules by joining two different carbon nanotubes, denoted as elbow structures, with matching radii and introduces the pentagon-heptagon pair [7–9]. Moreover, Dunlap [7–9] predicts that the molecule in all comprises twelve connecting sections occurring for the 360◦ turn and therefore, the tubule bend angle is 30◦ for each section. The energetic stability of molecules that are constructed based on the C60 fullerene and carbon nanotube structures are investigated by Ihara et al. [11], Itoh and Ihara [12] and Itoh et al. [13]. They find that these structures are more thermodynamically stable [10, 11] and such toroidal shapes are expected to be physically more interesting than those of the two original structures [12]. Although, these theoretically proposed structures have not been confirmed by experiment [19], they are believed to give rise to fascinating electrical, magnetic and elastic properties arising from the pattern of their hexagonal rings [10]. In this paper, we investigate two types of nanotoroidal shaped molecules. To start, we model the elbow structure mathematically using two distinct carbon nanotube sections as considered by Dunlap [7–9], which is shown in Section 3.1. The question arises as to the generality of the procedure, and whether or not we might determine other toroidal shaped molecules, such as ones constructed from three distinct nanotubes. Therefore, we present the elbow-connected nanotori formed from three distinct carbon nanotube sections in Section 3.2.

3.1 Nanotori Formed from Two Distinct Carbon Nanotubes The elbow structure is modelled by positioning atoms on ideal cylinders representing the two species of nanotube and then identifying the terminal atoms on each structure which bond with the corresponding terminal atoms on the other nanotube. In this study, only the variation in bond length is taken into account. The models are originally defined as centred on the origin and aligned with their axes as the zaxis. We then perform a translation of tube A in the negative z-direction by a length 1 and a corresponding translation of tube B in the positive z-direction by a length 2 . With the tubes so situated, we then perform a rotation of tube B by an angle ϕ around the x-axis, as shown in Figure 2. After the translations and rotation, we

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Fig. 2 Elbow formed from two carbon sections.

derive the following expression for the Euclidean distance between two connecting atoms,  |ai − bi | = (axi − bxi )2 + [ayi − (byi cos ϕ − (bzi + 2 ) sin ϕ)]2 1/2 + [(azi − 1 ) − ((bzi + 2 ) cos ϕ + byi sin ϕ)]2 , where the position vectors ai , and bi are for tubes A and B, respectively. With this distance between matching atoms so defined, we wish to minimise the variance of this distance from the bond length σ , as described in equation (1). Therefore, we seek to minimise f (1 , 2 , ϕ). Once the parameters 1 , 2 , ϕ are determined, the elbow unit can be obtained and by repeating such units, and the nanotori can be formed. In Figure 3, we present an example for a nanotorus comprised of the (3,3) the (5,0) nanotubes, and we refer the reader to [5] for the full details of this determinations. We also desire to determine a mean radius of the toroidal shapes c which is found to be given by c = [r1 sinh−1 (1 /r1 ) + r2 sinh−1 (2 /r2 )]/ϕ.

(2)

We comment that this procedure for the determination of the average parameter c is by no means unique, but appear as the most natural and simplest for the determination of a representative value.

3.2 Nanotori Formed from Three Distinct Carbon Nanotubes In this subsection, we investigate the elbow structure required for toroidal molecules by joining three distinct carbon nanotubes of lengths 21 , 22 and 23 and utilising the least squares bond length. The proposed model assumes that the basic repeating unit comprises tubes A and C as half unit lengths and tube B as one unit length.

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Fig. 3 C240 molecule with structure 5(3,3)18 5(5,0)30 .

Further, it is assumed that the origin O of a rectangular Cartesian coordinate system (x, y, z) is located at the central point of tube B, such that the axis of tube B is aligned along the z-axis, as illustrated in Figure 4a. In this case, we divide the system of tubes A, B and C to be two sub-systems of tubes A and B, and tubes B and C, as depicted in Figures 4b and 4c. By precisely the same considerations presented in Section 3.1, the Euclidean distances between tubes A and B, and tubes B and C are given by  |ai − bi | = [axi cos ϕ1 − (azi + 1 ) sin ϕ1 − bxi ]2 + (ayi − byi )2 1/2 + [(azi + 1 ) cos ϕ1 + axi sin ϕ1 − (bzi − 2A )]2 , and  |ci − bi | = [cxi cos ϕ2 − (czi + 3 ) sin ϕ2 − bxi ]2 + (cyi − byi )2 + [(czi + 3 ) cos ϕ2 + cxi sin ϕ2 − (bzi − 2B )]2

1/2

,

where the position vectors ai , bi and ci are for tubes A, B and C, respectively. In this case, we are seeking to minimise the following objective functions:

f (1 , 2A , ϕ1 ) = (|ai − bi | − σ )2 , g(2B , 3 , ϕ2 ) = (|ci − bi | − σ )2 . i

i

For the nanotorus, we require that an even number of elbow sections forms a symmetrical torus, so that the angles ϕ1 and ϕ2 must be constrained to the value ϕ1 + ϕ2 = 180◦/n where n is {2, 3, 4, . . .}. So that we are seeking to minimise the objective function F (1 , 2A , 2B , 3 , ϕ1 ) = f (1 , 2A , ϕ1 ) + g(2B , 3 , 180◦/n − ϕ1 ).

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Fig. 4 Basic double elbow unit formed from three distinct nanotube sections (a), and Cartesian coordinate for two single nanotube elbows (b) for tubes A and B, and (c) for tubes B and C.

By using the relation between the Cartesian and the toroidal coordinate systems, a mean torus radius c can be derived and is given by r1 sinh−1 (1 /r1 ) + r2 [sinh−1 (2A /r2 ) + sinh−1 (2B /r2 )] + r3 sinh−1 (3 /r3 ) . ϕ1 + ϕ2 (3) The full details of this study can be found in [2]; here we illustrate the nantoroidal shaped molecule 3(3,3)126(6,0)323(4,4)24 which is shown in Figure 5. We comment that the formula given by (3) provides the appropriate generalisation of that given by (2) for the case of two distinct tubes. c=

4 Joining between a Carbon Nanotube and a Flat Grapheme Carbon nanotubes of different radii frequently are seen to occur as a Y-junction. Instead of examining this problem, we might first assume that the radius of one tube is much larger than the radius of the other. This line of reasoning leads us to first consider the perpendicular joining a tube to a flat graphene sheet. Moreover, for future nanoelectromechanical signalling, graphene sheets may be needed as the platform to transmit signals to other materials through such joined carbon nanotubes. Thus connecting grapheme sheets with carbon nanotubes is an interesting problem with

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Fig. 5 Nanotorus formed from 3(3,3)12 6(6,0)32 3(4,4)24 where ϕ1 + ϕ2 = 60◦ .

potential applications. For this problem, we employ both least squares approaches given in Section 2 to present an example of joining a (6,0) carbon nanotube and a flat graphene sheet. For the variation in bond length and in a Cartesian coordinate system, the graphene sheet is assumed to be located in the (x, y) plane and the ith atom on the sheet is assumed to have the position vector ai = (axi + X, ayi + Y, 0). The sheet is allowed to move in both x- and y-directions by the distances X and Y , respectively, which can be either positive or negative. The position vector of the ith atom on the tube open end is assumed to be given by bi = (bxi , byi , ) where  is the spacing between the tube and the sheet in the positive z-direction. In addition, the tube can be rotated by an angle ϕ. The distance between the atom on the tube end and the atom on the sheet is then given by  |ai − bi | = [(axi + X) − (bxi cos ϕ − byi sin ϕ)]2 1/2 + [(ayi + Y ) − (bxi sin ϕ + byi cos ϕ)]2 + 2 ] . The aim is to determine the parameters X, Y ,  and ϕ by minimising the Euclidean distance from the carbon-carbon bond length σ = 1.42 Å. Since there are six atoms at the tube open end which need the other bond to complete the sp2 structure, so that we need to choose the defect on the graphene sheet which also has six atom locations that require the other bond to complete the sp2 network. Here, we decide the defect configuration which is shown in Figure 6a to join with the (6,0) carbon nanotube. By utilising the algebraic package MAPLE, all the parameter values are easily obtained and the three dimensional illustration is depicted in Figure 7a. Since the defect shown in Figure 6a has a six-fold symmetry, only one joining location needs to be examined. Following the variation in bond angle approach given in Section 2.2, we obtain all atom positions at the junction site which are described in Figure 6. In this case, we wish to minimise the spacing between the tube and the

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Fig. 6 Model formation for joining (6,0) tube with the most symmetric defect where (a) is for graphene sheet, and (b) and (c) are for the (6,0) tube.

Fig. 7 Three dimensional illustrations for (6,0) connection with graphene sheet by (a) variation in bond length and (b) variation in bond angle.

sheet L, the rotational of the tube ϕ, and the circular path of the joined atom θ , by fixing the bond angle of the sheet to be 120◦, and the bond angle of the tube to be 117.65◦ [6]. Using the optimisation package in MAPLE, we obtain L = 2.315 Å, ϕ = 30.73◦, and θ = 9.39◦, and the three dimensional figure obtained is depicted in Figure 7b.

5 Nanobuds Both the variation in bond length and the variation in bond angle approximations are considered in this section in order to construct the novel carbon nanostructure formed from a C60 fullerene and a carbon nanotube, namely the nanobud. Nanobuds were experimentally observed in a ferrocene-carbon monoxide system in 2007 by Nasibulin et al. [16, 17] from transmission electron microscope observations. Furthermore, they also propose a one-step continuous process for the synthesis of carbon nanotubes with covalently attached fullerenes. However, we comment that a dynamical process involving the combination of these two carbon nanostructures was first shown by Zhao et al. [20] in 2003. They focused on a fullerene being en-

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capsulated inside the nanotube, and a nanobud being formed during this process. Nanobuds are believed to be promising field-emission devices since the off-plane of the fullerenes on the mat of carbon nanotubes can increase the surface area [17]. We start by considering the variation in bond length approach. The centre of a C60 fullerene is assumed to be located at the origin, and the tube axis is assumed to be parallel with the z-axis. The ith terminal atom at a join location is defined by the position vectors ai and bi for the tube open end and the C60 , respectively. We note that the terminal atoms at the defect of the fullerene are numerically obtained by MAPLE. The spacing between the tube and the fullerene in the z-direction is assumed to be , and the tube can be rotated about the z-axis through an angle ϕ. Moreover, the tube is allowed to move in both x- and y-directions by distances X and Y , respectively. Consequently, the Euclidean distance between the atoms at the junction is then given by  |ai − bi | = [(X + axi cos ϕ − ayi sin ϕ) − bxi ]2 1/2 + [(Y + axi sin ϕ + ayi cos ϕ) − byi ]2 + ( − bzi )2 ] . In this study, the defect on the C60 fullerene occurs by removing one pentagonal ring and the corresponding five bonds. Therefore, there are five atom locations on the C60 which require another bond to complete the sp2 network, subsequently a (5,0) carbon nanotube is assumed to be connected at the junction. By using the algebraic package MAPLE, all parameter values are obtained and the three dimensional illustration is shown in Figure 8a. For the variation in bond angle, we assume that all bond lengths are fixed to be σ but we vary the bond angles at connection sites so as to minimise the least square deviations from the physical bond angles. The carbon-carbon bond length is taken to be σ = 1.42 Å, the bond angles of the pentagon and the hexagon for the fullerene are assumed to be 108◦ and 120◦, respectively, and the bond angles for the carbon nanotubes are taken from the new model of carbon nanotubes [6]. Following the same considerations as described in Section 2.2, all atom locations can be determined. Again, all parameters are numerically obtained using the minimisation package in MAPLE and the three dimensional illustration for the nanobud is depicted in Figure 8b.

6 Results and Discussion In terms of polygonal rings which occur at the connection site, we use Euler’s theorem to verify that the proposed structures are geometrically sound. Euler’s theorem states that F + V − E = χ, where F , V and E denote the numbers of faces, vertices and edges for the given polyhedron and χ is the Euler characteristic. We note that any surface which is

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Fig. 8 Three dimensional illustrations for a (5,0) tube connection with a defect on a C60 fullerene by (a) variation in bond length and (b) variation in bond angle.

homeomorphic to a sphere has an Euler characteristic of 2. The notation Pn is used to denote the number of n-gon sides, e.g. P5 is the number of pentagonal sides, and every atom is linked with three others in the sp2 structure, therefore we may deduce F = P4 + P5 + P6 + P7 + P8 ,

3V = 4P4 + 5P5 + 6P6 + 7P7 + 8P8 ,

and 2E = 4P4 + 5P5 + 6P6 + 7P7 + 8P8 . For χ = 2, the Euler’s theorem simplifies to become 2P4 + P5 − P7 − 2P8 = 12.

(4)

For example, a C60 fullerene is formed by pentagons and hexagons and there are precisely twelve pentagons required to close the spherical shape. We also note that the number of hexagons P6 does not appear in equation (4) and can take any value. There is only one pentagon-heptagon pair occurring at the elbow-connection formed from both two and three distinct nanotube sections, which is satisfied by (4). We comment that a carbon nanotube can be considered to be capped at one end with a hemispherical C60 fullerene comprising six pentagons. In order to maintain the Euler characteristic at the junction of the tube and the sheet, the connection must necessarily have six pentagons with six heptagons or an equivalent number of other polygons. As a result, the polygons which occur at the junction must satisfy −2P4 − P5 + P7 + 2P8 = 6. From the three dimensional figures shown in Figure 7, there are six heptagonal rings at the connection site which is verified by Euler’s theorem. Finally, in the case of nanobud, there are five heptagons at the junction which agrees with (4).

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A number of authors [4, 14, 15, 18] adopt a numerical minimum energy principle such that the bonded potential energy for small deformations is given by
  E= kr (r − r0 )2 + kθ (θ − θ0 )2 + kτ [1 − cos(nτ − τ0 )] , i

where kr , kθ and kτ are certain bond stretching, bending angle and torsional constants, respectively, r0 , θ0 and τ0 are equilibrium values of the bond length, bond angle and ideal phase angle for this bond type, respectively, and n is an integer. In general, our variation in bond length approach corresponds to taking only the bond stretching energy and similarly, the variation in bond angle approach corresponds to taking only the angle bending energy into account. Furthermore, according to [4, 14, 15, 18], in terms of the relative magnitudes of the force constants, the torsional term plays only a minor effect on the system and it may be neglected. Consequently, our simple mathematical approaches are closely related to the physical principle of the energy minimisation.

7 Summary The major contribution of this paper is to propose two simple least squares approximations for combining two carbon nanostructures. The variation in bond length approach is employed to minimise the Euclidean distance between two atoms at the junction from the carbon-carbon bond length which is taken to be σ = 1.42 Å. In this case, we allow the bond angles to vary. On the other hand for the variation in bond angle, we assume that all bond lengths are fixed at σ and we seek to minimise the bond angles at connection sites from the ideal bond angles. Furthermore, Euler’s theorem is employed to verify that the polygons which occur at the junctions are geometrical sound. In this paper, we have examined three combined carbon nanostructures which are nanotori, the joining between a nanotube and a graphene sheet and a nanobud. The minimisation processes are calculated utilising the algebraic package MAPLE and the resulting joined structures are shown in three dimensional figures. However, we comment that such theoretical structures have yet to be confirmed either experimentally or by molecular dynamics simulations. We comment that although these approaches appear to be purely geometric in nature, by trying to make each inter-atomic bond length and bond angle as close as possible to the ideal bond length and the ideal bond angle, we are implicitly taking some account of the requirement to minimise the energy.

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References 1. Baowan, D., Cox, B.J., Hill, J.M.: Two least squares analyses of bond lengths and bond angles for the joining of carbon nanotubes to graphemes. Carbon 45, 2972–2980 (2007). 2. Baowan, D., Cox, B.J., Hill, J.M.: Toroidal molecules formed from three distinct carbon nanotubes. J. Math. Chem. 44, 515–527 (2008). 3. Baowan, D., Cox, B.J., Hill, J.M.: Junctions between a boron nitride nanotube and a boron nitride sheet. Nanotech. 19, 075704 (2008). 4. Cornell, W.D., Cieplak, P., Bayly, C.I., Gould, I.R., Merz, K.M. Jr., Ferguson, D.M., Spellmeyer, D.C., Fox, T., Caldwell, J.W., Kollman, P.A.: A second generation force field for the simulation of proteins, nucleic acids, and organic molecules. J. Am. Chem. Soc. 117, 5179–5197 (1995). 5. Cox, B.J., Hill, J.M.: New carbon molecules in the form of elbow-connected nanotori. J. Phys. Chem. 111, 10855–10860 (2007). 6. Cox, B.J., Hill, J.M.: Exact and approximate geometric parameters for carbon nanotubes incorporating curvature. Carbon 45, 1453–1462 (2007). 7. Dunlap, B.I.: Connecting carbon tubules. Phys. Rev. B 46, 1933–1936 (1992). 8. Dunlap, B.I.: Relating carbon tubules. Phys. Rev. B 49, 5643–5650 (1994). 9. Dunlap, B.I.: Constraints on small graphitic helices. Phys. Rev. B 50, 8134–8137 (1994). 10. Ihara, S., Itoh, S.: Helically coiled and toroidal cage forms of graphitic carbon. Carbon 33, 931–939 (1995). 11. Ihara, S., Itoh, S., Kitakami, J.: Toroidal forms of graphitic carbon. Phys. Rev. B 47, 12908– 12911 (1993). 12. Itoh, S., Ihara, S.: Toroidal forms of graphitic carbon. II. Elongated tori. Phys. Rev. B 48, 8323–8328 (1993). 13. Itoh, S., Ihara, S., Kitakami, J.: Toroidal form of carbon C60 . Phys. Rev. B 47, 1703–1704 (1993). 14. Jin, Y., Yuan, F.G.: Simulation of elastic properties of single-walled carbon nanotubes. Comp. Sci. Technol. 63, 1507–1515 (2003). 15. Li, C., Chou, T.W.: A structural mechanics approach for the analysis of carbon nanotubes. Int. J. Solids Struct. 40, 2487–2499 (2003). 16. Nasibulin, A.G., Anisimov, A.S., Pikhitsa, P.V., Jiang, H., Brown, D.P., Choi, M., Kauppinen, E.I.: Investigations of nanobud formation. Chem. Phys. Lett. 446, 109–114 (2007). 17. Nasibulin, A.G., Pikhitas, P.V., Jiang, H., Brown, D.P., Krasheninnikov, A.V., Anisimov, A.S., Queipo, P., Moisala, A., Gonzalez, D., Lientschnig, G., Hassanien, A., Shandakov, S.D., Lolli, G., Resasco, D.E., Choi, M., Tomanek, D., Kauppinen, E.I.: A novel hybrid carbon material. Nature Nanotech. 2, 156–161 (2007). 18. Natsuki, T., Tantrakarn, K., Endo, M.: Efects of carbon nanotube structures on mechanical properties. Appl. Phys. A 79, 117–124 (2004). 19. Zhang, X.F., Zhang, Z.: Polygonal spiral of coil-shaped carbon nanotubules. Phys. Rev. B 52, 5313–5317 (1995). 20. Zhao, Y., Lin, Y., Yakobson, B.I.: Fullerene shape transformations via Stone–Wales bond rotations. Phys. Rev. B 68, 233403 (2003).

On the Eigenfrequencies of an Ordered System of Nanoobjects Victor A. Eremeyev and Holm Altenbach

Abstract A method is discussed to determine the eigenfrequencies of nanostructures (nanotubes, nanospheres, and nanocrystals) by measuring the eigenfrequencies of a ‘large system’ that consists of an array of vertically oriented similar nanotubes or nanocrystals equidistantly grown on a substrate. It is shown that the eigenfrequencies of a single nanoobject can be derived from the eigenfrequency spectra of the large (array-substrate) system and of the substrate. With other words, using experimental data for large systems one can determine the eigenfrequencies of a single nanoobject. The method can be also applied to systems of nanotubes grown in parallel to the substrate and to the systems of micro- and nanospheres. The modeling of nanocomposite plates using the direct approach to the shell theory is discussed. The effective stiffness tensors are considered. As an example, the eigenfrequencies of an array of ZnO micro- or nanocrystals and GaAs multiwalled nanotubes on a sapphire substrate are calculated.

1 Introduction The experimental determination of the mechanical characteristics of nanoobjects is today a challenging problem (see, among others, [6, 7, 14, 16, 17, 20]). One of the most efficient methods to determine elastic moduli in macromechanics is the measurement of the eigenfrequencies of an object. However, attempts to apply this approach to nanoobjects sometimes demonstrate difficulties. There are at least two problems lying at the interfaces between mechanics and experimental physics. The Victor A. Eremeyev South Scientific Center of RASci South Federal University, 8a Milchakova st., Rostov on Don 344090, Russia; e-mail: [email protected] Holm Altenbach Lehrstuhl für Technische Mechanik, Zentrum für Ingenieurwissenschaften, Martin-LutherUniversität Halle-Wittenberg, D-06099 Halle (Saale), Germany

R. Pyrz and J.C. Rauhe (eds.), IUTAM Symposium on Modelling Nanomaterials and Nanosystems, 123–132. © Springer Science+Business Media B.V. 2009

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Fig. 1 Example of nanoobject array: ZnO nanocrystals on a substrate (photos by courtesy of Konstantin Dvadnenko, South Scientific Center of RASci).

first one is the determination of the elastic moduli of nanoobjects when there is a possibility to measure the frequencies of the microsubstrate-nanoarray system and determine the elastic characteristics of the substrate (for example, from the eigenfrequencies of the free substrate). The second problem is how to extract the eigenfrequencies of nanoobjects from the eigenfrequency spectrum of the substrate-array system. The success in solving both problems directly depends on the experimental conditions, especially, on the way how nanoobjects are grown on the substrate and how the substrate with nanoobjects is fixed in the measuring device; as well as on the geometries, weights, and the elastic properties of the nanoobjects and the substrate. Thus, from the mechanical point of view, one should not only discuss the measured data, but also elaborate a suitable design of the experiments. Here we discuss the method of determining the eigenfrequencies of nanostructures (nanotubes and nanocrystals) from the measured eigenfrequencies of a large system comprising a highly ordered array of identical nanospheres, nanotubes or nanocrystals grown on a substrate proposed in [10–13]. The geometry of the considered nanostructures is presented in Figure 2. An example of such large system, the array of nanocrystals of zinc oxide, is presented in Figure 1. ZnO nanocrystals are of considerable interest for nanomechanics and nanophotonics and can be fabricated by different techniques. The array of GaAs, GeSi nanotubes growing horizontally on a substrate may be obtained by technology described in [15, 21]. The interest to

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Fig. 2 Nanostructures: (a) an array of nanotubes or nanocrystals on substrate; (b) an array of spherical micro- or nanoparticles; (c) an array of nanotubes lying on a substrate; (d) nanocomposite plate.

the arrays of hollow sphere-like nanoparticles is motivated by possible applications to modeling of fullerenes, dendrimers, micelles, vesicles, and liposomes. Some of them are widely used in modern medicine and farmacology, for example, in order to deliver encapsulated drugs to target organs in the organism. Using the asymptotic solutions based on the shell theory it is shown that from the spectrum of the large system (nanoobjects-substrate), the eigenfrequencies of an individual nanoobject can be derived. That means, the eigenfrequencies of a single nanoobject can be determined from experimental data obtained for the large system. The direct FEM calculation results also indicate the possibility of extracting the spectrum of nanoobjects from the spectrum of the large system. The oriented arrays of nanoobjects can be used for assembling of nanocomposite thin-walled structures, for example, plates (see Figure 2d). In the last section of this paper we discuss the methods of identifications of elastic properties of a nanocomposite plate using the approach [1–3, 23].

2 Governing Equations of the Plate and Shell Theory We apply the theory of plates and shells formulated earlier in [1–5, 22, 23]. From the direct approach point of view a plate or a shell is modeled as a material surface each particle of which has five degrees of freedom (three displacements and two

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rotations, the rotation about the normal to plate is not considered as a kinematically independent variable). Note that the application of the direct approach to the thinwalled nanostructures has advantages because it is free from the consideration of the nanostructure on the base of the three-dimensional continuum mechanics. The equations of motion are formulated as the Euler’s laws of dynamics: ∇ · T + q = ρ u¨ + ρ
1 · ϕ, ¨

∇ · M + T× + m = ρ
T1 · u¨ + ρ
2 · ϕ. ¨

(1)

Here T, M are the tensors of forces and moments, q, m are the vectors of surface loads (forces and moments), T× is the vector invariant of the force tensor, ∇ is the nabla (Hamilton) operator, u, ϕ are the vectors of displacements and rotations,
1 ,
2 are the first and the second tensor of inertia, ρ is the density (effective property of the deformable surface), (. . .)T denotes transposed and (·)• the time derivative. The geometrical equations are given as µ = (∇u · a)sym ,

γ = ∇u · n + c · ϕ,

κ = ∇ϕ.

(2)

a is the first metric tensor (plane tensor), n is the unit outer normal vector at the surface, c is the discriminant tensor (c = −a × n), µ, γ and κ are the strain tensors (the tensor of in-plane strains, the vector of transverse shear strains and the tensor of the out-of-plane strains), while tsym denotes the symmetric part. The boundary conditions are given by ν · T = f,

ν · M = l (l · n = 0)

or u = u0 ,

ϕ = ϕ 0 along S.

(3)

Here f and l are external force and moment vectors acting along the boundary of the plate S, while u0 and ϕ 0 are given functions describing the displacements and rotations of the plate boundary, respectively. ν is the unit outer normal vector to the boundary S (ν · n = 0). The relations (3) are the static and the kinematic boundary conditions. Other types of boundary conditions are also possible. The constitutive equations are given by the following relations: Strain energy of the deformable surface W : W (µ, γ , κ) =

1 1 1 µ··A··µ+µ··B··κ + κ ··C··κ + γ · ·γ +γ ·( 1 ··µ+ 2 ··κ). 2 2 2

A, B, C are fourth rank tensors,  1 ,  2 are third rank tensors,  is a second rank tensor of the effective stiffness properties. They depend on the material properties and the cross-section geometry. In the general case the tensors contain 36 different values – a reduction is possible assuming some symmetries. Constitutive equations • In-plane forces T·a=

∂W = A · ·µ + B · ·κ + γ ·  1 . ∂µ

(4)

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• Transverse forces T·n=

∂W =  · γ +  1 · ·µ +  2 · ·κ. ∂γ

(5)

MT =

∂W = µ · ·B + C · ·κ + γ ·  2 . ∂κ

(6)

• Moments

3 On the Modeling of Natural Oscillations of an Array of Nanoobjects For the analytical investigation of such complex structures as presented in Figure 1 we have to make some assumptions on the stress state and the possible deformations of the shell-like structure. Here we have been used the models of the beams, of spherical and cylindrical shells for the nanocrystals, the nanotubes and the nanospheres as well as the plate equations for the substrate. To avoid the awkward calculations we present here the final result: for all of three cases we may choose the geometry of the large system consisting of an array of nanoobjects in such a way that one may find the first eigenfrequencies of the nanoobject from its spectrum. The accuracy of determination of the first eigenfrequency for the vertically oriented ZnO crystals is about 5%, while for the GaAs nanotubes lying horizontally the accuracy is about 10–15% (for details, see [10–13]). Note that the behavior of the system with horizontally located nanotubes differs from the behavior of a similar system with vertically located nanotubes [10], as well as from the behavior a system of nanospheres [11], because the horizontally attached nanotubes change the effective stiffness of the plate and the plate with the horizontal nanotubes is anisotropic and inhomogeneous with respect to the effective properties. For an arbitrary stress-strain state we investigate the system consisting of a certain number of nanoobjects on the substrate in the framework of the threedimensional theory by means of the FEM. Since the materials under consideration (ZnO, GaAs, InAs, GeSi, etc.) have piezoelectric properties, these systems as a whole are a composite piezoelectric solids. Then for modeling of GaAs nanotubes and ZnO nanocrystals we used the constitutive equations of the theory of anisotropic electroelasticity [18, 19]. As an example of FEM modeling let us consider the array of horizontal GaAs nanotubes (Figure 2c). For the modal analysis of the corresponding boundary value problems we have used FE code ANSYS. Various numerical experiments are performed for different mesh, different types of finite elements, different numbers of nanotubes (from one to ten), different system geometries (the ratio of the thicknesses of the substrate and the nanofilm, the ratio of the nanotube radius to the substrate length, etc.), and various substrate bonding conditions. We consider the same materials of the substrate and the nanotubes used in [15, 21]. The properties of these

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Fig. 3 Sapphire substrate with three GaAs nanotubes: (a) natural mode localized in nanotubes; (b) natural mode corresponding to the first bending mode of the substrate.

materials are taken from [8], and the geometrical parameters are taken from [15,21]. The calculations show that for any bonding of the substrate, it is possible to choose the problem parameters in such a way that the natural frequencies of the nanotubes and the substrate can be extracted from the general spectrum of the large system. The numerical estimates confirm the results of the above theoretical analysis. The calculation results for the free sapphire substrate with three GaAs nanotubes are shown in Figure 3. Figure 3a corresponds to the natural vibrations localized in the nanotubes. Figure 3b shows the natural vibrations at the frequency corresponding to the first bending mode of the substrate vibrations. The eigenmodes for the nanospheres are presented in Figure 4. It is easy to see that we have approximately the same situation as shown in Figure 3. The spectrum of eigenmodes consists of modes which are localized in a sphere and the modes of the substrate. Let us note that the separation of the eigenmodes is fulfilled only for the first part of the spectrum when the localized motion of the array is possible. The high-oscillating modes do not correspond to any modes of a single nanoobject or the substrate. The high-oscillating modes depend on the interaction of the nanoarray and the substrate. Note that the vibration interaction between the substrate and nanotubes is stronger than that for a case of a vertical array of nanocrystals or nanospheres. The eigenfrequencies for different nanoarrays are summarized in Figure 5. Here the distribution of the natural frequencies ω over their ordinal numbers n is depicted. In Figure 5a the results of the modal analysis of the large system consisting of 8 ZnO nano-sized crystals grown on a sapphire substrate are presented, see [10]. The eigenfrequencies of three GaAs nanotubes attached to the sapphire substrate are given in Figure 5b, while in Figure 5c the eigenfrequencies of 33 hollow spheres made of a polymer are presented. The graphs show the presence of a plateau in the distributions. The length of the plateau depends on the number of nanoobjects in the array. The plateau corresponds to the eigenfrequency of one nanoobject. A big number of approximately equal eigenfrequencies gives us additional chance to find it by using of experimental measurements. On the other hand if one extends the graphs to the higher n-values the next plateaus will be disappeared. For example, the second plateau for the array of nanotubes is not strictly expressed. But for the first frequencies, the numerical coincidence of the eigenfrequencies of the large system

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Fig. 4 Eigenmodes of a substrate with nanospheres: (a) natural mode localized in one nanosphere; (b) natural mode corresponding to the first bending mode of the substrate.

with the respective partial eigenfrequencies of the substrate and single nanotube is satisfactory. From an engineering point of view, let us mention that the main restriction on application of the described method is the frequency range of measuring instruments. If the eigenfrequencies of the nanoobjects are too high, such frequencies cannot be registered. On the other hand, the method is the most effective and accurate method if the first eigenfrequencies of the nanoobjects are comparable with the first eigenfrequencies of the substrate. Thus, the governing factor in using this method is a good choice of proportions between both the geometric and the physical characteristics of the nanoobjects and the substrate.

4 Effective Properties of Nanocomposite Plates In [1–3, 23] a new approach to the determination of the effective stiffness tensors was proposed. The idea of this approach is based on the comparison of the exact solutions of the three-dimensional elasticity and the corresponding solutions of Equations (1)–(6) for several test problems of the elastostatics such as tension and bending, plane shear, and torsion. The method was applied to isotropic plates as well as to orthotropic and transversally isotropic material behavior. Now it is well known that the nanocomposites demonstrate mechanical properties which may be quite different from its macro-analogues, see e.g. [7, 17]. One of the reasons of

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Fig. 5 Distribution of the natural frequencies of the large system vs. their number n: (a) for ZnO nanocrystals; (b) for GaAs nanotubes lying horizontally (the squares are the natural frequencies corresponding to the vibrations of the substrate); (c) for spherical nano-membranes.

such difference is the surface tension, because these materials have a fractal-like surface. The influence of the surface tension on the elastic moduli was investigated by several authors, see for example [9]. From this point of view they may consider the nanocomposite based on an oriented array of nanocrystals or nanotubes as a transversally isotropic material. We propose to extend the procedure [1, 2, 23] to the case of such materials. Finally, one obtains the components of the effective stiffness tensors which depend not only on the bulk elastic properties and the thickness of the plate but also on the surface tension.

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5 Conclusion A direct approach method in the theory of shells for nano-sized thin-walled structures is introduced. The basic items of the theory are related to: • the formulation of all balances for a deformable directed surface (a priori twodimensional equations), and • the specific constitutive equations. The direct approach has an advantage to the modeling of nanostructures because it is not necessary to consider nanostructures as three-dimensional solids, while many of them are not existed in the bulk phase or the properties of the bulk phase are quite different from the properties of the nanoobjects. On the other hand the determination of the elastic stiffness tensors is essential for the applicability of the theory. The effective stiffness may be obtained by consideration of experimental data on eigenfrequencies. Here we discussed a possible method of determination of the eigenfrequencies of single nanoobjects by investigation the spectrum of the large system consisting of an array of identically nanoobjects grown on a substrate. The theoretical and numerical analysis performed for nanospheres, nanocrystals and nanotubes, shows that it is possible to determine the eigenfrequencies of microand nanoobjects in experiments using the measurement of eigenfrequencies of the substrate with an array of such nanodimensional objects bonded to the surface. The results of numerical calculations of the eigenfrequencies and the eigenmodes are presented. The identification procedure of the effective stiffness of nanocomposite plates is discussed. Acknowledgements The research work was partially supported by the Martin-Luther-University Halle-Wittenberg and the program of development of the South Federal University.

References 1. Altenbach, H.: An alternative determination of transverse shear stiffnesses for sandwich and laminated plates. Int. J. Solids Struct. 37, 3503–3520 (2000). 2. Altenbach, H.: On the determination of transverse shear stiffnesses of orthotropic plates. ZAMP 51, 629–649 (2000). 3. Altenbach, H., Eremeyev, V.A.: Direct approach based analysis of plates composed of functionally graded materials. Arch. Appl. Mech. 37, 775–794 (2008). 4. Altenbach, H., Zhilin, P.: A general theory of elastic simple shells. Usp. Mekh. 11, 107–114 (1988) [in Russian]. 5. Altenbach, H., Zhilin, P.: The theory of simple elastic shells. In: Kienzler, R., Altenbach, H., Ott, I. (Eds.), Critical Review of the Theories of Plates and Shells and New Applications, Lect. Notes Appl. Comp. Mech. Vol. 16, pp. 1–12. Springer, Berlin (2004). 6. Balzani, V., Credi, A., Venturi, M.: Molecular Devices and Machines. A Journey into the Nano World. Wiley, Weinheim (2003). 7. Bhushan, B. (Ed.): Springer Handbook of Nanotechnology. Springer, Berlin (2004). 8. Dargys, A., Kundrotas, J.: Handbook of Physical Properties of Ge, Si, GaAs, and InP, Sci. and Encyclopedia Publ., Vilnius (1994).

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9. Duan, H.L., Wang, J., Karihaloo, B.L., Huang, Z.P.: Nanoporous materials can be made stiffer than non-porous counterparts by surface modification. Acta Mater. 54, 2983–2990 (2006). 10. Eremeyev, V.A., Ivanova, E.A., Morozov, N.F., Soloviev, A.N.: Method of determining the eigenfrequencies of an ordered system of nanoobjects. Techn. Phys. 52, 1–6 (2006). 11. Eremeyev, V.A., Ivanova, E.A., Morozov, N.F., Strochkov, S.E.: The spectrum of natural oscillations of an array of micro- or nanospheres on an elastic substrate. Dokl. Phys. 52, 699–702 (2007). 12. Eremeyev, V.A., Ivanova, E.A., Morozov, N.F., Strochkov, S.E.: Natural vibrations of nanotubes. Dokl. Phys. 52, 431–435 (2007). 13. Eremeyev, V.A., Ivanova, E.A., Morozov, N.F., Strochkov, S.E.: Natural vibrations in a system of nanotubes. J. Appl. Mech. Techn. Phys. 49, 291–300 (2008). 14. Goddard, W.A., Brenner, D.W., Lyshevski, S. E., Iafrate, G.J. (Eds.): Handbook of Nanoscience, Engineering, and Technology. CRC Press, Boca Raton (2003). 15. Golod, S.V., Prinz, V.Ya., Mashanov, V.I., Gutakovski, A.K.: Fabrication of conducting GeSi/Si micro- and nanotubes and helical microcoils. Semiconductor Sci. Technol. 16, 181– 185 (2001). 16. Harik, V.M., Salas, M.D. (Eds.): Trends in Nanoscale Mechanics. Analysis of Nanostructured Materials and Multi-Scale Modeling. Kluwer, Dordrecht (2003). 17. Harris, P.: Carbon Nanotubes and Related Structures: New Materials for the Twenty-First Century. Cambridge University Press, Cambridge (2001). 18. Maugin, G.A.: Continuum Mechanics of Electromagnetic Solids. North-Holland, Amsterdam (1988). 19. Nowacki W.: Electromagnetic Effects in Deformable Solids. PWN, Warsaw (1983) [in Polish]. 20. Poole, C., Jr., Owens, F., Jr.: Introduction to Nanotechnology. Wiley, Hoboken NJ (2003). 21. Prinz, V.Ya.: A new concept in fabricating building blocks for nanoelectronics and nanomechanics devices. Microelectron. Engrg. 69, 466–475 (2003). 22. Zhilin, P.A.: Mechanics of deformable directed surfaces. Int. J. Solids Struct. 12, 635–648 (1976). 23. Zhilin, P.A.: Applied Mechanics. Foundations of the Theory of Shells. St. Petersburg State Polytechnical University, St. Petersburg (2006) [in Russian].

Monitoring of Molecule Adsorption and Stress Evolutions by In-situ Microcantilever Systems H.L. Duan, Y. Wang and X. Yi

Abstract In this paper, a theoretical analysis is given to two applications of microcantilever systems. The first is the detection of molecule adsorption by a two-layer composite cantilever consisting of a porous (nano- or micro-scale) film and a solid layer. In contrast to the classical cantilevers, the static deformation and the vibration frequency shift are greatly influenced by the density and sizes of the pores in the porous films. The second is that the cantilever can be used as a substrate to grow quantum dots (QDs). It is shown that a thin in-situ cantilever setup can monitor QD growth state by choosing the thickness of cantilever.

1 Introduction Numerous attempts have been made to improve the sensitivity of microcantilevers due to their potential as an extremely sensitive platform for chemical and biological detection [8, 11, 16]. We first analyze the static and dynamic properties of the twolayer composite cantilever consisting of a porous (nano- or micro-scale) film and a solid layer, and compare the sensitivity of this novel cantilever with that of the classical one. Then, we set up a connection between the surface stress in continuumlevel descriptions and the adsorption interactions in the molecular-level descriptions by the Lennard–Jones (L–J) potentials. This analysis provides the theoretical basis to clarify the mechanism behind cantilever-based detection. On the other hand, many theoretical and experimental investigations have been conducted to understand the growth mechanisms of quantum dots (QDs), in the H.L. Duan and Y. Wang Institute of Nanotechnology, Forschungszentrum Karlsruhe, P.O. Box 3640, 76021 Karlsruhe, Germany; and College of Engineering, Peking University, P.R. China; e-mail: [email protected] X. Yi College of Engineering, Peking University, P.R. China

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hope that such understanding will allow better control of their electronic properties [1, 3, 13]. To analyze the growth mechanisms and monitor the growth modes (e.g., the Frank–van der Merwe (FM), the Volmer–Weber (VW), the Stranski–Krastanow (SK) growth modes), we model the system where QDs are fabricated in a MBE chamber equipped with an in-situ cantilever measurement setup [7]. In this system, the cantilever is both used as a substrate of QD growth and a stress detector. We find that a thin in-situ cantilever setup can monitor QD growth state by choosing the thickness of cantilever.

2 Mechanics of Two-layer Microcantilevers 2.1 Static Deformation of Microcantilever Consider a cantilever including the upper and lower surfaces (subscripts u and l) and two bulk layers (the film f and substrate s, see Figure 1). In the film, there is an eigenstrain ε ∗ (z) which varies in the thickness direction of the cantilever. Surface stresses τu and τl exist on the upper and lower surfaces of the cantilever, respectively. τu = au + bu εu , τl = al + bl εl , where a denotes the constant surface stress and b is the surface modulus. According to the equilibrium equations of force and bending moment, we obtain the bending curvature κ [17]

κ=

6Ef


hf 0

zε∗ dz + 6hf bu ε∗ (hf ) − 6[hf (au + cbu ) − hs (al + cbl )] − 3c(Ef h2f − Es h2s )

6[hf (hf − hb )bu + hs (hs + hb )bl ] + h2f (2hf − 3hb )Ef + h2s (2hs + 3hb )Es (1)

where c is the uniform strain, and E and h denote the Young modulus and thickness of each layer, hb is the position of the bending axis, where the strain is equal to c. Eq. (1) is the first one to consider the influence of nonuniform eigenstrain, constant surface stress and surface modulus simultaneously. Under the conditions of hf / hs  1 and constant eigenstrain (or constant surface stress), Eq. (1) reduces to classical Stoney formula [14].

2.2 Dynamic Property of Microcantilever The ith mode resonance frequency f˜i of a two-layer cantilever can be expressed as    1 λi 2 T D ˜ , (2) fi = 2π L m + m

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Fig. 1 Schematic diagram of a two-layer cantilever.

where D  , m + m and L are the bending stiffness, effective mass and length of the cantilever, and λi is the eigenvalue satisfying cos λi cosh λi + 1 = 0. Using the vibration equation and the bending moment, the expression of D  can be obtained. D  depends on the modulus and thickness of each layer, and the surface modulus b, and is independent of the constant surface stress a. For the one-layer cantilever, D  is h4 E 2 + 4Eh3 (bu + bl ) + 12h2 bu bl . (3) D = 12(bu + bl + hE) If bu = bl , Eq. (3) reduces to the result of Gurtin et al. (1976).

2.3 Surface Stress Due to Adsorption Surface stress can arise from many interacting mechanisms of the adsorbates, including the electrostatic interaction, van der Waals forces, entropic effects, changes in the charge distribution of surface atoms, and so on [5, 12]. Many researchers develop sensors to quantify the specific chemical, physical, and biological interactions between adsorbed molecules/atoms [2,5]. However, the origin of the induced surface stress at the atomic/molecular level is not quantitatively elucidated. To investigate the origin of the surface stress and its effect on the static and dynamic properties of the sensors, it is important to establish the quantitative relation between the interactions of the adsorbates and the surface stress for the different mechanisms. Because the van der Waals (vdW) interaction between adsorbates is the major driving force for physisorption, we investigate this interaction in terms of the shortrange L–J potential, and derive the relations between the surface stress at the continuum level and the adsorption interaction at the atomic/molecular level. The vdW interaction between the adsorbates (2, 3, 5 and 6) and the surfaces atoms (1 and 4),

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and that between the adsorbates themselves are depicted by the L–J 6-12 type potential (see Figure 1). After obtaining the total potential energy Up (= Ui + Ue ), where Ui is the potential energy due to the vdW interaction, and Ue is the elastic energy of the cantilever, and the kinetic energies of adsorbates and the cantilever, we minimize the total potential energy and get the constant surface stress au . If we substitute the Lagrangian function (consisting of the vdW interaction potential, elastic energy and kinetic energies) into Hamilton’s equation, the surface modulus bu is obtained. au and bu can be expressed as [17]     3ρ 6ρ A1 B1 a = 7 2A2 + 4 − 13 2B2 + 7 (4) η re re η     3ρ 3ρ A1 2A1 2B1 7B1 b = − 7 14A2 − 4 + 5 + 13 52B2 − 7 + 8 η re re η re re where ρ is the number density, η and re are parameters related to distance between adsorbed molecules and cantilever surface, and A1 , B1 , A2 , and B2 are the L–J constants.

2.4 Numerical Evaluation There are many kinds of porous materials, such as bi-continuous structures and channel-array structures [4] and nanoporous materials obtained by dealloying [9]. To formulate the effects of porous film layer on the curvature and resonance frequency of the first novel cantilever, we replace the elastic parameters of the solid film by the corresponding effective moduli of the porous one. Additionally, the constant surface stress a and surface modulus b of the film surface (upper/lower) in Eqs. (1)–(3) should also be replaced by a p and b p (the corresponding quantities on the upper surface of porous films). For the cantilever having the micro- (or nano-) porous film with aligned cylindrical pores, a p = (1 − p)a and b p = (1 − p)b (p is the porosity). In addition to the surface stress of the upper/lower surface, the surface stress am and surface modulus bm of the nano-pore surfaces also make contributions. am and bm lead to an eigenstrain (self-strain) in the film layer, and bm makes a contribution to the effective modulus of the film layer, which cause a frequency shift. At a high porosity, the area of the upper/lower surface is very small compared with that of the nano-pore surfaces. Therefore, for the nano-porous films with a large porosity p, the contribution to the static bending from the film surface stress a p and film surface modulus bp is very small compared with that from am and bm , and can be ignored. Similarly, the contribution to the frequency shift from b p can also be ignored, when it is compared with that from bm . The curvature of two-layer cantilever due to the surface stress is plotted in Figure 2(a). Three types of gold film layer are considered: the solid gold film, and films with aligned cylindrical pores at micro-scale and nano-scale. The material parameters used in Figure 2(a) are as follows: Ef = 90 GPa, νf = 0.42, Es = 10 GPa,

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Fig. 2 (a) Variation of curvatures with log10 (hf / hs ) for silicon cantilevers with gold films (solid, micro-porous and nano-porous films); (b) Variation of the normalized frequency shift (fi /fi ) with the porosity p.

hs = 200 nm, p = 0.7, τ0 = 1 N/m, r0 = 4 nm, au = 1 N/m, λs = −2.7 N/m, µs = −2.6 N/m. From Figure 2(a), it is shown that there is little difference of the curvature between the cantilevers with the solid film and micro-porous film. However, the curvature of cantilevers with the nano-porous film is very larger than that with the solid film and micro-porous film. To study the sensitivity of the surface stress sensors, the normalized frequency shift fi /fi of a two-layer cantilever as a function of the porosity p is shown in Figure 2(b). The material parameters used in Figure 2(b) are as follows: bu = 0.5 N/m, Ef = 90 GPa, νf = 0.42, Es = 10 GPa, hs = 300 nm, hf = 30 nm, r0 = 2.5 nm. It is seen from Figure 2(b) that fi /fi of the cantilever with nano-porous film is always larger than that of the cantilever with the micro-porous and solid films, and it also increases with the increase of porosity p. From Figure 2(b), it can be seen that for a certain substrate thickness, fi /fi of cantilevers with nano-porous films is the largest, and that of classical solid cantilevers is the smallest. Fox instance, for p = 0.6 the cantilevers with nano-porous films display sensitivity enhancements up to about 30-fold and 22-fold, respectively, compared with the cantilevers having the solid films and micro-porous films.

3 Quantum Dot Growth on In-situ Cantilever Consider a QD growth system consisting of a cantilever (material B with lattice constant dB , the cantilever can be regarded as the substrate with a finite thickness ts ), wetting layer and pyramidal QDs (material A with lattice constant dA , see Figure 3(a)). Letting κ be the curvature and z0 the position of the neutral plane of the system, then the strain in the substrate and in the wetting layer will be, respectively, εs = κ(z − z0 ) and εf = κ(z − z0 ) + ε∗ . ε ∗ is the mismatch strain (ε∗ = (dA − dB )/dB ), which arises from the different lattice constants between the wetting layer and the substrate. To obtain simple analytical solutions, we assume

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Fig. 3 (a) QD growth system consisting of substrate with a finite thickness, wetting layer and pyramidal QDs; (b) RVE of QD growth system.

that the materials A and B are isotropic, and it has been validated by comparing isotropic and anisotropic solutions for semiconductor materials. The self-organization of QDs is a thermodynamic process in which the free energy of the system is minimum when QDs form [1, 10]. The free energy u of the system consists of the energies of the finite substrate (usub ), the wetting layer (uf ) and the island (uis ) u = uf + usub + uis (5) Moreover, the expressions of usub , uf and uis include the elastic energies, the wetting layer-substrate interaction energy, island-substrate interaction energy, islandisland interaction energy, and the surface energies [3, 13, 15]. We investigate a representative volume element (RVE) shown in Figure 3(b). The area of the element is D × D, the equivalent thickness of material deposited on the substrate is H and the total volume of deposited material is H D 2 . H is composed of thickness of wetting layer and the equivalent thickness of islands. We assume that the thickness of wetting layer is tf , the base size of the island is L, the tilt angle of the uniform island is ϕ, and the island coverage is q = L2 /D 2 . The total free energy of the system is functions of κ and z0 . We minimize total free energy u in respect to κ and z0 , and get two equations about them. The analytical solutions of κ and z0 can be obtained by solving these two equations, and they are functions of ts , tf , L, ϕ and q [15]. When QDs grow on a thin cantilever, the cantilever can be used to monitor the morphologies of QDs. Figure 4 shows the change of curvatures κ with island coverage q and island size L. It can be seen that when the cantilever is thin (Figure 4, ts = 0.5 µm), the curvature κ is very large, and it is easy to be measured. Therefore, to monitor the morphology of QDs with a high sensitivity, the thickness of the substrate (cantilever) should be properly chosen so that it is sensitive enough to detect the morphologies of QDs.

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Fig. 4 Variation of curvature κ with island coverage q and island size L (ϕ = 10◦ , tf = 1.4 nm, ts = 0.5 µm).

4 Conclusions We first investigate the static and dynamic properties of the composite cantilever consisting of a solid layer and a porous film. It is found that the curvature and the vibration frequency shift are greatly influenced by the density and sizes of the pores. Then, we give a theoretical framework to analyze the growth of quantum dots (QDs) on in-situ cantilever. It is shown that a thin in-situ cantilever setup can monitor QD growth state with high sensitivity by choosing the thickness of cantilever.

References 1. Chiu, C.H., Huang, Z., Poh, C.T., 2004, Formation of nanostructures by the activated Stranski– Krastanow transition method, Phys. Rev. Lett. 93, 136105-1–4. 2. Dareing, D.W., Thundat, T., 2005, Simulation of adsorption-induced stress of a microcantilever sensor, J. Appl. Phys. 97, 043526-1–5. 3. Daruka, I., Barabasi, A.L., 1997, Dislocation-free island formation in heteroepitaxial growth: A study at equilibrium, Phys. Rev. Lett. 79, 3708–3711. 4. Duan, H.L., Wang, J., Karihaloo, B.L., Huang, Z.P., 2006, Nanoporous materials can be made stiffer than non-porous counterparts by surface modification, Acta Mater. 54, 2983–2990. 5. Godin, M., 2004, Surface stress, kinetics, and structure of alkanethiol self-assembled monolayers. Ph.D. Thesis, McGill University, Canada. 6. Gurtin, M.E., Markenscoff, X., Thurston, R.N., 1976, Effect of surface stress on the natural frequency of thin crystals, Appl. Phys. Lett. 29, 529–530. 7. Hu, D.Z., 2007, Stress evolution during growth of InAs on GaAs measured by an in-situ cantilever beam setup. Ph.D. thesis, Humboldt University, Germany. 8. Huang, G.Y., Gao, W., Yu, S.W., 2006, Model for the adsorption-induced change in resonance frequency of a cantilever, Appl. Phys. Lett. 89, 043506-1–3. 9. Kramer, D., Viswanath, R.N., Weissmüller, J., 2004, Surface-stress induced macroscopic bending of nanoporous gold cantilevers, Nano Lett. 4, 793–796. 10. Liu, F., Huang, M.H., Rugheimer, P.P., Savage, D.E., Lagally, M.G., 2002, Nanostressors and the nanomechanical response of a thin silicon Film on an insulator, Phys. Rev. Lett. 89, 136101-1–4.

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11. Lu, P., Lee, H.P., Lu, C., O’Shea, S.J., 2005, Surface stress effects on the resonance properties of cantilever sensors, Phys. Rev. B 72, 085405-1–5. 12. Ren, Q., Zhao, Y.P., 2004, Influence of surface stress on frequency of microcantilever-based biosensors, Microsystem Technologies 10, 307–314. 13. Shchukin, V.A., Ledentsov, N.N., Kop’ev, P.S., Bimberg, D., 1995, Spontaneous ordering of arrays of coherent strained islands, Phys. Rev. Lett. 75, 2968–2971. 14. Stoney, G.G., 1909, The tension of metallic films deposited by electrolysis, Proc. Roy. Soc. A 82, 172–175. 15. Wang, Y., et al., 2008, Tuning and monitoring of quantum dot growth by an in-situ cantilever. Phys. Rev. B, submitted. 16. Wu, G., Ji, H., Hansen, K., Thundat, T., Datar, R., Cote, R., Hagan, M.F., Chakraborty, A.K., Majumdar, A., 2001, Origin of nanomechanical cantilever motion generated from biomolecular interactions, Proc. Natl. Acad. Sci. U.S.A. 98, 1560–1564. 17. Yi, X. et al., 2008, Mechanics of microcantilevers based on interactions of adsorbates and application to design of surface-enhanced sensors. J. Mech. Phys. Solids, submitted.

Using Thermal Gradients for Actuation in the Nanoscale E.R. Hernández, R. Rurali, A. Barreiro, A. Bachtold, T. Takahashi, T. Yamamoto and K. Watanabe

Abstract In this paper we will argue that temperature gradients show great potential to induce controlled motion of nanoscaled objects, and could be used in the design of novel nano machines performing useful tasks in that scale. The well known phenomenon of thermal electromigration is a manifestation of this effect; a temperature gradient along a conductor induces the migration of charge carriers towards the cool end of the conductor, and thus a charge current is produced. Examples of experiments and simulations will be shown where it is demonstrated that the same effects can achieve the controlled migration of larger objects, such as clusters, fullerenes and nanotubes, and ideas on how to harness and exploit these effects in nano-electromechanical systems (NEMS) will be put forward.

1 Introduction It has been recently shown experimentally that thermal gradients have great potential for inducing and controlling the motion of nanoscaled objects [1]. This phenomenon, well known for light objects such as electrons or ions, had not been contemE.R. Hernández Institut de Ciència de Materials de Barcelona (ICMAB-CSIC), Campus de Bellaterra, 08193 Barcelona, Spain; e-mail: [email protected] R. Rurali Departament d’Enginyeria Electrònica, UAB, 08193 Barcelona, Spain; e-mail: [email protected] A. Barreiro and A. Bachtold Centre d’Investigacions en Nanociencia i Nanotecnologia and Centro Nacional de Microelectrónica, Campus de Bellaterra, 08193 Barcelona, Spain; e-mail: [email protected] T. Takahashi, T. Yamamoto and K. Watanabe Department of Physics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan; e-mail: [email protected]

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Fig. 1 Schematic picture of the mobile element of the nanofabricated device. The long inner nanotube (shown in yellow) is suspended between the two electrodes; the gold platelet is attached to the outer nanotube (shown in red), which can slide down and rotate around the inner nanotube due to the low friction contact between nanotube walls.

plated for heavier objects such as entire molecules or clusters. In this talk we will summarise the experimental results to date, and will discuss simulation efforts that have on the one hand contributed to identifying the cause of the observed experimental behaviour, and are expected to lead to a detailed atomistic understanding of the process in the future, on the other.

2 Summary of Experimental Findings In a recent paper, Barreiro et al. [1] have shown that thermal gradients can make nano-scale objects move in a controlled way, in the direction of decreasing temperature, i.e. down the thermal gradient. Barreiro et al. constructed a prototype nanoscaled device in which a multi-walled carbon nanotube (MWCNT) is contacted between two gold electrodes, with a small gold platelet being attached to the MWCNT at approximately the mid point between the two electrodes, which will be referred to as the cargo. The outer layer or layers of the MWCNT are then selectively eliminated along the length of the tube, except below the platelet, by circulating a large current through the nanotube. This technique is known as electrical breakdown, and has been documented elsewhere [2–4]. The device is completed by a wet etching step, which has the result of suspending the nanotube and cargo between the two electrodes. A schematic picture of the resulting device can be seen in Figure (1). The electrical breakdown technique results in a device consisting of a mobile element (the shorter outer nanotube layer or layers plus the cargo) which can slide along and/or rotate around the suspended nanotube. This schematic picture can be experimentally confirmed by reversibly moving the mobile element with the tip of an atomic force microscope (AFM). In devices fabricated as described above, it was discovered that translational and/or rotational motion of the cargo (and presumably of the outer layer(s) to which

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it is attached could be induced by making an electrical current (≈ 0.1 mA) circulate between the electrodes through the suspended MWCNT. Motion of a translational and/or rotational nature was observed in 10 out of 16 fabricated devices. In most (4 out of 6) devices where no motion was observed, the resistance to the electrical conduction was larger than 1 M
, while in the devices displaying motion of the cargo the resistance varied between 35 and 680 k
. This is an important issue to which we will return below. In devices in which the electrical break down technique was not used (i.e. where the outer layer(s) were not partially ablated) motion was never observed. However, it was also found that the electrical current was not itself directly responsible for inducing the observed displacement of the mobile element, because upon reversing the polarity of the device (i.e. inverting the direction of the electron flow) the mobile element continued to move in exactly the same way as before, i.e. translating and/or rotating in the same direction. Nor could the rotational motion of the mobile element be accounted for by the magnetic field generated by the circulating current, as the rotation was seen to be either clockwise or anti-clockwise, depending on the device. Finally, it was found that the stray electric field present in the device could not explain either the observed motion. Whatever the field between the electrodes in the devices where motion was observed was also present in the highly resistive devices where no motion was seen. It would seem, therefore, that some how the resistance of the device holds the key to understanding the mechanism driving the observed motion. Given that neither the current flowing through the device nor the electrical field acting between the electrodes could explain the observed phenomena, Barreiro and co-workers formulated the hypothesis that the cargo moved under the influence of a thermal gradient. This hypothesis was considered after taking into account the following observations. Firstly, it was noticed that the temperature of the device increased substantially when the current was flowing through the MWNT, due to Joule heating. This was made evident by the fact that frequently the gold platelet on the mobile element was observed to melt. The melting temperature of gold is ≈ 1300 K, and while it is known that clusters normally have melting temperatures lower than that of the bulk, given the large size of the metal platelet in the experiments of Barreiro et al., it is likely that its melting temperature was not significantly reduced from that of the bulk metal. On the other hand, the electrodes, also made of gold, were never observed to melt, and given their massive nature it is most probable that they remained at temperatures not greatly above room temperature. Recent experiments by Begtrup et al. [5] indeed confirm this expectation. This implies that the suspended nanotube was subject to a large temperature gradient, with the temperature being highest (high enough to melt the gold platelet) close to the centre of the suspended tube, and lowest (close to room temperature) at the points of contact with the electrodes. Taking into account the typical suspended length of the nanotubes (≈ 1.5–2 µm), this implies a thermal gradient of up to 1 K/nm. The above hypothesis seemed to fit all the experimental observations, and could explain why no motion had been observed in the case of highly resistive devices. Indeed, in such devices the current circulating through the suspended MWNT is small,

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Table 1 Details of the simulated DWNT systems. Values separated by a slash correspond to the inner and outer tube, respectively. System

Length (nm)

Number of atoms

(8,2)@(12,8) (8,2)@(17,2) (10,10)@(15,15) (10,10)@(26,0) (17,0)@(26,0) (17,0)@(15,15) (12,9)@(17,14)

100.2 / 5.6 100.2 / 5.2 99.9 / 4.95 99.9 / 5.14 100.3 / 5.14 100.3 / 4.95 99.0 / 7.7

8568 / 912 8568 / 872 16160 / 1200 16160 / 1248 15912 / 1248 15912 / 1200 16872 / 1928

on account of the high resistivity, which in turn implies that the Joule heating of the tube is lower, perhaps too low to result in a thermal gradient large enough to induce the motion of the mobile element. Nevertheless, it was not directly possible to test the thermal gradient hypothesis experimentally, so in order to check its feasibility, a series of molecular dynamics simulations was carried out. As will be explained below, those simulations confirmed the hypothesis, and furthermore, gave indications on the atomistic details of the mechanism underlying the experimental observations. In the following section details of the atomistic simulations conducted by Barreiro et al. [1], as well as new ones reported here for the first time, are discussed.

3 Molecular Dynamics Simulations 3.1 Simulation Details In order to test the hypothesis of the thermal gradient being responsible for the observed motion of the mobile element, a series of MD simulations was performed in which double-walled carbon nanotubes (DWCNTs) were modelled. Each DWCNT consisted of a long inner tube and a short outer one; many different combinations of tube geometries were considered, but in all cases the inner tube was 100 nm long, while the outer nanotube had a length of 5 nm; these lengths varied slightly on account of the different unit-cell lengths along the nanotube axis. Each pair of nanotubes was selected with the condition that the difference in tube radii fell in the range of experimentally observed values in MWCNTs (3.4 Å). Details of the different DWCNT systems considered in the simulations are given in Table 1. There are two distinct types of interactions in the DWCNT systems that need to be taken care of, namely, the covalent bonding interactions responsible for the atomic arrangement in the walls of the nanotubes, and the long-range dispersive type interactions between atoms belonging to different tubes. The first type of interaction can be adequately modelled with the bond-order potential developed by Tersoff [6], while the second one can be described with a simple Lennard–Jones 12–6 potential. In choosing the parameters of the latter potential, we have followed Saito et al. [7].

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The DWCNT systems listed in Table 1 were simulated with the long inner tube subjected to a thermal gradient. In order to do this, the inner tube was first simulated on its own, with atoms at opposite ends constrained to remain at different temperatures. Typically, a few hundred atoms (e.g. 400 atoms at each end in the case of the (10,10) nanotube) would have their velocities scaled at regular intervals in order to fix the temperature at the ends. The remaining atoms were allowed to evolve in time undisturbed. The low temperature end was constrained to be at an average temperature of 300 K, while the hot end was scaled at different temperatures chosen to impose the desired thermal gradients. Specifically, we considered thermal gradients of 7, 4 and 1 K/nm, the latter roughly corresponding to that estimated to hold in the experiments. The equations of motion were numerically integrated using the Velocity Verlet algorithm [8] using a time step of 2 fs. In preparation for the production run, the inner tube was simulated separately for 200 ps, so as to stabilise the desired thermal gradient. Subsequently, the outer tube was introduced into the simulations, with initial atomic velocities corresponding to the Maxwell–Boltzmann distribution at the temperature of the mid point of the long nanotube. So as to not bias the dynamics of the outer tube in any direction, care was taken to ensure that the net centre-ofmass velocity of the outer tube was zero. With the systems prepared in this way, simulations of the DWCNTs were performed for up to 400 ps, during which the dynamics of the inner and outer tube were monitored.

3.2 Results In all the simulations that we have performed the spontaneous displacement of the outer tube down the thermal gradient was observed. In some cases the observed dynamics of the outer tube as mostly of a translational nature, or of a rotational nature, though normally the motion contained both components. In Figure 2 we illustrate the displacements of the centre of mass of the outer nanotube as a function of time for the combination of (12,9)@(17,14) nanotubes. The results were obtained from three different simulations with different imposed thermal gradients, namely 1, 4 and 7 K/nm. As can be seen, in all three cases the outer nanotube displaces toward the cool region of the inner nanotube, but the speed of the displacement is larger the larger the temperature gradient. While calculations such as those illustrated in Figure 2 confirm that thermal gradients are indeed capable of inducing the motion (both translational and rotational) of the outer tube in DWCNT systems, as observed in the experiments of Barreiro et al. [1], the question arises as to whether it is possible to use the same effect to induce motion of other objects, located inside, rather than outside, the nanotube. There have been previous simulation reports [9, 10] that indicate that indeed gold clusters encapsulated inside SWCNTs can migrate along the inside of the nanotube under the influence of a thermal gradient. Pursuing this idea further, we have conducted simulations of C60 fullerenes encapsulated in different SWCNTs, all having

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Fig. 2 Displacements of the centre of mass of the outer (17,14) nanotube along the inner (12,9) tube, under three different thermal gradients.

similar diameter to the (10,10) SWCNT, namely 1.4 nm. This diameter has been chosen because it is known to be that leading to an optimal encapsulation energy of C60 ; diameters narrower than this lead to an endothermic encapsulation, while for wider diameters the encapsulation is barely exothermic. Specifically, we have considered the encapsulation of C60 in nanotubes (10,10), (17,0), (14,5) and (12,8). In Figure 3 we have plotted the displacement of the centre of mass of the encapsulated C60 cluster in the (10,10) nanotube as a function of simulation time, for the different temperature gradients considered. As can be seen, the fullerene, which has its initial position in the middle of the nanotube, is accelerated towards the cold side of the nanotube, reaching the end, some 50 nm away from the initial position, in about 200 ps, for all temperature gradients considered, namely 1, 4 and 7 K/nm. Once the fullerene reaches the end of the nanotube it bounces back. This is because the open end of the nanotube poses a potential energy barrier for the motion of the cluster; if the cluster was to go out of the nanotube, its potential energy would increase dramatically, as it would loose the stabilising dispersion interactions with the wall of the nanotube. As a result, reaching the end of the nanotube the fullerene collides with a potential energy barrier that makes it bounce back inside the nanotube, moving against the thermal gradient. However, the same mechanism that initially speeds up the fullerene from rest and forces it to move down the thermal gradient now slows it down until it stops completely, and then accelerates it again in the direction from hot to cold, until again the cluster collides with the potential energy barrier at the end of the nanotube. This process is repeated several times in the time span of our sim-

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Fig. 3 Displacements of the centre of mass of a C60 cluster encapsulated inside a (10,10) nanotube subject to different thermal gradients.

ulation, namely 1.2 ns, without there being much difference, qualitatively, for the results obtained with different thermal gradients. Similar results, varying only in the detail, but not in the essentials, have been observed for C60 in the other nanotubes considered. Thus far we have discussed the use of molecular dynamics simulations to confirm that indeed thermal gradients are capable of inducing the motion of nano-scaled objects along one-dimensional tracks such as carbon nanotubes; the results discussed above do confirm that it is possible to move, in a controlled fashion, nanoscaled objects both outside (outer tubes) or inside (encapsulated fullerenes) by using thermal gradients. However, we would like to obtain a better understanding of the underlying mechanism inducing this motion. In particular, it would be interesting to know how the different phonon bands of the nanotube participate in the process of inducing the motion of the mobile objects, be them inside or outside the nanotube. To this end we have started to perform a series of simulations which we detail below. Because carbon nanotubes have large unit cells, they also have large numbers of phonon bands; even SWCNTs with relatively simple structures such as the nonchiral (n, n) and (n, 0) nanotubes have many bands. For example, the (10,10) nanotube, which has 40 atoms in its unit cell, has 120 phonon modes, the majority of which, in fact all but four, are optical phonon modes. A fraction of the phonon dispersion diagram for the (10,10) nanotube is displayed in the left panel of Figure 4. Presumably, not all phonons contribute in equal measure to inducing the motion of the mobile objects that we have observed in our simulations above. For exam-

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ple, it is easy to imagine that the totally symmetric radial breathing mode (RBM) of the SWCNT will have a strong influence, because in this mode the atoms of the SWCNT wall vibrate perpendicularly to the wall, and thus the diameter of the SWCNT oscillates in breathing fashion. A phonon wave packet with a large RBM component travelling down the tube from the hot region toward the cold one will effectively push the mobile element, transferring some momentum to it. On the other hand, other phonons in which the distortion does not involve a noticeable diameter change will presumably have a smaller effect. In order to analyse this question, we have commenced a series of simulations in which a wave packet constructed from a phonon of the SWCNT, with a certain amplitude in real space, according to the following formula:
2 2 uνink0 = A
νi e(k−k0 ) /2σ eı˙(kxn −ων (k)t ) (1) k

where uink0 is the displacement of atom i in cell n according to the phonon band ν at wave vector k0 with amplitude A and
is the projection of the polarisation vector of phonon band ν at k0 onto atom i. The phonon frequency is ω, and σ is a parameter determining the spread of the phonon wave packet. Using Equation (1) above it is possible to prepare a phonon wave packet in a SWCNT, and follow its time evolution in a conventional molecular dynamics simulation [11]. In particular it is possible to direct the packet against a target, such as a fullerene inside the SWCNT, or an outer tube, positioned some way down the length of the tube, and analyse the collision process. Simulations like these are expected to be very helpful in revealing the mechanism of the transfer of energy from the tube vibrational modes to the different degrees of freedom of the mobile object. In the right panel of Figure 4 we illustrate preliminary results of such a simulation. A wave packet is constructed from the lowest optical phonon of a (10,10) nanotube (highlighted in red on the left hand side panel of Figure 4) with a k0 vector in the middle of the Brillouin zone. The phonon wave packet moves from left to right along the nanotube with a speed corresponding to the phonon group velocity of the band at k0 . Eventually it reaches a point where a C60 fullerene is encapsulated. The largest part of the wave packet is transmitted beyond the point where the C60 molecule is located, apparently oblivious to its presence, but there is a small reflection, and also a part of the transmitted wave packet moves with a slower speed after the collision, indicating that some transfer of energy to the fullerene has taken place. Nevertheless, in this case the C60 remains relatively unaffected by the passing of the wave packet. Simulations like this one, currently being carried out, are expected to clarify the underlying physics of the thermal gradient induced motion reported by Barreiro et al. [1].

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Fig. 4 The left panel shows the dispersion relation for the low energy phonon bands of a (10,10) SWCNT calculated with the Brenner [12] potential. The lowest optical band, shown in red, was used to construct the phonon wave packet shown on the right panel.

4 Conclusions In this talk we have shown simulation results aimed at obtaining a better understanding of the motion of nanoscaled objects induced by thermal gradients. Qualitatively, the process is clear: a net flow of phonons is established from the hot region towards the colder one when a one-dimensional object such as a nanotube or nanowire is subject to a thermal gradient. This current of phonons is capable of interacting with and transferring energy to other objects such as encapsulated clusters or other layers of the nanotube, inducing them to move in the direction of the phonon current. However, the atomistic details of this process remain to be fully clarified. We are working along these lines with simulation methods based on molecular dynamics, and hope to be able to offer a complete picture in the near future. Acknowledgements We acknowledge financial support from the Spanish Ministry of Science through grants FIS2006-12117-C04 and TEC2006-12731-C02-01 and a Ram’on y Cajal Fellowship (Rurali), form the AGAUR through grant 2005SGR683, from the EU through grant FP6-IST021285-2 and an EURYI award (Bachtold), the Japan Science and Technology Corporation (JSTCREST) and the HOLCS project of the Japan Ministry of Education, Culture, Sports, Science and Technology.

References 1. A. Barreiro, R. Rurali, E.R. Hernández, J. Moser, T. Pichler, L. Forró, A. Bachtold, Science 320, 775 (2008). 2. P.G. Collins, M.S. Arnold, P. Avouris, Science 292, 706 (2001). 3. P.G. Collins, M. Hersham, M. Arnold, R. Martel, P. Avouris, Phys. Rev. Lett. 86, 3128 (2001). 4. B. Bourlon, D.C. Glattli, B. Plaçais, J.M. Berroir, C. Miko, L. Forró, A. Bachtold, Phys. Rev. Lett. 92, 026804 (2004).

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5. G.E. Begtrup, K.G. Ray, B.M. Kessler, T.D. Yuzvinsky, H. García, A. Zettl, Phys. Rev. Lett. 99, 155901 (2007). 6. J. Tersoff, Phys. Rev. B 37, 6991 (1988). 7. R. Saito, R. Matsuo, T. Kimura, G. Dresselhaus, M.S. Dresselhaus, Chem. Phys. Lett. 348, 187 (2001). 8. M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, (1987). 9. P.A.E. Schoen, J.H. Walther, S. Arcidiacono, D. Poulikakos, P. Koumoutsakos, Nano Lett. 6, 1910 (2006). 10. P.A.E. Schoen, J.H. Walther, D. Poulikakos, P. Koumoutsakos, Appl. Phys. Lett. 90, 253116 (2007). 11. P.K. Schelling, S.R. Phillpot, P. Keblinski, Appl. Phys. Lett. 80, 2484 (2002). 12. D.W. Brenner, Phys. Rev. B 42, 9458 (1990).

Systematic Design of Metamaterials by Topology Optimization Ole Sigmund

Abstract Metamaterials are engineered materials with properties usually not seen in nature. This paper reviews the authors work in the field of metamaterial design by the topology optimization method. Examples include the optimization of elastic materials with negative Poisson’s ratio and thermal expansion coefficient, electromagnetic band gap materials and electromagnetic metamaterials with negative permittivity.

1 introduction There exist many definitions of the term “Metamaterials”. Especially amongst electrical engineers the topic is currently very hot due to the possibility of synthesizing materials with negative refractive index, in turn paving the way for super lenses and cloaks [14]. More generally, metamaterials may be characterized as ordered composites with exceptional properties usually not encountered in nature. Clever physicists and engineers have come up with many intriguing metamaterial structures by intuition and experience. However, when it comes to taking full advantage of the possibilities of metamaterials, systematic design methods are required. One possible systematic design method for metamaterial design is the topology optimization method, originally developed for structural mechanics problems [1, 3]. The topology optimization method is based on repeated finite element analyses, sensitivity analyses and material redistribution steps and makes use of math programming techniques in order to obtain fast convergence. Material type or density in each element of the finite element discretized base cell serve as design variables and hence, the design freedom is huge. The topology optimization method was first applied to material design, i.e. inverse homogenization by the author [15, 16]. FigOle Sigmund Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, Nils Kopples Alle, Building 404, 2800 Lyngby, Denmark; e-mail: [email protected]

R. Pyrz and J.C. Rauhe (eds.), IUTAM Symposium on Modelling Nanomaterials and Nanosystems, 151–159. © Springer Science+Business Media B.V. 2009

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Fig. 1 The inverse homogenization problem. White arrows indicate the conventional forward homogenization approach, and black arrowe indicate the inverse homogenization approach.

ure 1 illustrates the idea. In conventional homogenization approaches one takes out the smallest repetitive unit of a periodic material, i.e. the base cell, and derives the effective properties by a number of finite element analysis test cases depending on the physical problem and dimension. In the inverse homogenization approach the goal is to extremize the effective properties and this is obtained by optimizing the material distribution in the base cell such that the resulting periodic materials has the wanted properties. The paper reviews the authors work in the field of metamaterial design (or inverse homogenization) by the topology optimization method. Examples include the optimization of elastic material with negative Poisson’s ratio and thermal expansion coefficient, electromagnetic band gap materials and electromagnetic metamaterials with negative permittivity. The paper is organized as follows. In Section 2, the topology optimization problem formulation is described. In Section 3, the method is applied to design problems in elasticity, in Section 4 to electromagnetic band gap structures and in Section 5 to electromagnetic metamaterial design. Section 6 summarizes the paper.

2 The Inverse Homogenization Problem Ideally, we would like to determine the pointwise material distribution in a base cell that optimizes the design goal (e.g. minimum Poisson’s ratio) with constraints on for example volume fraction and/or isotropy. In practise, the true pointwise material distribution cannot be found but one can discretize the base cell with a large number of finite elements and let the material type or density in each element be a variable. Denoting the vector of density design variables as ρ , the optimization problem may be defined as

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min : (Cij∗ kl (ρ ρ )) ρρ

s.t. :

N


ve ρe /V0 = vT ρ /V0 ≤ f

e=1

: gi (Cij∗ kl (ρ ρ )) ≤ gi∗ , i = 1, ..., M  : ρe =

0 (material 1) 1 (material 2)

: K(ρ ρ )Ui = Fi ,

(1)

, e = 1, ..., N

i = 1, . . . , m

where (Cij∗ kl )

is a function of the homogenized material (elasticity) tensor (for example the effective Poisson’s ratio), ve the volume of element e, V0 the total volume of the base cell, f the volume fraction constraint, gi (Cij∗ kl (ρ ρ )) and gi∗ constraint values and limit (e.g. isotropy) and K(ρ ρ )Ui = Fi denotes the finite element system for the different load cases used in the homogenization. For this discrete optimization problem the design variables may take the value 0 or 1 (white and black, cf. Figure 2 left), yielding the element material properties Cij kl (ρe ) = (1 − ρe )Cij1 kl + ρe Cij2 kl ,

(2)

where Cij1 kl and Cij2 kl are the material tensors of two constituent materials (one of which could be void). For a fine discretization it is not possible to solve the stated discrete optimization problem efficiently. Instead, we convert the discrete optimization problem into a continuous one by allowing the design variables to take any value between 0 and 1 (cf. Figure 2 right) min : (Cij∗ kl (ρ ρ )) ρρ

s.t. :

N


ve ρe /V0 = vT ρ /V0 ≤ f

e=1

: gi (Cij∗ kl (ρ ρ )) ≤ gi∗ , i = 1, ..., M

(3)

:0≤ρ ≤1 : K(ρ ρ )Ui = Fi ,

i = 1, . . . , m

In this way, we may use sensitivity analysis and math-programming to obtain speedy convergence. In practise, we use adjoint sensitivity analysis to obtain the gradients (see e.g. [22]) and the Method of Moving Asymptotes [28] for the design updates. Of course there is now a risk of ending up with intermediate values of the design variables, however, in order to avoid this one may use different penalization schemes as discussed in [2]. For example one may use the SIMP approach where the density variables are raised to a power p > 1 that makes intermediate density values uneconomical and hence forces the design to discrete solutions despite

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Fig. 2 Discretized unit cell. Left: Discrete 0-1 variables and Right: Continuous variables. White denotes material 1 and black denotes material 2.

of the continuous variables. The penalized element material interpolation function now reads p Cij kl (ρe ) = (1 − ρe )p Cij1 kl + ρe Cij2 kl . (4) In most cases, the described optimization procedure converges in a couple of hundred iterations (homogenization evaluations). With an exchange of homogenization approach, the method can immediately be applied to other design problems in e.g. band gap engineering and electromagnetic meta-material design as will be demonstrated in the following sections.

3 Elastic Materials with Negative Coefficients The first applications of the inverse homogenization approach as described above were truss-like material structures as presented in [15, 16]. Later, the method was extended to continuum type elasticity problems [12, 17, 18], multimaterial problems [6, 21, 22], thermoelastic problems [21, 22] and problems involving piezoelectricity [23–25]. The method has also been applied to combined elasticity and permeability problems [9, 17, 29] and lately to combined fluid permeability and structural problems [8]. Figure 3 (left) shows the result of the minimization of Poisson’s ratio with a constraint on minimum bulk modulus value and isotropy (from [12]). The resulting structure (shown as a 5 by 5 base cell super cell) is seen to consist of a triangle structure where horizontal compression causes vertical closing of the triangles and thereby negative Poisson’s ratio. The material has been build and tested in various length scales. The insert shows a micro machined glass sample with cell size 50 µm [12]. Figure 3 (right) shows the result of a minimization of the isotropic thermal expansion coefficient with a constrain on minimum bulk modulus. Here, the material interpolation law was extended with an extra design variable allowing for the distribution of three materials (void, solid 1 and solid 2) in the base cell [21,22]. The two

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Fig. 3 Left: Topology optimized negative Poisson’s ratio materials (from [12]). Right: Topology optimized negative thermal expansion material (from [21, 22]).

solid materials have high and low (but both positive) expansion coefficients and the resulting structure (shown as a 3 by 3 super cell) has effectively negative thermal expansion coefficients. In [22], the obtained effective properties are shown to be very close to theoretical bounds on the attainable thermoelastic properties. In fact, improved theoretical bounds were inspired by the numerical results [7].

4 Electromagnetic Band Gap Materials For certain periodic material configurations waves in certain frequency ranges cannot propagate. Such materials are called band gap materials and the phenomenon exists for all physical wave propagation problems (e.g. elasticity, acoustics, photonics, etc.). The topology optimization method can be applied to such problems by substituting the static homogenization procedure by eigenvalue analyses based on Floquet–Bloch wave theory as described in several publications [4, 5, 10, 11, 19, 20]. The objective function for band gap optimization problems may be the maximization of the relative band gap size min : ωn+1 (k, ρ ) − max : ωn (k, ρ ) ωn k k = 2 ωn0 min : ωn+1 (k, ρ ) + max : ωn (k, ρ ) k

(5)

k

where ωn (k, ρ ) is the n’th eigenfrequency for a value of the wavevector k on the boundary of the irreducible Brillouin zone (cf. Figure 4 top). The band gap optimization problem is quite complex and involves a challenging sensitivity analysis due to multiple eigenvalues and constraints.

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Fig. 4 Top: Irreducible Brillouin zone and band diagram for periodic wave propagation problems. Bottom: Composite pictures showing topology optimized maximum band gap materials as well as the geometrically optimized partitions (from [19]).

A recent example of topology optimization for photonic band gap structures was presented in [19]. An example of the results presented in the paper is shown in Figure 4 (bottom). The paper proves that optimal planar microstructures for TE and TM polarized light follow a simple geometric rule and that the overall optimal structures correspond to a closed-walled hexagonal structure and a triangular pattern of circular rods for the two polarization cases, respectively. In conclusion, the long lasting quest for optimal planar photonic band gap structures has come to an end with the results presented in the paper.

5 Electromagnetic Metamaterials The final example is concerned with the design of optical or microwave materials with negative permittivity. Homogenization formulations for such problems are still in their infancy [13, 26, 27] but for now we use an approach suggested in the former of the three papers.

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Fig. 5 Top: Design domain for transmission design problem. Center: Initial topology with negative permittivity. Bottom: Optimized topology with negative permittivity.

Based on a transmission study (cf. Figure 5 top), S-parameters may be extracted and the effective refractive index may be calculated as   1 1 2 n∗ = cos−1 (1 − S11 S22 + S21 ) (6) kd 2S21 and the impedance is found as  z∗ =

2 (1 + S11 )(1 + S22 ) − S21 2 (1 − S11 )(1 − S22 ) − S21

(7)

From these values, the effective permittivity may be found as  ∗ = n∗ z∗ and the effective permeability as µ∗ = n∗ /z∗ . Again, gradients may be found very efficiently using the adjoint method.

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The goal of the optimization problem is here to minimize the real part of the effective permeability by distributing a high permittivity material in air. A known negative permeability solution is the circular rod in air shown in Figure 5 (center). Depending on starting guess, it may not be possible to come up with a negative permeability solution if one starts on the wrong side of the positive peak in µ . Therefore, a practical solution may be to start by minimizing the imaginary part of µ, i.e. µ since this value does not exhibit a positive peak. Based on this optimized topology one may proceed by minimizing the real part of the permeability µ . The result of this process is shown in Figure 5 (bottom). The resulting topology is a hollow ring structure and it is interesting to note from the dispersion diagram that the optimized structure apart from having a better µ than the known design also has a larger negative permeability bandwidth.

6 Conclusions This paper has presented a review of the author’s (and other researchers) work in adapting the topology optimization method to systematic metamaterial design. Acknowledgements This work received support from the Eurohorcs/ESF European Young Investigator Award (EURYI, www.esf.org/euryi) through the grant “Synthesis and topology optimization of optomechanical systems”, the Danish Research Council for Technology and Production Sciences within the TopAnt project (http://www.topant.dtu.dk), the New Energy and Industrial Technology Development Organization project (NEDO, Japan), and from the Danish Center for Scientific Computing (DCSC).

References 1. M.P. Bendsøe and N. Kikuchi. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 71(2):197– 224, 1988. 2. M.P. Bendsøe and O. Sigmund. Material interpolation schemes in topology optimization. Archives of Applied Mechanics, 69(9-10):635–654, 1999. 3. M.P. Bendsøe and O. Sigmund. Topology Optimization – Theory, Methods, and Applications. Springer Verlag, Berlin, 2004. 4. S.J. Cox and D.C. Dobson. Maximizing band gaps in two-dimensional photonic crystals. SIAM Journal for Applied Mathematics, 59(6):2108–2120, 1999. 5. S.J. Cox and D.C. Dobson. Band structure optimization of two-dimensional photonic crystals in h-polarization. Journal of Computational Physics, 158(2):214–224, 2000. 6. L.V. Gibiansky and O. Sigmund. Multiphase elastic composites with extremal bulk modulus. Journal of the Mechanics and Physics of Solids, 48(3):461–498, 2000. 7. L.V. Gibiansky and S. Torquato. Thermal expansion of isotropic multiphase composites and polycrystals. Journal of the Mechanics and Physics of Solids, 45(7):1223–1252, 1997. 8. J.K. Guest and J.H. Prevost. Optimizing multifunctional materials: Design of microstructures for maximized stiffness and fluid permeability. International Journal of Solids and Structures, 43(22-23):7028–7047, 2006.

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9. J.K. Guest and J.H. Prevost. Design of maximum permeability material structures. Computer Methods in Applied Mechanics and Engineering, 196(4-6):1006–1017, 2007. 10. S. Halkjær, O. Sigmund, and J.S. Jensen. Inverse design of phononic crystals by topology optimization. Zeitschrift für Kristallographie, 220(9-10):895–905, 2005. 11. S. Halkjær, O. Sigmund, and J.S. Jensen. Maximizing band gaps in plate structures. Structural and Multidisciplinary Optimization, 32(4):263–275, 2006. 12. U.D. Larsen, O. Sigmund, and S. Bouwstra. Design and fabrication of compliant micromechanisms and structures with negative Poisson’s ratio. IEEE Journal of Microelectromechanical Systems, 6(2):99–106, 1997. 13. J.-M. Lerat, N. Malljac, and O. Acher. Determination of the effective parameters of a metamaterial by field summation method. Journal of Applied Physics, 100(8):084908, 2006. 14. J.B. Pendry, D. Shurig, and D.R. Smith. Controlling electromagnetic fields. Science, 312(5781):1780–2, 2006. 15. O. Sigmund. Materials with prescribed constitutive parameters: An inverse homogenization problem. International Journal of Solids and Structures, 31(17):2313–2329, 1994. 16. O. Sigmund. Tailoring materials with prescribed elastic properties. Mechanics of Materials, 20:351–368, 1995. 17. O. Sigmund. On the optimality of bone microstructure. In P. Pedersen and M.P. Bendsøe (Eds.), Synthesis in Bio Solid Mechanics, pages 221–234. IUTAM, Kluwer, 1999. 18. O. Sigmund. A new class of extremal composites. Journal of the Mechanics and Physics of Solids, 48(2):397–428, 2000. 19. O. Sigmund and K. Hougaard. Geometrical properties of optimal photonic crystals. Physical Review Letters, 100(15):153904, April 18 2008. 20. O. Sigmund and J.S. Jensen. Systematic design of phononic band gap materials and structures by topology optimization. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 361:1001–1019, 2003. 21. O. Sigmund and S. Torquato. Composites with extremal thermal expansion coefficients. Applied Physics Letters, 69(21):3203–3205, 1996. 22. O. Sigmund and S. Torquato. Design of materials with extreme thermal expansion using a three-phase topology optimization method. Journal of the Mechanics and Physics of Solids, 45(6):1037–1067, 1997. 23. O. Sigmund, S. Torquato, and I.A. Aksay. On the design of 1-3 piezocomposites using topology optimization. Journal of Materials Research, 13(4):1038–1048, 1998. 24. E.C.N. Silva, J.S.O. Fonseca, and N. Kikuchi. Optimal design of piezoelectric microstructures. Computational Mechanics, 19(5):397–410, 1997. 25. E.C.N. Silva, J.S.O. Fonseca, and N. Kikuchi. Optimal design of periodic piezocomposites. Computer Methods in Applied Mechanics and Engineering, 159(2):49–77, 1998. 26. D.R. Smith and J.B. Pendry. Homogenization of metamaterials by field averaging (invited paper). Journal of the Optical Society of America B: Optical Physics, 23(3):391–403, 2006. 27. D.R. Smith, D.C. Vier, T. Koschny, and C.M. Soukoulis. Electromagnetic parameter retrieval from inhomogeneous metamaterials. Physical Review E – Statistical, Nonlinear, and Soft Matter Physics, 71(3):1–11, 2005. 28. K. Svanberg. The Method of Moving Asymptotes – A new method for structural optimization. International Journal for Numerical Methods in Engineering, 24:359–373, 1987. 29. S. Torquato, S. Hyun, and A. Donev. Optimal design of manufacturable three-dimensional composites with multifunctional characteristics. Journal of Applied Physics, 94(9):5748– 5755, 2003.

Modeling of Indentation Damage in Single and Multilayer Coatings J. Chen and S.J. Bull

Abstract In many coating applications damage resistance is controlled by the mechanical properties of the coating, interface and substrate. As coatings become thinner and more complex, with multilayer and graded architectures now in widespread use, it is very important to obtain the mechanical properties (such as hardness, elastic modulus, fracture toughness, etc.) of individual coating layers for use in design calculations and have failure-related design criteria which are valid for such multilayer systems. Nanoindentation testing is often the only viable approach to assess the damage mechanisms and properties of very thin coatings (< 1 µm) since it can operate at the required scale and provides fingerprint of the indentation response of the coating/substrate system. Finite element analysis of indentation load displacement curves can be used to extract materials properties for design; as coating thicknesses decrease it is observed that the yield strength required to fit the curves increases and scale-dependent materials properties are essential for design. Similarly the assessment of fracture response of very thin coatings requires modeling of the indentation stress field and how it is modified by plasticity during the indentation cycle. An FE approach using a cohesive zone model has been used to assess the locus of failure and demonstrates the complexity of adhesive failure around indentations for multilayer coatings.

1 Introduction The assessment of the mechanical properties of coatings and thin films becomes increasingly difficult as the thickness of the coating is reduced and it is often only by the use of indentation techniques that meaningful data can be obtained. In the deforming volume size range where length-scale effects are observed this can be a J. Chen and S.J. Bull Chemical Engineering and Advanced Materials, Newcastle University, Newcastle Upon Tyne NE1 7RU, UK; e-mail: [email protected]

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J. Chen and S.J. Bull Table 1 Structure of the multilayer coatings investigated. Position in the coating stack

Layer material

Nominal thickness (nm)

Outermost barrier coating Outer AR coating Protective layer Wavelength selective layer Inner AR layer Inner barrier layer Substrate

TiOx Ny SnO2 ITO Silver ZnO TiOx Ny Float glass (air side coated)

10 40 2 10 10 20 42000000

particular problem since it is almost impossible to gain direct evidence for the operating deformation mechanisms in a thin coating (< 200 nm) without performing experiments in the transmission electron microscope where the deforming geometries are not the same as usually seen in indentation tests. A particular issue arises in the mechanical design of multilayer solar control coatings on architectural glass based on a 10nm layer of silver surrounded by antireflection and barrier layers to produce a total coating thickness of the order of 100 nm. With typical commercial indenter tips having a radius of 50 nm or more indentation tests will produce a composite response from all the coating layers and the substrate. Modeling of the indentation response is essential to determine what controls deformation in such multilayer coatings. One concern is the validity of continuum mechanics approaches when very thin coating layers are produced. Another concern is how fracture may be assessed, be it through thickness or adhesive failure. Again modeling approaches can be used to gain mechanistic understanding and develop design rules. Some different modeling approaches which can be used to give useful information for multilayer optical coatings on glass are discussed in this paper.

2 Experimental and Modeling Approaches 2.1 Solar Control Coating Architecture and Failure Modes Experiments were carried out on float glass, coated with a multilayer stack of silver and metal oxides in a solar control configuration; the total thickness of the coating stack is approximately 100 nm. The coating consists of a 10 nm silver layer surrounded by SnO2 and ZnO anti-reflection coatings and TiOx Ny barrier layers; the layer structure and nominal coating thicknesses are presented in Table 1. A thin conducting ITO layer is used to prevent the silver layer from oxidation during the subsequent deposition of tin oxide. The coatings were produced by sputtering in a commercial coating plant at Pilkington Technical Centre (Lathom, UK) using the same process parameters as

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used for commercially available solar control coatings from the same manufacturer (available with the trade name Optitherm). The major mechanical failure modes of such coatings are scratches during delivery of the coated glass leading to coating detachment and through-thickness fracture. The nanoindentation test can be used to simulate the plastic deformation and fracture observed in such a transit scratch.

2.2 Indentation Testing Indentation experiments were performed using Hysitron Triboindenter fitted with a new Berkovich indenter (tip-end radius 50 nm). The system hardness and elastic modulus were measured by the standard Oliver and Pharr method [1] since these materials do not display significant pile-up or sink-in. Measurements were made at a range of contact scales under load or displacement control to match the modeling approaches used.

2.3 Finite Element Modeling of Indentation Finite element simulations have been carried out using the commercially available software ANSYS 9.0 using the approach described previously for bulk materials [2]. For plastic response assessment the coating stack detailed in Table 1 was modeled but for adhesion assessment a simpler stack consisting of a 10 nm Ag layer surrounded by ZnO layers of equal thickness was adopted. In all cases bonding between the coating and substrate was assumed to be perfect (i.e. no delamination at the interface). The contact depth was directly measured from the surface contour of the finite element simulations. Inward lateral displacement correction [3] has been used when deriving the hardness.

2.4 Cohesive Zone Model (CZM) at the Interface The cohesive zone model describes the traction-separation relation between two surfaces and is based on a surface potential [4]. The traction is defined as derivative of the cohesive surface potential and the normal work of fracture is the area under the traction-separation law. It is not easy to obtain the parameters in the general form of this model so an irreversible bi-linear cohesive zone model was introduced [5] as depicted in Figure 1. This bilinear CZM is embedded in the indentation finite element model to assess the potential for interfacial failure in this study. Although there is no numerical instability in the case of smooth separation, there is a potential convergence problem in the quasi-static finite element simulation of crack initiation [6]. However, it has been shown that the introduction of a small viscosity (or

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Fig. 1 Schematic of the traction-separation relationship in the bilinear cohesive zone model.

artificial damping coefficient) [5, 7] in the constitutive equations for the cohesive surface will avoid the numerical instability caused by crack initiation. In this study, the damping coefficient was set to 0.01 times the minimum time increment which gives a numerically stable and accurate solution.

3 Results and Discussion 3.1 Validation of FE Models The elastic contact between a sphere and flat infinite surface simulated by finite element has been verified by the Hertz contact solution. The nonlinear large deformation simulated by finite element analysis has been validated in two different ways. As discussed in [2], the same input values have been used in the program, and the output data such as load-displacement curve, and plastic deformation zone radius are in excellent agreement with measured values. Furthermore, for simulation of the indentation of a material similar to SiC with an input value of Young’s modulus (E) 450 GPa, calculation of the Young’s modulus from the simulated load displacement curve using the Oliver and Pharr method [1] gives 443.7 GPa with a deviation of less than 2%. The input yield strength was 12.4 GPa (deformation assumed to be elasticperfectly-plastic) and the expected hardness will be around 30 GPa according to the expression and finite element simulations proposed in [8]. The value determined in this study is 30.5 GPa which is also in good agreement with the expected value. In order to parameterise the cohesive zone model estimates of the work of separation and the interfacial strength are required. Based on experimental data [8] and

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Fig. 2 Experimental and modeled load-displacement curves for evaporated and sputtered aluminium (open symbols load control, closed symbols displacement control, solid line experimental data).

theoretical calculations [9] we assume that the work of separation is 0.1 J/m2 , at the lower end of the observed range. For a typical weak interface, it is usually assumed the interfacial strength is less than 10% of the yield strength of a ductile material; thus we set the interfacial strength to 0.05 GPa. During the deposition process, the limited adatom mobility, surface contamination or water adsorption may result in defects at the interface. In this study, we assume there is a 10 nm defect 100 nm away from indenter. Initially the cohesive zone model was validated by modeling a simple double cantilever beam (DCB) test of an ITO coating on a glass substrate. This test can be analysed analytically to determine fracture toughness; the calculated fracture toughness based on the analytical theory agrees very well with the numerical simulation and both are identical to the input toughness value.

3.2 Scale Effects in Modeling of Nanoindentation Load-Displacement Curves Figure 2 shows the modeled load-displacement curves compared to experimental data for aluminium thin films. The fit is good whether the modeling is under load or displacement control. Depending on the coating thickness or the indenter penetration it is necessary to change the yield stress used in the modeling to maintain a high quality fit (Figure 3). The increase in yield stress with reduction in contact scale or film thickness implies that scale-dependent plastic properties are required – provided that the materials properties are changed the continuum mechanics approach works for coating thicknesses as low as 10 nm if the grain size of the coating (typically 1–2× coating thickness) is considerably smaller than the deforming volume. Indentation size effects are dislocation-related in metal films [10] with models based on geometrically necessary dislocations or discontinuous deformation explaining the increase in hardness as the contact scale is reduced. However, it is also

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Fig. 3 Variation of yield stress with coating thickness for aluminium.

necessary to consider the time dependent behaviour of the coating (e.g. by introducing creep into the simulation [11]) if the yield stress data from tests on coatings produced by different methods are to be rationalized (Figure 3). For the oxide coatings on glass tested here the majority of the layers are amorphous and dislocation mechanisms are less significant. For this reason the scaledependent mechanical response is not so marked and can be ignored with reasonable accuracy. Nanoindentation tests at low penetration may be simulated by molecular dynamics [12] or discrete dislocation dynamics methods [13] to identify the operating deformation mechanisms. However, the contact scales simulated may be smaller than those generally obtained in experiment and the indenter displacement rates are very high so quantitative agreement with experiment is not always very good. Plasticity is also overestimated in MD simulations unless the stress relaxation due to the elastic hinterland around the modeled region is included as in hybrid MD/FE models [12]. For design purposes the increase in yield strength with decrease in contact scale is advantageous because it usually means that smaller structures remain elastic in service. However, when plastic deformation cannot reduce contact stresses there can be an increased chance of fracture and alternative modeling approaches are necessary.

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Fig. 4 Delamination at the upper interface during loading at a penetration of 22.5 nm. Extrusion of silver results in expansion in the region marked by the arrow to than the original Ag thickness.

3.3 Interfacial Failure in Silver Containing Multilayers When indenting ZnO/Ag/ZnO coatings the delamination nucleates at the upper Ag/ZnO interface during loading when the penetration exceeds 1.2 times the thickness of the cap layer. When indenting a sandwich stack with a very soft Ag layer between two harder and stiffer oxide layers, the silver coating undergoes extrusionlike flow from beneath the contact and the maximum plastic strain moves laterally with the flow as the size of the contact increases. This will enhance the bending effect on the upper ZnO layer resulting eventually in delamination (Figure 4). During indenting, the Ag layer underneath the indenter has been significantly thinned and the build up of extruded material at the edge of contact increases the effective silver thickness which introduces tensile stress outside the contact and produces a wedge opening effect on the crack. Therefore, the observed buckling height increases when coating extrusion is more extensive (i.e. for a softer Ag layer). The crack tip closest to the axis of symmetry remains just outside the contact patch as the indentation load increases. The normalized crack length increases with contact depth then decreases in all cases (Figure 5). The yield strength of Ag does not significantly affect the delamination length. However, if we replace the Ag layer by ZnO without changing the interfacial properties, no delamination nucleation was observed. This implies that the plastic deformation and the resultant severe extrusion of silver is the key factor controlling delamination nucleation. During unloading it was observed that a double pinned blister was formed for the coated systems indented to a penetration exceeding the total thickness of the coat-

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Fig. 5 The normalized delamination crack length as a function of contact depth during indentation.

ings. The unloading adhesion failure first nucleates at the lower Ag/ZnO interface when the maximum penetration exceeds the total thickness of the coatings. The delamination nucleates at some distance from the axis of symmetry and extends away from it. The plasticity of Ag does not significantly affect the lateral position where the delamination nucleates. The coating buckles away from the surface during the whole unloading procedure. The crack propagation in softer Ag is more extensive which results in a larger buckling height. The previous delamination cracking at the upper interface produced during loading is suppressed during unloading. When unloading to a certain depth, additional delamination cracking nucleates at the upper interface but with limited length. The extent of this delamination is constrained by the buckling of the coatings underneath it. The final crack length of the delamination at the upper interface is almost independent of the yield strength of the Ag layer regardless of the crack propagation conditions.

4 Conclusions Finite element simulations can be used to assess the nanoindentation response of nanostructured multilayer thin films provided that scale-dependent mechanical properties are used. For single layer metal coatings which are polycrystalline a strong indentation size effect is observed and the scale-dependence of mechanical properties is significant. In the case of oxide coatings on glass which are pre-

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dominantly amorphous there is relatively little size effect. For soft metals, creep processes can be critical in dictating performance and successful multiscale models must bridge both length and timescales. The cohesive zone model embedded in finite element simulations has been used to assess nucleation and propagation of delamination around an indentation in a ZnO/Ag/ZnO multilayer on glass. When the indenter penetration exceeds the total thickness of the coatings, all the coatings and substrate have been plastically deformed. The severe plastic deformation in the Ag layer imposes a bending effect on the upper ZnO layer which causes delamination nucleation during loading. During unloading, the stresses change from compressive to tensile in some regions outside the contact zone and these tensile stresses cause further delamination. This study shows that the delamination processes occurring around the indent in a multilayer coating are complex and depend critically on the mechanical properties of the component layers and their adhesion as well as on the penetration of the indenter. Acknowledgements This research was supported by The Engineering and Physical Sciences Research Council (EPSRC) through the multiscale modelling initiative.

References 1. Oliver, W.C. and Pharr, G.M. (1992). Improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments, J. Mater. Res. 7, 1564– 1580. 2. Chen, J. and Bull, S.J. (2006). A critical examination of the relationship between plastic deformation zone size and Young’s modulus to hardness ratio in indentation testing, J. Mater. Res. 21, 2617–2627. 3. Hay, J.C., Bolshakov, A. and Pharr, G.M. (1999). A critical examination of the fundamental relations used in the analysis of nanoindentation data, J. Mater. Res. 14, 2296. 4. Needleman, A. (1987). A continuum model for void nucleation by inclusion debonding, J. Appl. Mech. 54, 525–531. 5. Alfano, G. and Crisfield, M.A. (2001). Finite element interface models for the delamination analysis of laminated composites: Mechanical and computational issues, Int. J. Numer. Meth. Engrg. 50, 1701–1736. 6. Abdul-Baqi and Van der Giessen, E. (2001). Delamination of a strong film from a ductile substrate during indentation unloading, J. Mater. Res. 16, 1396–1407. 7. Gao, Y.F. and Bower, A.F. (2004). A simple technique for avoiding convergence problems in finite element simulations of crack nucleation and growth on cohesive interfaces, Model. Simul. Mater. Sci. Engrg. 12, 453–463. 8. Barthel, E., Kerjan, O., Nael, P. and Nadaud, N. (2005). Asymmetric silver to oxide adhesion in multilayers deposited on glass by sputtering, Thin Solid Films 473, 272–277. 9. Lin, Z. and Bristow, P.D. (2007). Microscopic characteristics of the Ag(111)/ZnO(0001) interface present in optical coatings, Phys. Rev. B 75, 205423. 10. Bull, S.J. (2003). On the origins and mechanisms of the indentation size effect, Z. Metallkunde 94, 787–792. 11. Soare, S., Bull, S.J., O’Neill, A., Wright, N., Horsfall, A. and dos Santos, J.M.M. (2004). Nanoindentation assessment of aluminium metallisation; the effect of creep and pile-up, Surf. Coat. Technol. 177–178, 497–503.

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12. Mcgee, E., Kenny, S.D. and Smith, R. (2007). Multiscale modelling of nanoindentation, Int. J. Mat. Res. 98, 430–437. 13. Nibur, K., Akasheh, F. and Bahr, D. (2007). Analysis of dislocation mechanisms around indentations through slip step observations, J. Mater. Sci. 42, 889–900.

Reverse Hall–Petch Effect in Ultra Nanocrystalline Diamond Ioannis N. Remediakis, Georgios Kopidakis and Pantelis C. Kelires

Abstract We present atomistic simulations for the mechanical response of ultra nanocrystalline diamond, a polycrystalline form of diamond with grain diameters of the order of a few nm. We consider fully three-dimensional model structures, having several grains of random sizes and orientations, and employ state-of-the-art Monte Carlo simulations. We calculate structural properties, elastic constants and the hardness for this material; our results compare well with experimental observations. Moreover, we verify that this material becomes softer at small grain sizes, in analogy to the observed reversal of the Hall–Petch effect in various nanocrystalline metals. The effect is attributed to the large concentration of grain boundary atoms at smaller grain sizes. Our analysis yields scaling relations for the elastic constants as a function of the average grain size.

1 Introduction Most solids are polycrystalline, having grains in the micrometer to millimeter range. As the percentage of atoms residing on grain boundaries is negligible, the polycrystallicity only marginally affects the properties of the material. In particular, mechanical properties of such solids usually depend on bulk properties of the ideal material and the concentration of various defects, such as cracks, dislocations, vacancies and interstitials; usually the grain size is of minor importance. There are, however, properties where grain size plays a key role, hardness being one of them. The Hall–Petch Ioannis N. Remediakis and Georgios Kopidakis Department of Materials Science and Technology, University of Crete, 71003 Heraklion, Greece; e-mail: [email protected] Pantelis C. Kelires Department of Physics, University of Crete, 71003 Heraklion, Greece; and Department of Mechanical Engineering and Materials Science and Engineering, Cyprus University of Technology, 3603 Limassol, Cyprus

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law states that the hardness (H ) of polycrystalline metals increases with decreasing average grain size (d), being a linear function of d −n , where n > 0. The effect is attributed to the impedance of dislocation motion due to grain boundaries [1]. Modern nanotechnology makes it possible to synthesize nanocrystalline solids, i.e. polycrystalline solids with average grain sizes of a few nm. Such materials offer new possibilities for technological applications, mainly due to their unique mechanical properties [15]. The Hall–Petch law dictates that a nanocrystalline solid should have huge hardness, usually much higher than its usual polycrystalline phase. While this is true in most cases, Cu was found to become softer with decreasing grain size in the range between 3 and 7 nm [19]. This was called the “reverse Hall–Petch effect”. Later, it was found that many materials posses a “strongest size” [25], which turned out to be around 15 nm for Cu [20]. A similar effect has been observed recently for BN [4]. The existence of a “strongest size” suggests that the mechanism of undertaking mechanical load should be different in the nano-world. In this region, the presence of dislocations no longer governs the mechanical response on the material: as dislocations are extended defects, they cannot reside in the limited space of nano-grains. Any external mechanical load will be primarily undertaken by sliding along grain boundaries [19, 23, 24]. This, in turn, is a direct consequence of the enormous concentration of grain boundary atoms in a nanocrystalline material. Imagine for example a cube of side d, containing N × N × N atoms. The fraction of surface atoms is roughly proportional to 6N 2 /N 3 = 6/N ≈ 1 nm/d, ranging from 1 per million when d is of the order of mm, to 30% when d is around 3 nm. Contrary to a large number of studies for metals, very few studies have addressed the mechanical properties of semiconductors and insulators as a function of their grain size, although several pioneer workers have examined the mechanical response of nanocrystalline ceramics [3, 21]. Covalent solids are characterized by their open structures and the strong directionality of their bonds. Such bonds should not allow easy sliding along grain boundaries. At the same time, bonds on grain boundaries will be considerably weaker than those in the bulk of grains, due to the loss of the ideal local bonding geometry [10]. Recent studies in BN [4], together with the wellestablished results for various metals, suggest that the latter is true, and that the reverse Hall–Petch effect might be universal. To check whether this is the case, we study diamond, which comprises the ideal test suite for this purpose. C atoms form perhaps the most directional bonds known, as indicated by the supreme hardness and shear modulus of diamond. In addition to being strong, C-C might in some cases break and form a more stable structure, as threefold sp2 C atoms are more stable than fourfold sp3 ones, the later being responsible for mechanical failure in carbon-based materials [6, 17]. Ultra nanocrystalline diamond (UNCD) is a polycrystalline carbon-based material, with grain diameters mostly between 2 and 5 nm [8]. It is a low-cost material with a potential for a wide range of applications due to its unique mechanical and electronic properties [14]. Despite the strong directional C-C bonds, resulting in inhomogeneity at the atomic scale, the material can be considered as isotropic at larger

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scales, as no particular orientation for the grain boundaries in UNCD seems to be favoured in the experiment [8] or simulations [13, 26].

2 Computational Method We use fully three-dimensional atomistic models for UNCD, having grains of different random size and orientations separated by random grain boundaries. The simulations were performed using a continuous-space Monte Carlo method. We employ the many-body potential of Tersoff [22], which provides a very good description of the structure and energetics for a wide range of carbon-based materials [11,12]. This method, although considerably demanding computationally, allows for great statistical accuracy, as it is possible to have samples at full thermodynamic equilibrium. Such accuracy is necessary in order to capture all hybridizations of C. We model UNCD by a periodic repetition of cubic supercells that consist of eight different regions (grains). The number of grains in the unit cell guarantees the absence of artificial interactions between a grain and its periodic images. The grains have random shapes and sizes, and are filled with atoms in a randomly oriented diamond structure. The method we use is identical to the method used by Schiøtz et al. [19, 20]. To achieve a fully equilibrated structure for each grain size, we perform four steps: first, the structure is compressed and equilibrated at constant volume at 300 K, in order to eliminate large void regions near some grain boundaries, that are an artifact of the randomly generated structure. In the second step, we anneal the system at 800 K allowing volume relaxation and quench down to 300 K. Third, we anneal once more, at 1200 K this time, in order to ensure full equilibration. Fourth, we fully relax the structure at 300 K allowing for changes in both volume and shape of the unit cell.

3 Structure and Elastic Moduli The relaxed structure for a typical sample is shown in Figure 1. The grain boundaries are a few atomic diameters across, in accordance with experiments showing widths of 0.2–0.5 nm [8]. Atoms at grain boundaries are either three-fold coordinated or form bonds at different lengths or angles from those observed in diamond. The structural and elastic properties for characteristic samples are summarized in Table 1. The fraction of the three-fold atoms in the samples is about 1/10 for grain sizes between 3.5 and 4.5 nm; in experiment, it was observed that the fraction of atoms residing at grain boundaries is close to 10% for similar crystallite sizes [8]. The density of UNCD increases with increasing grain size, as both the percentage of sp2 atoms and the concentration of voids is decreased. The cohesive energy decreases with increasing grain size, suggesting that most grain boundary atoms should be considered as defective ones.

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Fig. 1 Relaxed model structure for Ultra-Nanocrystalline Diamond, with an average grain size of 4.4 nm. Atoms are coloured according to their number of neighbours (z) and average angle between their bonds (θ). Atoms in the diamond structure (z = 4, cos θ = −1/3) are coloured gray; all other atoms are coloured black. Table 1 Properties of some characteristic UNCD samples at 300 K: average grain size (d, in nm); number of atoms in the simulation cell (N); percentage of three-fold atoms in the cell (N3 , at %); mass density (ρ, in g/cc); cohesive energy (Ec , in eV per atom); bulk (B), Young’s (E) and shear (G) moduli (all in GPa); Vickers hardness (H ), as estimated from the theory of Gao et al. [7]. For comparison, the corresponding values for single-crystal diamond, calculated with the same method [11], are shown in the last row. d 2.4 3.4 4.4 ∞

N

N3

ρ

Ec

B

E

G

H

18,528 53,494 116,941 ∞

26 12 9 0

3.22 3.30 3.40 3.51

–7.06 –7.10 –7.15 –7.33

323 363 384 443

808 939 987 1066

373 439 461 485

89.8 90.9 91.6 94.2

The bulk modulus of UNCD decreases with decreasing grain size, despite the increase in the fraction of energetically favorable sp2 atoms. Although they have lower energy, sp2 C atoms are actually easier to deform compared to sp3 ones. This can be demonstrated by employing the concept of local bulk moduli [12]. A similar analysis of our samples yields the average local bulk modulus of sp2 atoms to be around 250 GPa, while the average bulk modulus of sp3 atoms is around 420 GPa. This agrees very well with experimental observations for UNCD where the grain boundaries have been found to have much lower local bulk moduli than the bulk of grains [16].

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Fig. 2 Cohesive energy (Ec ), bulk modulus (B), Young’s modulus (E) and shear modulus (G) of UNCD as a function of the average grain size. Solid lines in the upper panels are fits to the simulations. The triangle in the graph for E represents an experimental measurement [5].

4 Hardness Decrease of all elastic moduli with decreasing average grain size suggests that the hardness of the material should also drop with decreasing grain size, as the hardness of many materials is proportional to the Young’s or shear modulus [2]; in particular, the hardness of all known carbon-based materials has been found to be between 10 and 16% of the Young’s modulus [18]. Elastic moduli are reliable probes of hardness for nanocrystalline solids, as the later cannot contain extended defects, such as cracks or dislocations, that have characteristic lengths exceeding the size of the grains. To get a quantitative description of hardness, we use the theory of Gao et al. [7], who correlated the Vickers hardness of covalent crystals with the electron density per bond and the energy gap of the material. The hardness of a complex material is the geometrical mean of the values of hardness for each subsystem. Here, we consider each individual pair of neighbouring C atoms as a subsystem. The density of valence electrons in a particular bond can be obtained from the bond length and the coordination numbers of the two atoms that participate in this bond. The calculated hardness of UNCD is shown in Figure 3, demonstrating the existence of the reverse Hall–Petch effect for this material.

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Fig. 3 Estimated hardness of UNCD as a function of the average grain size. Hardness is calculated based on the theory of Gao et al. [7] for covalent solids.

5 Scaling Laws for the Properties of UNCD To understand the mechanism behind softening of UNCD, as well as other materials, at low grain sizes, we consider the different types of atoms and local bonding geometries that exist in a nanocrystalline material. The main dramatic change that takes place in a polycrystalline material as its grain size enters the nm regime is the increase in the fraction of atoms residing near grain boundaries. Atoms at grain faces, edges or vertexes, as well as atoms near other discontinuities, will naturally form bonds that are weaker than those formed by atoms in the bulk. Such weaker bonds will then bend or stretch with greater ease, compared to the bonds in the crystalline region. This explains the softening of polycrystalline solids when the grain size is at the nanometer range. For much larger grain sizes, the number of grainboundary atoms will be negligible compared to the number of bulk atoms; in this regime, the behavior of the material under mechanical load will be mostly determined by bulk defects, such as dislocations. To make this picture quantitative, let us divide the atoms in the polycrystalline material into three categories: 1. Atoms deep inside the grains, forming bonds that are similar to those in the single-crystal material. The number of such atoms is proportional to d 3 , where d is the average grain size. 2. Atoms near the grain boundaries; these behave similarly to surface or interface atoms. The number of such atoms is proportional to d 2 . 3. Atoms near grain boundary edges; these are similar to kink surface atoms, or atoms near dislocation cores. The number of such atoms is proportional to d.

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Of course, there will be other types of atoms, such as vertex atoms or atoms near topologic defects, but their number will be much smaller than the numbers of atoms falling in one of the aforementioned categories. The cohesive energy of the solid will be the sum of the energies of the three different atom types, multiplied by their respective numbers, and divided by the total number of atoms, which is proportional to d 3 . Therefore, the cohesive energy should be described by a function of the form E c = E0 + a/d + b/d 2 , where a and b are constants, and E0 is the cohesive energy of the single crystal. Indeed, such a function fits our data perfectly, the rms error being less than 0.5%. Moreover, E0 is found to be −7.31 eV, very close to the calculated cohesive free energy of diamond at 300 K, which is −7.33 eV. As B is proportional to the second derivative of the total energy with respect to the system volume, it can also be decomposed into contributions from bulk, interface and vertex atoms. As shown in Figure 2, a quadratic function of 1/d fits the results of the simulation very nicely. The constant value, 467 GPa, corresponding to the ideal monocrystalline solid, is only 5% off the calculated value for diamond (see Table 1). Such a decomposition of the total bulk modulus to a sum of atomic-level moduli has been used previously, in order to investigate the rigidity of amorphous carbon [12]. A similar scaling law should also hold for the mass density of UNCD as a function of grain size, assuming that the volume per atom is different for atoms in grain boundaries and atoms in the bulk of grains. Fitting our data to a quadratic form of 1/d gives ρ = 3.6 − 1.2/d + 0.4/d 2. Again, the agreement of the constant value with the calculation for ideal diamond (Table 1) is very good (3%). Hardness is related to the electron density according to Gao et al.; the local electron density is proportional to the local mass density, as all C atoms have the same number of electrons. Therefore, hardness should also be decomposed into contributions from different kinds of atoms. As shown in Figure 3, the hardness of UNCD can be fitted to a quadratic form of 1/d. Moreover, the constant term, showing the limit of hardness as d goes to infinity, coincides with the hardness of diamond at 300 K, calculated using the same method (see Table 1). The Young’s and shear moduli of UNCD will not necessarily follow the same scaling law. As both moduli are related to bond bending, the nature of the interatomic bonds is perhaps equally important to the concentration of these bonds. We tried to fit our simulation data to a quadratic form of 1/d. Although the fit does not look disappointing, the rms error in the fits were significantly higher than those for the fits of Ec , B, ρ or H . Moreover, the constant values deviate from the properties of diamond by more than 20%. However, even such a poor agreement between model and simulation provides extra evidence that our model has some solid basis.

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6 Conclusions Using ultra-nanocrystalline diamond (UNCD) as a prototype for a polycrystalline covalent solid with grains at the nanometer region, we have observed softening of the material as the grain size decreases, in analogy with the reverse Hall–Petch effect observed in nanocrystalline metals. The effect is attributed to the increasing fraction of grain-boundary atoms as the grain size decreasing. A simple quadratic form in 1/d, where d is the average grain size, suffices to provide excellent fit of our results for cohesive energy, mass density, bulk modulus and estimated hardness, while it yields the correct values for bulk diamond. The measured Young’s modulus of UNCD is reproduced well by the simulations. Our results provide further evidence that softening at low grain sizes might be a universal property for nanocrystalline solids. Acknowledgements The authors are grateful to Professor Jacob Schiøtz who shared his programs for generating models of nanocrystalline solids, and acknowledge inspiring discussions with Dr. Maria Fyta. This work was supported by a grant from the Ministry of National Education and Religious Affairs in Greece through the action “EEAEK” (programme “YAOPA”).

References 1. Bata V, Pereloma EV (2004) An alternative physical explanation of the Hall–Petch relation. Acta Mater 52:657–665. 2. Brazhkin VV, Lyapin AG, Hemley RJ (2002) Harder than diamond: Dreams and reality. Philos Mag A 82:231–253. 3. Demkowicz MJ, Argonz AS, Farkas D, Frary M (2007) Simulation of plasticity in nanocrystalline silicon. Philos Mag 87:4253–4271. 4. Dubrovinskaia N, Solozhenko VL, Miyajima N, Dmitriev V, Kurakevych OO, Dubrovinsky L (2007) Superhard nanocomposite of dense polymorphs of boron nitride: Noncarbon material has reached diamond hardness. Appl Phys Lett 90:101912. 5. Espinosa HD, Peng B, Moldovan N, Friedmann TA, Xiao X, Mancini DC, Auciello O, Carlisle J, Zorman CA, Merhegany M (2006) Elasticity, strength and toughness of single crystal silicon carbide, ultrananocrystalline diamond and hydrogen-free tetrahedral amorphous carbon. Appl Phys Lett 89:073111. 6. Fyta MG, Remediakis IN, Kelires PC, Papaconstantopoulos DA (2006) Insights into the fracture mechanisms and strength of amorphous and nanocomposite carbon. Phys Rev Lett 96:185503. 7. Gao FM, He JL, Wu ED, Liu SM, Yu DL, Li DC, Zhang SY, Tian YJ (2003) Hardness of covalent crystals. Phys Rev Lett 91:015502. 8. Gruen DM (1999) Nanocrystalline diamond films. Annu Rev Mater Sci 29:211–259. 9. Kaner RB, Gilman JJ, Tolbert SH (2005) Materials science – Designing superhard materials. Science 308:1268–1269. 10. Keblinski P, Phillpot SR, Wolf D, Gleiter H (1999) On the nature of grain boundaries in nanocrystalline diamond. Nanostruct Mater 12:339–344. 11. Kelires PC (1994) Elastic properties of amorphous-carbon networks. Phys Rev Lett 73:2460– 2463. 12. Kelires PC (2000) Intrinsic stress and local rigidity in tetrahedral amorphous carbon. Phys Rev B 62:15686–15694.

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13. Kopidakis G, Remediakis IN, Fyta MG, Kelires PC (2007) Atomic and electronic structure of crystalline-amorphous carbon interfaces. Diam Relat Mater 16:1875–1881. 14. Krauss AR, Auciello O, Gruen DM, Jayatissa A, Sumant A, Tucek J, Mancini DC, Moldovan N, Erdemir A, Ersoy D, Gardos MN, Busmann HG, Meyer EM, Ding MQ (2001) Ultrananocrystalline diamond thin films for MEMS and moving mechanical assembly devices. Diam Relat Mater 10:1952–1961. 15. Meyers MA, Mishra A, Benson DJ (2006) Mechanical properties of nanocrystalline materials. Progr Mater Sci 51:427–556. 16. Pantea C, Zhang J, Qian J, Zhao Y, Migliori A, Grzanka E, Palosz B, Wang Y, Zerda TW, Liu H, Ding Y, Stephens PW, Botez CE (2006) Nano-diamond compressibility at pressures up to 85 GPa, in 2006 NSTI Nanotechnology Conference and Trade Show, pp 823–826. 17. Remediakis IN, Fyta MG, Mathioudakis C, Kopidakis G, Kelires PC (2007) Structure, elastic properties and strength of amorphous and nanocomposite carbon. Diam Relat Mater 16:1835– 1840. 18. Robertson J (2002) Diamond-like amorphous carbon. Mat Sci Eng R 37:129–281. 19. Schiøtz J, Di Tolla FD, Jacobsen KW (1998) Softening of nanocrystalline metals at very small grain sizes. Nature 391:561–563. 20. Schiotz J, Jacobsen KW (2003) A maximum in the strength of nanocrystalline copper. Science 301:1357–1359. 21. Szlufarska I, Nakano A, Vashishta P (2005) A crossover in the mechanical response of nanocrystalline ceramics. Science 309:911–914. 22. Tersoff J (1988) Empirical interatomic potential for carbon, with applications to amorphouscarbon. Phys Rev Lett 61:2879–2882. 23. Van Swygenhoven H, Spaczer M, Caro A, Farkas D (1999) Competing plastic deformation mechanisms in nanophase metals. Phys Rev B 60:22–25. 24. Yamakov V, Wolf D, Phillpot SR, Mukherjee AK, Gleiter H (2004) Deformation-mechanism map for nanocrystalline metals by molecular-dynamics simulation. Nat Mater 3:43–47. 25. Yip S (1998) Nanocrystals – The strongest size. Nature 391:532–533. 26. Zapol P, Sternberg M, Curtiss LA, Frauenheim T, Gruen DM (2002) Tight-binding moleculardynamics simulation of impurities in ultrananocrystalline diamond grain boundaries. Phys Rev B 65:045403.

Elastic Fields in Quantum Dot Structures with Arbitrary Shapes and Interface Effects H.J. Chu, H.L. Duan, J. Wang and B.L. Karihaloo

Abstract Elastic fields in quantum dot (QD) structures affect their physical and mechanical properties, and they also play a significant role in their fabrication. The elastic fields in QD structures may be induced by mismatches in the coefficients of thermal expansion and the lattice constants of species, by defects, and by external loading. The calculation of the elastic fields in QD structures is complicated by several factors: by the complex shapes of QDs; by the anisotropy of the material species; and by the interface effects at the nano scale. In this paper we present a general approach to the calculation of the elastic fields in QD structures of arbitrary shape. This approach can also deal with the anisotropy of the QD material, the nonuniformity of its composition, the mismatch in the elastic constants of the matrix and the QD, and the interface effect. The effects of these factors on the elastic fields are depicted by analytical and numerical results.

1 Introduction Semiconductor quantum dots (QDs) have attracted considerable experimental and theoretical interest in recent years, since quantum-dot devices offer a bright prospect H.J. Chu College of Hydraulic Science and Engineering, Yangzhou University, P.R. China H.L. Duan Institute of Nanotechnology, Forschungszentrum Karlsruhe, Germany; and LTCS and College of Engineering, Peking University, Beijing 100871, P.R. China J. Wang LTCS and College of Engineering, Peking University, Beijing 100871, P.R. China; e-mail: [email protected] B.L. Karihaloo School of Engineering, Cardiff University, Queen’s Buildings, P.O. Box 925, The Parade, Cardiff CF24 0YF, UK

R. Pyrz and J.C. Rauhe (eds.), IUTAM Symposium on Modelling Nanomaterials and Nanosystems, 181–189. © Springer Science+Business Media B.V. 2009

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for improved electronic and optical properties arising from the zero-dimensional quantum confinement of carriers. The stress/strain field in them is essential to understanding the physical and mechanical properties of QDs, such as optical properties, piezoelectric properties, degeneration and mechanical stability. The elastic field also plays a significant role in the fabrication of closely stacked, self-assembled and vertically aligned QDs. These fields in QD structures may be induced by mismatches in the coefficients of thermal expansion and the lattice constants of species, by defects, and by external loading. Some analytical approaches (e.g., [1, 2, 4, 5, 9]), finite element computations (e.g., [8,10]), and atomistic simulations (e.g., [13]) have been carried out to reveal the elastic fields and their effect on the electronic properties of QD structures. However, at the nano-scale, the interface effects at the interfaces of QDs and the matrix for buried QDs may become significant, and also, QDs may be elastically anisotropic and have distinct elastic constants from those of the matrix. Therefore, the calculation of the elastic fields in QD structures is complicated by many factors: by the complex shapes of QDs, e.g., polyhedral such as truncated pyramids; by the anisotropy of the material species; and by the interface effects at the nano-scale, which include interface mixing, and the existence of an interphase and interface stress. In this paper, we present a general approach that can deal with these factors.

2 Elastic Fields in QD Structure with Non-Uniform Composition Experiments and theoretical simulations show that the composition of QD structures is not uniform [14, 15]. It has been reported that Ge atoms can diffuse into the third [11, 12, 20] and fourth [16, 23] layers in Ge/Si nanostructures. The crystal growth is a thermodynamic process involving atom adsorption, desorption and diffusion as a result of which a non-uniform composition in the interface region between the QD and matrix is almost unavoidable. Consider an arbitrarily shaped quantum dot embedded in an infinite matrix with a non-uniform composition. The initial isotropic misfit strain induced by the lattice or thermal mismatch depending on the composition of the structure is denoted by ε0 (x). For lattice mismatch, ε0 (x) is calculated by Vegard’s law [21]. Based on the assumption that the QD and matrix are isotropic and have identical elastic constants, the stress fields in the QD structure are [1, 2] σ (x) = −

E (T1 + T2 − T3) 4π(1 − ν)

with T1 = 4πε0 (x)H (x)I
[ε0 (x
)A(x, x
)] ⊗ dS(x
) T2 = ∂

(1)

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183

A(x, x
) ⊗ [∇x
ε0 (x
)] dV (x
)

T3 =

(2)



where

 xi − xi
e , r = (xi − xi
)(xi − xi
). i r3 The subscripts obey the usual convention of summation. E and ν denote the Young modulus and Poisson ratio, respectively.  denotes the region occupied by the QD. The first term (T1) is in an analytical form and only depends on the initial misfit strain at the concerned point inside the QD but vanishes outside of it. The second term (T2) is an integral over the interface between the QD and matrix, and the third term (T3) is an integral over the volume of the QD. When the distribution of the initial misfit strain is linear, namely, its gradient k = ∇x ε0 (x) is a constant, we get A(x, x
) =

T1 = 4πε0 (x)H (x)I
[ε0 (x
)A(x, x
)] ⊗ dS(x
) T2 =


∂

T3 = ∂

dS(x
) ⊗k r

(3)

Eqs. (1) and (3) show that the stress in the QD structure with a linearly graded composition can be expressed through integrals over the interface between the QD and matrix, avoiding volume integration. Hence, the stress field in an arbitrarily shaped QD structure can be obtained by simple numerical calculations. From Eq. (1), the hydrostatic stress at any point in the QD structure can be derived: σii =

−2E ε0 (x)H (x) 1−ν

(4)

Obviously, the hydrostatic stress at any point x inside the QD structure is proportional to the initial misfit strain at this point. This result is consistent with the previous reports for QDs with a uniform composition (e.g., [6, 10, 17]). Based on the above expressions, Chu and Wang [1] calculated the strain distribution in a buried cuboidal QD with a linearly graded composition, and found that the non-uniform composition has a significant influence on the strain field inside and around the QD, but less so on points far away from the QD region. The elastic fields are sensitive to the height of the QD for both uniform and linearly graded composition in the growth direction.

3 Elastic Fields in Heterogeneous and Anisotropic QD Structures In the previous section, the QD and the matrix are both assumed to be isotropic and have the same elastic constants. However, the real semiconductor QD structures

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are often anisotropic and the physical properties of the QD and the matrix are not the same. In this section, a perturbation theory, which was developed by Wang and Chu [22], is outlined to calculate the strain distribution in the QD structure taking into account the influence of the anisotropic properties of the QD and matrix. Consider an arbitrarily shaped anisotropic QD in an alien infinite anisotropic matrix under the initial misfit strain ε∗ inside the QD occupying the region . The body force is denoted by gi . The elastic problem in this QD structure is referred to as the original problem. In order to solve the elastic fields in the QD structure, a comparison material is introduced. The comparison material is an infinite, homogeneous, and isotropic medium with an eigenstrain strain ε∗ prescribed in a region  which has the same shape and size as the QD in the original problem. This problem, referred to as the comparison problem, is an Eshelby inclusion problem. In the QD region, the equilibrium equation in the original problem is 



∗ Cij kl (uk,l − εkl ) QD

,j

+ gi = 0

(5)

QD

where uk and Cij kl denote the components of the displacement and of the stiffness tensor of the QD, respectively in the original structure. The corresponding equation in the comparison problem is  0  ∗ Cij kl (u0k,l − εkl ) ,j + gi = 0 (6) where u0k and Cij0 kl denote the components of the displacement and of the stiffness tensor of the matrix, respectively, in the comparison problem. Subtracting Eq. (5) from Eq. (6), we have  QD  ∗ Cij0 kl uk,lj + (Cij kl − Cij0 kl )(uk,l − εkl ) ,j = 0

(7)

where uk = uk − u0k . By the same method, we can get   Cij0 kl uk,lj + (Cijmkl − Cij0 kl )uk,l ,j = 0

(8)

where Cij0 kl denote the components of the stiffness tensor of the matrix. The stress continuity condition at the interface between the QD and the matrix in the original problem is QD − ∗ Cijmkl u+ (9) k,l nj = Cij kl (uk,l − εkl )nj where nj denotes the direction cosine of the unit normal vector to the interface and it points from the QD to the matrix. The corresponding stress continuity condition in the comparison problem is 0− 0 ∗ Cij0 kl u0+ k,l nj = Cij kl (uk,l − εkl )nj

Subtracting Eq. (9) from Eq. (10), we get

(10)

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+ + 0 m 0 0 Cij0 kl u− k,l nj − Cij kl uk,l nj = (Cij kl − Cij kl )uk,l nj − (Cij kl − Cij kl ) QD

∗ (u− k,l − εkl )nj

(11)

Combining Eqs. (7), (8) with Eq. (11), it is seen that the displacement difference um is the solution of the Eshelby inclusion problem under body forces QD

fi

  fim = (Cijmkl − Cij0 kl )uk,l ,j ,

 QD  ∗ = (Cij kl − Cij0 kl )(uk,l − εkl ) ,j ,

and the interface traction − 0 ∗ Ti = (Cijmkl − Cij0 kl )u+ k,l nj − (Cij kl − Cij kl )(uk,l − εkl )nj . QD

Therefore,
um = E−

Gmi fim dV (x
)


+ 

QD Gmi fi dV (x
)




Gmi Ti dS(x
)

+

(12)

∂

where Gmi denotes the components of the Green function for the comparison material. By Gauss theorem, the above expression can be simplified as follows:
um = Gmi,j (Cij0 kl − Cijmkl )εkl dV (x
) E−




∗ Gmi,j (Cij0 kl − Cij kl )(εkl − εkl ) dV (x
) QD

+ 

For simplicity, Eq. (13) is rewritten as
∗ um = Gmi,j (Cij0 kl − Cij kl )(uk,l − εkl ) dV (x
)

(13)

(14)

E

∗ denote the components of the stiffness tensor and of the eigenwhere Cij kl and εkl QD ∗ strain of the real material in the original problem. If x
∈ , Cij kl = Cij kl and εkl m ∗ is equal to the misfit strain inside the QD. Otherwise, Cij kl = Cij kl and εkl = 0. It should be mentioned that Eq. (14) can be easily generalized to deal with the elastic field in a QD structure with a non-uniform eigenstrain, i.e.,
   ∗
um = Gmi,j Cij0 kl − Cij kl (x
) uk,l − εkl (x ) dV (x
) (15) E

∗ (x
) depend on the composition at the position of point x
. where Cij kl (x
) and εkl ∗ denotes the elastic strain in the original From Eq. (14), it is clear that uk,l − εkl problem. An iterative process can be constructed to obtain
∗
unm = Gmi,j (Cij0 kl − Cij kl )(un−1 k,l − εkl ) dV (x ) E

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n = 1, 2, . . .

(16)

Based on Eq. (16), we can get u1m ,u2m , . . . , and hence u1m ,u2m , . . . . The displacement unm may be called the nth-order approximate solution. Using the above perturbation theory, Wang and Chu [22] calculated the strain field in a truncated pyramidal Ge/Si QD structure. The numerical results show that: (1) the iterative algorithm for calculating the elastic field in the studied Ge/Si QD structure converges quickly. The third-order approximate solution almost coincides with the second-order approximate solution; (2) the hydrostatic strain is not constant inside the QD and not zero outside it, if the anisotropic properties of the QD structure and the difference between the elastic constants of the QD and the matrix are considered. This feature is different from that obtained by the isotropic approximate solution; and (3) the influence of the anisotropic properties of the QD structure is significant for the considered case.

4 Perturbation Theory for Interface Effect 4.1 Eshelby Formalism for a Spherical QD with Interface Stress Recently, the effects of surface and interface energy and stress on the elastic fields of heterogeneous media containing nano-inhomogeneities have received much attention (e.g., [7,18,19]. As the size of QDs is in the nano-scale, the interface effects including the interface mixing, interphase, and the interface stress may become significant. Thus, in this section, we shall generalize the above perturbation theory to the case when there is an interphase between the dot and the matrix. However, as the interface stress model and the interphase model are two common models simulating the interface effects at the nano-scale, we will recapitulate the Eshelby formalism for a spherical nano-inhomogeneity with the effect of interface stress [7], before presenting the generalized theory. The interior Eshelby tensor S(x) for a spherical nano-inhomogeneity of radius R embedded in an infinite matrix with the interface stress can be accurately expressed as [7] α(x)lκ + β(x)lµ S(x) = S(∞) + (17) R where S(∞) is the classical Eshelby tensor without the interface stress effect; α and β are two fourth-order tensors, and lκ and lµ are two intrinsic length scales determined by the elastic constants of the interface and the matrix. Eq. (17) reveals several salient features different from the classical Eshelby formalism with a perfect interface bonding. First, the elastic field becomes non-uniform even for a uniform eigenstrain; second, the elastic field becomes size dependent; and third, the Eshelby tensor and thus the elastic field will approach the classical results as the size of the inhomogeneity increases. These features are likely to exist in a QD structure.

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4.2 Interface Perturbation Theory For heterogeneous materials such as QD structures, there often exists an interphase between two constituents. In general, the influence of an interphase on the properties of a nano-structured material is much larger than that of a macrostructure or microstructure. In this section, the perturbation theory is generalized to take into account the interface effect [3]. Consider a QD nanostructure which is divided into three regions, namely, the QD region , interphase region δ and matrix region. According to Eq. (14), we have
um = Gmi,j (Cij0 kl − Cijmkl )uk,l dv(x
) E−−δ




+ 

∗
Gmi,j (Cij0 kl − CijQD kl )(uk,l − εkl ) dv(x )




(18)

∗ Gmi,j (Cij0 kl − Ciji kl )(uk,l − εkl ) dv(x
)

+ δ

where Ciji kl denote the components of the stiffness tensor of the interphase. Assume that the interphase is thin of constant thickness δ. The third term on the right-hand side in Eq. (18) becomes

δ→0 ∗ ∗ Gmi,j (Cij0 kl − Ciji kl )(uk,l − εkl ) dv(x
) −→ − Gmi,j Cijiskl (uk,l − εkl ) dS(x
) δ

∂

(19)

where Cijiskl = Ciji kl δ. Hence,
um = E

∗ Gmi,j (Cij0 kl −Cij kl )(uk,l −εkl ) dv(x
)−


∂

∗ Gmi,j Cijiskl (uk,l −εkl ) dS(x
)

(20)

Based on Eq. (20), an iterative form can be constructed, i.e.,
(n) ∗
um = Gmi,j (Cij0 kl − Cij kl )(un−1 k,l − εkl ) dv(x ) E




− ∂

∗
Gmi,j Cijiskl (un−1 k,l − εkl ) dS(x )

(n) unm = un−1 m + um

(21)

It should be mentioned that the mechanical properties of an interface or interphase are often expressed by a mathematical constitutive relation. For example, s s s σαβ = Cαβςξ εςξ

(α, β, ς, ξ = 1, 2)

(22)

s , Cs s where σαβ αβςξ and εςξ (α, β, ς, ξ = 1, 2) denote the interface stress, interface modulus and interface strain. Under the mathematical interface constitutive relation,

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Eq. (21) becomes
u(n+1) = u(n) m m +


Gmi,j (Cij0 kl − Cij kl )uk,l dv(x
) (n)

E

is Gmα,β Cαβςξ uς,ξ dS(x
) (n)

− ∂

= unm + u(n+1) un+1 m m

(23)

The third term on the right-hand side in the first expression in Eq. (18) or Eq. (23) denotes the influence of the interface or interphase on the displacement difference between the original problem and comparison problem. Eq. (21) or Eq. (23) is the basic formula of the interface perturbation theory. It is pointed out that the planar interface effect of a nanostructure can be studied by the interface perturbation theory. This is different from the Eshelby formalism with the interface effect depicted above. By the iterative formulas in Eq. (21), the strain field in a cubic Ge/Si semiconductor QD structure is calculated (Chu, 2006). The numerical results show that: (1) the iterative algorithm for calculating the elastic field in the studied structure is convergent, and the third-order approximate solution nearly coincides with the second-order approximate solution; (2) the relative difference of the strain at the center of the QD induced by the interface is about 5.5%; and (3) the size-dependent interface effect is confirmed, i.e., the influence of the interphase on the elastic field inside the cubic QD structure increases with the decrease of its size.

5 Conclusions In this paper we have presented a theoretical approach for the calculation of the elastic fields in QD structures. This approach can deal with the characteristics of real QD structures, which includes their non-uniform composition, irregular shape, anisotropy, the difference between the elastic constants of the QD and the matrix, and the effects of the interface stress and an interphase. The numerical results show that these factors have a significant effect on the elastic fields and that the proposed approach is efficient. Acknowledgements This work is supported by the National Natural Science Foundation of China under grant Nos. 10602050 and 10525209.

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References 1. Chu H.J. and Wang J., 2005, Strain distribution in arbitrarily shaped quantum dots with nonuniform composition, J. Appl. Phys. 98: 034315. 2. Chu H.J. and Wang, J., 2005, An approach for calculating strain distributions in arbitrarily shaped quantum dots. Chin. Phys. Lett. 22: 667–670. 3. Chu H.J., 2006, Mechanics of semiconductor quantum dot structures. PhD Thesis, Peking University. 4. Davies J.H., 2003, Elastic field in a semi-infinite solid due to thermal expansion or a coherently misfitting inclusion, J. Appl. Mech. 70: 655–660. 5. Downes J.R. and Faux D.A., 1995, Calculation of strain distributions in multiple-quantumwell strained-layer structures, J. Appl. Phys. 77: 2444–2447. 6. Downes J.R., Faux D.A., and O’Reilly E.P., 1997, A simple method for calculating strain distributions in quantum dot structures, J. Appl. Phys. 81: 6700–6702. 7. Duan H.L., Wang J., Huang Z.P., and Karihaloo B.L., 2005, Eshelby formalism for nanoinhomogeneities, Proc. R. Soc. A 461: 3335–3353. 8. Freund L.B. and Johnson H.T., 2001, Influence of strain on functional characteristics of nanoelectronic devices, J. Mech. Phys. Solids 49: 1925–1935. 9. Gosling T.J., and Willis J.R., 1995, Mechanical stability and electronic properties of buried strained quantum wire arrays, J. Appl. Phys. 77: 5601–5610. 10. Grundmann M., Stier O., and Bimberg D., 1995, InAs/GaAs pyramidal quantum dots: Strain distribution, optical phonons, and electronic structure, Phys. Rev. B 52: 11969–11981. 11. Gunnella R., Castrucci P., Pinto N., Davoli I., Sébilleau D., and Crescenzi M.D., 1996, Xray photoelectron-diffraction study of intermixing and morphology at the Ge/Si(001) and Ge/Sb/Si(001) interface, Phys. Rev. B 54: 8882–8891. 12. Ikeda A., Sumitomo K., Nishioka T., Yasue T., Koshikawa T., and Kido Y., 1997, Intermixing at Ge/Si(001) interfaces studied by surface energy loss of medium energy ion scattering, Surf. Sci. 385: 200–206. 13. Makeev M.A., Wenbin Yu, and Madhukar A., 2004, Atomic scale stresses and strains in Ge/Si(001) nanopixels: An atomistic simulation study, J. Appl. Phys. 96: 4429–4443. 14. Migliorato M.A., Cullis A.G., Fearn M., and Jefferson J.H., 2002, Atomistic simulation of strain relaxation in Inx Ga1−x As/GaAs quantum dots with nonuniform composition, Phys. Rev. B 65: 115316. 15. Migliorato M.A., Cullis A.G., Fearn M., and Jefferson J.H., 2002, Atomistic simulation of Inx Ga1−x As/GaAs quantum dots with nonuniform composition, Phys. E 13: 1147–1150. 16. Patthey L., Bullock E.L., Abukawa T., Kono S., and Johansson L.S.O., 1995, Mixed Ge-Si dimer growth at the Ge/Si(001)-(2×1) surface, Phys. Rev. Lett. 75: 2538–2541. 17. Pearson G.S. and Faux D.A., 2000, Analytical solutions for strain in pyramidal quantum dots, J. Appl. Phys. 88: 730–736. 18. Sharma P. and Ganti S., 2004, Size-dependent Eshelby’s tensor for embedded nano-inclusions incorporating surface/interface, J. Appl. Mech. 71: 663–671. 19. Sharma P., Ganti S., and Bhate, N., 2003, Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl. Phys. Lett. 82: 535–537. 20. Uberuaga B.P., Leskovar M., Smith A.P., Jónsson H., and Olmstead M., 2000, Diffusion of Ge below the Si(100) surface: theory and experiment, Phys. Rev. Lett. 84: 2441–2444. 21. Vegard L., 1921, The constitution of the mixed crystals and the filling of space of the atoms. Z. Physik 5: 17–26. 22. Wang J. and Chu H.J., 2006, A perturbation theory for calculating strain distributions in heterogeneous and anisotropic quantum dot structures. J. Appl. Phys. 100: 053520. 23. Yeom H.W., Sasaki M., Suzuki S., Sato S., Hosoi S., Iwabuchi M., Higashiyama K., Fukutani H., Nakamura M., Abukawa T., and Kono S., 1997, Existence of a stable intermixing phase for monolayer Ge on Si(001), Surf. Sci. 381: L533.

Numerical Modelling of Nano Inclusions in Small and Large Deformations Using a Level-Set/Extended Finite Element Method J. Yvonnet, E. Monteiro, H. Le Quang and Q.-C. He

Abstract In this work, a numerical procedure is proposed to model nano inclusions in both small and large elastic deformations in a continuum framework. An extended finite element method is combined with the level-set method. The interface between the matrix and the inclusions is modelled as an imperfect interface using the Laplace–Young model and is described implicitly through arbitrary mesh using a level-set function. Associated weak forms are derived in small and large deformations. As no mesh of the interface is needed, arbitrary inclusions shapes or distributions can be studied. The advantage of the a continuum approach, in contrast with the molecular dynamics approach, is that a large number of nano inclusions can be modelled at low computational costs, to determine the effective properties of a material containing arbitrarily distributed nano inclusions. The proposed procedure allows modelling size dependent effective properties in nanomaterials. Numerical examples in small and large elastic deformations are proposed to demonstrate the capabilities of the method.

1 Introduction In recent year, it has been shown that nanomaterials can lead to substantial advances in the increase of engineering materials properties [6]. Nanomaterials cover two scales: the atomistic scale, where the continuum mechanics and physics can no longer be used, and the scale of a representative volume element of the material, which size can be one or several orders higher than the nano inclusions characteristic length. At this scale, the Molecular Dynamics approach involves way too many atoms and leads to a dramatic increase of the computational costs. However, it is desirable to carry out computations on large or randomly distributed representative J. Yvonnet, E. Monteiro, H. Le Quang and Q.-C. He Laboratoire Simulation et Modélisation Multi Echelles, FRE 3160 CNRS, Université Paris-Est, 5 Bd. Descartes, Marne-la-Vallée Cedex, France; e-mail: [email protected]

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volume elements, in order to compute the effective properties of the material. To avoid this issue, it is needed to develop continuum models for nanosystems, in order to use fast methods such as the finite element method, to allow computations on representative volume elements that contain a large number of inclusions. One major effect associated with the nanoscale is the surface effect. At this scale, the ratio between surface and volume is so high that the surface energy, which is usually neglected in microscopic materials, becomes dominant. Such surface energy can be modelled by an imperfect interface between two media. In particular, it has been shown that the Laplace–Young model can accurately reproduce the effects associated with the nanoscale in a continuum framework. In most of the existing works, analytical models are proposed to compute the effective properties of nanomaterials and are thus limited to simple geometries and distributions of nano inclusions. To explore more complex cases, finite element simulations are required. However, one limitation is the meshing of the interface, which is required to compute the surface energy integral term in the weak form, and which can be burdensome for complex geometries/distributions. In this work, we propose to remove this locking by using a combined eXtended Finite Element Method (XFEM) [2,7] and a level-set approach. We formulate the problem in both small and large elastic deformations, and illustrate the method through several numerical examples.

2 Constitutive Equations 2.1 Small Deformations We consider a domain
in
d , d being the dimension of the domain. Without loss of generality, we assume that an
interface  divides
into several domains
(i) ,
 (i) i = 1, . . . , M such that
= i=1,...,M
, i=1,...,M
(i) = ∅. Let ∂
(i) be the external boundary of
and  ∂
the external boundary
of each subdomain
(i) . Here we assume that ∂
∂
(i) = ∅ and thus  = i=1,...,M ∂
(i). Let n be the unit vector normal to  an n(i) the unit vector normal to ∂
(i) pointing into
(i) . Thus, n(i) is also the normal to  pointing into the domain
(i) . For the sake of simplicity, we consider in the following only two domains. The equilibrium equations are then given by div(σ (i) ) + b = 0 in
(i) ,   divs (σ s ) = −[[t]] = σ (2) − σ (1) n(1) on 

(1) (2)

In the above, σ denotes the bulk Cauchy stress tensor and b is a volumetric force term. Eq. (1) is associated with bulk equilibrium, while Eq. (2) refers to the Laplace– Young equation resulting from the interface equilibrium. In particular, σ s = Pσ P is the surface stress tensor, where P(x) = 1 − n(x) ⊗ n(x) describes the projection on the plane tangent to  at x, and divs (.) is defined by

Numerical Modelling of Nano Inclusions in Small and Large Deformations

divs (T) = ∇(T) : P,

193

(3)

for any differentiable second-order tensor T. The boundary conditions are described by  σ n = −F on ∂
F (4) u = u¯ on ∂
u where F and u¯ are prescribed tractions and displacements, and ∂
u and ∂
F are the disjoint and complementary Dirichlet and Neumann boundaries, respectively. According to the coherent interface model, we have the following relations: [[u]] = 0 on ,

[[ε]] = a ⊗ n + n ⊗ a on 

(5)

and [[εs ]] = [[εPε]] = 0 on 

(6)

where ε is the infinitesimal strain tensor, and a is a real-valued vector. Here we assume that the solid undergoes small displacements. In the context of a linear elastic model, the bulk constitutive law is governed by the classical Hook’s law. According to Cammarata [3], the surface stress σ s is related to the surface-strain energy by the Schuttleworth’s equation that leads, in the context of isotropic linear elastic interfaces, to   σ s = σ 0 + CS : ε S , CijS kl = λS Pij Pkl + µS Pik Pj l + Pil Pj k (7) where λS and µS are constants characterizing the interface  and σ 0 = τ 0 P, τ 0 being the strain-independent surface/interfacial tension. With this model, the weak form is finally given by   ε(δu) : C : ε(u)d
+ Pε(δu)P : CS : Pε(δu)Pd








=

δu · bd
+




δu · F d + ∂
F

Pε(δu)P : σ 0 d.

(8)



2.2 Large Deformations In this section we assume that the solid undergoes large elastic deformations. In that context,
denotes the reference configuration of the solid. The problem is defined by Div(T(i) ) + B = 0 in
(i) (9)  TN = −g on ∂
F (10) u = u¯ on ∂
u

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where T, N and g denote the first Piola–Kirchhoff stress tensor, the normal unit vector to the interface in the reference configuration, and a prescribed force on the Neumann boundary in the reference configuration, respectively. We have [[F]] = a ⊗ N on 

(11)

where F = 1 + ∇X u is the deformation gradient tensor. Generalizing the Laplace– Young equation leads to Divs (Ts ) = Divs (PTP) = −[[TN]] on 

(12)

The weak form is obtained in a similar manner than in the context of small deformation:    S T : ∇(δu)d
+ TS : ∇X (δu)d = δu · gd (13)


where

S (·) ∇X



∂
F

= ∇X (·)P. Bulk and interface constitutive laws are expressed by T=

∂W b s ∂W s ,T = ∂F ∂Fs

(14)

where W b and W S denote bulk strain and surface strain dependent energies densities, respectively. In the present work, a compressible Gent [5] model was adopted for both W b and W S :

µ I1 − D µ d(J 2 − 1) − 2(d + 1)(J − 1) + (15) W = − Jm ln 1 − 2 Jm 2 where µ, Jm and d are material parameters, and J = det(F). The constant D = 2 for two-dimensional bulk constitutive law and D = 1 for two-dimensional surface constitutive law. The right choice of the constitutive law for treating nanomaterials is at the present moment an open question. We present here only the mathematical and numerical tools.

3 XFEM/Level-Set Discretization 3.1 Evaluation of the Unit Vector Normal to the Interface The domain
is discretized by n nodes ni that do not necessarily match the interface . Here we use a mesh of triangles, whereas other types of elements can be employed. Regular meshes can then be adopted for parallelepipedic domains, even if the interface has a complex geometrical shape. In the present context,  is defined as the zero level-set of a function φ(x), whose value is known at every node ni , i.e. φ(xi ) = φi . Wherever needed, the components of n(x) can be evaluated by

Numerical Modelling of Nano Inclusions in Small and Large Deformations

n(x) = where ˜ i= ∇ φ(x)

˜ ∇ φ(x) , ˜ ∇ φ(x)

n

∂Nj (x) j =1

∂xi

195

(16)

φj .

(17)

Here Nj (x) are the standard finite element shape functions, φj are the nodal value of the level-set function, and n is the number of nodes of the elements. The above approximation for n(x) can then be used to evaluate the components of the projector P needed in the weak forms (8) or (13).

3.2 Discrete System For the coherent interface model, the displacements must be continuous at the interface whereas the strains must follow the relations described in Eq. (5). These conditions can be enforced by superposing to the standard finite element field an enrichment term that possesses the above continuity conditions (XFEM method [2, 7]). In this context, the approximation is defined at a particular point x lying in an element
e by n m

Ni (x)ui + Nj (x)ψ(x)ai (18) uh (x) = j =1

i=1

where Ni (x) are the standard finite element shape functions associated with the nodes ni of the elements, Nj (x) are the shape functions of the nodes of the elements whose support are cut by the interface (see [10] for more details) and ψ(x) is a function with the required continuity. To meet the conditions (5), we use the enriched approximation proposed in [8]. The numerical integration in the bulk is performed by using Gauss integration on subtriangles near the interface [10]. The specificity of the present problem is the presence of the internal virtual work term related to the implicit surface (not discretized by nodes) in Eqs. (8) and (13). To evaluate the associated surface integral, we first approximate the interface by piece-wise linear segments in 2D, and triangular facets in 3D. We then perform a Gauss integration on each edge/facet of the interface approximation. On substituting the trial and test functions from Eq. (18) in Eq. (8), and using the arbitrariness of nodal variations, the following discrete system of linear equations is obtained: (K + KS )q = f where



(19)



KI J =


BTI C(i) BJ d
,

KSIJ =



BTI MTp CS Mp BJ d

(20)

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Fig. 1 Zero level-set function of a spherical inclusion in a non-conforming tetrahedral mesh (left) and normalized effective bulk modulus for different void radii (right).





f=

N bd
+


 N Fd +

T

T

∂
F



BT MTp σ 0 d

(21)

where B is a matrix of shape functions derivatives. In the above equations, C(i) is the bulk stiffness matrix associated with the elasticity tensor of phase i. To determine the phase associated with a particular bulk integration point, we simply use the sign of φ(x) interpolated by the linear finite element discretization. The matrix Mp is constructed such that  s = Mp . For large deformations, the discrete set of nonlinear equations obtained by introducing (18) in (13) is solved by a Newton–Raphson procedure.

4 Numerical Examples 4.1 Spherical Nano Void A spherical nanovoid is modeled in a cubic box by the level-set method. The interface is imperfect and is of the Laplace–Young type (2). For spherical void with a coherent surface, the effective modulus can be evaluated according to [4]. The used numerical parameters can be found in [10]. In Figure 1 we compare the normalized effective reference bulk modulus for a volume fraction f = 0.5 for different radii of the spherical void. Excellent agreement with the reference solution is noticed and the size effects due to the interfacial energy are clearly shown.

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Fig. 2 Zero level-set associated with random nanostructures (left); statistical convergence of the effective bulk modulus for various void radii (right).

4.2 Random Nanostructure Next we explore the effective properties of an aluminium material containing randomly distributed nanopores with constant radii, in order to investigate its sizeeffects on effective properties. For this purpose, we use 30 circular voids randomly distributed, by choosing the size of the square domain such that the volume fraction is f = 0.3 and we vary the radius of the pores. A uniform mesh of 80 × 80 nodes is used, and the level-set function was utilized to model the nanopores (see [10] for more details).

4.3 Large Deformations In this example, a two-dimensional representative volume element containing a cylindrical inclusion with surface strain dependent energy in the form of Eqs. (12)– (15) is considered. We impose uniform homogeneous deformation in one direction. In Figure 3 (left), we compute the energy with respect to deformation and compare the response with the one of a material without surface energy. The radius of the inclusion is 0.67 nm and the volume fraction is 0.125. The material parameters are µM = 357, µI = 714 MPa, d M = d I = d S = 5, J M = J I = J S = 100 and µS = 28.5, where the superscripts M, I and S refer to the matrix, inclusion, and interface, respectively. In Figure 3 (right) we compare the average energies of the RVE for different inclusions sizes. In both examples, surface and size effects can be observed.

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Fig. 3 Energy with respect to the strain in large deformations; comparison between a material containing an interface witout surface energy and with surface energy (left); comparison of the response of a material with interface energy for different inclusion sizes, at same volume fractions (right).

5 Conclusion In this work, a numerical procedure has been proposed to compute the overall elastic properties of nanomaterials and nanostructures with surface/interface effects. For this purpose, the coherent interface model has been adopted, leading to an additional stiffness matrix. To handle efficiently complex and arbitrary nano-inhomogeneities only through regular meshes, we have developed a level-set approach in tandem with an extended finite element method. The proposed approach has also been employed to determine the effective elastic moduli of materials with randomly distributed nanopores. However, it is worth noting that this model is not appropriate to model thermal problems at the nanoscale [1], as the Laplace–Young may not be valid physically in the thermal context. Nevertheless, it can be fully applied to microscopic particular composites with highly conducting (HC) interface. Examples of applications of the proposed framework for determining the effective properties of composites with HC interface can be found in our recent work [11]. Acknowledgements The support this work enjoys from EDF is gratefully acknowledged.

References 1. Balandin A.A., Laearenko O.L., Mechanism for thermoelectric figure-of merit enhancement in regimented quantum dot superlattices, Appl. Phys. Lett. 82(3), 415–417 (2003). 2. Belytschko T., Black T., Elastic crack growth in finite elements with minimal remeshing. Int. J Numer. Methods. Eng. 45(5), 601–620 (1999). 3. Cammarata R.C., Surface and interface stress effects in thin films. Progr Surface Sci. 46, 1–38 (1994). 4. Duan H.L., Wang J., Huang Z.P., Karihaloo B.L., Size dependent effective elastic constants of solids containing nanoinhomogeneities with interface stress. J. Mech. Phys. Solids 53(7),

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1574–1596 (2005). 5. Horgan C.O., Saccomandi G., Constitutive models for compressible nonlinearly elastic materials with limiting chain extensibility. J. Elasticity 77, 123–138 (2004). 6. Miller R.E., Shenevoy V.B., Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11(3), 139–147 (2000). 7. Moës N., Dolbow J., Belytschko T., A finite element method for crack growth without remeshing. Int. J. Numer. Methods. Eng. 46(1), 131–156 (1999). 8. Moës N., Cloirec M., Cartraud P., Remacle J.-F., A computational approach to handle complex microstructure geometries. Comput. Methods Appl. Mech. Eng. 192, 3163–3177 (2003). 9. Sukumar N., Chopp DL., Moës N., Belytschko T., Modeling holes and inclusions by level sets in the extended finite-element method. Comput. Meth. Appl. Mech. Eng. 190, 6183–6200 (2001). 10. Yvonnet, J., Le Quang H., He Q.-C., An XFEM/level-set approach to modelling surface/interface effects and to computing the size-dependent effective properties of nanocomposites. Comput. Mech. 42, 119–131 (2008). 11. Yvonnet J., He Q.-C., Toulemonde C., Numerical modelling of the effective conductivities of composites with arbitrarily shaped inclusions and highly conducting interface. Compos. Sci. Technol. 68, 2818–2825 (2008).

Thermo-Elastic Size-Dependent Properties of Nano-Composites with Imperfect Interfaces H.L. Duan, B.L. Karihaloo and J. Wang

Abstract We study the thermo-elastic properties of heterogeneous materials with imperfect interfaces. The interface between the matrix and second phase inhomogeneity is imperfect with either the displacement or the stress experiencing a jump across it. We demonstrate that it is theoretically possible to design nano-porous materials by adjusting the pore surface elastic parameters, porosity and size of the nanopores so that the resulting effective coefficients of thermal expansion take on specific positive, zero or even negative values. Moreover, we show that when imperfect interfacial bonding conditions are taken into account, the effective properties of heterogeneous materials containing inhomogeneities become dependent upon the size of the inhomogeneities. This size-dependence is shown to be captured by simple scaling laws in terms of some intrinsic length scales which emerge automatically depending upon the type of the interface imperfection.

1 Introduction We study the thermo-elastic properties of heterogeneous materials with imperfect interfaces. The effect of imperfect interfaces in heterogeneous materials on their properties (elastic moduli, thermal conductivity, coefficient of thermal expansion, etc.) is often simulated by an interface stress model (ISM; stress discontinuity), and a linear spring model (LSM; displacement discontinuity) for the elastic (or thermoH.L. Duan LTCS and College of Engineering, Peking University, Beijing 100871, P.R. China; and Institute of Nanotechnology INT, Forschungszentrum, 76021 Karlsruhe, Germany B.L. Karihaloo School of Engineering, Cardiff University, Queen’s Buildings, The Parade, Cardiff CF24 3AA, UK; e-mail: [email protected] J. Wang LTCS and College of Engineering, Peking University, Beijing 100871, P.R. China

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elastic) problems, and a high conducting (HC) or a low conducting (LC) interface model for conductivities. Surfaces have a profound effect on the properties of nano-structured and heterogeneous materials due to the large ratio of surface/interface atoms to the bulk. For the nano-structured materials, the interface between the matrix and second phase inhomogeneity is imperfect with the stress experiencing a jump across it. This is representative of nano-inhomogeneities in a matrix. The effective coefficient of thermal expansion (CTE) of a solid containing pores or cracks at the macroscale is always equal to that of the parent material (e.g., [1]). For nano-structured materials, in which a large fraction of atoms resides within a few atomic layers near the surface, the surface effect on their properties can be significant [2], and the CTE may be different from the parent bulk material. Nanoporous materials with arrays of pores of nano-size possessing a large surface area provide an obvious opening for designing functional nanoporous/cellular materials with new physical and chemical properties. We show that by adjusting pore surface elastic parameters, porosity and size of nanopores, the effective CTEs of nano-porous materials can be precisely tailored to a specific positive, negative or even zero value. Moreover, we show that when imperfect interfacial bonding conditions are taken into account, the effective properties of heterogeneous materials containing inhomogeneities become dependent upon the size of the inhomogeneities. This size-dependence is shown to be captured by simple scaling laws in terms of some intrinsic length scales which emerge, automatically depending upon the type of the interface imperfection.

2 Interface Conditions for Elastic Properties The boundary-value problems of thermo-elasticity with a uniform temperature change are described by the usual basic equations for the bulk materials (including equilibrium equation, constitutive equations, and strain-displacement relations for the matrix and the inhomogeneity) and the interface conditions, which will be described below.

2.1 Linear Spring Interface Model We start the analysis with the widely used linear spring interface model, in which the interface bonding conditions between two constituents in a heterogeneous medium are described by the following equations (e.g., [3, 4]): [σ ] · n = 0,

[u] = β · σ · n + γ T

(1)

where [·] represents the discontinuity of a quantity across the interface, β is a second-order tensor, β = βn n ⊗ n + βs s ⊗ s + βt t ⊗ t, βn , βs and βt represent

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the interface elastic parameters in the normal and tangential directions, respectively, n is a unit vector normal to the interface, and s and t represent the two orthogonal unit vectors in the tangent plane of the interface. γ = γ n + γs s + γt t is a displacement-temperature vector of the interface, and γ , γs and γt represent the interface thermo-elastic parameters in the normal and tangential directions, respectively [3]. Generally, the linear-spring model can be used to simulate a thin and compliant interphase (e.g., [4, 5]). In this case, the interface parameters (βn , βs and βt ) and the interface thermo-elastic parameter (γ ) can be expressed in terms of the modulus, thickness, and the coefficient of thermal expansion of the interphase. The relations in Eq. (1) and the conventional equilibrium equations, constitutive equations, and strain-displacement relations for the bulk material constitute the basic equations for solving thermo-elastostatic problems of heterogeneous materials with the linear spring interface model.

2.2 Interface Stress Model For nano-structured [2] and nanochannel-array materials with a large ratio of the surface/interface to the bulk, the surface/interface stress effect can be substantial. Thus, materials such as thin films, nanowires, nanotubes and nanoporous materials may exhibit unusual properties not noticed at the macro-scale. As small devices and nanostructures are all pervasive, and the elastic constants of materials are a fundamental physical property, it is important to understand and predict the sizeeffect in elastic properties of materials at the nano-scale. Based upon the idea that a nanostructure=bulk+surface, i.e., a surface has its own elastic constants and a CTE distinct from those of the bulk, Pathak and Shenoy [6] used the free energy density to develop a continuum theory and studied the size-dependence of CTEs of nanostructures. The continuum theory is illustrated on a nanoslab and is compared with full-scale molecular dynamics simulations (which serve as numerical experiments); excellent agreement is found. Thus, the following analysis for nanoporous materials is conducted within the continuum formalism. The basic equations of surface thermo-elasticity consist of the generalized Young-Laplace equation for solids and the constitutive equation of the surface. Analogous to the above linear spring interface model, the interface conditions of the interface stress model are depicted by the following relations [7]: [u] = 0, [σ ] · n = −∇S · τ

(2)

where n is the unit normal vector to the surface, τ is the surface stress tensor, ∇S · τ denotes the surface divergence of a tensor field τ [8]. The surface constitutive equation is (e.g., [9]) τ = 2µS ε S + λS (trεS )1 − T D0

(3)

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where λS and µS are the surface moduli, and 1 is the second-order unit tensor in a two-dimensional space. T is the change of temperature. D0 is the stress temperature tensor of the surface. For an isotropic case, D0 = d0 I, d0 = αS κS , where αS is the CTE of the surface, κS = 2(µS + λS ), and I is the second-order unit tensor in a three-dimensional space. Eqs. (2) and (3), when augmented by the conventional equilibrium equations, strain-displacement relations, and thermo-elastic constitutive equations for the matrix, constitute the basic set of equations for solving boundary value problems of thermo-elasticity of nanoporous materials.

3 Predicting and Tailoring the CTE of Nano-porous Materials Consider a solid with aligned cylindrical nanopores. The cylindrical nanopores with the same radii can be arbitrarily distributed in the matrix. If they are randomly distributed or placed in a hexagonal array, the effective behaviour of the solid is transversely isotropic, and can be characterized by five effective elastic constants. The effective CTE is also transversely isotropic, which can be expressed by two scalars αeT and αeL . We choose a cylindrical coordinate system (ρ, φ, z) such that the z-axis is parallel to the aligned cylindrical nanopores and the ρ − φ plane is perpendicular to them. We prove the existence of an axi-symmetric loading path, which includes a uniform temperature change T , such that it creates a uniform deformation field in the RVE, i.e., the following strain field: ερρ = e0 ,

εφφ = e0

εzz = ε0

(4)

where e0 and ε0 are two constants. Under the axi-symmetric deformation mode (Eq. (4)), the boundary conditions (Eqs. (2) and (3)) at the surfaces of nanopores become
 
 λS + 2µS λS κS αS  2λm + 2µm − e0 + λm − ε0 − 3κm αm − T = 0 a a a (5) where a is the radius of the nanopores, κm , λm and µm are the bulk modulus and the Lamé constants , and αm is the CTE of the matrix. Eq. (5) means that the homogeneous deformation field will prevail if the external loading, specified by e0 , ε0 and T , is properly selected. This particular solution can be viewed as a characteristic field in various admissible deformation modes [10]. Using Eqs. (2)–(5), and the expressions of the volume average stress and strains with the surface stress effect [11], the relation between the effective CTE (αeT and αLT ) and the effective elastic moduli (ke , le and ne ) can be obtained: 2ke αeT + le αeT − 3αm (1 − f )κm − f αSaκS S ke − (1 − f )(λm + µm ) − f λS +2µ 2a

=

3αm κm − λm + µm −

αS κS a λS +2µS 2a

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Fig. 1 Normalized αeT /αm as a function of C and D.

of a porous material, the coating material should have a higher stiffness than the matrix and a negative CTE. Research on materials with negative CTE has attracted the attention of many researchers. At present, three families of materials, ZrW2 O8 , ZrP2 O7 and Sc2 (WO4 )3 have displayed unusual negative CTEs [14, 15]. Moreover, the surface elastic constants and CTEs can be altered by, e.g., chemical modification of the pore surface. In this way, it may be possible to obtain any effective CTEs. It is seen from Eqs. (6) and (7) that the CTE of a material with coated pores only depends on the ratio t/a, and not on the absolute size of the pores. Therefore, the application of a coating is not limited to nano-porous materials. It can be used to change the CTE of micro- and macro-porous materials. Materials with unusual physical/mechanical properties such as negative or zero CTE [16], negative Poisson ratio [17, 18] and negative stiffness [19] have been designed in the past by manipulating their microstructure. The results of the present paper provide an opening for designing porous materials with a specific CTE by means of pore surface manipulation. Materials with low coefficients of thermal expansion (CTEs) are vital in high-precision optical devices and sensors whose properties do not degrade as the temperature varies. Pyrex glass is perhaps the best known example of this type [20]. Those with a tunable CTE matching that of another material are of importance in electronic and biomedical applications. In particular, light-weight porous materials with vanishingly low CTEs are ideal for aerospace applications.

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4 Scaling Laws for Elastic Properties 4.1 Linear Spring Interface Model When we study the effective elastic properties of a composite consisting of a continuous isotropic matrix with shear modulus µm and bulk modulus km and spherical particles or cylindrical fibres with shear modulus µp and bulk modulus kp , three intrinsic length scales will emerge for the LSM, namely, lr = µm βn , lθ = µm βt , lα = γ /α m , where we have assumed βs = βt , and αm is the coefficient of thermal expansion of the matrix. The non-dimensional properties of the concerned material can then be expressed as functions of the non-dimensional variables lr /L, lθ /L and lα /L, where L is the characteristic size of the heterogeneous material, say, the radius of the particles/fibres. Next we expand the functions in Taylor series of these variables. When these variables are small such that the terms of the order two and higher can be neglected and only the linear terms are retained, these functions can then be expressed in a simple form as follows [3]: H (∞) 1 = 1 + (αr lr + βθ lθ ) H (L) L

(8)

where H /(L) denotes the property corresponding to a characteristic size L, and H (∞) denotes the same property when L → ∞ or, equivalently, when the interface effect is vanishingly small. αr and βθ are two non-dimensional parameters [3]. The effective CTE of the heterogeneous medium obeys a scaling law similar to that in Eq. (8), but involving the two length scales lr and lα [3].

4.2 Interface Stress Model For the interface stress model, the emerging intrinsic length scales are lλ = λs /µm and lµ = λµ /µm . Thus, it has been shown that the size-dependence of many properties of heterogeneous media with the surface/interface stress effect, including the Eshelby tensor and the stress concentration tensor for inhomogeneities, the effective elastic constants and the effective CTE of heterogeneous media, can be accurately depicted by the following scaling law [3, 5]: H (L) 1 = 1 + (αλ lλ + βµ lµ ) H (∞) L

(9)

where αλ and βµ are two non-dimensional parameters [3]. Scaling laws in Eqs. (8) and (9) not only accurately predict the size-dependence, but also provide benchmarks for checking experimental measurements and numerical computations of the properties of materials.

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5 Conclusions We demonstrate that it is theoretically possible to design nano-porous materials by adjusting the pore surface elastic parameters, porosity and size of the nanopores so that the resulting effective coefficients of thermal expansion take on desired positive, zero or even negative values. We also indicate two routes to achieving this in practice on nano-cellular materials. It is interesting to point out that as the interface imperfections simulated by the two types of interface model (ISM/HC and LSM/LC) have opposite physical interpretations, the corresponding scaling laws (8) and (9) also have a formal mathematical reciprocity.

References 1. Hashin, Z. Analysis of composite materials – A survey, J. Appl. Mech. 50, 481–505, 1983. 2. Gleiter H, Weissmüller J, Wollersheim O, Wurschum R. Nanocrystalline materials: A way to solids with tunable electronic structures and properties?, Acta Mater. 49, 737–745, 2001. 3. Duan HL, Karihaloo BL. Thermo-elastic properties of heterogeneous materials with imperfect interfaces: generalized Levin’s formula and Hill’s connections, J. Mech. Phys. Solids 55, 1036–1052, 2007. 4. Hashin, Z. Thin interphase/imperfect interface in elasticity with application to coated fiber composites, J. Mech. Phys. Solids 50, 2509–2537, 2002. 5. Wang J, Duan HL, Zhang Z, Huang ZP. An anti-interpenetration model and connections between interphase and interface models in particle-reinforced composites, Int. J. Mech. Sci. 47, 701–718, 2005. 6. Pathak S, Shenoy VB. Size dependence of thermal expansion of nanostructures, Phys. Rev. B 72, 113404-1–4, 2005. 7. Gurtin ME, Murdoch AI. A continuum theory of elastic material surfaces, Arch. Rat. Mech. Anal. 57: 291–323, 1975. 8. Duan HL, Wang J, Huang ZP, Karihaloo BL. Eshelby formalism for nano-inhomogeneities, Proc. Roy. Soc. A 461, 3335–3353, 2005. 9. Murdoch AI. Some fundamental aspects of surface modelling, J. Elasticity 80, 33–52, 2005. 10. Dvorak GJ. On uniform fields in heterogeneous madia, Proc. Roy. Soc. A 431, 89–110, 1990. 11. Duan HL, Wang J, Huang ZP, Karihaloo BL. Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress, J. Mech. Phys. Solids 53, 1574– 1596, 2005. 12. Duan HL, Wang J, Karihaloo BL, Huang ZP. Nanoporous materials can be made stiffer than non-porous counterparts by surface modification, Acta Mater. 54, 2983–2990, 2006. 13. Levin VM. Thermal expansion coefficient of heterogeneous materials, Mech. Solids 2, 58–61, 1967. 14. Mary TA, Evans JSO, Vogt T, Sleight AW. Negative thermal expansion from 0.3 to 1050 Kelvin in ZrW2 O8 , Science 272, 90–92, 1996. 15. Ernst G, Broholm C, Kowach GR, Ramirez AP. Phonon density of states and negative thermal expansion in ZrW2 O8 , Nature 396, 147–149, 1998. 16. Sigmund O, Torquato S. Design of materials with extreme thermal expansion using a threephase topology optimization method, J. Mech. Phys. Solids 45, 1037–1067, 1997. 17. Lakes R. Foam structure with a negative Poisson’s ratio, Science 235, 1038–1040, 1987. 18. Phan-Thien N, Karihaloo BL. Materials with negative Poisson’s ratio: A qualitative microstructural model, J. Appl. Mech. 61, 1001–1004, 1994.

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19. Lakes RS, Lee T, Bersie A, Wang YC. Extreme damping in composite materials with negative stiffness inclusions, Nature 410, 565–567, 2001. 20. Evans JSO. Negative thermal expansion materials, J. Chem. Soc. Dalton Trans. 19, 3317– 3326, 1999.

Modeling the Stress Transfer between Carbon Nanotubes and a Polymer Matrix during Cyclic Deformation C.C. Kao and R.J. Young

Abstract Raman spectroscopy was used in this study to investigate the cyclic deformation behavior of the single-walled carbon nanotubes (SWNTs)/epoxy composites. The stress transfer between the nanotube and epoxy resin has been followed through the stress-induced variation of the G’ Raman band position of the nanotubes. A hysteresis loop was found between the loading and unloading cycles and its size decreased with the increase of the deformation cycles. The energy dissipated in the composite and at the interface between the nanotube and matrix has been modeled from the loop area. The amount of interface damaged for each loading cycle was further predicted from the estimated dissipation energy.

1 Introduction Considerable research has concentrated on using nanotubes as reinforcement to enhance the mechanical properties of polymer composites due to the remarkable properties of nanotubes, such as a high aspect ratio (up to 104 ), high tensile strength (50 GPa) [1] and high modulus (about 1 TPa) [2]. In order to gain improved mechanical properties, the degree of the adhesion between the nanotubes and polymeric matrix plays a crucial role, as well as being an important parameter for understanding stress transfer in the nanotube composites. Failure of the interface introduces little or no stress transfer and the mechanical properties of the composites may be inferior to those of the matrix. The degree of the interfacial adhesion between nanotubes and matrix has been determined by several methods reported elsewhere [3–18]. The pull-out of the nanotubes from the polymer matrix by using an AFM tip is one of these methods [3, 4]. The pull-out energy can be estimated from the force recorded by a cantilever and the C.C. Kao and R.J. Young Materials Science Centre, School of Materials, University of Manchester, Grosvenor Street, Manchester, M1 7HS, UK; e-mail: [email protected]

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embedded length of the nanotube in matrix. Another approach can be performed by utilizing mechanical testing on nanotube composites [5, 6]. The energy dissipated during the deformation process can be monitored. Other methods, including Raman spectroscopy [7–10], fragmentation of nantoubes in a polymer [11, 12], pull-out of nanotubes bridged across defects in a polymer [13–16] and simulation [17,18], have been also used to investigate the stress transfer behavior of nanotubes composites. The mechanical properties, such as Young’s modulus, tensile strength and interfacial shear strength, can also be estimated. This study aims to investigate cyclic deformation behavior of the nanotube composites by using Raman spectroscopy. The stress-sensitive G’ Raman band was used to monitor the mechanical response of the composites. The energy dissipated in the nanotube composites and at nanotube-polymer interface was modeled using various assumptions.

2 Experimental The nanotube composites consisted of purified HiPco single-walled carbon nanotubes (SWNTs), supplied by CNI Company in Houston, USA, and an epoxy resin mixed with 50 parts by weight of Araldite LY 5052 epoxy resin and 19 parts by weight of Araldite HY 5052 hardener. As to the nanotube dispersion process, about 0.1 wt.% of the nanotubes was first suspended in the hardener and put in a sonic bath for 3 hours at room temperature. Subsequently, another 3 hours of sonication was applied before magnetic stirring the mixture for 2 hours. The epoxy resin was then added into the well-dispersed mixture before degassing in a vacuum chamber for an hour. The liquid mixture was then spread into a flat square mould and cured at room temperature for 7 days. The G’ nanotube Raman band was followed using a Renishaw 1000 Raman microprobe system with the 633 nm red line of a 20 mW He-Ne laser and a chargecoupled device (CCD) as a detector. A 50× lens was used to focus the laser on the surface of the sample to give the spot size of 2 µm. All the Raman spectra collected were curve-fitted with Lorentizian routines. The cold-cured composites produced were cut into rectangular beams (10 × 65 ± 1.0 mm bars of 2.0 ± 0.1 mm thickness) and inserted into a four-point bending rig located on the Raman microscope stage to perform the loading and unloading test. A total of five Raman spectra were collected at each strain interval, of about 0.05% strain, using 10 seconds exposure time. The sample was deformed until 1.0% strain was reached and then unloaded back to 0% strain. A total of five loading and unloading cycles were performed in this study. The incident laser polarization was fixed aligned parallel to the tensile axis during the test. The laser spot was focused at the same area after the sample was loaded or unloaded. The results are shown as the average value of the G’ band position collected at each strain interval.

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(a)

(b) Fig. 1 Shift of the G’ band position with axial tensile strain for the initial two cycles [(a) cycle 1 and (b) cycle 2] of loading and unloading test. Note that the rest of cycles behaved similarly to cycle 2.

3 Results and Discussion Figure 1 shows the variation of the G’ band position with applied strain under for initial two loading and unloading cycles. It should be pointed out that the subsequent third to fifth cycles are not shown in here as they all behave similarly to the second loading cycle. It can be seen from the figure that the G’ band position shifts to

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Fig. 2 Shift of G’ band position with axial strain for each loading cycle.

lower wavenumber with applied tension. The strain-shift behavior of the G’ band position exhibits three different regions as the loading data shown in Figure 1a. A linear shift of the G’ band with strain is observed in the range of 0 to 0.4% strain which indicates the elastic deformation of the composites. The behavior becomes non-linear over 0.4% strain and tends to reach a plateau region, where there is no more G’ band shift, over 0.8% strain. The non-linear behavior may indicate the change of the stress transfer between the matrix and nanotubes while the plateau region appears no or little stress transfer. The origin of the non-linear behavior may due to the presence of the slippage taking place at the nanotube-nanotube interface in bundles [8] and/or the nanotube-matrix interface starting to undergo damage. In the comparison between loading and unloading cycles, the strain-shift behavior of the G’ band is non-linear for the loading cycles and approximately linear for the unloading cycles. Comparing each loading cycle, it can be seen that the plateau region only occurs in the first loading cycle as shown in Figure 1a. It may indicate the most of interface damaged occurs in this cycle. In addition to the comparison between the loading and unloading cycles, a resulting loop is observed and its size appears to decrease with following cycles.

3.1 Residual Stress Figure 2 shows the deformation behavior at each loading cycle. It can be seen that the loading behavior stabilizes after the third loading cycle. An increase in the initial G’ band position is found to increase with each loading cycle indicating the devel-

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Fig. 3 Residual stresses observed during cyclic deformation.

opment of residual compressive stress following each cycle. The residual stress is quantified by using a universal stress shift rate of −5 cm−1 /GPa of the G’ band for carbon fibers [9] and is shown in Figure 3. It should be noted that the residual stress at each loading cycles was estimated relative to first loading cycle. Figure 3 shows that the residual stress increases dramatically over the initial cycles (cycle 1 and cycle 2) and tends to be stable in the following cycles. The maximum residual stress observed is about 0.3 GPa.

3.2 Energy Dissipated during Cyclic Deformation Energy Dissipated in the Nanotube Composite Figure 4 demonstrates the estimation of the energy dissipated in the nanotube composite during the loading and unloading test. The variation in the estimated energy dissipated from the loading and unloading curves was calculated to be ±10%. The stress was estimated by using the universal G’ band stress shift rate for carbon fibers [9]. The loading result was curve fitted using a polynomial equation whereas the unloading results were fitted by a linear equation. The areas under these loading and unloading data were measured by an integral of their curve-fitted equations from 0 to 1.0% strain. The dissipation energy per unit volume of the nanotube composite was therefore estimated by subtracting the integrated areas under the unloading curve from the loading curve which is equivalent to the resulting loop area shown

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Fig. 4 Derived nanotube stress-strain curves for the calculation of the energy dissipated during loading and unloading of the nanotube composites. Table 1 Energy dissipated per unit volume of the composite estimated. Loading cycles

Loading energy (MJm−3 )

Unloading energy (MJm−3 )

Energy dissipated (MJm−3 )

Cycle 1 Cycle 2 Cycle 3 Cycle 4 Cycle 5

10.8 9.7 10.0 10.2 10.4

6.0 6.9 8.1 8.1 8.4

4.8 2.8 1.9 2.1 2.2

in Figure 4 [19]. The results estimated for each cycle are listed in Table 1 and the energy dissipated is shown in Figure 5. Results show that the energy is dissipated largely in the initial two cycles and tends to reach a constant value in the remaining cycles. The dissipation energy is about 4.8 MJm−3 for the first cycle and drops by about 40% for the second cycle. Subsequently, the dissipation energy becomes lower, about a 30% decrease, in the third cycle and constant afterward. The constant dissipation energy may be due to the dynamic friction at the damaged nanotube-matrix interface.

Energy Dissipated at the Interface Several assumptions have to be made prior to modeling the energy dissipated at the interface. First of all, the composites produced are assumed to contain a uniform distribution of isolated nanotubes. Secondly, no rotation of the nanotube occurs when

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Fig. 5 Energy dissipated per unit volume of composite as a function of increasing loading and unloading cycles.

the nanotube composite is deformed. Finally, the dimensions of the individual nanotube are assumed to be 1 nm in diameter with 1 µm in length in the epoxy resin for simplifying the estimation. Since the density of the epoxy resin, about 1 gcm−3 [20], is similar to the density of the HiPco SWNTs, 1.3 gcm−3 [21], the volume fraction of the nanotubes in the composite is about 0.1% with approximately 0.1% by weight of nanotubes loaded in the epoxy resin. Therefore, the amount and the total nanotube surface area of nanotubes in unit volume of the composite can be estimated. Since the yield strain and stress of epoxy resin are about 3% and 40 MPa [22], the epoxy resin is expected to undergo elastic deformation up to 1.0% strain. Furthermore, it has been reported that the slippage of the nanotube-nanotube interface inside the nanotube bundle is not responsible for the loop between the loading and unloading cycles [23]. Therefore, the energy dissipation is mainly attributed from the interface between the nanotubes and the epoxy resin. Thus, the interface dissipation energy is determined from the energy dissipated per unit volume of the composite divided by the total nanotube surface area per unit volume. Figure 6 shows the energy dissipated at the nanotube-matrix interface during loading and unloading tests. The energy dissipated is significant for the initial two cycles and tends to be constant in last three cycles.

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Fig. 6 Energy dissipated per unit area of the nanotube-matrix interface as a function of the loading and unloading cycles. Table 2 Length of the interface damage estimated (assuming individual nanotubes). Loading cycles

Interface damaged (%)

Cycle 1 Cycle 2 Cycle 3 Cycle 4 Cycle 5

4.5 2.7 1.9 2.0 2.1

Prediction of the Extent of Damage Interface The study [3] of nano “pull-out” of an individual SWNT bundle has revealed that it required 25.6 J/m2 to pull out a SWNT rope from an epoxy resin. By comparing the pullout energy with the interface dissipation energy obtained in this study, the percentage of the nanotube-matrix interface damaged can be estimated approximately. The results are shown in Table 2 and Figure 7, respectively. It can be seen that the interface damage decreases following each loading cycle. The model can also be used to predict the damage interface between the nanotube bundle and matrix. The nanotube bundles are usually in the form of hexagonal close packed geometry so that the round shape in the cross section is presented. The approximate interface damage therefore can be evaluated from the diameter and length of the bundles. The interface damage between the nanotube bundle and matrix estimated is larger than the individual nanotube one (if the same concentration of the nanotubes in matrix is considered) due to less interfacial area to the matrix to sustain and share the load during deformation.

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Fig. 7 Amount of interface damage (estimated assuming individual SWNTs) as a function of cycles of loading and unloading.

4 Conclusions Stress-induced G’ Raman band shifts have been used to monitor the cyclic deformation behavior of nanotube composites. The strain-shift of G’ band position showed three types of behavior which is related to the interface between the nanotube and matrix under deformation process. A hysteresis loop between loading and unloading results has been found and decreased with an increase of loading cycle. The energy dissipated in the nanotube composite and at the nanotube-matrix interface has been modeled from the loop area with several assumptions being made. Furthermore, the extent of interface damage under this cyclic deformation test has also been estimated.

References 1. Yu, M.F., Files, B.S., Arepalli, S., Ruoff, R.S.: Tensile loading of ropes of single wall carbon nanotubes and their mechanical properties. Phys. Rev. Lett. 84, 5552–5555 (2000). 2. Krishnan, A., Dujardin, E., Ebbesen, T.W., Yianilos, P.N., Treacy, M.M.J.: Young’s modulus of single-walled nanotubes. Phys. Rev. B 58, 14013–14019 (1998). 3. Cooper, C.A., Cohen, S.R., Barber, A.H., Wagner, H.D.: Detachment of nanotubes from a polymer matrix. Appl. Phys. Lett. 81, 3873–3875 (2002). 4. Barber, A.H., Cohen, S.R., Kenig, S., Wagner, H.D.: Interfacial fracture energy measurement of multi-walled carbon nanotubes pulled from a polymer matrix. Comp. Sci. Tech. 64, 2283– 2289 (2004).

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5. Suhr, J., Koratkar, N.A., Keblinski, P., Ajayan, P.: Viscoelasticity in carbon nanotube composites. Nat. Mater. 4, 134–137 (2005). 6. Koratkar, N.A., Suhr, J., Joshi, A., Kane, R.S., Schalder, L.S., Ajayan, P.M., Bartolucci, S.: Characterizing energy dissipation in single-walled carbon nanotube polycarbonate composites. Appl. Phys. Lett. 87, 063102-1-3 (2005). 7. Schalder, L.S., Giannaris, C., Ajayan, P.M.: Load transfer in carbon nanotube epoxy composites. Appl. Phys. Lett. 73, 3842–3844 (1998). 8. Ajayan, P.M., Schalder, L.S., Giannaris, C., Rubio, A.: Single-walled carbon nanotubepolymer composites: Strength and weakness. Adv. Mater. 12, 750–753 (2000). 9. Cooper, C.A., Young, R.J., Halsall, M.: Investigation into the deformation of carbon nanotubes and their composites through the use of Raman spectroscopy. Comp. A 32, 401–411. (2001) 10. Kao, C.C., Young, R.J.: A Raman spectroscopic investigation of heating effects and the deformation behaviour of epoxy/SWNT composites. Comp. Sci. Tech. 64, 2291–2295 (2004). 11. Lourie, O., Cox, D. M., Wagner, H. D.: Buckling and collapse of embedded carbon nanotubes. Phys. Rev. Lett. 81, 1638–1641 (1998). 12. Wagner, H.D., Lourie, O., Feldman, Y., Tenne, R.: Stress-induced fragmentation of multi-wall carbon nanotubes in a polymer matrix. Appl. Phys. Lett. 72, 188–190 (1998). 13. Qian, D., Dicky, E.C., Andrews, R., Rantell, T.: Load transfer and deformation mechanisms in carbon nanotube-polystyrene composites. Appl. Phys. Lett. 76, 2868–2870 (2000). 14. Ye, H., Lam, H., Titchenal, N., Gogotsi, Y., Ko, F.: Reinforcement and rupture behavior of carbon nanotubes-polymer nanofibers. Appl. Phys. Lett. 85, 1775–1777 (2004). 15. Malik, S., Rösner, H., Hennrich, F., Böttcher, A., Kappes, M., Beck, M.T., Authorn, M.: Failure mechanism of free standing single-walled carbon nanotube thin films under tensile load. Phys. Chem. Chem. Phys. 6, 3540–3544 (2004). 16. Lu, J.P.: Elastic properties of carbon nanotubes and nanoropes. Phys. Rev. Lett. 79, 1297–1300 (1997). 17. Wagner, H.D.: Nanotube-polymer adhesion: A mechanism approach. Chem. Phys. Lett. 361, 57–61 (2002). 18. Frankland, S.J.V., Harik, V.M.: Analysis of carbon nanotube pull-out from a polymer matrix. Surf. Sci. 525, L103–L108 (2003). 19. Dieter, G.E.: Mechanical Metallurgy. SI Metric Edn., McGraw-Hill Book Company (1998). 20. Manufacture Datasheet. Huntsman, UK. 21. Zhou, X., Shin, E., Wang, K.W., Bakis, C.E.: Interfacial damping characteristics of carbon nanotube-based composites. Comp. Sci. Tech. 64, 2425–2437 (2004). 22. Young, R.J., Lovell, P.A.: Introduction to Polymers, 2nd edn. Stanley Thornes (2000). 23. Kumar, R., Cronin, S.B.: Raman scattering of carbon nanotube bundles under axial strain and strain-induced debundling. Phys. Rev. B 75, 115421-1-4 (2007).

Atomistic Studies of the Elastic Properties of Metallic BCC Nanowires and Films Pär A. T. Olsson and Solveig Melin

Abstract In this paper a systematic study of the surface influence on the elastic properties of nanosized iron and tungsten wires and films is performed. Single crystal defect-free nanowires and nanofilms are examined through molecular statics simulations, and the concepts of surface energy and third order elastic constants are used in an attempt to describe the elastic properties. For structures where the relaxation strains are small in magnitude, reasonable agreements between the continuum mechanical solutions and the simulations are obtained. For structures where the relaxation strains are significant it is shown that third order elastic continuum theory is not sufficient to describe the elastic properties; in fact, sometimes it actually increases the discrepancies between the simulated and predicted results.

1 Introduction Due to recent developments in nanotechnology, the interest in surfaces influence on the elastic properties has increased substantially. Introduction of surfaces implies breaking of atomic bonds and deviations in the energy of the atoms close to the surface. Consequently, surface atoms display elastic properties different from the bulk. Hence, for structures such as nanowires and nanofilms, where the fraction of surface atoms is not by any means negligible, the influence becomes substantial and macroscopic continuum mechanical generalizations to the nanoscale may result in inaccuracies. Experiments have indicated that the size dependence of Young’s modulus for metallic nanosized structures varies between materials and can either increase [5] or decrease [12, 13] with decreasing size. Different numerical techniques have been employed to simulate the elastic properties of single-crystal nanostructures. For inPär A.T. Olsson and Solveig Melin Division of Mechanics, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden; e-mail: [email protected]

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stance, molecular dynamics simulations have shown that, depending on the crystal orientation, the stiffness may either increase or decrease. It has also been shown that the choice of interatomic potential may influence the results, since, for pair potentials, the stiffness decreases with decreasing size for all orientations, whereas for multibody potentials, such as EAM or Finnis–Sinclair potentials, it may increase as well as decrease with decreasing size in accordance with results from ab initio simulations [10, 15, 20, 22]. It has also been reported that the surface stresses give rise to relaxation strains, resulting in properties deviating from that of the ideal bulk [10, 15, 21]. In fact, for gold there has been reports of surface stresses sufficiently large to induce phase transformations [6]. In this paper we study the elastic properties of single crystal, defect-free, nanowires and nanofilms through molecular statics (MS) simulations. We use the concepts of surface energy and third order elastic constants in an attempt to describe the elastic properties.

2 Continuum Mechanical Description Following the framework developed by Dingreville et al. [8], based on the concept of surface free energy, it is assumed that the area specific density of surface free energy can be expressed as
1
= wdV (1) A0 V0 where w is the volume specific excess energy density, A0 is the reference surface area and V0 the reference volume of the body. Further, it is assumed that we can s expand the surface free energy in a series of the surface strain, αβ , 1 s s s s ) =
0 +
αβ αβ +
αβκλ αβ κλ +
(αβ 2 1 s s s +
αβκλγ η αβ κλ γ η + · · · 6

(2)

where
0 ,
αβ ,
αβκλ and
αβκλγ η denote the Lagrangian surface energy, surface stress tensor, second order surface elastic tensor and third order surface elastic tensor. The Greek subscripts range from 1 to 2 and repeated subscripts are summed. Analogously, for the volume specific bulk energy density, W , we assume that it can be expanded in terms of the bulk strain, ij , 1 1 W (ij ) = W0 + Cij kl ij kl + Cij klmn ij kl mn + · · · 2 6

(3)

where W0 is the volume specific potential energy of the bulk, Cij kl the second order elastic constants and Cij klmn the third order elastic constants in the Lagrangian description and the Roman subscripts range from 1 to 3. The total strain energy, U , can be written as a sum of bulk and surface contributions as

Atomistic Studies of the Elastic Properties of Metallic BCC Nanowires and Films

(a)

223

(b)

Fig. 1 Geometries of the studied structures (a) film, (b) wire.







ij

U= V0

0

dW dkl dV + dkl


A0


0

s κλ

d
dαβ dA dαβ

(4)

The relaxation strains, ij∗ , balancing the surface stresses, can be determined by minimizing the total strain energy, i.e. by solving  1 dU  =0 (5) V0 dij ij = ∗ ij

From this it is possible to calculate the second order elastic tensor about the self equilibrium state  1 d 2 U  ˜ Cij kl = (6) V0 dij2 ij = ∗ ij The films that have been studied have two identical surfaces parallell with the x1 x2 -plane, cf. Figure 1(a), and in the simulations we assume that the in plane dimensions of the films are extended infinitely, i.e. periodic boundary conditions are employed in the in plane directions. Consequently, we remove any artifacts from finite in plane dimensions. In this study, films with three different crystallographic orientations are considered, with the orthogonal (x1 , x2 , x3 ) direc¯ and tions in Figure 1(a) corresponding to ([100], [010], [001]), ([110], [001], [110]) ¯ [11¯ 1]). ¯ The wires we are studying have four free surfaces with nor([110], [112], mals in the x2 and x3 directions, cf. Figure 1(b), where the surfaces are pairwise identical. In order to remove the influence from finite length we impose periodic boundary conditions in the x1 direction. For the wires, we study two different crystallographic orientations, with the orthogonal (x1 , x2 , x3 ) directions in Figure 1(b) ¯ corresponding to ([100], [010], [001]) and ([110], [001], [110]).

3 Simulation Procedure To simulate the interaction between atoms we use slightly modified EAM potentials for iron and tungsten [14]. The potential energy, , of an ensemble of atoms is given

224

P.A.T. Olsson and S. Melin Table 1 Second and third order elastic constants of iron and tungsten (GPa).

Fe Experimental [1] Fe Potential W Experimental [19] W Potential

Fe Experimental (77K) [2] Fe Experimental (273K) [2] Fe Potential W Potential

by =

 i

C11

C12

C66

239.55 239.56 532.55 531.65

135.75 135.75 204.95 204.93

120.75 120.75 163.13 163.11

C111

C112

C123

C144

C155

C456

–2876 –2705 –2860.0 –5456.3

–542 –626 –629.5 –1473.8

–747 –575 –640.2 –1571.4

–869 –836 –647.3 –1658.3

–533 –531 –605.8 –1478.8

–557 –721 –625.0 –1554.3

F (ρi ) +

 1  V (rij ) + M(Pi ) 2 i

where ρi =

j  =i



f (rij )

(7)

i

(8)

j  =i

Pi =

 (f (rij ))2

(9)

j  =i

where rij denotes the interatomic distance, V (rij ) the pair interaction, F (ρi ) is the embedding energy of atom i due to the electron density ρi and M(Pi ) is a modification term added to the original EAM potential to improve the otherwise linear superposition of the atomic electron density, as proposed in [9]. The second and third order elastic coefficients of the ideal bulk for the current iron and tungsten potentials are seen in Table 1 along with experimental data in Voigt notation (11 → 1, 22 → 2, 33 → 3, 23 → 4, 13 → 5 and 12 → 6 [11]). As can be seen in Table 1, the second order elastic constants of the interatomic potentials are in good agreement with experimental data. It can also be observed that the third order elastic constants for the iron potential are in reasonably good agreement with experimental data. When calculating the surface energy, the surface stress and the second order surface elastic tensor we use the semi-analytical approach suggested in [7] with minor modifications. Instead of assuming constant elastic properties throughout the height of the film, we calculate the local elastic constants for each atom and assume linear elastic plane stress for each atom. This is, however, an approximation since the atoms away from the bulk does not necessary display homogeneous strains within the interaction range. Furthermore, we approximate the third order surface elastic constants by assuming that the strains satisfy the constraints of linear elastic plane stress. It is very important to realize that this assessment of the third order surface elastic constants is inconsistent, since we use linear elasticity to approxi-

Atomistic Studies of the Elastic Properties of Metallic BCC Nanowires and Films

225

Table 2 Surface elastic properties for (100) surfaces, in Voigt notation and in units of J/m2 , calculated using a variation of the semi-analytical method by Dingreville and Qu [7], except for the values within brackets which were calculated using the method by Shenoy [17, 18].
0 Fe 1.6300 (1.6300) W 2.8720 (2.8720)
11


12


66

Fe –10.574 –8.5470 –11.734 (–10.963) W –20.220 –17.185 –19.299 (–20.548)


1


2

2.1500 (2.1500) 3.3413 (3.3413)

2.1500 (2.1500) 3.3413 (3.3413)


111


112


166

31.632 14.590 (31.297) 98.994 86.432 (138.51)

48.806 156.57

Table 3 Surface elastic properties for (110) surfaces, in Voigt notation and in units of J/m2 , calculated using a variation of the semi-analytical method by Dingreville and Qu [7], except for the values within brackets which were calculated using the method by Shenoy [17, 18].
0 Fe 1.4867 (1.4867) W 2.6381 (2.6381)
111


112


1


2


11

–1.0291 (–1.0291) –1.2518 (–1.2518)

1.1067 (1.1067) 1.9682 (1.9682)

–4.6095 2.4575 (–4.5878) –12.359 7.7837 (–12.141)


122


166


222

2.5761

–0.6611 –1.5254 (–2.1599) –29.314 –18.218 (–19.012)

Fe 39.865 –13.157 5.4898 (36.190) W –19.485 –28.546 –21.589 (–3.6135)

–8.7300


12


22


66

–1.5077 –0.9589 (–1.4863) –2.5129 –2.0000 (–2.3994)


266

mate non-linear elasticity. However, for small strains it may be considered to be approximately accurate. In Tables 2, 3 and 4 we give the calculated surface properties for (100), (110) and (111) surfaces, respectively, for the iron and tungsten potentials in Voigt notation. For comparison with these semi-analytical results, we perform molecular static strain meshing simulations in the same manner as proposed by Shenoy [17, 18]. This method is a straining scheme where structures of different heights are exposed to increments of strain, and for each increment, the potential energy is minimized. Thereafter, the energies are fitted to a polynomial of the strains from which the bulk and surface contributions are extracted. Some results from the strain meshing simulations are given within the brackets in Tables 2, 3 and 4. When calculating the relaxation strains and the elastic properties around the self equilibrium state, we let the structure undergo an initial relaxation by coupling Berendsen barostats [3] to the Parinello–Rahman parallelepiped [16] in the periodic directions of the simulation cell with the pressure tensor set to zero while, simultaneously, using a potential energy minimizer [4] to obtain a state of balanced stresses

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Table 4 Surface elastic properties for (111) surfaces, in Voigt notation and in units of J/m2 , calculated using a variation of the semi-analytical method by Dingreville and Qu [7], except for the values within brackets which were calculated using the method by Shenoy [17, 18].
0 Fe 1.8310 (1.8310) W 3.2697 (3.2697)
111


112


1


2


11

–0.4561 (–0.4561) –0.5246 (–0.5246)

–0.4561 (–0.4561) –0.5246 (–0.5246)

–13.976 –3.8164 (–15.426) –30.190 –6.5004 (–30.579)

–13.976 –5.0798 (–15.426) –30.190 –11.845 (–30.577)


116


122


126


166


222


226

11.258

0.0205

69.015

–41.01

–0.1308

217.34

241.10 0.01729 (249.70) 682.51 –0.0655 (611.86)

Fe 194.74 57.488 0.0238 (198.98) W 537.91 104.58 –0.1961 (459.17)


12


22


66


266 22.752 71.995

and equilibrium. From this state, the simulation cell is subjected to increments of uniform strain around the balanced equilibrated state. During each strain increment, the energy minimizer finds a state of equilibrium. Thereafter, polynomial fits of the potential energies as functions of strain are found, from which the elastic properties are extracted.

4 Results and Discussion In Tables 2, 3 and 4 we compare the semi-analytical results for the surface elastic properties with results from the strain meshing scheme. The surface energy and the surface stress components are indistinguishable between the two methods. The second order surface elastic constants are also in quite good agreement. There are, however, some discrepancies, especially for iron (111) surfaces where the discrepancies are of order 10%. Considering the assumptions that are made when approximating the third order surface elastic constants, the resulting third order surface elastic constants are, however, for most of the cases in reasonable agreement between the two methods. The largest discrepancies can be found for tungsten films, while for iron a quite reasonable agreement between the two methods is obtained. When studying the elastic properties of films a suitable quantity to identify is the biaxial stiffness, Y , i.e. the stiffness associated with a film being subjected to equal normal strains in both in plane directions. In Figures 2(a) and 2(b) the results from simulations, linear elastic calculations as well as calculations including third order elasticity for iron and tungsten, respectively, using data from Tables 1–4, are displayed. For (110) and (111) surfaces it can be seen that non-linear elasticity improves the continuum mechanical results, i.e. including third order elasticity brings the curve closer to the markers from the simulated results. However, for (100) surfaces non-linear elasticity actually increases the discrepancies and the linear elastic predictions are in better agreement with the simulations, both for iron and tungsten.

Atomistic Studies of the Elastic Properties of Metallic BCC Nanowires and Films

800 700 Y [GPa]

1200

(111) (110)

Y [GPa]

Fig. 2 Biaxial modulus for films of (100), (110) and (111) surfaces for (a) iron, (b) tungsten. The solid lines correspond to nonlinear elastic theory, the dashed lines correspond to linear elastic theory and the markers come from simulations.

600 500

227

(100)

400

(110) 1150 (100) 1100 (111) 1050 1000

300 0

5 h0 [nm]

950 0

10

5 h0 [nm]

(a)

10

(b)

2

0.3

1

0.2

−1 −2

ε∗αβ [%]

ε∗αβ [%]

ε∗αβ [%]

0

0

0.1

−1

−3 0

5 h0 [nm]

0

−2 0

10

(a)

5 h0 [nm]

10

−0.1 0

(b)

5 h0 [nm]

10

(c)

1 0.2

−0.5 −1

ε∗αβ [%]

0.5 ε∗αβ [%]

ε∗αβ [%]

0

0 −0.5

0

−1 −1.5 0

5 h0 [nm]

(d)

10

0

0.1

5 h0 [nm]

(e)

10

0

5 h0 [nm]

10

(f)

Fig. 3 Relaxation strains of (a) (100), (b) (110), (c) (111) iron surfaces and (d) (100), (e) (110), (f) (111) tungsten surfaces. The solid, dashed and dashed dotted lines corresponds to predictions ∗ ,  ∗ and  ∗ , respectively. The circles, squares and diamonds corresponds to the results of of 11 22 12 ∗ ,  ∗ and  ∗ , respectively. the simulations of 11 22 12

Moreover, it can be observed that the linear elastic biaxial stiffnesses for (100) and (111) tungsten surfaces overlap and that all the biaxial stiffnesses for tungsten converge toward similar values which is quite reasonable since tungsten single crystals are close to isotropic. In Figures 3(a)–3(f) the in plane relaxation strains of various surfaces are given, and it can be concluded that the predicted in plane relaxations

240 220 200 180 160 140 120

[110]

450 E [GPa]

Fig. 4 Young’s modulus for wires with [100] and [110] directions for (a) iron, (b) tungsten. The solid lines correspond to nonlinear elastic theory, the dashed lines correspond to linear elastic theory and the markers come from simulations.

P.A.T. Olsson and S. Melin

E [GPa]

228

[100]

2

4 6 h0 [nm]

8

10

300 0

10

1 [110]

0

[110]

0 [100]

−1

ε* [%]

ε* [%]

5 h0 [nm]

(b)

1

−2

[100]

−1 −2

−3 −4 0

[110]

350

(a)

Fig. 5 Relaxation strains for wires with [100] and [110] directions for (a) iron, (b) tungsten. The solid lines correspond to nonlinear elastic theory, the dashed lines correspond to linear elastic theory and the markers come from simulations.

[100]

400

5 h0 [nm]

(a)

10

−3 0

5 h0 [nm]

10

(b)

for all the surfaces are in very good agreement with the results from the simulations. Due to symmetry the transversal normal strains for the (100) surfaces are identical, and for (111) surfaces the transversal normal strains are identical due to the apparent linear elastic isotropy of those surfaces, cf. Table 4. Furthermore, all the in plane shear strains are zero for all of these low index surfaces and the strains are the smallest for (111) surfaces, approximately one tenth as compared to the others. For nanowires we choose the heights and widths so that they are approximately equal. The quantity subject to our investigation is Young’s modulus, E, which can be seen in Figures 4(a)–4(b) along with the relaxation strains in Figures 5(a)–5(b). Wires with the length direction [100] display much larger relaxation strains than those with [110]. It can also be observed that the agreement between continuum solutions and simulations is much better for [110] than for [100]. For [110] the linear as well as the nonlinear elastic continuum predictions are in reasonable agreement with the simulations. However, for [100] wires, when incorporating third order elastic constants into the continuum mechanical solution, the prediction is that the nanowires stiffen, whereas for linear elasticity it is predicted that the nanowires weaken. From the simulations we get both; iron stiffens while tungsten weakens and, actually, neither the linear nor the nonlinear elastic continuum solutions are in that good agreement with the simulations.

Atomistic Studies of the Elastic Properties of Metallic BCC Nanowires and Films

250

500

E [GPa]

450 E [GPa]

Fig. 6 Young’s modulus for bulk with [100] and [110] directions for (a) iron, (b) tungsten as functions of 11 . The solid and dashed lines correspond to nonlinear elastic theory for [100] and [110] directions, respectively, and circles and diamonds come from simulations of [100] and [110] directions, respectively.

229

200

400 350

150 −4

−2 ε [%]

(a)

0

300

−4

−2 ε [%]

0

(b)

In order to study the Young’s modulus behavior for the bulk as function of relaxation strains simulations were performed with periodic boundary conditions in all directions and barostats coupled to the appropriate directions. In Figures 6(a) and 6(b) plots of bulk Young’s modulus are given for iron and tungsten, respectively, both from simulations and from nonlinear elastic calculations. It can be seen that there is an agreement between the simulations and the calculations around the zero strain mark. But, when the strain increases in magnitude, the discrepancies between the two increase. In fact, the simulations show that for tungsten [100] bulk Young’s modulus first increase with increasing strain magnitude but then reaches a peak and decreases, whereas the nonlinear continuum calculations display a monotonic increase with increasing strain magnitude. In fact, for most of the strains, the ideal bulk values are closer to those of the simulations than the predicted nonlinear elastic solution which probably explains the massive discrepancies in Figure 4(b) between the simulations and the nonlinear continuum solution. For iron [100] bulk Young’s modulus it can be seen that both simulations and nonlinear continuum solutions display monotonous stiffening with increasing strain magnitude. However, the discrepancies between the two increase with increasing strain magnitude which is in agreement with the results in Figure 4(a). For the [110] directions there is a good agreement between the result from the simulations and the continuum solutions in the regions of strain that are of interest, in agreement with the results in Figures 4(a) and 4(b).

5 Conclusions In this work the elastic properties of metallic nanosized BCC films and wires have been studied. In an attempt to predict the elastic properties we utilized the concepts surface energy and third order elastic constants. This approach has shown beneficial whenever the relaxation strains are small in magnitude, but for less close packed directions, where the relaxation strains are significant, linear elastic continuum me-

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chanics have actually lead to more reliable predictions for some structures. However, Young’s modulus behavior for [100] iron nanowires could not have been predicted without higher order elastic theory. Comparisons between simulations and continuum mechanical calculations of the bulk reveal that, for small relaxation strains, there is a good agreement between the two. However, the discrepancies increase with relaxation strains and the deviations in such situations are quite significant, which is an indicator that it may not sufficient with third order elastic constants to describe the elastic properties. Acknowledgements This work has been supported by the Swedish Research Council. The simulations were performed using the computational resources at LUNARC, Center for Scientific and Technical Computing, Lund University.

References 1. J.J. Adams, D.S. Agosta, R.G. Leisure, H. Ledbetter, J. Appl. Phys. 100, 113530 (2006). 2. A.G. Avery, A.K. McCurdy, in Landolt–Börnstein – Group III Condensed Matter, D.F. Nelson (Ed.), p. 649. Springer-Verlag, New York (1992). 3. H.J.C. Berendsen, J.P.M. Postma, W.F. van Gunsteren, A. DiNola, J.R. Haak, J. Chem. Phys. 81, 3684 (1984). 4. E. Bitzek, P. Koskinen, F. Gähler, M. Moseler, P. Gumbsch, Phys. Rev. Lett. 97, 170201 (2006). 5. S. Cuenot, C. Frétigny, S. Demoustier-Champagne, B. Nysten, Phys. Rev. B 69, 165410 (2004). 6. J. Diao, K. Gall, M.L. Dunn, J. Mech. Phys. Solids 52, 1935 (2004). 7. R. Dingreville, J. Qu, Acta Mater. 55, 141 (2007). 8. R. Dingreville, J. Qu, M. Cherkaoui, J. Mech. Phys. Solids 53, 1827 (2005). 9. W. Hu, X. Shu, B. Zhang, Comp. Mater. Sci. 23, 175 (2002). 10. H. Liang, M. Upmanyu, H. Huang, Phys. Rev. B 71, 241403(R) (2005). 11. M.A. Meyers, K.K. Chawla, Mechanical Behavior of Materials, Prentice-Hall, Upper Saddle River, NJ (1999). 12. S.G. Nilsson, X. Borrisé, L. Montelius, Appl. Phys. Lett. 85, 3555 (2004). 13. S.G. Nilsson, E.L. Sarwe, L. Montelius, Appl. Phys. Lett. 83, 990 (2003). 14. P.A.T. Olsson, unpublished communication. 15. P.A.T. Olsson, S. Melin, C. Persson, Phys. Rev. B 76, 224112 (2007). 16. M. Parinello, A. Rahman, J. Appl. Phys. 52, 7182 (1981). 17. V.B. Shenoy, Phys. Rev. B 71, 094104 (2005). 18. V.B. Shenoy, Phys. Rev. B 74, 149901(E) (2006). 19. G. Simmons, H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook, 2nd edn. The MIT Press, Cambridge (1971). 20. F.H. Streitz, K. Sieradzki, R.C. Cammarata, Phys. Rev. B 41, 12285(R) (1990). 21. P. Villain, P. Beauchamp, K.F. Badawi, P. Goudeau, P.-O. Renault, Scr. Mater. 50, 1247 (2004). 22. L.G. Zhou, H. Huang, Appl. Phys. Lett. 84, 1940 (2004).

Advanced Continuum-Atomistic Model of Materials Based on Coupled Boundary Element and Molecular Approaches Tadeusz Burczy´nski, Wacław Ku´s, Adam Mrozek, Radosław Górski and Grzegorz Dziatkiewicz

Abstract The paper contains a description of a multiscale algorithm based on the boundary element method (BEM), coupled with a discrete atomistic model. The discrete model uses empirical pair-wise potentials and the Embedded Atom Method (EAM) to compute interaction forces between atoms. The Newton–Raphson method with the backtracking algorithm is applied to solve a nonlinear system of equations of the nanoscale model. The continuum domain is modeled using BEM with subregions. Some numerical results of simulations at the nanoscale are shown to examine the presented technique.

1 Introduction Recently, multiple-scale models of engineering materials have been developed to address the coupling of different length scales for various applications. The strength and stiffness of engineering materials are affected by the characteristics at various length scales. Atomic defects such as vacancies and dislocations play a role at the atomic scale, while characteristics of grain boundaries at the micro- or meso-scales contribute to the material strength. In order to understand and predict mechanical behaviours of engineering materials it is necessary to incorporate all those characteristics in different length scales.

Tadeusz Burczy´nski Department for Strength of Materials and Computational Mechanics, Silesian University of Technology, 18a Konarskiego St., 44-100 Gliwice, Poland; and Department of Artificial Intelligence, Institute of Computer Modelling, Cracow University of Technology, 24 Warszawska St., 31-155 Cracow, Poland; e-mail: [email protected] Wacław Ku´s, Adam Mrozek, Radosław Górski and Grzegorz Dziatkiewicz Department for Strength of Materials and Computational Mechanics, Silesian University of Technology, 18a Konarskiego St., 44-100 Gliwice, Poland

R. Pyrz and J.C. Rauhe (eds.), IUTAM Symposium on Modelling Nanomaterials and Nanosystems, 231–240. © Springer Science+Business Media B.V. 2009

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Most of the multi-scale models considered two neighbouring length scales, while some other examines bridged more than two length scales. A recent survey on multiscale modelling was provided in [5]. It summarized and compared various coupling techniques between the atomic model and a continuum model. In all papers the continuum model is considered in the framework of the finite element method. This work deals with a multiscale algorithm based on the boundary element method (BEM) with subregions [1,4], coupled with a discrete atomistic model. It is a developed version of the own approach presented previously in [2]. In this technique, the material behaviour at the atomic level can be simulated and the total number of degrees of freedom is reduced, because in most cases only a small part of the multi-scale model contains atoms. The discrete scale is modelled using empirical Lennard–Jones and Morse potentials. These two-body potentials are sufficient for benchmark problems. For more realistic simulations of metallic materials, atomic interactions are obtained using the Embedded Atom Method [6]. The presented multiscale algorithm can be used for many different types of interatomic potentials. In the applied EAM, potential, embedding energy and electron density functions are prepared using results data from ab initio computations.

2 The Continuum Model Consider a continuum model of material on the microscale level as an elastic medium which occupies a domain
bounded by a boundary . The field of displacement u(x) is described by Navier–Lamé equation: (λ + µ) grad div u + µ∇ 2 u = 0

(1)

where λ and µ are the Lamé parameters characterizing the material medium. Eq. (1) should be completed with boundary conditions: u(x) = uo (x),

x ∈ u

t(x) = σ n = to (x),

x ∈ t

(2)

where σ = (σij ) is a stress tensor, n is a unit normal vector outward to the boundary , u and t are boundaries, where displacements and tractions are prescribed. To solve the boundary-value problem given by Eqs. (1) and (2), the boundary element method is used [1]. The method allows formulating the following vector boundary integral equation:

c(x)u(x) + T(x, y)u(y)d(y) = U(x, y)t(y)d(y) (3) 



where U(x, y) and T(x, y) are the fundamental solutions of elastostatics, u and t are vectors of displacements and tractions, respectively, c is a free term coefficient matrix which depends on the boundary geometry.

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To solve Eq. (3) the boundary  of the continuum domain
is discretized using three node quadratic boundary elements. Second order shape functions are used to approximate boundary tractions and displacements. The algebraic counterpart of Eq. (3) is given in matrix form: [H]{u} = [G]{t}

(4)

where the matrices [H] and [G] depend on boundary integrals of the fundamental solution U and T, respectively. Vectors {u} and {t} contain nodal values of boundary displacements and tractions, respectively. The boundary element method with subregions [4] is applied in proposed multiscale algorithm. This approach is very convenient for building an interface area, which couples a discrete atomistic structure with a continuum domain. The construction of the interface is presented in the fourth section of this paper. The final set of equations for the whole continuum structure in the BEM with subregions can be formulated by assembling the set of equations for each subregion using compatibility of displacements and equilibrium of tractions between zones. The final system of equations for structure with
i and
j domains is ⎧ i ⎫ u ⎪  ⎪ ⎪   i  i ⎨ ij ⎪ ⎬ Gi 0 u t H H ij −Gij 0 = (5) ij j j 0 H j i Gj i H j ⎪ t t 0 G ⎪ ⎪ ⎩ j ⎪ ⎭ u where the superscripts i and j denote the quantities corresponding to domain
i and
j , respectively.

3 The Discrete Atomic Model The discrete atomic model is applied to simulate deformations of the atomistic lattice under loads. This model is based on the equilibrium equations of atomic interaction forces. The equilibrium state of the lattice corresponds to the minimal value of the total potential energy of the atomic structure. The potential energy is described using different equations depending on a distance between each 2, 3 or many atoms. The parameters of the equations are computed to provide best fit to various properties of a material (equilibrium lattice constant, elastic constants, etc.). The parameters of the potentials equations are prepared on the base of ab initio calculations [6]. The best results can be obtained performing whole computations using ab initio approach, but the computer cost of such approach would be very large. To describe the potential energy and interactions between atoms the empirical potentials can be used:

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Fig. 1 Interatomic potentials.

• Lennard–Jones 2-body potential:

 (rij ) = 4ε

σ rij

12

 −

σ rij

6  (6)

• Morse 2-body potential:   (rij ) = ε e2α(r0−rij ) − 2eα(r0−rij )

(7)

• EAM potential: (rij ) =

1 V (rij ) + F (ρi ) 2

where ρi =



ρ(rij )

(8)

j  =i

where (rij ) denotes a pair potential energy, rij is the distance between i-th and j -th atom, r0 is an equilibrium bond length, σ is a collision diameter, and ε is a dissociation energy. V (rij ) is a potential energy for core to core repulsion between two atoms. F is a so-called embedding energy as a function of the host density ρi induced at site i by all other atoms in the lattice. Parameter α in Eq. (7) means an inverse length scaling factor. The second term in Eq. (8) represents many-body interactions in the atomistic system. Thus, the EAM is kind of “glue model” potential, and describes behavior of the metallic material at the nanolevel more accurately than standard pair-wise interactions. The potentials shown in Figure 1 are applied and used in numerical examples. The potential’s parameters are taken from [3, 6, 8]. The interaction forces (Figure 2) between each pair of atoms in the lattice are computed as the derivative of an interatomic potential respect to the distance between two atoms:

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235

Fig. 2 Forces acting between i-th and j -th atom.

Fig. 3 Initial and displaced positions of the two atoms.

fij = −

∂(rij ) nrij , ∂rij

fj i = −fij

(9)

Consider two atoms i and j in the initial placement X at distance Rij (Figure 3). Each atom gets displacement u (due to acting external forces) and finally the atoms occupy positions x, rij denotes the resultant distance vector: rij = Rij + uij ,

uij = uj − ui

(10)

The interaction force between atoms in displaced position can be written as fij (rij ) = fij (rij )

rij rij

(11)

After substitution (9) and (10) into above equation, the following equilibrium equation can be formulated: fij (rij )

uij Rij − fij (rij ) =0 rij rij

or in matrix form (for two-dimensional case): ⎤⎛ ⎡ ⎞ ⎛ uix fix k 0 −k 0 ⎥⎜ ⎢ ⎟ ⎜ ⎢ 0 k 0 −k ⎥ ⎜ uiy ⎟ ⎜ fiy ⎥⎜ ⎢ ⎟=⎜ ⎜ ⎢ −k 0 ⎟ ⎜ k 0 ⎥ ⎦ ⎝ uj x ⎠ ⎝ fj x ⎣ 0 −k 0 k ujy fjy

(12)

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

where k =

fij (rij ) rij

(13)

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Fig. 4 The coupled BEM-atomic multiscale model.

This system of equations describing one atomic bonding, is nonlinear and must be transformed into the form, which can be solved using an iterative method. After some transformations, system of equations can be expressed as ⎧ ⎧ ⎫ ⎫ (uix − uj x ) − (xi − xj ) ⎪ L1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (uiy − ujy ) − (yi − yj ) ⎪ ⎨ L2 ⎪ ⎬ ⎬ ≡k (14) L(u) = 0 where L(u) = ⎪ ⎪ L3 ⎪ (uj x − uix ) − (xj − xi ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ ⎭ L4 (ujy − uiy ) − (yj − yi ) The system of nonlinear eqs. (14) is solved iteratively, using the Newton–Raphson method: !−1 un+1 = un − L (un ) L(un ) (15) The Jacobian matrix L and the vector L are computed for all atoms, which interact with others in a circular area. The cut-off radius is defined as a multiplicity of the lattice constant. After aggregation of L and L, the constraints are applied using elimination method. The main concept is to assume some initial positions of molecules (e.g. an undeformed lattice) and obtain final, stable equilibrium configuration of atoms with appropriate boundary conditions. To improve stability and convergence of the Newton–Raphson scheme, especially when initial guess is not sufficiently close to root or more complicated potential functions are used, the backtracking algorithm is applied [7].

4 Multiscale Model The schema of the multiscale model is shown in Figure 4. The discrete atomic model occupies only rather small area of the model, where the simulation at the nanoscale should be performed. The rest of the structure is modelled by BEM.

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Fig. 5 The algorithm of solving coupled atomistic-BEM model.

The interface domain contains so-called embedded atoms, overlapped on the subregion of the continuum domain. The coordinates of these atoms are equal to the corresponding nodes of boundary elements. Boundary conditions are applied on the continuum model. The algorithm of solving the coupled multiscale model is presented in Figure 5. In the first step BEM model is solved. Displacements of the interface subregion are obtained and introduced as initial displacements of the atomic lattice. In the next step, equilibrium positions of the atoms are computed, using the method described in previous chapter. Finally, forces acting on interface atoms are computed and introduced as a tractions nodal values to the BEM model. These computations are repeated iteratively until the stop condition is satisfied. The stop condition is executed when the difference between displacements of the embedded atoms during two iterations is less than an admissible value.

5 Numerical Examples In order to present the application of the multiscale algorithm, some numerical tests are performed. The discrete lattice with 68 aluminium atoms is presented in Figure 6L. The middle column of atoms is constrained (as marked in Figure 6) and the distance between two nearest atoms is equal to 3A (lattice constant for aluminium, r0 = 2.89A [8]). In this test, the equilibrium state is obtained using three different types of interatomic potentials: Morse (Figure 6R), EAM (Figure 7L) and Lennard–Jones (Figure 7R). In each case, the proper equilibrium state of the lattice is achieved and the average bond length corresponds to the specific potential’s parameters and theirs minima.

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Fig. 6 Discrete atomic lattice: initial state (Left) and equilibrium state, computed using Morse potential (Right).

Fig. 7 Equilibrium states of atomic lattice computed using Embedded Atom Method (Left) and Lennard–Jones potential (Right).

Some differences in the final geometry are caused by different shapes and characters of used potentials (Figure 1). The next example shows a two-scale plate model of the plate with a rectangular notch (Figure 8). The left side of the plate is constrained and the shear load is applied on the opposite side. Dimensions of the plate are 40 × 27 [nm]. The continuum model contains 113 quadratic elements. The discrete model contains 884 atoms and the atomic lattice has randomly added imperfections. The Lennard–Jones potential parameters (σ = 0.2575 nm,  = 0.1699 nm*nN) are taken from [8]. Results of the numerical simulations are presented in Figure 8b. The atoms moved to the new equilibrium state. The opening of nano-cracks at the corners of the notch can be observed.

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Fig. 8 The plate with U-notch under the shear load: (a) initial equilibrium, (b) equilibrium state under shear load.

6 Final Remarks Presented multiscale algorithm gives possibility of analysis the material behavior at micro and nanoscale, e.g.: slips, crack behaviour and fractures. The more realistic results for metallic materials can be obtained using the EAM potential. The presented examples can be treated as a benchmark problems. The convergence of the Newton–Raphson method and the total number of iterations strongly depend on the initial position of the atomic structure. The efficiency of the Newton–Raphson routine can be significantly improved by the backtracking algorithm. Acknowledgements The research is financed from the Polish science budget resources as a research project (2007–2010).

References 1. Burczy´nski, T.: The Boundary Element Method in Mechanics, WNT, Warsaw (1995). 2. Burczy´nski,T., Mrozek, A., Ku´s, W.: A computational continuum-discrete model of materials. Bulletin of the Polish Academy of Sciences, Technical Sciences 55(1), 85–89 (2007). 3. Girifalco, L.A., Weizer, V.G.: Application of the Morse potential function to cubic metals. Physical Review 114(3), 687–690 (1959). 4. Górski, R., Fedelinski, P.: Analysis, optimization and identification of composite structures using boundary element method. Journal of Computational and Applied Mechanics 6(1), 53– 65 (2005).

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5. Liu, K., Karpov, E.G., Zhang S., Park, H.S.: An introduction to computational nanomechanics and materials. Computer Methods in Applied Mechanics and Engineering 193, 1529–1578 (2004). 6. Mishin, Y., Farkas, D.: Interatomic potentials for monoatomic metals from experimental data and ab initio calculations. Physical Review 59(5), 3393–3407 (1999). 7. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flanery, B.P.: Numerical Recipes. The Art of Scientific Computing, Cambridge University Press, Cambridge (2007). 8. Sunyk, R., Steinmann, P.: On higher gradients in continuum-atomistic modeling. International Journal of Solids and Structures 40, 6877–6896 (2002).

Finite Element Modelling Clay Nanocomposites and Interface Effects on Mechanical Properties Julian Y.H. Chia

Abstract The use of modelling to understand the science of engineering strong and tough materials through clay nanocomposite technology is potentially beneficial and needs to be realised. In this paper, the use of continuum finite element method as a start point to model clay/epoxy nanocomposites and its mechanical properties is explored. Its computation cost is cheap; a full three-dimensional continuum representative volume element (RVE) model to investigate the effects of interfaces on the mechanical properties of nanocomposites is new; and the use of continuum finite element method at the nano-scale with all its advantages and short-comings needs understanding. This paper briefly introduces the approach to developing RVE models of nanocomposites consisting of Montmorillonite clay nanofillers that are randomly orientated and randomly embedded in an epoxy matrix. Thereafter, the mechanics of interface failure such as particle splitting (or debonding) and its effect on the mechanical properties such as stress-strain behaviour and strength of the nanocomposite is discussed.

1 Introduction The benefit of using modelling to investigate the mechanisms controlling the mechanical properties of clay nanocomposites is gradually being realised. Modelling mechanical stiffness of clay nanocomposites is well documented. For example, Fornes and Paul [1], Wang and Pyrz [2] and Sheng et al. [3] showed that the key factors enhancing the stiffness of clay nanocomposite are the clay particle’s high modulus, high aspect ratio, volume fraction, and morphology. Concerning the latter factor, it was predicted that highest stiffening of the nanocomposite is achieved when the clay particle has least silicate layers or the gallery thickness separating Julian Y.H. Chia Institute of Materials Research and Engineering, A*STAR (Agency for Science, Technology and Research), 3 Research Link, Singapore 117602; e-mail: [email protected]

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the silicate layers in the clay particle is largest, i.e. exfoliated clay particles stiffen the nanocomposite more than the intercalated clay particles. Recently, Hbaieb et al. [4] and Chia et al. [5] have shown that the stiffness in clay nanocomposites is also affected by the formation of clay clusters in the nanocomposite and current analytic models such as the Mori–Tanaka model do not accurately account its effect on stiffness. These predicted effects of morphologies of the nano clay on stiffness of the nanocomposite is strongly dependent on the interface properties of the clay particle and particle-matrix interface. The review by Zeng et al. [6] on multiscale modelling nanocomposites illustrates the use of modelling to investigate the interface thermodynamics and kinetics on clay morphology formations. The ability to link the interface properties and clay morphologies at the molecular scale to the bulk mechanical properties of a clay nanocomposite is however still at an infant stage. This author believes that literature on using modelling to investigate the effects of interfaces on mechanical properties such as strength and fracture toughness of clay nanocomposites is new and needs to be established. Hence the motivation for this paper. In this paper, a three-dimensional finite element representative volume element (RVE) modelling approach to consider the properties of the nanoclay interfaces, the nanoclay and the matrix is described to form a starting point to investigate bulk mechanical properties such as strength, fracture toughness and stress-strain behaviour of clay/epoxy nanocomposites. Specifically, clay splitting occurring at the gallery layer of the nano clay particles is modelled and its effect on these mechanical properties is investigated. The occurrence of clay splitting in clay nanocomposite has been observed by Wang et al. [7] and Kim et al. [8]. However, the theory on clay splitting and its ability to function as a toughening or strengthening mechanism still needs further understanding. A finite element modelling method is used in this work because its computation cost is cheaper than multiscale or molecular modelling methods, a full three-dimensional continuum RVE model to investigate the effects of interfaces on the mechanical properties of nanocomposites is new; and the use of continuum finite element method at the nanoscale with all its advantages and short-comings needs understanding.

2 RVE Modelling Clay/Epoxy Nanocomposites The RVE model of a clay/epoxy nanocomposite consists of a dominant epoxy matrix region that is embedded with many clay particle regions. The clay particles are assumed here to be circular-disc shaped to facilitate generation of the RVE model. The clay particles are homogeneously dispersed in the matrix, randomly orientated, non-intersecting, perfectly bonded with the matrix, and is periodic. The random orientation of the clay particles is achieved using a random number generator. The non-intersecting clay particles are successfully generated through satisfying the intersection criteria of (a) two obliquely positioned clay particles and (b) two coplanar clay particles. Figures 1 and 2 show the position vectors and rec-

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Fig. 1 The position vectors and position system of two obliquely positioned clay particles.

Fig. 2 The position vectors and position system of two coplanar clay particles.

tangular coordinate system associated to these intersection criteria, respectively. The criteria describing the intersection of two obliquely positioned clay particles are |S¯1 | ≤ dcrit

and |S¯2 | ≤ dcrit

(1)

and |c¯ld | ≤ ycrit.

(2)

In Equation (1), the symbols |S¯1 | and |S¯2 | denote the shortest distance between the centroid of the two clay particles to the intersection line L formed by the plane of the particles, and the symbol dcrit denotes the minimum distance required by both particles to intersect each other when the centroid of both particles is viewed from the plane normal to the intersection line L. In Equation (2), the symbol |c¯ld |

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Fig. 3 Scheme to model an intercalated clay particle: (a) the idealised clay particle with diameter d, silicate thickness ds and gallery thickness dg , (b) FE representation of the idealised clay particle in which only one gallery layer is physically modelled and the rest of the galleries and silicate layers are idealised into two effective clay layers, and (c) the cohesive traction-separation laws modelling the gallery layer.

denotes the separation distance of the end points of the position vectors S¯1 and S¯2 on the intersection line L and symbol ycrit denotes a critical distance. The derivation and explicit definition of both symbols dcrit and ycrit are not included here due to space limitation and is documented in [9]. The criteria describing the intersection of two coplanar clay particles are |ux
| ≤ d,

|uz
| ≤ d

and |uy
| ≤ t.

(3)

Here, the symbols d and t denote the diameter and thickness of the clay particles and the symbols |ux
|, |uy
|, |uz
| denote the components of the separation vector u¯ defined in the local (x
, y
, z
) coordinate system of the coplanar particles. The clay particles are represented in the RVE model following the discrete scheme shown in Figure 3. For a clay particle with N number of silicate layers, we simplify the structure of a clay particle to consist of only two effective clay layers and a cohesive gallery layer. The cohesive gallery layer allows splitting to occur within the clay particle. The effective clay layer is treated as isotropic elastic and its Young’s modulus Eec is approximated as Eec =

Eg (N − 1)dg Es Nds + . Nds + (N − 1)dg Nds + (N − 1)dg

(4)

Here, Es and Eg denote the Young’s modulus of the silicate and gallery, respectively. For Equation (1) to be valid, the Poisson’s ratio of the gallery is assumed identical to the silicate. Eg is assumed identical to the Young’s modulus of the epoxy Em due to intercalation of epoxy molecules in the gallery. The cohesive gallery behaviour and failure is described by an uncoupled traction-separation law where the scalar cohesive energy of the gallery
0 is defined as:

Finite Element Modelling Clay Nanocomposites


0 =

245

1 σˆ δ 2

(5)

Here, the symbol σˆ denotes the damage initiation stress in the gallery, and the symbol δ denotes the resultant separation distance of the gallery. Full damage of the gallery (or clay splitting) occurs when
0 of the gallery is satisfied. The epoxy matrix is treated as isotropic elastic-plastic and its uniaxial true stress– true strain response during tensile loading is ε= and is

σ Em

when σ < σy


n σ σ + εy α ε= Em σy

when σ ≥ σy .

(6)

(7)

Here, the symbols ε and σ denote strain and stress in the matrix, εy and σy denote the matrix strain and stress at yield, n is a shape parameter and Em is the Young’s modulus of the matrix. Equation (6) also describes the unloading or reloading behaviour of the matrix up to its last known yield point. The three-dimensional expression of this isotropic elasto-plastic stress-strain relations is described in [10]. The RVE model is subjected to a series of three-dimensionally applied displacement boundary conditions, ranging from hydrostatic tension state to pure shear state, to investigate the effect of clay splitting on the stress-strain response and strength of clay/epoxy nanocomposites. The RVE models is generated using in-house FORTRAN codes and implemented and solved using the commercial finite element package ABAQUS [10].

3 Results & Discussions The predicted tensile hydrostatic stress-strain response of the RVE model of a clay/epoxy nanocomposite is shown in Figure 4. Initially, its mean stress is linearly proportional to its volumetric strain. The proportionality constant relating the mean stress and the volumetric strain is the bulk modulus of the nanocomposite. Localised plastic zones are observed to develop in the matrix around the clay particles even though the average stress and strain state of the RVE model is hydrostatic. This occurrence is caused by the large mismatch in modulus between the clay particles and the epoxy matrix. As the applied strain increases, clay splitting is observed to initiate upon reaching a critical stress and the subsequent stress-strain response of the nanocomposite becomes non-linear. The complete splitting of the first damaged clay particle is observed to occur at a very late stage in the nanocomposite stressstrain response and using a completely split (damaged) clay particle to indicate the onset of damage in the clay/epoxy nanocomposite is shown to be inappropriate. Instead, partial splitting or damage initiating in the clay particles is shown to provide a better indication of the damaged state of the clay nanocomposite. Based on this

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Fig. 4 The predicted hydrostatic stress-strain curve of a clay/epoxy nanocomposite; the corresponding damage state in a single clay particle compared with the damage state of the entire nanocomposite; the contour plots of the growing plastic zones in the epoxy matrix and damage evolving in the clay particles.

observation, the damage state of the nanocomposite, D, defined as the ratio of the number of damaged finite element over the total number of finite elements forming the gallery layer in the clay particles, is shown to have a sigmoidal relation with the volumetric strain of the nanocomposite. Also, the sigmodal growth of D is shown to have caused the non-linear stress-strain behaviour of the nanocomposite. Figure 5 shows the damage stress state of a clay/epoxy nanocomposite RVE model ranging from pure hydrostatic tension state to pure shear state. The clay volume fraction of the unit cell is 1%, the cohesive energy of the gallery layer is varied from 0.1
0 to 2.0
0 where
0 has a value of 0.41 J/m2 , and the RVE model

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Fig. 5 The damage stress surface of an epoxy/clay nanocomposite with clay particles having different gallery cohesive energies.

is allowed to undergo total damages of not more than 0.1D. The predicted damage stress surfaces of the nanocomposite are observed to be dependent on
0 , pressuredependent, i.e. σq is a function of σ¯ , and non-smooth. Between the hydrostatic tension state and biaxial tension state, the damage stress surface is described by an elliptical stress criterion
2
 σq 2 σ¯ + = 1. (8) σ¯ 0 σq0 Here, the symbols σ¯ 0 and σq0 represent the damage stress at the hydrostatic state and pure shear state, and the symbols σ¯ and σq represent the mean stress and equivalent stress of the RVE model. Between the biaxial tension state and pure shear state, the damage stress surface is described by a second-order polynomial stress criterion σq = σq0 + a σ¯ + bσ¯ 2 ,

(9)

where the symbols a and b are constants. The transition in the damage stress surfaces is also observed to occur when the elliptical stress surface described by Equation (8) and the second-order polynomial stress surface described by Equation (9) intersects with the pressure-independent yield surface of the matrix. This observation indicates that occurrence of plasticity in the matrix of the nanocomposite caused the damage stress surface of the nanocomposite to be non-smooth.

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4 Conclusion A finite element RVE modelling method of investigating the effect of interfaces on mechanical properties such as stress-strain behaviour and strength of clay/epoxy nanocomposites has been demonstrated. The understanding gained shows continuum modelling method can serve as a reasonable start point to elucidate the complex mechanics of clay nanocomposites. Acknowledgements Discussions with Dr Kais Hbaieb and Dr Brian Cotterell on the development of finite element RVE models of clay nanocomposite are gratefully acknowledged.

References 1. Fornes TD and Paul DR: Modeling properties of nylon 6/clay nanocomposites using composite theories. Polymer 44, 4993–5013 (2004). 2. Wang J and Pyrz R: Prediction of the overall moduli of layered silicate-reinforced nanocomposites – Part I: Basic theory and formulas. Compos Sci Technol 64, 925–934 (2004). 3. Sheng N, Boyce MC, Parks DM, Rutledge GC, Abes JI, Cohen RE: Multiscale micromechanical modeling of polymer/clay nanocomposites and the effective clay particle. Polymer 45, 487–506 (2004). 4. Hbaieb K, Wang QX, Chia YHJ, Cotterell B: Modelling stiffness of polymer/clay nanocomposites. Polymer 48, 901–909 (2007). 5. Chia JYH, Hbaieb K, Wang QX: Finite element modelling epoxy/clay nanocomposites. Key Eng Mater 334–335, 785–788 (2007). 6. Zeng QH, Yu AB, Lu GQ: Multiscale modeling and simulation of polymer nanocomposites. Prog Polym Sci 33, 191–269 (2008). 7. Wang K, Chen L, Wu JS, Toh ML, He CB, Yee AF: Epoxy nanocomposites with highly exfoliated clay: Mechanical properties and fracture mechanisms. Macromol 38, 788–800 (2005). 8. Kim GM, Lee DH, Hoffmann B, Kresslerd J, Stöppelmann G: Influence of nanofillers on the deformation process in layered silicate/polyamide-12 nanocomposites. Polymer 42, 1095– 1100 (2001). 9. Chia JYH: Mathematical basis for developing 3-dimension unit cell models of nanocomposites containing randomly dispersed-orientated circular-disc-shaped particles. Institute of Materials Research and Engineering, A-STAR, WBS Code: IMRE/07-1R0223 Internal Report (Nov 2007). 10. ABAQUS Version 6.7, SIMULIA, Dassault Systems.

Small Scale and/or High Resolution Elasticity I. Goldhirsch and C. Goldenberg

Abstract A general exact formulation of elasticity as a macroscopic theory with resolution that can be chosen, is presented. The theory is fully compatible with the classical theory of elasticity for coarse resolutions but exhibits differences at fine resolutions. For instance, a correction term to the classical expression for the elastic energy that stems from the work of the fluctuating displacements (in disordered systems) is obtained. Some applications and open problems are discussed.

1 Introduction The classical theory of elasticity is an important and successful tool for the description of the mechanics of solids, with applications ranging from micron-sized systems to buildings. An interesting application is soil mechanics and granular materials in general (which possess an elastic phase). In recent years there have been two major developments which required the revisiting of the foundations of elasticity. The first is the development of nanoscale materials for whose mechanics one needs continuum descriptions on rather small scales. The second is the increased interest in granular matter. The common feature of these disparate fields is the lack of scale separation between the relevant macroscopic and microscopic (grains’ or atomic distances) scales. Elasticity is a coarse-grained (or averaged) theory that is formally valid on (spatial and temporal) scales that are sufficiently large with respect to the microscopic scales. This fact notwithstanding, classical elasticity and its extensions are Isaac Goldhirsch School of Mechanical Engineering, Faculty of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel; e-mail: [email protected] Chay Goldenberg Laboratoire de Physique et Mécanique des Milieux Hétérogènes (CNRS UMR 7636), ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France; e-mail: [email protected]

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commonly used for the description of nanoscale systems (for recent work, see e.g., [4, 5, 12, 19, 22, 25]). In some cases, e.g. in the presence of defects [4], the elastic approach is less successful. The effective elastic properties of small objects can also exhibit a dependence on the size of the object, which may be explained by the increased importance of surface effects in these systems [18, 20, 21]. Motivated by the above mentioned developments we set out to study the limitations of elasticity and suggest possible extensions when small scales are concerned. We found it rather surprising that derivations of elasticity (unlike, for example, hydrodynamics) on the basis of microscopic interactions could be found only for (near) homogeneously strained lattice configurations [6], i.e. no basic justification for the elasticity of disordered solids or granular matter is known. The work presented below aims to fill this gap and presents some unexpected findings concerning small scale elasticity.

2 Continuum Mechanics: Brief Review and Notation Setting The equations of continuum mechanics (see e.g., [23]) are based on the conservation of mass, momentum and energy. These are commonly expressed as differential equations in the spatial coordinate, r, and time, t, for the (mass) density field, ρ(r, t), the momentum density field, p(r, t), and the energy density field, e(r, t). These equations (in the absence of body forces and sources or sinks of mass and energy) read: ρ˙ = −div(ρV), ∂ p˙ α = − [ρVα Vβ − σαβ ], ∂rβ ∂ e˙ = − [Vβ e − Vα σαβ + Qβ ], ∂rβ

(1) (2) (3)

where Greek indices denote Cartesian coordinates (the Einstein summation convention is used), and the explicit dependence of the fields on r and t has been omitted. Here σ is the stress tensor, and Q denotes the heat flux. As Eqs. (1–3) are obtained from fundamental conservation laws, their validity is very general. The standard continuum description of solids requires the consideration of the notion of deformation (with respect to a reference state), expressed in terms of the displacement and strain fields [23]. Let the Lagrangian coordinate, R, denote the position of a material particle at time t = 0, and let r(R, t) denote its position at time t. The displacement field is defined by u(R, t) ≡ r(R, t) − R. The velocity field (by definition) is related to the displacement by V(R, t) ≡

∂u(R, t) . ∂t

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The Eulerian description employs r and t, instead of R and t as the independent variables. Since material behavior is assumed to be invariant to uniform translations and rotations constitutive descriptions of solids are typically given in terms of the strain tensor, which is defined in terms of gradients of the displacement field. For small deformations, it is sufficient to use the linear strain tensor. Note that (∂uα )/(∂Rβ ) = (∂uα /∂rβ ) to linear order in the gradients. The description of large deformations (which will not be considered here) typically requires nonlinear measures for the strain. In the framework of linear elasticity, the stress field is linearly and locally dependent on the strain, hence the constitutive relation [15] σαβ = Cαβγ δ γ δ , where Cαβγ δ is the tensor of elastic constants. When thermal effects are neglected (i.e., the internal energy is purely mechanical), the energy density is given by e=

1 1 1 Cαβγ δ αβ γ δ = σαβ αβ = σ : . 2 2 2

(4)

3 Coarse Graining In this section, we derive the equations of continuum mechanics from microscopic considerations [2, 13, 26]. Consider a system of particles, indexed by Roman subscripts {i}) of masses {mi }, the corresponding center of mass positions and velocities at time t, being: {ri (t)} and {vi (t)}, respectively, where vi (t) ≡ r˙ i (t). Below only spatial coarse graining is invoked, see [10] for a spatio-temporal coarse-graining formulation. Note that although the present article focuses on quasi-static behavior, the derivation of the continuum mechanical equations presented here is general. Particle rotations and torques are not treated here, although the formalism described below can be extended to include them in a straightforward manner. Such an extension may be required for granular materials due to the presence of friction. The microscopic mass density at a point r at time t, ρ mic (r, t), is defined by
mic ρ (r, t) ≡ i mi δ[r − ri (t)], where δ(r) is the delta function. This definition complies with the requirement that the integral of the mass density over a volume equals the mass contained in this volume. Below we omit the explicit dependence on time in some of the expressions. The coarse grained mass density is defined as a convolution of the microscopic density with a coarse graining scalar, non-negative function φ(R) whose spatial integral is unity (normalization), which possesses a single maximum at R = 0, and has a well-defined width w (one useful choice is a Gaussian):   ρ(r, t) = drφ(r − r )ρ mic (r , t) = mi φ[r − ri (t)]. (5) i

The mass conservation (continuity) equation (Eq. 1) can be derived by taking the time derivative of the coarse grained mass density:

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 ∂  ∂ρ(r, t) ∂φ(r − ri ) = mi φ(r − ri ) = − mi r˙iβ ∂t ∂t ∂rβ i

i

∂pβ (r, t) ∂  =− mi viβ φ(r − ri ) = − , ∂rβ ∂rβ

(6)

i


where p(r, t) ≡ i mi viβ φ(r − ri ) is the coarse grained momentum density. The velocity field is defined by V(r, t) ≡ p(r, t)/ρ(r, t). Notice that the velocity is meaningful only as a coarse grained field, as it is not a density of a physical entity. Substituting the definition of V in Eq. (6), one obtains the equation of continuity, Eq. (1). The momentum equation, Eq. (2), is derived as follows: ∂pα (r, t) ∂  = mi viα φ(r − ri ) ∂t ∂t i

=



mi v˙iα φ(r − ri ) +

i

=

 i

=

 ij

=



mi viα

i

∂  fiα φ(r − ri ) − mi viα viβ φ(r − ri ) ∂rβ i

∂  fij α φ(r − ri ) − mi viα viβ φ(r − ri ) ∂rβ i

1 ∂  fij α [φ(r − ri ) − φ(r − rj )] − mi viα viβ φ(r − ri ) 2 ∂rβ ij

=−

∂φ(r − ri ) ∂t

1 ∂  fij α rijβ 2 ∂rβ ij

i



1

ds φ(r − ri + srij ) −

0

∂  mi viα viβ φ(r − ri ), (7) ∂rβ i

where we denoted mi v˙iα = fiα , fi being the force on particle i, and used fij = −fj i , where fij is the force exerted on particle i by particle j , and the following identity:  φ(r − ri ) − φ(r − rj ) =

1

rijβ ds 0

∂ φ(r − ri + srij ). ∂rβ

The latter corresponds to a specific choice of the integration path from ri to rj (a straight line); in [24] it is shown that symmetry requirements render this choice unique. Define the fluctuating velocity field of particle i with reference to the coarse  (r, t) ≡ v (t) − V (r, t). Substituting v + V for v in graining point, r, and time, t: v
i i i i Eq. (7) and using the identity i mi vi (r, t)φ(r − ri ) ≡ 0 one obtains the equation for the momentum field, Eq. (2), where the stress field is identified as σαβ (r,t) = −

1 fij α (t)rijβ (t) 2 i,j



1 0

ds φ(r − ri + srij )

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−



  mi viα (r, t)viβ (r, t)φ(r − ri ).

(8)

i

The first term on the right hand side of Eq. (8) is known as the “contact stress” while the second term is the “kinetic stress” (which vanishes for quasi-static deformations). For large coarse graining scales, w, compared to the separation of interacting particles the above expression reduces to the well-known Born–Huang formula. The equation for the energy density can be obtained in a similar way. Assume, for simplicity, that the forces are derived from a potential (the derivation can be extended to non-conservative forces), (rij ). The energy density is given by e(r, t) ≡

1 1  mi vi2 φ(r − ri ) +

(rij )φ(r − ri ). 2 2

(9)

i,j ;i =j

i

By taking the time derivative of the above expression and rearranging terms, much like in the derivation of equation for the momentum density, one obtains the continuum mechanical equation for the energy density. The heat flux is identified as the following expression in terms of the microscopic entities: Qβ (r, t) =

1  2 viβ {mi vi + (rij )}φ(r − ri ) 2 i  1 1   fij α rijβ (viα + vj α ) ds φ(r − ri + srij ). + 4 0

(10)

ij

This completes the derivation “from microscopics” of the equations of continuum mechanics, which also provides explicit microscopic expressions for the stress tensor, σ (Eq. 8) and the heat flux, Q (Eq. 10). Note that the resulting expressions are exact. They are useful for analyzing the results of particle-based simulations, experimental results (which provide details) and derivations of constitutive relations (an example of which is given in Section 3.2).

3.1 Displacement and Strain The simplest, and most commonly used, approach for relating the particle displacements and macroscopic strain [3, 8, 9] is to adopt the kinematic assumption of affinity, also referred to as the mean field or Voigt assumption, according to which the relative displacement of each pair of particles is determined by the macroscopic 0 , where u = u − u is the relative displacement strain tensor, : uij α = αβ rijβ ij i j between particles i and j and r0ij = r0i −r0j . While this assumption is exact for a uniformly strained lattice configuration, it cannot be correct for a disordered system [1], such as a structural glass or a typical granular solid, since the relative particle displacements obtained in this manner give interparticle forces which are not generally

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compatible with mechanical equilibrium. Improvements over the Voigt assumption were suggested e.g., in [14, 16]. Beyond the inapplicability to disordered systems, the above mentioned approaches contradict the basic notion that the description in terms macroscopic fields, typically obtained from an averaging procedure, involves the loss of information, so that the microscopic fields cannot be recovered. Furthermore they are inconsistent with continuum mechanics due to the fact in these theories the the time derivative of the strain does not equal (to linear order), the rate of strain tensor. In addition, the neglect of the fluctuations in the particle displacements in mean field theories comprises an uncontrolled approximation (which leads to inaccurate estimates for the constitutive coefficients, see e.g., [17]). To overcome the shortcomings of the approaches described above, we propose a microscopic definition [13] for the displacement (hence, strain) fields. Following Section 2, the material particle’svelocity is defined as V(R, t) = ∂u (R, t)/∂t. It therefore follows that u(R, t) = 0t V(R, t  )dt  , where the integration is performed along the trajectory of the material particle at R (notice that Lagrangian coordinates are used here). Using now the microscopic expression for p one obtains:  t
   i mi vi (t )φ[r(R, t ) − ri (t )] 
u(R, t) ≡ dt . (11)   0 j mj φ[r(R, t ) − rj (t )] Invoking integration by parts in Eq. (11), this expression can be separated into a trajectory-independent part and a “correction”:
i mi uiα (t)φ[r − ri (t)] uα (r, t) =
j mj φ[r − rj (t)]  t ∂ φ[r(R, t  ) − ri (t  )] dt  mi uiα (t  ) 
−  ) − r (t  )] ∂t m φ[r(R, t j 0 j j i  t  1 ∂  = ulin mi viβ (r, t  )uiα (r, t  )φ[r − ri (t  )]dt  , (12) α + ρ ∂r β 0 i

where we have reverted to an Eulerian representation. The expression
i mi ui (t)φ[r − ri (t)] ulin (r, t) ≡
j mj φ[r − rj (t)]

(13)

involves only the overall displacements of the particles between the initial and final times, ui (t), and is therefore independent of the detailed particle trajectories. The fluctuating displacements are denoted by ui (r, t) ≡ ui (t) − ulin(r, t); note that ∂ui /∂t = vi , since ui is defined with respect to ulin and not the exact displacement, u. The correction ulin − u can be easily shown to be of second order in the strain (therefore ulin is sufficient for the description of linear elasticity). Following the above arguments and straightforward algebra, the linear strain tensor is given by

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    1 ∂ ∂ lin 0 0 φ[r − ri ] , αβ (r, t) = mi mj φ[r−rj ] uij α (t) + uijβ (t) 2ρ 2 (r, t) ∂rβ ∂rα ij

where use has been made of the fact that to linear order in uij , one can replace ri in the argument of the coarse graining function, φ, by its initial value, r0i .

3.2 Microscopic Derivation of Linear Elasticity For sake of simplicity, let the particle interactions correspond to linear springs: fij = −Kij (|rij | − lij )ˆrij . For an unstressed reference configuration, in which all particle pairs are at their equilibrium separation (|r0ij | = lij , where the superscript 0 denotes the reference configuration), the linearized force is given by fij = −Kij (ˆr0ij ·uij )ˆr0ij . As discussed in Section 3.1, the assumption of affinity cannot be exact in disordered systems, and typically leads to incorrect predictions of the elastic moduli [7, 17]. Other methods are therefore required in this case. Note that linearity of the interactions alone does not guarantee the local constitutive relation that defines elasticity. Consider the contact stress for harmonically interacting particles, to linear order in the relative particle displacements {uij }: lin σαβ (r, t) =

1 0 0 Kij rˆij0 α rijβ rˆij γ uij γ 2 ij



1 0

ds φ[r − r0i + sr0ij ].

(14)

The stress field is a specific average over the interparticle displacements, involving the stiffnesses {Kij } as well as the geometry specified by the contact vectors rˆ 0ij α . The strain comprises a different average over the interparticle displacements. They are not manifestly proportional, and it is not a-priori clear that the stress field can be expressed as a linear and local functional of the strain field. In order to obtain such a relation, consider a (say, spherical) volume, , whose linear dimension, W , is much larger than the coarse graining scale, w, and let r be an interior point of  which is ‘far’ from its boundary. Let upper case indices denote the particles in the exterior of  which interact with particles inside . Since the system is linear, there exists a (discrete) Green’s function G such that uiα = GiαJβ uJβ for i ∈  (Einstein summation implied). Define Lij αJβ ≡ GiαJβ − Gj αJβ . It follows that uij α = Lij αJβ uJβ . Under rigid translation (all {uJ } equal), uij = 0, hence one can write uij α = Lij αJβ [uJβ − uβ (r)], where u(r) is the macroscopic displacement field at the point r. Upon adding and subtracting uβ (rJ ), one obtains: uij α = Lij αJβ [uβ (rJ ) − uβ (r)] + Lij αJβ [uJβ − uβ (rJ )].

(15)

The term [uβ (rJ ) − uβ (r)] is the difference in the macroscopic displacement fields over a distance of order W  w (which scales like  W where  is the typical magnitude of the strain). The term [uJβ − uβ (rJ )] is a fluctuating displacement,

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being the difference between the macroscopic displacement at rJ , and the actual displacement of particle J . This difference scales as the individual interparticle particle displacement uI J (I being any boundary particle close to J ), a scale we denote by u . Clearly, u scales as  d where d is the mean nearby interparticle separation). Since the fluctuating displacement [uJβ − uβ (rJ )] is a (nearly) random vector field (for disordered systems) its contribution to uij in Eq. (15) should scale like the square root of the number of terms in the sum (which is the number of particles on the surface of , proportional to (W/d)D−1 in D dimensions), while the terms [uJβ − uβ (rJ )] are highly correlated (assuming a sufficiently smooth u(r), whose gradient depends on the coarse graining scale w), so that the contribution of the first sum should be proportional to the number of terms, and not its square root. Therefore the ratio of the second term of the right hand side of Eq. (15) to the first scale like   1−D   D+1 (W/d)D−1 2 2  d d W d = = ,  W (W/d)D−1 W d W so that the second term is subdominant when W is sufficiently large. The first term equals, to leading order in the gradients: uβ (rJ ) − uβ (r) 

∂uβ (r) (rJ γ − rγ ). ∂rγ

(16)

Substituting Eq. (16) in Eq. (14), we obtain: ⎧ ⎫  1 ⎨1  ⎬ ∂u (r) µ 0 0 σαβ (r)  Kij Lij γ J µ (rJ0 ν − rν )ˆrij0 α rijβ rˆij γ dsφ[r − r0 + sr0ij ] ⎩2 ⎭ ∂r ν 0 ij

invoking the symmetry of the Green’s function, one can replace ∂uµ (r)/∂rν by µν (r), yielding ⎧ ⎫  1 ⎨1  ⎬ 0 0 Kij Lij γ J µ (rJ0 ν − rν )ˆrij0 α rijβ rˆij γ dsφ[r − r0 + sr0ij ] µν (r) σαβ (r)  ⎩2 ⎭ 0 ij

(17) i.e., the elastic tensor is given by Cαβµν =

1 0 0 Kij Lij γ J µ (rJ0 ν − rν )ˆrij0 α rijβ rˆij γ 2 ij



1 0

ds φ[r − r0 + sr0ij ].

(18)

The elastic moduli depend, in principle, both on position and resolution (through the coarse graining function φ). The expression, Eq. (18), is not convenient since it depends on the microscopic Green’s function, but suffices to demonstrate that a local linear relation exists between the stress and the strain, subject to the following restrictions: (1) the strain components need to be small (  1), otherwise the dependence of the stress on the interparticle displacement is nonlinear (invalidating

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the harmonic approximation); (2) the scale of the description is sufficiently large (W  d), else the displacement fluctuations may not be negligible; and (3) the strain gradients on the scale W should be small (|W ∇α ∇β u|  1).

3.2.1 Elastic Energy In the quasistatic limit, the energy density reduces to the potential energy density. To lowest nonvanishing order in the strain one therefore obtains: eel (r, t) =

1 Kij (ˆr0ij · uij )2 φ[r − ri (t)]. 4

(19)

i,j

One way to calculate the above expression is to first express the standard expression for the energy density in terms of the microscopic entities and try to relate it to Eq. (19). It is convenient to calculate 1 ∂ (σ lin ulin) 2 ∂rβ αβ α instead (use has been made of the symmetry of the stress tensor and the equilibrium condition). Due to space limitations only the result of the calculation is presented here:  1 1 1 ∂  0 eel (r, t) = σ lin :  lin − fij α uiα rijβ ds φ[r − r0i + sr0ij ]. (20) 2 4 ∂rβ 0 ij

Therefore, the coarse grained elastic energy density is not given by (σ lin /2) :  lin as assumed in continuum theories; the second term on the right hand side of Eq. (20) provides a correction to the classical expression. This correction represents the divergence of the fluctuating part of the work of the interparticle forces (i.e., the work done on the fluctuating part of the displacement). As this term is a divergence of a flux, its average over a volume , whose linear dimension is W , is proportional to W D−1 , while the average over the first term is proportional to W D . Therefore their ratio tends to zero as 1/W , i.e., the average of (σ lin/2) :  lin over a sufficiently large volume (not its ‘local value’) is the elastic energy density. It should be emphasized that both terms in Eq. (20) are O( )2 , so that the correction does not represent higher orders in σ or in , but rather the fact that for a non-affine deformation, part of the microscopic energy is created by work that involves the fluctuating displacements. Classical elasticity is of course obtained in the limit of coarse resolutions.

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4 Conclusion The number of potential applications of the above formulation is enormous. We have applied it to the study of granular systems and obtained results for the response of such systems to external forces as well as an elucidation of the nature of force transmission in these systems, which has been a matter of some controversy [11]. We are currently looking at experimental data on granular systems near the jamming transition. The correction term to the classical expression for the elastic energy raises the possibility (considered in the realm of granular systems so far) that additional characterizing fields are needed in order to properly close the constitutive relations for random systems, much like in Grad’s approach to kinetic theory. The suggestions of a static granular temperature comprise attempts in this direction. This extra term has also been shown to be strongly correlated with incipient failure and plasticity. Similar and more questions pertain of course to nanoscale systems, where quantum effects may play an important role as well. Acknowledgements One of us (IG) gratefully acknowledges partial support from the Israel Science Foundation (ISF), grant no. 689/04, the German-Israeli Science Foundation (GIF), grant no. 795/2003, and the US-Israel Binational Science Foundation (BSF), grant no. 2004391.

References 1. Alexander S (1998) Amorphous solids: Their structure, lattice dynamics and elasticity. Phys Reports 296:65–236. 2. Babic M (1997) Average balance equations for granular materials. Int J Eng Sci 35:523–548. 3. Bathurst RJ, Rothenburg L (1988) Micromechanical aspects of isotropic granular assemblies with linear contact interactions. J Appl Mech 55:17–23. 4. Berber S, Tománek D (2004) Stability differences and conversion mechanism between nanotubes and scrolls. Phys Rev B 69:233,404. 5. Bhushan B, Agrawal G (2002) Mechanical property measurements of nanoscale structures using an atomic force microscope. Nanotech 13:515–523. 6. Born M, Huang K (1988) Dynamical Theory of Crystal Lattices. Clarendon Press, Oxford. 7. Chang CS, Liao CL (1994) Estimates of elastic modulus for media of randomly packed granules. Appl Mech Rev 47:197–206. 8. Chang CS, Misra A (1990) Application of uniform strain theory to heterogeneous granular solids. J Eng Mech 116:2310–2328. 9. Digby PJ (1981) The effective elastic moduli of porous granular rocks. J Appl Mech 48:803– 808. 10. Glasser BJ, Goldhirsch I (2001) Scale dependence, correlations, and fluctuations of stresses in rapid granular flows. Phys Fluids 13:407–420. 11. Goldenberg C, Goldhirsch I (2005) Friction enhances elasticity in granular solids. Nature 435:188–191. 12. Goldfarb I, Banks-Sills L, Eliasi R, Briggs G (2002) Finite element analysis of CoSi2 nanocrystals on Si(001). Interface Science 10:75–81. 13. Goldhirsch I, Goldenberg C (2002) On the microscopic foundations of elasticity. Eur Phys J E 9:245–251. 14. Kruyt NP, Rothenburg L (1996) Micromechanical definition of the strain tensor for granular materials. J Appl Mech 118:706–711.

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15. Landau L, Lifshitz E (1986) Theory of Elasticity, 3rd Edition. Pergamon, Oxford. 16. Liao CL, Chang TP, Young DH, Chang CS (1997) Stress-strain relationship for granular materials based on the hypothesis of best fit. Int J Solids Struct 34:4087–4100. 17. Makse HA, Gland N, Johnson DL, Schwartz LM (1999) Why effective medium theory fails in granular materials. Phys Rev Lett 83:5070–5073. 18. Miller RE, Shenoy VB (2000) Size-dependent elastic properties of nanosized structural elements. Nanotech 11:139–147. 19. Qian D, Wagner GJ, Liu WK, Yu MF, Ruoff RS (2002) Mechanics of carbon nanotubes. Appl Mech Rev 55:495–533. 20. Roukes M (2001) Plenty of room indeed. Sci Am 285:48–57. 21. Sharma P, Ganti S, Bhate N (2000) Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl Phys Lett 82:535–537. 22. Srivastava D, Wei C, Cho K (2003) Nanomechanics of carbon nanotubes and composites. Appl Mech Rev 56:215–230. 23. Truesdell C (1965) The classical field theories. In: Flügge S (ed) Encyclopedia of Physics, vol III/1, Springer-Verlag, Berlin, p 1. 24. Wajnryb E, Altenberger AR, Dahler JS (1995) Uniqueness of the microscopic stress tensor. J Chem Phys 103:9782–9787. 25. Xiao T, Liao K (2003) Non-linear elastic response of fullerene balls under uniform and axial deformations. Nanotech 14:1197–1202. 26. Zhu HP, Yu AB (2002) Averaging method of granular materials. Phys Rev E 66:021,302.

Multiscale Molecular Modelling of Dispersion of Nanoparticles in Polymer Systems of Industrial Interest Maurizio Fermeglia and Sabrina Pricl

Abstract Atomistic-based simulations such as molecular mechanics (MM), molecular dynamics (MD), and Monte Carlo-based methods (MC) have come into wide use for material design. Using these atomistic simulation tools, we can analyze molecular structure on the scale of 0.1–10 nm. However, difficulty arises concerning limitations of the time and length scale involved in the simulation, particularly when nanoparticles are involved in the system. Although a possible molecular structure can be simulated by the atom-based simulations, it is less realistic to predict the mesoscopic structure with nanoparticles defined on the scale of 100–1000 nm. For the morphology on these scales, mesoscopic simulations are available as alternatives to atomistic simulations allowing to bridge the gap between atomistic and macroscopic simulations for an effective material design. In this contribution, a hierarchical procedure for bridging the gap between atomistic and macroscopic (FEM) modeling passing through mesoscopic simulations will be presented and applications of systems with nanoparticles will be discussed.

1 Introduction The main goal of computational materials science is the rapid and accurate prediction of properties of new materials before their development and production. In order to develop new materials and compositions with designed new properties, it is essential that these properties can be predicted before preparation, processing, and characterization. Polymers are complex macromolecules whose structure varies from the atomistic level of the individual backbone bond of a single chain to the scale of the radius of gyration, which can reach tens of nanometres. Polymeric structures in melts, blends Maurizio Fermeglia and Sabrina Pricl Department of Chemical, Environmental and Raw Materials Engineering, University of Trieste, Piazzale Europa 1, 34127 Trieste, Italy; e-mail: [email protected]

R. Pyrz and J.C. Rauhe (eds.), IUTAM Symposium on Modelling Nanomaterials and Nanosystems, 261–270. © Springer Science+Business Media B.V. 2009

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and solutions can range from nanometre scales to microns, millimetres and larger. The corresponding time scales of the dynamic processes relevant for different materials properties span an even wider range, from femtoseconds to milliseconds, seconds or even hours in glassy materials, or for large scale ordering processes such as phase separation in blends. No single model or simulation algorithm currently available can encompass this range of length and time scales. In order to simulate a polymeric system one must consider models that range from those including quantum effects and electronic degrees of freedom to chemically realistic, classical models. One of the most important issue in computational materials research is the multiscale simulation, namely the bridging of length and time scales, and the linking of computational methods to predict macroscopic properties and behaviour from fundamental molecular processes [1, 2]. Despite the advances made in the modelling of the structural, thermal, mechanical, and transport properties of materials at the macroscopic level (finite element analysis of complicated structures), there remains tremendous uncertainty about how to predict many properties of industrial interest containing nanoparticles dispersed at nanoscale level. In this paper, we will show hierarchical procedures for bridging the gap between atomistic and macroscopic (FEM) modelling passing through mesoscopic simulations. In particular, we will present and apply to some cases of industrial interest the concept of ‘message passing multiscale modelling’. Examples considered will be (i) mesoscale simulation for diblock copolymers with dispersion of nanoparticles and (ii) polymer-carbon nanotubes system. The strategy described in this paper is based on an overlapping array of successively coarser modelling techniques. At each plateau (a range of length and time scales), the parameters of the coarse description are based on representative results of the immediately finer description, as it will be explained in the following paragraphs.

2 Multiscale Molecular Modeling Molecular modelling and simulation combines methods that cover a range of size scales in order to study material systems. These range from the sub-atomic scales of quantum mechanics (QM), to the atomistic level of molecular mechanics (MM), molecular dynamics (MD) and Monte Carlo (MC) methods, to the micrometer focus of mesoscale modelling. Quantum mechanical methods have undergone enormous advances in the past ten years, enabling simulation of systems containing several hundred atoms. Molecular mechanics is a faster and more approximate method for computing the structure and behaviour of molecules or materials. It is based on a series of assumptions that greatly simplify chemistry, e.g., atoms and the bonds that connect them behave like balls and springs. The approximations make the study of larger molecular systems feasible, or the study of smaller systems, still not possible with QM methods, very fast. Using MM force fields to describe molecular-level

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Fig. 1 Multiscale molecular modeling: characteristic times and distances.

interactions, MD and MC methods afford the prediction of thermodynamic and dynamic properties based on the principles of equilibrium and non-equilibrium statistical mechanics [3]. Mesoscale modelling uses a basic unit just above the molecular scale, and is particularly useful for studying the behaviour of polymers and soft materials. It can model even larger molecular systems, but with the commensurate trade-off in accuracy [4, 5]. Furthermore, it is possible to transfer the simulated mesoscopic structure to finite elements modelling tools for calculating macroscopic properties for the systems of interest [6]. Figure 1 shows the class of models that are available at each single scale. There are many levels at which modelling can be useful, ranging from the highly detailed ab-initio quantum mechanics, through classical molecular modelling to process engineering modelling. These computations significantly reduce wasted experiments, allow products and processes to be optimized, and permit large numbers of candidate materials to be screened prior to production. QM, MM and mesoscale techniques cover many decades of both length and time scale, and can be applied to arbitrary materials: solids, liquids, interfaces, self-assembling fluids, gas phase molecules and liquid crystals, to name but a few. There are a number of factors, however, which need to be taken care of to ensure that these methods can be applied routinely and successfully. First and foremost of course are the validity and usability of each method on its own, followed by their interoperability in a common and efficient user environment. Of equal importance is the integration of the simulation methods with experiment. Multiscale simulation can be defined as the enabling technology of science

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and engineering that links phenomena, models, and information between various scales of complex systems. The idea of multiscale modelling is straightforward: one computes information at a smaller (finer) scale and passes it to a model at a larger (coarser) scale by leaving out, i.e., coarse-graining, degrees of freedom. The ultimate goal of multiscale modelling is then to predict the macroscopic behaviour of an engineering process from first principles, i.e., starting from the quantum scale and passing information into molecular scales and eventually to process scales. Thus, based on accurate QM calculations, a force field (FF) is determined, which includes charges, force constants, polarization, van der Waals interactions and other quantities that accurately reproduce the QM calculations. With the FF, the dynamics is described with Newton’s equations (MD), instead of the Schrödinger Equation. The MD level allows predicting the structures and properties for systems much larger in terms of number of atoms than for QM, allowing direct simulations for the properties of many interesting systems. This leads to many relevant and useful results in materials design; however, many critical problems in this filed still require time and length scales far too large for practical MD. Hence, the need to model the system at the mesoscale (a scale between the atomistic and the macroscopic) and to pass messages from the atomistic scale to the mesoscale and to the macro scale. This linking through the mesoscale in which the microstructure can be described is probably the greatest challenge to developing reliable first principles methods for practical materials’ design applications. Only by establishing this connection from micro scale to mesoscale it is possible to build first principles methods for describing the properties of new materials and (nano) composites. The problem here is that the methods of coarsening the description from atomistic to mesoscale or mesoscale to continuum is not as obvious as it is in going from electrons to atoms [2]. For example, the strategy for polymers seems quite different than for metals, which seem different from ceramics or semiconductors. In other words, the coarsening form QM to MD relies on basic principles and can be easily generalized in a method and in a procedure, while the coarsening at higher scales is system specific. This paper focuses the attention to the prediction of macroscopic properties of polymer systems and polymer nano-structured materials with particular attention to interface phenomena. Finite element analysis software is used to this aim, but the physical properties input to such programs are consequence of the microstructure at few nanometre scale. Multiscale simulation poses great challenges for polymer materials than for metallic and ceramic systems due to the larger range of length and time scales that characterize macromolecules. Although each tool performs calculations using only one technique, the output from one level can be used directly as input for another, allowing an off-line bridging of length and time scales. To achieve what its referred to as ‘seamless zooming’, namely the ability to spawn higher resolution simulations using more detailed methods where needed, will require additional theoretical and computational advances. Along similar lines, offline multiscale simulations of nanofilled polymers using coarse-grained molecular dynamics, mesoscopic time dependent Ginsburg–Landau theory (TDGL), and macroscopic continuum finite element techniques have been carried out. Significant

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Fig. 2 Coarse grained modeling from atomistic model.

advances in uniquely mapping atomistic models of polymers onto coarse-grained models [7, 8], have been made in recent years, in some cases providing nearly exact quantitative agreement between the two models for certain quantities, but these mappings, too, are performed off-line, and the various methods are not linked within a single simulation. Scale integration in specific contexts in the field of polymer modelling can be done in different ways. Any ‘recipe’ for passing information from one scale to another (upper) scale is based on the definition of multiscale modeling which consider ‘objects’ that are relevant at that particular scale, disregard all degrees of freedom of smaller scales and summarize those degrees of freedom by some representative parameters. All approaches are initially based on the application of a force field that transfers information from quantum chemistry to atomistic simulation. From atomistic simulation to mesoscale one can use a traditional approach based on the estimation of the characteristic ratio, the Kuhn length, and the Flory–Huggins interaction parameter [8]. This approach for determining the input parameters for mesoscale simulation is based on the following information: (i) the bead size and Gaussian chain architecture, (ii) the bead mobility M, and (iii) the effective Flory– Huggins χ interaction parameters. With this approach, the Flory–Huggins χ parameters between two components of the coarse-grained molecular models in the mesoscopic simulation are estimated through the atomistic simulation, and a mesoscopic structure is predicted using these parameters. Mesoscopic simulations are performed using a coarse-grained molecular model as shown in Figure 2: the particle in mesoscopic simulation is related to a group of several atoms in the atomistic simulation. Mesodyn and DPD mesoscale theory and simulation protocols are fully described in the literature [4, 5]. The traditional approach can be enhanced and improved by considering the detailed structure at the interface polymer-nanofiller [7]. If one resorts to a particle based method for describing the system at mesoscale, atomistic MD simulation gives the necessary details of the interface with a particular attention to the bind-

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Fig. 3 From mesoscale morphology to FEM analysis.

ing energies among components. Mapping of the binding energies on mesoscale beads by means of a combinatorial approach to repulsive parameter for particles is then carried out and the system is simulated at mesoscale. If both particle based and field based methods are to be used at mesoscale, then an hybrid method can be adopted [9] in which particles are treated as described above and field interaction is calculated from pair-pair distribution function. Mesoscale simulation typical result is the morphology and the structure of the matter at nanoscale level at the desired conditions of temperature, composition and shear. For the representation of flow of polymeric materials on a processing scale, one must employ a hydrodynamic description and incorporate phenomena occurring on mesoscopic to macroscopic length and time scales. For example, to capture the nonNewtonian properties of polymer flow behaviour one can either use special models for the materials stress tensor, or obtain it from a molecular simulation using the instantaneous flow properties of the hydrodynamic fields as input. In the area of highperformance materials and devices, polymer composites are finding a widespread application, and the modelling of these materials was until recently done primarily through finite element methods (FEM), and are beyond the realm of application of molecular modelling approaches. Nonetheless, a real problem in using FEM is the definition of the physical property of a complex material such as a polymer blend with phase segregation and/or a polymer with micro inclusion of nanosized platelets [5, 6]. Mesoprop technique is a method based on finite elements for estimating properties of a complex material starting from the density distribution at mesoscale. The method uses the results of a mesoscale simulation under the form of three dimensional density maps, and transforms such information into a fixed grid that is used for the integration of the equations to determine macroscopic properties. Palmyra is a different method that allows the simulation at FEM level with a variable grid methodology that allows to extend the size of the system studied.

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Figure 3 shows how the mapping from mesoscale to macroscale is done. At FEM level each finite element corresponds to one phase, with property tensor Pi , at mesoscale (MesoDyn or DPD) each element contains mixture of phases, with concentration Ci . It is necessary to perform a geometry mapping by converting MesoDyn cubic elements to Palmyra tetrahedrons. Once this is done, Laplace equation is solved directly for obtaining direct properties such as electric conductance, diffusion, permeability. Local deformation allows the calculation of mechanical properties. Integration between these methods (from mesoscale to macroscale) is of paramount importance for the estimation of the properties of the materials, as it will be described in the following paragraphs with some illustrative examples of application.

3 Applications of Multiscale Modeling In the following paragraphs, some applications of the message passing multiscale modelling described in the previous sections will be presented and discussed.

3.1 Mesoscale Simulation for Diblock Copolymers with Dispersion of Nanoparticles Mixing microphase-separating diblock copolymers and nanoparticles can lead to the self-assembly of organic/inorganic hybrid materials that are spatially organized on the nanometer scale. Controlling particle location and patterns within the polymeric matrix domains remains, however, an unmet need. Computer simulation of such systems constitutes an interesting challenge since an appropriate technique would require the capturing of both the formation of the diblock mesophases and the copolymer-particle and particle-particle interactions, which can affect the ultimate structure of the material. In this example [10] we discuss the application of dissipative particle dynamics (DPD) to the study of the distribution of nanoparticles in lamellar and hexagonal A-B diblock copolymer matrices. The DPD parameters of the systems were calculated according to a multiscale modeling approach, i.e., from lower scale (atomistic) simulations. In agreement with some experimental evidence, we found that, depending on the nature and type of nanoparticle covering (e.g., only A- or B-type covering), the particles can segregate into the centers of the corresponding compatible domains (these being lamellae or cylinders), forming nanowire-like structures that extend throughout the material (Figure 4). In effect, the interplay between micro phase separation and favorable interactions do result in the self-assembly of spatially ordered nanocomposites. Should these particles be, for instance, metals or semiconductors, these systems could constitute a sort of nanoelectrode array, which could be utilized to fabricate organized nanodevices. On

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Fig. 4 Distribution of a (or B) covered nanoparticles in diblock copolymers: nanoparticles are located in the center of each domain; left: distribution of the nanoparticles with respect to the copolymer domains; right: position of the center of mass of the nanoparticles.

Fig. 5 Distribution of A and B equal coverage nanoparticles in diblock copolymers: nanoparticles are located at the interface of the domains; left: distribution of the nanoparticles with respect to the copolymer domains; right: 3D representation of the nanoparticles at the interface.

the other hand, for a different covering type (e.g., A6B6), the particles segregate at the interfaces (figure 5) instead of the centers of the lamellae (or cylinders). In these cases, for instance, if the copolymer matrix was to be dissolved from the system, the remaining inorganic phase could give origin to a non-porous material, with a regular arrangement of uniform pores, which could find applications, for instance, in separation or catalytic processes. The results also indicate that the morphologies of the organic/inorganic hybrid materials can be tailored by adding particles of specific size and chemistry. The findings highlight the fact that, in such complex mixtures, it is not simply the ordering of the copolymers that templates the spatial organization of the particles: the particles do not play a passive role and can affect the self-assembly of the polymeric chains. In fact, we detected a phase transition from the hexagonal to lamellar morphology induced by a non-selective (i.e., A6B6(h)) block-particle interaction, indication that the particles actively contribute to the determination of the system structure.

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In conclusion, the proposed multiscale computational approach, which combines atomistic and mesoscale simulations, can yield important information for the design of systems with desired morphology for novel applications.

3.2 Polymer-Carbon Nanotubes System Carbon Nanotubes (CNT) are interesting for several applications in different fields: structural, electromagnetic, chemical and mechanical. They have already been used as composite fibres in polymers and concrete to improve the mechanical, thermal and electrical properties of the bulk product. In this example we show one application of computer simulations involving CNT, namely the simulation at mesoscale level of the structure of a polymer CNT system. For simulating the mesoscale morphology of the system, information at atomistic level is necessary. This information is obtained in a ‘multiscale’ fashion. Firstly the same model of CNT developed for the atomistic simulation of the gas adsorption is used to determine the Flory– Huggins interaction parameters of the CNT for different diameters. This is obtained directly from the definition of cohesive energy by running two different simulation, one of the CNT in the bundle and another one with a single CNT isolated. The difference in the energy is the so-called ‘bundling-debundling energy’. For the polymer the traditional approach to calculate the Flory–Huggins interaction parameter in MD and MC is followed [11]. A comparison of the interaction parameters obtained shows that only CNT of a given diameter are soluble in a given polymer. The calculations enable to predict the CNT diameter to be used for obtaining a homogeneous mixture. The Flory–Huggins interaction calculated by molecular modelling are then used at mesoscale level with a particle based method (DPD) for determining the mesoscopic structure of a CNT– polymer system. Recently these approaches has been used [12] for simulating the morphology of a polymer CNT system and simulate with FEM code the electric conductance as a function of copolymer morphology.

4 Conclusions In this paper we have introduced the concept of multiscale molecular modelling and discussed the general guidelines for its implementation. The multiscale molecular modelling is applied in many fields of the material science, but it is particularly important in the polymer science, due to the wide range of phenomena accruing at different scales (from quantum chemistry to the mesoscale) influencing the final property of the materials. In this context, multiscale molecular modelling can play a crucial role in the design of new materials whose properties are influenced by the structure at nanoscale.

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Multiscale molecular modelling can also be successfully used in process system engineering for the estimation of properties for pure components and mixtures in all cases in which the component data banks properties and interaction parameters of constitutive equations are missing. Examples have been reported in this paper showing the methodology and describing the simulation protocols. A general good agreement in the comparison with experimental literature data of mechanical properties and morphologies is obtained, thus showing that the multiscale molecular modelling is a mature tool that may be used in the design and development of new coatings. Advances in computational materials science in general will continue to facilitate the understanding of materials and materials processing, the prediction of properties and behaviour, and the design of new materials and new materials phases, thus facilitating the application of process system engineering to more sophisticated and innovative processes. Acknowledgements The authors thank Paola Posocco and Marek Maly for the calculations and fruitful discussions.

References 1. J.C. Charpentier, Chem. Engrg. Sci. 57, 4667–4690 (2002). 2. S.C. Glotzer, W.P. Paul, Annu. Rev. Mater. Res. 32, 401–436 (2002). 3. M.P. Allen and D.J. Tildesley, Molecular Simulations of Liquids, Oxford University Press, Oxford (1987). 4. J.G.E.M. Fraaije, B.A.C. van Vlimmeren, N.M. Maurits, M. Postma, O.A. Evers, C. Hoffman, P. Altevogt, G. Goldbeck-Wood, J. Chem. Phys. 106, 4260–4269 (1997). 5. R.D. Groot, P.B. Warren, J. Chem. Phys. 107, 4423–4435 (1997). 6. A.A. Gusev, Macromolecules 34, 3081–3093 (2001). 7. G. Scocchi, P. Posocco, M. Fermeglia, S. Pricl, J. Phys. Chem. B 111, 2143–2151 (2007). 8. M. Fermeglia, S. Pricl, Progr. Organic Coatings 58, 187–199 (2007). 9. J.G.E.M. Fraaije, personal communication (2007). 10. M. Maly, P. Posocco, S. Pricl, M. Fermeglia, Ind. Engrg. Chem. Res. 47, 5023–5038 (2008). 11. R. Toth, A. Coslanich, M. Ferrone, M. Fermeglia, S. Pricl, S. Miertus, E. Chiellini, Polymer, 45, 8075–8083 (2004). 12. A. Maiti, J. T. Wescott, G. Goldbeck-Wood, Int. J. Nanotechnology 2, 3 (2005).

Structural-Scaling Transitions in Mesodefect Ensembles and Properties of Bulk Nanostructural Materials Modeling and Experimental Study O.B. Naimark and O.A. Plekhov

Abstract Comparative analysis of thermodynamic properties of coarse grain and bulk nano-crystalline titanium prepared by the method of severe plastic deformation (SPD) allowed the explanation of nano-crystalline state as specific transformation in the grain boundary defect ensemble – structural-scaling transition. By the analogy with the first kind of phase transition the anomaly of mechanical energy absorption under transition to the fine grain state was predicted theoretically and supported experimentally under realization of cyclic loading of coarse grain and fine grain titanium using the high resolution infra-red camera.

1 Introduction Bulk nanostructural materials (BNM) prepared by the methods of intensive plastic deformation are characterized by specific state of ensemble of grain-boundary defect ensemble with long range spatial correlation that provides the unique mechanical behavior of this class of materials [1, 2]. Physical properties of BNM are caused by the length and the development of grain boundaries, which for the grain sizes about 10–100 nm consist of 10–50% of atoms of material. As the consequence, the transition to bulk nanocrystlalline state is characterized by pronounced scaling effects and the change of material properties linking with the decreasing of particle size and the increasing role of grain boundary defects. Most important in the study of physics of BNM is the question about the existence of sharp boundary between bulk polycrystalline and nanocrystalline state, i.e. the existence of area below some characteristic grain size, where the properties are characteristic for nanocrystalline solid. There is also thermodynamic statement

O.B. Naimark and O.A. Plekhov Institute of Continuous Media Mechanics, Russian Academy of Sciences, 1 Acad. Korolev., 614013 Perm, Russia; e-mail: [email protected]

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of problem concerning the analogy of transition from polycrystalline to nanocrystalline state and the first kind of phase transition [3]. This question can be considered as the key point and the unique properties of nanostructural materials can be understood and used in the application under resolving of mentioned fundamental problems. An effective method for the study of material properties under transition to nanocrystalline state is the analysis of energy absorption mechanisms.

2 Structural-Scaling Transitions and Nanocrystalline State of Solid Statistical theory of collective behavior of mesodefects was proposed in [4] and allowed the establishment of new class of critical phenomena in dislocation subsystem – structural-scaling transitions. Characteristic feature of this class of critical phenomena that is typical for out-of-equilibrium systems with mesodefects is the existence of additional order parameter – structural-scaling parameter, which depends on scale parameters of polycrystalline solid (grain size) and correlation radius of mesodefects interaction. Statistically based thermodynamics and phenomenology for the out-of-equilibrium states allowed us to establish new class of critical phenomena related to collective behavior of defects named as structural-scaling transitions and to propose the explanation of transition from conventional polycrystalline to the bulk nanocrystalline state. The characteristic feature of new type of criticality is the existence of additional order parameter, which depends on the heterogeneity scale (grain size) of material and correlation radius of interaction between defects. Results of statistical theory were used for the formulation of statistically based thermodynamics and the phenomenology of polycrystalline and nanocrystalline states. Phenomenological approach represents the generalization of the Ginzburg–Landau theory for the out-of-equilibrium systems with mesodefects and allowed the definition of characteristic solid states (quasi-brittle, ductile and fine grain states) in terms of collective mesodefect modes [5]. It was shown the links of kinetics of these modes with the mechanisms of structural relaxation under plastic shear, damage localization. Qualitative difference of polycrystalline and bulk nanocrystalline states is the consequence of the scaling transition under the pass of some characteristic grain size. This transition is analogous with the first kind of phase transition and the grain refining leads to the degeneration of collective orientation mode of defects from the auto-solitary shear mode providing the plastic relaxation to the formation at the pass of ‘scale’ critical point the quasi-periodic finite-amplitude sublattice of mesodefects. Qualitative change of the types of collective modes leads to the change of global symmetry properties of out-of-equilibrium system ‘solid with mesodefects’ and different mechanisms which responsible for structural relaxation and plastic flow, diffusion properties, damage-failure transition. The scaling regularities at the critical point of transition from polycrystalline to bulk nanocrystalline states provide the violation of the Hall–Petch law for the stress threshold under the

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grain refining and the anomaly of energy absorption under mechanical loading (for instance in fatigue test) was predicted theoretically.

3 Anomaly of Mechanical Energy Absorption in Bulk Nanocrystalline Titanium The mechanisms of energy absorption can be studied using the presentation of outof-equilibrium potential for solid with mesodefects F [4] F =

1 1 1 A(δ, δ∗ )p2 − Bp4 + C(δ, δC )p6 − Dσp + χ(∇l p)2 , 2 4 6

and the equations describing the kinetics of accumulation of grain-boundary defects in terms of parameter of defect density tensor p (defect induced strain) and structural-scaling parameter δ

 dp ∂p ∂ 3 5 = −Lp A(δ, δ∗ )p − Bp + C(δ, δC )p − Dσ − χ , (1) dt ∂xl ∂xl
 dδ 1 ∂A 2 1 ∂C 6 = −Lδ p − p , (2) dt 2 ∂δ 6 ∂δ where A, B, C, D and χ are the material parameters characterizing the structure state induced by mesodefects; δc and δ∗ are critical values of structural-scaling parameter separating quasi-brittle (δ < δc ), ductile (δc < δ < δ∗ ) and nanocrystalline (δ > δ∗ ) material states; Lp and Lδ are the kinetic coefficients. Study of characteristic stages of fatigue failure considering the solid as the outof-equilibrium system in the conditions of mesodefect induced structural-scaling transitions was carried out to analyze ‘in-situ’ the energy dissipated part by the infrared thermography method that allowed to establish the qualitative difference in the relaxation mechanisms characteristic for conventional polycrystalline and bulk nanocrystalline states [4]. Kinetics of structural-scaling transitions is determined by the types of collective modes in mesodefect ensembles and qualitative transformation of these modes under transition from polycrystalline to bulk nanocrystalline state that provides qualitative change of relaxation mechanisms responsible for the plastic flow. It was shown in [3–5] that the transition from conventional polycrystalline to bulk nanocrystalline state is accompanied by the change of the types of collective modes (Figure 1): from the auto-solitary (S2 , δc < δ < δ∗ ) to the steady-state localized modes creating quasi-periodic mesodefect sub-lattice (S1 , δ > δ∗ ). Blow-up collective modes (S3 , δ < δc ), for quasi-brittle and auto-solitary modes for ductile states have the threshold features of generation and the transition to quasiperiodic modes, that are characteristic for mesodefect sub-lattice of fine grain states,

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Fig. 1 Collective modes of mesodefects for polycrystalline (S2 , S3 ) and fine crystalline (S1 ) states.

leads to the disappearance of the stress threshold and a violation of the Hall–Petch law. It is well known that the energy balance under the plastic flow consists of both dissipative and configuration parts. The dissipative part leads to a temperature increase, the configuration part is determined by the level of stored energy related to structural stresses of multiscale dislocation sub-structures. The plastic flow for polycrystalline materials is realized as the scaling transition from fine to rough dislocation sub-structures up to the scale, when dislocation sub-structures loose the shear mobility and initiate damage localization scenario. Kinetics of damage localization proceeds according to the generation of blow-up collective modes of mesodefect (Figure 1, S3 mode) and the ‘induction’ time for this mode nucleation determines the life-time of materials [5]. The scaling transitions mentioned are not characteristic for bulk nanostructural materials and the deformation proceeds due to the formation of dislocation sublattice. According to this scenario the transition to failure is realized without pronounced damage localization, but due to the critical cluster formation linking the damaged sub-lattice areas over the entire bulk of specimen. This evolution of dislocation sub-structures for conventional polycrystalline and bulk nanocrystalline materials has qualitative similarity with the first kind of phase transformation in the terms of spinodal decomposition of out-of-equilibrium metastable systems [5]. Qualitative different features of damage-failure transitions for polycrystalline and bulk nanocrystalline materials are revealed in mechanisms of mechanical energy absorption that was studied under the cycle loading.

4 Infra-Red Study of Characteristic Stages of High Cycle Fatigue Comparative analysis of energy dissipation under cycle loading of polycrystalline and bulk nanocrystalline titanium was carried using previously tested expressmethod of the endurance limit estimation to combine the fatigue load and infrared monitoring of surface temperature [6, 7]. Main idea of mentioned method is to

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Table 1 Mechanical properties of coarse and fine grain titanium Grade 2. Type of processing

Yield strength, σB , MPa

Yield stress, σ0.2 , MPa

Elongation, δ, %

Initial state (grain size ∼25 µm) ECAP + ‘warm’ rolling (grain size ∼0.3 µm)

440 1090 ± 20

370 980 ± 20

38 13 ± 1

establish the correlations between the energy dissipation rate and transition to the final fatigue stage for different values of preloaded stress [8, 9]. Titanium specimens Grade 2 in conventional polycrystalline and nanocrystalline state were studied. The refining of initial grain structure was realized under intensive plastic deformation (equal canal angular pressing – ECAP). Mechanical characteristics of titanium in both states are presented in Table 1. The resonance testing machine (Vibrofon) was used to provide uni-axial cycle loading in the preloaded state varying the mean stress for each portion of cycle load with fixed amplitude. Infrared camera CEDIP Jade III (resolution 25 mK at 300 K) was used to measure surface temperature. The load history includes the cycle blocks for 30 000 cycles with asymmetry parameter 0.1. The preloaded stress increased for the following block and the stress increment was 10 MPa. The temperature increment was measured by the infra-red camera for each block. Between the blocks the specimens were unloaded and the thermodynamic equilibrium with environment was provided under transition to the following block testing. Experimental results are presented in Figure 2 for titanium specimens with conventional polycrystalline and fine grain structure. For the preloaded stress exceeding 80 MPa (stress amplitude 145 MPa) titanium demonstrates nonlinear two-stage dissipation growth. The cross-section point of two linear asymptotic corresponds to the fatigue limit for polycrystalline titanium. Infra-red analysis showed that cycle load of sub-microcrystalline titanium is characterized by the qualitative difference of the mechanisms of energy dissipation. For small amplitudes the mean temperature in fine-grain titanium exceeds insignificant the temperature of polycrystalline titanium. The approaching of preloaded stress to the fatigue limit leads to qualitative change of this scenario: the temperature increment in fine grain titanium is essentially low than in polycrystalline titanium. The stably linear history for fine grain titanium is extended into the stress area exceeding the fatigue limit. The mean temperature on the surface of fine grain titanium specimen is stabilized after 20 000 cycles that reflects the ability of fine grain structure to stabilization of thermodynamic state of grain boundary defects sub-lattice that is in the correspondence with theoretical prediction concerning the evolution of grain boundary defects sub-lattice along ‘thermodynamic branch’ with homogeneous growth of the mean size of grain boundary defects with the stress. Failure of fine grain titanium has quasi-brittle character for the stress amplitude that exceeds the fatigue limit on ∼40%.

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Fig. 2 The average temperature of titanium specimens with polycrystalline and submicrocrystalline structure under cycle loading versus preloaded stress.

Fig. 3 Temperature rate versus preloaded stress for polycrystalline and nanocrystalline titanium.

5 Discussion Intensive plastic deformation provides the refining of grains and the formation of sub-microcrystalline out-of-equilibrium structures with high density of the lattice and grain boundary defects with long-ranged correlation properties of structural stresses. These features allow the consideration of bulk nanocrystalline state as metastable one in the vicinity of some critical point related to the scaling order parameter. Interpretation of deformation and damage-failure transition in polycrys-

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talline materials in terms of structural-scaling transition and the definition of outof-equilibrium potential in terms of defect density tensor and structural-scaling parameter explain the transition from conventional polycrystalline state to the bulk nanocrystalline state. The link of this transition with qualitative transformation of collective mesodefect modes: from auto-solitary one for polycrystalline state to the finite amplitude steady-state collective modes creating the mesodefect sub-lattice under the pass of critical value of structural-scaling parameter related to the grain size. This transformation realizes in the course of qualitative change of scenario of spinodal decomposition in the metastability area of mentioned order parameters. Qualitatively this scenario has the analogy with the first kind of phase transformation and the anomaly of energy absorption was predicted theoretically under the approaching to the critical value of scaling parameter (critical size of grains). The important consequence of this qualitative changes is the transition from the multiscale evolution of mesodefect sub-system in the form of collective auto-solitary mode (with the following damage localization with the blow-up kinetics) to the formation of mesodefect sub-lattice evolving homogeneously in the bulk of fine grain material up to the cluster formation entering over the material. As was shown in [3] the rate of energy accumulation in polycrystalline materials reaches the maximum value (∼60% of total energy for deformation) at the vicinity of yield stress and monotonically decreases during the hardening. With the grain refining the character of energy accumulation undergoes qualitative changes. The dependence of the energy accumulation in sub-microcrystalline titanium looses mentioned maximum and approaches to 35% of total deformation energy at the strain 9% [4]. Experimental results (Figures 2 and 3) allow us to conclude that dissipated energy and strength depend significantly on the grain size. Damage accumulation and transition to failure do not reveal the qualitative change of the asymptotic on characteristic curves and conventional criteria (fatigue strength) cannot be introduced. The growth of mean temperature during the deformation of fine grain specimens (and the intensity of dissipation sources) is proportional to the square stress amplitude (deformation energy) for all specimens studied. This fact illustrates the ability of fine grain materials to initiate effectively the configuration channel of structural relaxation involving all material volume. As far as structural relaxation in polycrystalline materials proceeds due to the formation of multiscale dislocation sub-structures (shear bands, shear localization areas) with numerous ordered dislocations (dislocation pile-ups), the formation of grain boundary defect sub-lattice in fine grain materials is the consequence of degeneration of collective orientation modes of mesodefects. The degeneration of orientation metastability appears under the pass of structural-scaling parameter the δc critical value. Qualitative change of the types of collective modes and disappearance of threshold stresses associated with orientation transition in mesodefect substructures allow us to propose the explanation the violation of the Hall–Petch law in terms of qualitative different evolution of mesodefect ensembles.

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References 1. Valiev, R.Z.: Nanostructuring of metals by severe plastic deformation for advanced properties. Nature Materials 3, 511–516 (2004). 2. Valiev, R.Z., Langdon, T.G.: Principles of equal-channel angular pressing as a processing tool for grain refinement. Progress in Materials Science 51, 881–981 (2006). 3. Naimark, O.B.: On topological transitions in ensembles of grain boundary defects and physics of nanocrystalline state. Physics of Metals and Material Science 84, 327–337 (1997). 4. Naimark, O.B.: Defect induced transitions as mechanisms of plasticity and failure in multifield continua. In: G. Capriz, P. Mariano (Eds.), Advances in Multifield Theories of Continua with Substructure, pp. 75–114. Birkhauser, Boston (2003). 5. Naimark, O.B.: Collective properties of defects ensemble and some nonlinear problems of plasticity and failure. Physical Mesomechanics 4, 45–72 (2003). 6. Luong, M.P.: Infrared thermographics scanning of fatigue in metals. Nuclear Engineering and Design 158, 363–376 (1995). 7. Plekhov, O.A., Saintier, N., Palin-Luc, T., Uvarov, S.V., Naimark, O.B.: Theoretical analysis, infrared and structural investigation of energy dissipation in metals under and cyclic loading. Material Science and Engineering A 462, 367–369 (2007). 8. Oliferuk, W., Beygelzimer, Y., Maj, M., Synkov, S., Reshetov, A.: Energy storage rate in tensile test of ultrafine grained titanium produced by twist extrusion. In Proceedings of 35th Solid Mechanics Conference, Krakow, pp. 329–330 (2006). 9. Plekhov, O.A., Santier, N., Naimark, O.: Experimental study of energy accumulation and dissipation in iron in an elastic-plastic transition. Technical Physics 52(9), 1236–1238 (2007).

Modeling Electrospinning of Nanofibers T.A. Kowalewski, S. Barral and T. Kowalczyk

Abstract A fast discrete model for the simulations of thin charged jets produced during the electrospinning process is derived, based on an efficient implementation of the boundary element method for the computation of electrostatic interactions of the jet with itself and with the electrodes. Short-range electrostatic forces are evaluated with slender-body analytical approximations, whereas a hierarchical force evaluation algorithm is used for long-range interactions. Qualitative comparisons with experiments is discussed.

1 Motivation Electrospinning is a simple and relatively inexpensive mean of producing nanofibers by solidification of a polymer solution stretched by an electric field. Such fibers find applications in a variety of areas, including wound dressing [4], drug or gene delivery vehicles [19], biosensors [16], fuel cell membranes and electronics [15]. Recently electrospinning has been revitalized and successfully applied to the production of nanofibrous scaffolds for tissue-engineering processes [2], which constitute one of its most promising application. A conventional electrospinning setup consists of a spinneret with a metallic needle, a syringe pump, a high-voltage power supply and a grounded collector. A polymer solution is loaded into the syringe and driven through the needle at a steady and controllable feed rate by the pump, forming a droplet at the tip of the needle. A high voltage (typically up to 30 kV) is applied between the tip and a grounded collector. When the electric field strength overcomes the surface tension of the droplet an electrified liquid jet is formed. The jet is then elongated and whipped continuously by electrostatic repulsion (bending instability), describing a chaotic spiraling motion on its way to the collecting electrode. Although the process may appear T.A. Kowalewski, S. Barral and T. Kowalczyk IPPT-PAN, Swietokrzyska 21, Warsaw, Poland; e-mail: {tkowale,sbarral,tkowalcz}@ippt.gov.pl

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simple the achievement of stable operation is not an easy task. Properties that are known to significantly affect the electrospinning process are the polymer molecular weight, the molecular-weight distribution, the architecture (branched, linear, etc.) of the polymer, as well as the rheological and electrical properties of the solution (viscosity, conductivity, surface tension, etc.). In addition, the operating conditions such as electric potential, flow rate, distance from the needle tip to the collection plate, ambient parameters (temperature, humidity), and geometry of the collecting target play a crucial role in controlling the electrospinning characteristics [1, 17, 23, 24]. Because each material demands a different optimization procedure, the development of theoretical and numerical models of electrospinning appears thus highly desirable. The physical and mathematical description of the electrospinning process remains, however, a distant target. Despite several parametric studies performed in various experimental configurations [1, 11, 23] it appears difficult to formulate consistent scaling laws for electrospinning. The influence of the electric field and of the solution conductivity on the fibers quality was exhaustively investigated by Arayanarakul et al. [1], who concluded that higher polymer solution conductivity improves the spinning process but has only a marginal impact on the fiber size. Broadly speaking, higher applied electric potential, higher electric conductivity and viscosity of the polymer solution appear to improve fiber uniformity. Although the great role played by the electric field and the solution conductivity is conform to the intuition, experiments performed with similar configurations but different polymer solutions exhibit at times opposite relationships [1,17]. The validation of theoretical models with experimental data is thus far from trivial [22], and it has proved so far impossible to relate actual fiber morphology to the predicted characteristics [8]. The early stage of electrospun jets has been investigated in several onedimensional theoretical and numerical works [6, 9, 10, 13, 21], which have emphasized the importance of viscoelastic rheology. By contrast, very few studies have considered the unstable part of the jet, in part because of the difficulties inherent to its three-dimensional and unsteady character. Hohman et al. [13] have derived a linear stability theory aimed at predicting the onset of a bending mode in a straight jet, but its relevance to the actual phenomenon remains difficult to assess. A qualitative description of this instability has been given by Yarin et al. [24], using a simple discrete model consisting of point charges connected by dumbbell elements. The latter model appears to provide a reasonable explanation for the spiraling motion of the jet, but suffers from mathematical inconsistencies incurred by the discretization of the fiber into point-charges. Although this problem was remedied by accounting for the actual electrostatic form factors between two interacting sections of a charged fiber [17], electrostatic interactions have usually been accounted for via strong approximations [9, 10, 13]. One serious concern relates to the evaluation of short-range interactions, which in the case of standard discrete integration methods require very dense grids due to the large contribution of short-range electrostatic interactions within distances of the order of the fiber radius [14, 17]. The fiber radius being about 103 –105 times smaller than the macroscopic scales of interest, it appears most desirable to devise a discrete model that exploits the slenderness of

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the fiber to evaluate short-range interactions in an efficient manner. Likewise, the computation of long-range electrostatic interactions can easily become intractable due to the O(n2 ) operation count for a pairwise evaluation of interactions between n elements. A third issue with current numerical models is the assumption of a static external electric field, whereas in reality the external field is modulated by the net space charge of the fiber so as to keep constant the potential over the electrodes. This latter issue can be effectively addressed by use of a boundary element method (BEM). In essence, the boundary element method is a statement of the electrostatic problem (Poisson equation) in terms of boundary integrals; as such, it involves only the discretization of boundary surfaces, which in our case would be the electrode surfaces and the outer shell of the fiber. Although BEM-based models have been investigated in the context of electrospraying [5,14,18,25], no attempt has been made to cure the aforementioned inefficient evaluation of short-range and long-range interactions. An efficient handling of long-range interactions would, however, greatly benefit to both electrospraying and electrospinning simulations. For this, a variety of mesh-based and particle-based methods exist that can theoretically achieve O(n log n) operation count, or even O(n) for the fast multipole method (FMM). Particle-based methods are clearly at an advantage in the problem at hand, where the computation domain is three-dimensional but is only sparsely populated by a one-dimensional object (the fiber). Most fast particle-based algorithms are based on either treecode hierarchical algorithms [3] or on the FMM [12]. In either cases, distant particles are clustered and approximated as a truncated multipole expansion (of fixed order in the case of the treecode method, and of adaptive order in the FMM). The treecode algorithm considers particle-cluster interactions and achieves O(n log n) complexity, while FMM considers cluster-cluster interactions and can compute forces with O(n) complexity. At low n, however, and when the required accuracy on the computed forces is low, the treecode is usually found to outperform FMM due to a much smaller prefactor. Furthermore, the theoretical advantage of the O(n) force evaluation of the FMM is somewhat mitigated by the fact that for non-uniform spatial distribution of particles – such as a fiber in 3D space – the charge clusterization step still requires O (n log n) operations. Last and foremost, the treecode algorithm is simpler to implement and parallelize. It is worth noting that treecode accelerated BEMs have been recently investigated in other contexts, such as plasma physics [7].

2 Experiment Electrospinning was performed inside a custom-made polycarbonate chamber of approximately 1 cubic meter volume. The spinneret consists of a 2 mm long grounded syringe needle of internal diameter 0.35 mm, mounted vertically on an electrically insulated stand placed 15 cm above the collecting electrode. The spinneret needle was positively biased with a high voltage power supply. A flat cooper grid (310 mm × 240 mm) with a small cage (75 mm × 80 mm × 50 mm) was used as ground

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electrode, as described in previous reports [16,17]. The voltage applied between the spinneret and the ground electrode was varied from 1 kV up to 30 kV. The capillary needle was connected through PTFE tubing to a plastic syringe filled with the spinning solution. A constant volume flow rate was maintained using the syringe pump. In our study, a 3% solution of poly(ethylene oxide) (PEO) of molecular weight 400 kDa in 40% ethyl alcohol/water mixture was used to analyze electrospinning process. The effect of solution electrical conductivity was investigated with three inorganic salts (NaCl, LiCl, NH4 Cl) individually dissolved in the PEO solution to attain molar salt concentration in the range 0.001–0.02 M, which corresponds to a 15-fold variation of the solution conductivity. The electrospinning process was observed using high speed camera (hs1200.pco) and analyzed by evaluating the geometry of the fiber coils created by the electrified polymer jet. The spiral envelope angle, θ , and the length of the straight jet segment, L, were obtained from the high-speed camera images and their dependence on the applied voltage and solution composition was evaluated and correlated with microscopic images of the electrospun nanofibers. The pure PEO solution used in the experiments exhibits a bending instability threshold at about 3kV. Farther increase of the voltage elongates the straight part of the jet and leads to the development of relatively stable, large amplitude looping. The straight part increases from about 0.5 cm to over 2.5 cm. The looping cone initially increased with the voltage but above 5kV the spiral envelope angle systematically decreased. It is in agreement with our previous findings [17]. This general trend usually did not change after increasing the electrical conductivity of the solution (compare Table 1). A similar behavior of the jet could be observed for nearly all investigated PEO-inorganic salt solutions, i.e. the straight portion of the jet increased and the looping cone decreased with the voltage. However, increased salt concentration (and hence solution conductivity) only slightly decreased the cone angle θ for two salts NaCl and NH4 Cl, whereas the opposite behavior was observed for LiCl. The jet straight part L increased with the salt concentration only for NaCl, whereas it decreased for the two other salts. Relatively small changes of the jet spiralling geometry may suggest simmilar elongation rate of the fiber. In fact, images of collected fibers analysed under optical and TEM microscopes could not give any clear evidence that solution conductivity changes fiber size or morphology. Dispersion of the collected data underline complexity of the problem. Increasing solution conductivity by salt additives likely modifies other polymer properties such as surface tension, viscosity, and rheology. However, according to the data collected by Arayanarakul et al. [1] and in regard to the range of salt concentration used, only a slight decrease of the surface tension (from 31 to about 28 mN/m) and viscosity (from 206 to 190 mPa/s) can be expected. Hence, it is difficult to draw definite conclusions on the effects of solution conductivity, i.e. charge density carried by the fiber.

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Table 1 Variation of jet straight part L and cone angle θ for different salt concentrations c (and corresponding conductivity K) and applied voltage Ug ; data for 3% PEO water/alcohol solution at 25◦ C. #

salt

c [M] K [µS/cm]

Ug [kV]

θ [deg]

L [mm]

85 112 71.3 63.3 58.7

7.1 23.1 25.2 17.4 11.8

1

–

0.0 17.1

3 5 10 15 20

2

NaCl

0.01 399.5

10 15 20

86.7 62.7 52.7

26.5 38.0 39.7

3

NaCl

0.02 741.3

10 15 20

61.7 59.0 50.0

24.6 34.6 42.5

4

LiCl

0.01 280.6

10 15 20

70.3 48.3 39.3

34.2 40.2 46.6

5

LiCl

0.02 551.9

10 15 20

71.7 63.3 69.3

22.8 10.5 18.0

6

NH4 Cl

0.01 457.8

10 15 20

75.7 64.3 59.0

32.1 32.7 33.8

7

NH4 Cl

0.02 906.1

10 15 20

84.0 71.7 60.0

24.0 30.5 31.2

3 Model 3.1 Governing Equations The model is, essentially, a time-dependent three-dimensional generalization of known slender models [6, 9, 10, 13], with the following differences: (i) the electric field induced by the generator and by the charges on the fiber is explicitly resolved, instead of being approximated from local parameters [6, 9, 10]. (ii) electrical conductivity is neglected. Indeed, the convection of surface charges is believed to strongly overcomes bulk conduction at locations distant from the Taylor cone by a few fiber radii [6]; since we are mostly interested in the description of the bending instability, this assumption appears reasonable.

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Fig. 1 Fiber discretized into finite length elements Qk+1/2 with collocation nodes Nk localized at the element interfaces.

(iii) for the sake of simplicity, polarization effects are neglected. (iv) the polymer is considered a viscoelastic Maxwell fluid, as in [20]. Evaporation has been systematically neglected in the literature, except in [24] where an ad-hoc evaporation model was used. Following Fridrikh et al. [11], we shall also neglect evaporation even in the whipping part of the jet, by arguing that it occurs mostly after the fiber has been maximally stretched. The mass conservation equation accordingly reads ∂a 2 ∂(va 2 ) + = 0, (1) ∂t ∂ξ where a is the fiber radius, v is the velocity of the fiber and ξ is the arc length along the fiber. The force balance per unit of fiber length reads in turn ρπa 2 x¨ =

∂(πa 2 τ ˆt + πγ a ˆt) + λE, ∂ξ

(2)

where ρ is the mass density, x¨ is the acceleration vector, γ is the surface tension coefficient, ˆt is the local unit vector tangent to the fiber, λ is the linear charge density, ¯ is the dielectric constant of the medium (air), E is the electric field and τ is the viscoelastic stress. The stress is given by a Maxwell viscoelastic constitutive equation:
 τ τ˙ = G ε − , (3) µ where is G the elastic modulus, µ the fluid viscosity and ε the Lagrangian axial strain ∂ x˙ · ˆt. (4) ε≡ ∂ξ

3.2 Discretization and Electrostatic Solver A Lagrangian discrete model is used. The fiber is first decomposed into discrete charged elements Qi+1/2 , the length of which is typically much greater than the

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fiber radius, but smaller than other characteristic lengths of interest (such as the curvature radius). The equation of motion (2) is then resolved at the interfaces of each element (collocation nodes Ni , see Figure 1), where all forces are priorly evaluated. After displacements have been calculated, the radius in the central sections of elements Qi+1/2 , is straightforwardly obtained since the volume of each element is conserved. Likewise, the linear charge density in the center section is computed by assuming that the total charge of each element is conserved. The scheme is thus intrinsically mass and charge conserving (but is not momentum conserving). The forces at each collocation node are computed from the following discrete form of Eq. 2: 1 π(a 2 τ )i+1/2 − π(a 2τ )i−1/2 + πγ (ai+1/2 − ai−1/2 ) ˆti 1 ρπai2 (|x − x | + |x − x |) i i−1 i+1 i 2
 1 γ τi + κi nˆ i + λi Ei . + ρ ai

x¨ i =

(5)

The values ai , τi , λi required at the collocation nodes are linearly interpolated from those computed in the central section of the neighboring elements. The local tangent vector ˆti and curvature vector κ nˆ i are computed from the approximate osculating circle defined by (Ni−1 , Ni , Ni+1 ). Time integration is realized with a classical leapfrog scheme, which presents the advantage over other second order schemes by requiring only one evaluation of force term per time step: n+1/2

x˙ i

xn+1 i

n−1/2

= x˙ i =

xni

+ x¨ ni t,

(6)

n+1/2 + x˙ i t,

(7)

where upper indices refer to time. Eq. (5) shows that the accelerations x¨ ni can be explicitly obtained from the positions xni at the same instant, except for the contribution of the viscoelastic stress as the latter integrates a memory effect. A discretization of the stress equation (3) which preserves second order time accuracy of the scheme can nonetheless be found using n+1 n σi+1/2 − σi+1/2

t

n+1/2

= G εi+1/2 −

1 n n+1 (σ + σi+1/2 ), 2µ i+1/2 n+1/2

n+1/2 εi+1/2

(8) n+1/2

n+1 n+1 1 (xni+1 − xni + xi+1 − xi )(˙xi+1 − x˙ i = n+1 2 2 (xni+1 − xni + xn+1 ) i+1 − xi

)

,

(9)

n+1 since Eq. (8) is easily recast into a time-explicit expression for σi+1/2 . Dynamic refinement is used in simulations to maintain the size of elements below a prescribed characteristic length max ; whenever an element is elongated beyond max , it is split into two elements, each containing half the charge and mass of the initial element.

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Fig. 2 Idealized electrostatic configuration: the potential is prescribed between the tip of the needle and a grounded infinite plane. The needle electrode is modeled as a point electrode, the charge of which is computed such as to always maintain the prescribed potential at the needle tip. Image charges are used to implement the potential condition φ = 0 on the infinite plane.

The needle and the grounded collector are idealized by a point-charge/plate capacitor configuration, whereas the point-charge is meant to qualitatively reproduce the field lines concentration that exists close to the needle. The potential is prescribed between the location at which the fluid is introduced (the tip of the needle) and the infinite plane where the fiber is collected. A practical way to resolve the electrostatic field in such a configuration is the method of images (see Figure 2), whereas fictive mirror charges are placed symmetrically to the ground plane. The charge of the capacitor is computed at any instant to satisfy the prescribed potential φ = Ug at the inlet. The electrostatic field induced by the fiber can be computed by considering that the charges are distributed on the centerline of the fiber, as demonstrated by Homann et al. [13]. This alleviates the need for complex form factors, as used previously by

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the authors and others [5, 14, 17]. Even so, the computation of electrostatic interactions between the discrete charge elements Qi remains a complex problem. First, if a typical BEM implementation with pairwise evaluation of electrostatic interactions is used, the problem scales as O(n2 ) where n is the number of discrete elements. A more efficient, albeit more complex method is the hierarchical force calculation algorithm (treecode) [3], which complexity scales as O(n log n). Charge elements are in that case recursively clustered and the monopole coefficients (charge and center of charge) of the clusters are computed. The field at a location Ni is then computed by considering only the largest clusters which are sufficiently well separated [12], that is, for which the distance di,C between the collocation node Ni and the charge center of the cluster satisfies RC < α, (10) di,C where α < 1 determines the accuracy of the force evaluation and RC is the radius of a sphere containing all the charges of the cluster. Value α = 0.8 was used in our computations. A typical implementation of the treecode or FMM involves the recursive decomposition of the space domain into cubic cells, where each parent cell contains up to 8 children cubic cells, until each cell contains a single charge. Hence, the root of the oct-tree contains all charges of the domain and its leaves contain the charges. In our case, however, it is possible to create a hierarchized tree at a much lower costs, since the fiber approximately organizes the charges by nearest neighbors. At each time step, a binary tree is thus constituted by recursively grouping neighbors two by two, as illustrated by Figure 3, calculating at each level the smallest enclosing spheres that contain the cluster pairs. The root of the tree is a sphere that contains all charges, and its leaves are the elements Qi+1/2 . Unlike the classical treecode clusterization algorithm, this method creates a fully balanced tree and consequently easily lends itself to parallelization. To evaluate the field at a given collocation node, condition (10) is first tested on the root cluster, and, whenever it fails, on the two subclusters recursively until clusters which are sufficiently separated are found. If a leaf element is reached which does not satisfy condition (10), the interaction is considered a close interaction and near-field expressions must be used instead of the monopole approximation. Otherwise, far field expression are used based on the monopole approximation of the cluster. It must be underlined, however, that since charges are assumed to lay on the centerline but induce a field on the outer shell of the fiber, one must account for a small correction to the classical potential and Coulomb’s law: φC→i =

1 qC  , 4π ¯ d 2 + a 2

(11)

1 qC u , 2 + a 2 C→i 4π ¯ di,C i

(12)

i,C

EC→i =

i

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Fig. 3 Illustration of the clustering method used in our treecode implementation: neighbor charges are recursively grouped two-by-two and the smallest enclosing spheres are calculated at each clustering level, until all charges are contained in a binary tree which root is a sphere that contains all charges.

which is to be contrasted with the laws in 1/d and 1/d 2 for monopole-monopole interaction. In the above, q stands for the total charge of the cluster and u for the unit vector pointing to the collocation node. Obviously, the above formulae are exact only for a monopole that lies along the axis of the ring where the field is evaluated. However, the slenderness of the fiber implies that nearby charges are located close to the axis; therefore, this condition is violated only when the cluster is far (d  a), in which case the error made is negligible since the formula reduces to Coulomb’s law (which is appropriate when d  a). As mentioned earlier, condition (10) may not be fulfilled by nearby elements. This happens in particular for elements Qi−1/2 and Qi+1/2 , which are contiguous to node Ni . In such case, the following near-field analytical expressions are used: 

 λC ξi − ξC0 ξi − ξC1 asinh C→i = − asinh , (13) 4π ¯ ai ai ⎡

(E · ˆt)C→i

⎤ ξC1 − ξC0 λC ξC1 − ξC0 λC ⎢ 1+ ⎥ 1− λC ⎢ 2 λC 2 λC ⎥ ⎢ ⎥ = −   ⎥

4πai ¯ ⎢ ⎣ ξi − ξC1 2 ξi − ξC0 2 ⎦ 1+ 1+ ai ai

 dλC  ξi − ξC1 ξi − ξC0 dξ + asinh − asinh , 4π ¯ ai ai

(14)

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⎡

ˆ C→i (E · n)

289

⎤

ξi − ξC1 ξi − ξC0 ⎥ ⎥ ai ai ⎥ −   ⎥

ξi − ξC1 2 ξi − ξC0 2 ⎦ 1+ 1+ ai ai 

 κλC ξi − ξC0 ξi − ξC1 + asinh − asinh , 4π ¯ ai ai

⎢ κλC ⎢ ⎢ = 4π ¯ ⎢ ⎣

(15)

which are leading order approximation for the potential and field generated by a slender, curved line charge on a neighboring ring. Here, λC is the linear charge density of the line charge, ξi the curvilinear location of the location where the field and potential are evaluated, ξC0 and ξC1 the locations of the end points of the line charge, and κ the curvature of the fiber. Boundary conditions implemented at the inlet (tip of the needle) prescribe the volume flow rate Qv , the surface charge density σ0 and the initial fiber radius a0 . The initial stress is set to zero. It must be noted, however, that boundary conditions are notoriously difficult to implement in a consistent way in electrospinning models. This is even more true in the present case where electrical conductivity is neglected and where the charge density is not computed self-consistently. We observe a clear influence of both the boundary conditions and the discretization parameters at the inlet where strong gradients take place, which clearly challenge the assumption of slenderness. Additional investigations will be required in the future to derive boundary conditions that avoid such singularities. A small random perturbation to the position of each element introduced at the inlet is imposed, so as to initiate the bending instability. The magnitude of this perturbation has no notable influence on the simulation results, provided that it is small enough.

3.3 Simulation Results Behavior of the code was tested using parameters typical for the experiment. Several test runs performed confirmed general ability of the code to replicate our previous findings [17], i.e. increased electrical potential, solution viscosity and elastic modulus decreased jet sweeping amplitude, effectively seen as decreasing of the spiral cone. Here, we show an example of the simulation results obtained by varying fiber charge density and the voltage. Although it is not possible to directly integrate the liquid electrical conductivity into the model at this point, we shall assume that the conductivity mainly affects the electrical current and thus the charge density carried by the fiber. The simulations shown were performed for the following process parameters: surface tension γ = 0.02 N/m, elastic modulus G = 10 kPa, viscosity µ = 10 Pa·s, mass density ρ = 1000 kg/m3, volume flow rate Qv = 1 mm3 /s, initial fiber radius a0 = 150 µm, distance between needle tip and collector 20 cm, and distance between point charge and needle tip 1 cm. Typical jet paths are plotted

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Fig. 4 Simulated jet paths at Ug = 10 kV (upper row) and 15 kV (bottom row), for two charge densities 10 C/m3 (left) and 20 C/m3 (right).

on Figure 4. As we may find the model is unable to predict initial straight part of the jet. Hence we may only compare geometry of the spiral cone. We may conclude that higher charge density evidently increases amplitude of the jet sweeps. It is only partly consistent with the experimental findings (compare Table 1), where the cone angle first increases for lower salt concentrations but again decreases for higher values. Obviously, secondary effects due to the salt-polymer interactions may play a role for higher concentrations.

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4 Conclusions In this paper we have investigated dynamics of electrospun fibers within the slenderbody approximation proposing numerically consistent model based on hierarchical charge clustering. A versatile boundary value method is implemented to enforce fixed-potential boundary conditions, allowing realistic electrode configurations to be investigated. The model is tested against experiments performed for different polymer-salt solutions. Due to the several experimental and theoretical limitations the model outcome predicts only qualitatively observed experimentally dependence of the jet geometry on the solution conductivity. Further investigations are necessary to elucidate sources of inconsistencies between the experimental findings and the modeling. Acknowledgements This work was supported by the Polish Ministry of Science Grant No. N508 031 31/1740. The authors thank Diana Lamparska for her invaluable help in experimental work.

References 1. Arayanarakul, K., Choktaweesap, N., Aht-ong, D., Meechaisue, C., Supaphol, P.: Effects of poly(ethylene glycol), inorganic salt, sodium dodecyl sulfate, and solvent system on electrospinning of poly(ethylene oxide). Macromol. Mater. Eng. 291, 581 (2006). 2. Arumuganathar, S., Jayashinghe, S.N.: Living scaffolds (specialized and unspecialized) for regenerative and therapeutic medicine. Biomacromolecules 9, 759 (2008). 3. Barnes, J., Hut, P.: A hierarchical O(NlogN) force-calculation algorithm. Nature 324, 446 (1986). 4. Bhattarai, S.R., Bhattarai, N., Yi, H.K., Hwang, P.H., Kim, H.Y.: Novel biodegradable electrospun membrane: Scaffold for tissue engineering. Biomaterials 25, 2595 (2004). 5. Carretero-Benignos, J.A.: Numerical simulation of a single emitter colloid thruster in the pure droplet cone-jet mode. Ph.D. thesis, Massachusetts Institute of Technology (2005). 6. Carroll, C.P., Joo, Y.L.: Electrospinning of viscoelastic Boger fluids: Modeling and experiments. Phys. Fluids 18, 053102 (2006). 7. Christlieb, A.J., Krasny, R., Verboncoeur, J.P., Emhoff, J.W., Boyd, I.D.: Grid-free plasma simulation techniques. IEEE. Trans. Plasma Sci. 34, 149 (2006). 8. Dayal, P., Kyu, T.: Dynamics and morphology development in electrospun fibers driven by concentration sweeps. Phys. Fluids 19, 1011061 (2007). 9. Feng, J.J.: The stretching of an electrified non-Newtonian jet: A model for electrospinning. Phys. Fluids 14, 3912 (2002). 10. Feng, J.J.: Stretching of a straight electrically charged viscoelastic jet. J. Non-Newtonian Fluid Mech. 116, 55 (2003). 11. Fridrikh, S.V., Yu, J.H., Brenner, M.P., Rutledge, G.C.: Controlling the fiber diameter during electrospinning. Phys. Rev. Lett. 90, 144502 (2003). 12. Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73, 325 (1987). 13. Hohman, M.M., Shin, M., Rutledge, G., Brenner, M.P.: Electrospinning and electrically forced jets. I. Stability theory. Phys. Fluids 13, 2201 (2001). 14. Khayms, V.: Advanced propulsion for microsatellites. Ph.D. thesis, Massachusetts Institute of Technology (2000).

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15. Ko, F.K., El-Aufy, A., Lam, H.: Wearable Electronics and Photonics, chap. Electrostatically Generated Nanofibres for Wearable Electronics, pp. 13–40. Woodhead Publishing Ltd., Cambridge, UK (2005). 16. Kowalczyk, T., Nowicka, A., Elbaum, D., Kowalewski, T.A.: Electrospinning of bovine serum albumin. Optimization and use for the production of biosensors. Biomacromolecules 9, 2087 (2008). 17. Kowalewski, T.A., Blonski, S., Barral, S.: Experiments and modelling of electrospinning process. Bull. Polish Acad. Sci. 53, 385 (2005). 18. Lopez-Herrera, J.M., Ganan-Calvo, A.M., Perez-Saborid, M.: One-dimensional simulation of the breakup of capillary jets of conducting liquids. Application to E.H.D spraying. J. Aerosol Sci. 30, 895 (1999). 19. Luu, Y.K., Kim, K., Hsiao, B., Chu, B., Hadjiargyrou, M.: Development of a nanostructured DNA delivery scaffold via electrospinning of PLGA and PLAPEG block copolymers. J. Control. Release 89, 341 (2003). 20. Reneker, D.H., Yarin, A.L., Fong, H., Koombhongse, S.: Bending instability of electrically charged liquid jets of polymer solutions in electrospinning. J. Appl. Phys. 87, 4531 (2000). 21. Spivak, A.F., Dzenis, Y.A.: Asymptotic decay of radius of a weakly conductive viscous jet in an external electric field. Appl. Phys. Lett. 73, 3067 (1998). 22. Thompson, C.J., Chase, G.G., Yarin, A.L., Reneker, D.H.: Effects of parameters on nanofiber diameter determined from electrospinning model. Polymer 48, 6913 (2007). 23. Yang, Y., Jia, Z., Li, Q., Guan, Z.: Experimental investigation of the governing parameters in the electrospinning of polyethylene oxide solution. IEEE Trans. Dielectr. Electr. Insul. 13, 580 (2006). 24. Yarin, A.L., Koombhongse, S., Reneker, D.H.: Bending instability in electrospinning of nanofibers. J. Appl. Phys. 89, 3018 (2001). 25. Yoon, S.S., Heister, S.D., Epperson, J.T., Sojka, P.E.: Modeling multi-jet mode electrostatic atomization using boundary element methods. J. Electrostat. 50, 91 (2001).

Use of Reptation Dynamics in Modelling Molecular Interphase in Polymer Nano-Composite J. Jancar

Abstract In polymer matrix composites, exhibiting heterogeneous structure at multiple length scales, the interphase phenomena at various length scales were shown to be of pivotal importance for the control of the performance and reliability of such structures. At the micro-scale, the interphase is modelled as a 3D continuum with some average properties most commonly resulting from data fitting procedures. Number of continuum mechanics models was derived over the last 50 years to describe the stress transfer between matrix and an individual fiber, considering the interphase with various chemical structure and thickness the third component of the model composite characterized by some average shear strength, τa , with realtively good success. The observed strong thickness dependence of the elastic modulus of the interphase with thickness smaller than 500 nm suggested presence of its underlying nano-scale molecular sub-structure. On the nano-scale, the discrete molecular structure of the polymer has to be considered. At this length scale, the continuum mechanics can only be used for materials with characteristic length scale greater than approximately 20 nm. Below 20 nm, continuum mechanics becomes not valid and gradient-strain elasticity along with molecular dynamics approach has to be used. The segmental immobilization seems to be the primary mechanism controlling the behavior of nano-scale “interphase”. Modified reptation model was used to describe the dynamics of chains near a solid nano-particles and to explain the peculiarities in the viscoleastic response of polymer nanocomposites. These results reflecting the discrete molecular nature of the nano-scale interphase can be used in gradient-strain elasticity models. Experimental results obtained for model nanocomposites were used to support theoretical predictions.

J. Jancar Institute of Materials Chemistry, Brno University of Technology, Czech Republic; e-mail: [email protected]

R. Pyrz and J.C. Rauhe (eds.), IUTAM Symposium on Modelling Nanomaterials and Nanosystems, 293–301. © Springer Science+Business Media B.V. 2009

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1 Introduction It has been proposed that advanced light weight structures can be manufactured employing the design of polymer composites on various length scales from macro- to nano-scale. Reliable models capable of linking large-scale mechanical properties of polymer nano-composites with their nano-scale structure are scarce and not generally accepted. However, the need to establish reliable prediction algorithms for large-scale mechanical properties required for structural applications of nano-scale engineered composite materials is highly desired [1]. In the fiber reinforced polymer matrix composites composites (FRCs), exhibiting heterogeneous structure at multiple length scales, the interphase phenomena at various length scales were shown to be of pivotal importance for the control of the performance and reliability of such structures. Various models based on continuum mechanics were used to describe effects of the macro- and meso-scale interphase on the mechanical response of laminates and large FRC parts, satisfactorilly. At the micro-scale, the interphase is considered a 3D continuum characterized by a set of average properties (elastic moduli, yield strength, toughness, etc.) and uniform thickness, t. Number of continuum mechanics models was derived over the last 50 years to describe the stress transfer between matrix and individual fiber with realtively good success. In these models, the interphase was characterized by some average shear strength, τa , and elastic modulus, Ea [2]. On the other hand, models for tranforming the properties of the micro-scale interphase around individual fiber into the mechanical response of macroscopic multifiber composite have not been generally successfull. The strong thickness dependence of the elastic modulus of the micro-scale interphase suggested presence of its underlying sub-structure [3, 4]. On the nano-scale, the term interphase, originally introduced for continuum matter, has to be re-defined to include the discrete nature of the matter. The segmental immobilization resulting in retarded reptation of chains caused by interactions with solid surface seems to be the primary phenomenon which can be used to re-define term interphase on the nano-scale [5, 6]. The very nature of chain and segment relaxations has to be re-examined since the presence of solid particles with size equal to the portion of the radius of gyration, Rg , resulted in perturbation of chain statistics and dynamics on a large scale. In most cases, there are no ‘bulk’ chains in the nanocomposites containing more than 3–5 vol.% of the nano-sized filler, hence, all the chains are more or less in the ‘interphase’ [7–11]. In small volume fraction nanocomposites commonly considered (vf ≤ 0.05), the effect of filler volume becomes of second order importance due to the large surface to volume ratios and the surface effects are of primary importance. This should be reflected in modified predictive models where the vf used as the main structural variable should be replaced with specific surface area per unit weight of composite, Af [6]. All the experiments measuring mechanical properties of nano-composites are performed using macro-scale specimens, however, attempts are made to interpret these data in terms of nano-structural features. One has to be very cautious in doing so, since the traditional continuum mechanics approach has to be modified substantially in order to account for the discontinuous nature of matter at the nano-scale.

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Fig. 1 Schematic visualization of the micro- to nano-scale transition (left) and the shape of a 1000 unit individual chain (right) from MD simulation.

When considering the molecular nature of the reinforcement at the nano-scale, the limits of validity of size independent continuum mechanics should be determined, since the discrete nature of the matter at the nano-scale prohibits simple scaling down the micro-mechanics models [12–17]. It has also been shown that in polymers below 10 nm, Bernoulli–Euler continuum elasticity becomes not valid and higher order elasticity along with the proposed reptation dynamics approach can provide suitable means for bridging the gap in modelling the transition between the mechanics of continuum matter at the micro-scale and mechanics of discrete matter at the nano-scale. The physical reasons for the expected breakdown of continuum elasticity on the nano-scale include increasing importance of surface energy due to appreciable surface to volume ratio, the discrete molecular nature of the polymer matrix resulting in non-local behavior in contrary to local character of classical elasticity, presence of nano-scale particles with the length scale similar to the radius of gyration of the polymer chains and internal strain due to molecular motion within a non-primitive lattice. Quantum confinement effects can also play a role inducing a strain field on the nano-scale without presence of external loading, however, its importance is limited to the size range below 2 nm.

2 Micro- and Nano-Scale Interphase Good succes has been achieved in describing the role of the interphase in stress transfer from the matrix to the fiber using model single fiber composites. From the simple Kelly–Tyson model, to the various lap shear models, to the numerical F.E.A. models, the approach based on the continuum mechanics has been employed [18, 19, 22–27, 33]. Even though the molecular structure of the interphase has been anticipated in many papers, with some exceptions [20,21,32], the main effort has been devoted to the relationship between the type, thickness and deposition

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Fig. 2 Visualization of the micro-scale interphase [1].

conditions of the fiber coating and the average shear strength of the interphase, τa , measured in a simple test employing model single fiber composite [28–31]. Stiff, well bonded interphases provide very efficient stress transfer, less water diffusion, however, promote brittle failure, thus, limit the damage tolerance of the FRC structures. Low modulus tough interphases slightly reduce the effectiveness of stress transfer and may be less resistant to water attack, however, they provide significant enhancement of damage to lerance of the FRC part [34]. Moreover, the tough interphases are less sensitive to the direction of the external loading compared to the stiff ones. Research of the nano-scale interface/interphase phenomena in true nanocomposites has resulted in renewed interest in basic principles of polymer physics capable of describing the changes in chain statistics and dynamics caused by presence of nanoparticles with size similar to that of the polymer chain [36–40]. One has to keep in mind, however, that when the length scale considered reaches few nanometers, which is equal to the size of individual polymer chains, the very term ‘interphase’ becomes ill-defined due to the fact that the discrete nature of the matter has to be taken into account. Originally, the term interphase has been defined in the framework of continuum mechanics as a layer or series of layers, respectively, with some average property or gradient of properties, respectively, occuring between the matrix bulk and fiber or filler particle. However, the continuum elasticity in the Bernoulli–Euler form may no longer be valid on the nano-scale due to very large non-locality in elastic response of heterogeneous systems with coordinated movement of large number of atoms, such as observed in polymer nanocomposites [35]. Polymers are unique systems with macroscopic viscoelastic response driven by the relaxation processes on the molecular level. These relaxation processes represent particular molecular motions occurring in some characteristic volume, Vc . The

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Fig. 3 Visualization of the nano-scale interphase [1].

Vc depends on the type of the relaxation process and temperature. The characteristic volumes vary from 10−3 nm3 for localized bond vibrations to 106 nm for the non-local normal mode of relaxation. In the case of the non-local normal mode of relaxation, its characteristic volume is the upper limit for Vc displaying strong dependence on the chain size. The characteristic time, τc , for each particular relaxation process varies from 10−14 s for bond vibrations above Tg to the infinitely long times below Tg . Thus, the macroscopic viscoelastic response of a polymer is a manifestation of a range of molecular relaxations localized in some characteristic volume and the rate of the relaxation mode is indirectly proportional to its locality [1].

3 Retarded Reptation Model In some sense, the effects observed in nanocomposites resemble behavior of colloidal systems and, in some cases, behavior of thin polymer films. In comparison to colloids, the long chain character of the ‘liquid’ phase, however, greatly complicates any theoretical treatment compared to the low molecular weight ‘liquid’ phase in colloids. Reducing the size of rigid inclusions from micro- to nano-scale is accompanied by 2–3 orders of magnitude increase in the internal contact area between the chains and the inclusions. At the same time, particle diameter becomes comparable to the radius of gyration of a regular polymer matrix chain. Thus, the moment of innertia of the chain and the particle become similar altering the chain dynamics substantially. For a nanofiller with the specific surface area of approximately 120 m2 /g, adding approximately 2 vol.% of nano-particles, the average interparticle distance is reduced below 2 radii of gyration, 2Rg , of the chain [41]. It has been shown [6]

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that in the case of polyvinylacetate of Mw = 9 × 104 , almost all the chains are in contact with the solid surface, possess reduced segmental mobility at temperatures T ≥ Tg when the filler-matrix internal area reaches about 42 m2 /g [42]. Below Tg , main chain segmental mobility is frozen and only secondary low temperature side chain mobility can be affected by the nano-filler. In the molten state above Tg , the conformation statistics of chains in near solid surface can be altered from Gaussian random coil to Langevin coil above Tg and this phenomenon can be transformed into the behavior of nanocomposites also upon solidification below Tg . In order to characterize the reduction in chain mobility in an entangled melt quantitatively, one can use the characteristic reptation relaxation time, τrep [43]. The τrep is given for an entangled chain as L2 ∼ NL2 , τrep ∼ = = Dc D0

(1)

where L is the length of the reptation path, N is the number of monomer units in a chain, Dc and D0 are diffusion constants of a chain and a monomer, respectively. The terminal relaxation time of a chain in a neat polymer melt can be expressed in a number of ways [43–45]. One of them is in the following form: τrep =

b 4 ζ0 N 3 , π 2 kB T aT2

(2)

where ζ0 is the monomer friction coefficient, b is the length of the statistical segment, kB is the Boltzmann constant, T is absolute temperature, aT is the effective tube diameter. In order to describe the change in reptation dynamics of the chains as a function of nanoparticle volume fraction, percolation model was used. At the percolation threshold, physical network formed by interconnection of immobilized chains on individual nanoparticles penetrates the entire sample volume. In this case, only physical ‘cross-links’ are considered and the terminal relaxation time reaches the value characteristic for the life time of the physical filler-polymer bond. Thus, the relaxation time near the percolation threshold is expressed in the form:
rec adv τcomposite = τrep νeff

∗ νeff − νeff ∗ 1 − νeff

b ,

(3)

∗ is the critical effective filler volume fraction [5]. ν ∗ is a sum of the where νeff eff filler volume fraction and the volume fraction of immobilized chains and was shown to be equal to 0.04 for PVAc-HAP nanocomposites at 90◦C. All the chains were immobilized at the internal contact area of 42 m2 per 1 g of the nano-composite.

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4 Conclusions Existing continuum mechanics models provide satisfactory means to relate structure and properties of micro-scale interphase to the stress transfer from matrix to the individual fiber. On the nano-scale, the segmental immobilization seems to be the primary mechanism in controlling properties of chains in the vicinity of nanoparticles. Reptation model and simple percolation model were used to describe immobilization of chains near solid nano-particles and to explain the peculiarities in the viscoleastic response of polymers near solid surfaces of large polymer-inclusion contact area. For a common polymer, all the chains in the composite are immobilized when the polymer-filler internal contact area reaches 42 m2 . In polymers at very low temperatures, the classical Bernoulli–Euler continuum elasticity becomes not valid below approximately 5 nm. The size of this characteristic volume increases with increasing temperature and reaches approximately 20 nm near Tg . Higher order strain elasticity along with molecular dynamics approach has to be used as the bridging law to connect behavior of the discrete matter at nano-scale with mechanical response of continuous matter at larger length scales. Acknowledgements Financial support from the Czech Ministry of Education, Youth and Sports under grant MSM 0021630501 is greatly appreciated.

References 1. Jancar J (2008), Review of the role of the interphase in the control of composite performance on micro- and nano-length scales, J. Mater. Sci., in print. 2. DiBenedetto AT (2001), Tailoring of interfaces in glass fiber reinforced polymer composites: A review, Materials Science and Engineering A 302, 74–82. 3. Jancar J (2006), Effect of interfacial shear strength on the mechanical response of polycarbonate and PP reinforced with basalt fibers, Comp. Interfaces 13, 853–864. 4. Jancar J (2006), Hydrolytic stability of PC/GF composites with engineered interphase of varying elastic modulus, Comp. Sci. Technol. 66, 3144–3152. 5. Kalfus J, Jancar J (2007), realxation processes in PVAc-HA nanocomposites, J. Polym. Sci.: Part B: Polym. Phys. 45, 1380–1388. 6. Kalfus J, Jancar J (2007), Viscoelastic response of nanocomposite poly(vynil acetate) hydroxyapatite with varying particle shape – Dynamic strain softening and modulus recovery, Polym. Compos. 28, 743–747. 7. Vacatello M (2002), Chain dimensions in filled polymers: An intriguing problem, Macromolecules 35, 8191–8193. 8. Vacatello M (2002), Molecular arrangements in polymer-based nanocomposites, Macromol. Theory Simul. 11, 757–765. 9. Vacatello M (2003), Phantom chain simulations of polymer-nanofiller systems, Macromolecules 36, 3411–3416. 10. Vacatello M (2003), Predicting the molecular arrangements in polymer-based nanocomposites, Macromol. Theory Simul. 12, 86–91. 11. Starr FW, Schröder TB, Glotzer SC (2002), Molecular dynamics simulation of a polymer melt with a nanoscopic particle, Macromolecules 35, 4481–4492.

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Appendix 1 Presentations without Paper

Concurrent and Sequential Multi-Scale Simulations of Friction and Contact Mechanics M. Müser Molecular Dynamics Simulations of the Mechanical Properties of NanotubeReinforced Composite Materials M. Griebel Nanotube and Nanocomposite Mechanics H.D. Wagner On Issues in Multi-Scale Modeling of Damage in Heterogeneous Solids R. Talreja Surface/Edge Induced Intrinsic Size-Dependent Properties of Nanowires T-Y. Zhang, M. Luo and W.K. Cha QM/MM Hybrid Simulation of Bio-Nanosystem Immobilization on Various Substrates Y-P. Zhao, Z. Yang and J. Yin Search for a Source of Cavitation in Plasticity of Crystalline Polymers A. Galeski, A. Pawlak and A. Rozanski Transforming Nanoparticles – Experiments and Modeling F.D. Fischer and D. Vollath Atomistic Description of Nanoisland Growth: Co on Single Crystal Cu Surfaces L. Diekhöner, N.N. Negulyaev, V.S. Stepanyuk, P. Bruno, P. Wahl and K. Kern Understanding Brittle Fracture in Nanostructured Silicon Carbide by Atomistic Simulations L. Colombo

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Presentations without Paper

Deformation and Failure Modes of a Single Nanofiber C.T. Lim Multiscale Modeling of Interface Fracture A. Siddiq and S. Schmauder Molecular Simulations of Deformation, Flow and Physical Aging in Glassy Solids J. Rottler

Appendix 2 Scientific Programme

Monday 19 May, 2008 K.Y. Volokh – Multiscale failure modeling: From atomic bonds to hyperelasticity with softening R.K. Kalia, A. Nakano and P. Vashishta – Multimilion-to-bilion atom molecular dynamics simulations of deformation, fracture and nanoductility in silica glass M. Müser – Concurrent and sequential multi-scale simulations of friction and contact mechanics M. Griebel – Molecular dynamics simulations of the mechanical properties of nanotube-reinforced composite materials H.D. Wagner – Nanotube and nanocomposite mechanics Ł. Figiel, F.P.E. Dunne and C.P. Buckley – Multiscale modelling of layered silicate/PET nanocomposites during solid-state processing R. Talreja – On issues in multi-scale modeling of damage in heterogeneous solids K. Jolley and S.P.A. Gill – Modelling transient heat conduction at multiple length and time scales: A coupled non-equilibrium molecular dynamics/continuum approach V.B.C. Tan, M. Deng, T.E. Tay and K.M. Lim – Multiscale modeling of amorphous materials with adaptivity P.K. Valavala and G.M. Odegard – Thermodynamically-consistent multiscale constitutive modeling of glassy polymer materials L.C. Zhang – Effective wall thickness of single-walled carbon nanotubes for multiscale analysis: the problems and a possible solution T-Y. Zhang, M. Luo and W.K. Chan – Surface/edge induced intrinsic size-dependent properties of nanowires R. Pyrz and B. Bochenek – Discrete-continuum transition in modelling nanomaterials Tuesday 20 May, 2008 B. Palosz – Looking beyond limitations of diffraction methods of structural analysis of nanocrystalline materials G. Winther – Multiscale modeling of mechanical anisotropy in deformed metals

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G. Geandier, O. Castelnau, E. Le Bourhis, P.-O. Renault and Ph. Goudeau – Micromechanical modelling of the size-dependent elastic behavior of multilayer thin films: Comparison with in situ data from X-ray diffraction J.M. Hill – Geometry and mechanics of carbon nanotubes and gigahertz nanooscillators V.A. Eremeyev and H. Altenbach – On the eigenfrequencies of an ordered system of nano-objects H.L. Duan, J. Weissmuller, Y. Wang and X. Yi – Monitoring of molecule absorption and stress evolutions by in situ microcantilever systems Y-P. Zhao, Z. Yang and J. Yin – QM/MM hybrid simulation of bio-nanosystem immobilization on various substrates E.R. Hernandez – Using thermal gradients for actuation at the nanoscale O. Sigmund – Systematic design of nano-photonic crystals and meta-materials J. Chen and S.J. Bull – Modelling of indentation and scratch damage in multilayer coatings and bulk materials F.D. Fischer and D. Vollath – Transforming nanoparticles – experiments and modelling L. Diekhöner, N.N. Negulyaev, V.S. Stepanyuk, P. Bruno, P. Wahl and K. Kern – Atomistic description of nanoisland growth: Co on single crystal Cu surfaces Wednesday 21 May, 2008 L. Colombo – Understanding brittle fracture in nanostructured silicon carbide by atomistic simulations I.N. Remediakis – Atomistic models for the mechanical response of nanomaterials C.T. Lim – Deformation and failure modes of a single nanofiber H.J. Chu, H.L. Duan, J. Wang and B.L. Karihaloo – Elastic fields in quantum dot structures with arbitrary shapes and interface effects J. Yvonnet, H. Le Quang and Q.-C. He – Thermo-mechanical numerical modelling of nano-inclusions with arbitrary shapes H.L. Duan, B.L. Karihaloo and J. Wang – Thermo-elastic size-dependent properties of nanocomposites with imperfect interfaces R.J. Young, S. Cui, I. Kinloch, Ch.C. Kao, S. Eichhorn and P. Kannan – Modelling the stress transfer between carbon nanotubes and a polymer matrix A. Siddiq and S. Schmauder – Multiscale modeling of interface fracture P. Olsson, C. Persson and S. Melin-Petersson – A study of the elastic properties of iron nanowires T. Burczynski, W. Kus and A. Mrozek – Advanced continuum-atomistic model of materials based on coupled boundary element and molecular approaches J.Y.H. Chia – Finite element modeling nanocomposites and interface effects on mechanical properties I. Goldhirsch – Small scale and/or high resolution elasticity M. Fermeglia – Enthalpic and entropic effects of nanoparticles in polymer matrices: Industrial applications

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Thursday 22 May, 2008 O.B. Naimark – Structural-scaling transitions in mesodefect ensembles and properties of bulk nanostructural materials – Modeling and experimental study J. Rottler – Molecular simulations of deformation, flow and physical aging in glassy solids T.A. Kowalewski, S. Barral and T. Kowalczyk – Modeling electrospinning of nanofibres J. Jancar – Use of reptation dynamics in modeling molecular interphase in polymer nanocomposite A. Rozanski, A. Galeski and J. Golebiewski – Low density polyethylenemontmorillonite nanocomposites for film blowing

Appendix 3 List of Participants

Bogdan Bochenek Cracow University of Technology Poland [email protected] Steve Bull University of Newcastle United Kingdom [email protected] Tadeusz Burczynski Silesian University of Technology Poland [email protected] O. Castelnau Université Paris 13, Institut Galilée France [email protected] Julian Y.H. Chia Institute of Materials Research and Engineering Singapore [email protected] Luciano Colombo University of Cagliari Italy [email protected] Huiling Duan Forschungszentrum Karlsruhe GmbH Germany [email protected]

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L. Diekhöner Aalborg University Denmark [email protected] Victor A. Eremeyev RaSci South Federeral University Russia [email protected] Maurizio Fermeglia University of Trieste Italy [email protected] Ł. Figiel University of Oxford United Kingdom [email protected] F.D. Fischer Montan universität Leoben Austria [email protected] A. Galeski Polish Academy of Sciences Poland [email protected] E. Piorkowska-Galeska Polish Academy of Sciences Poland [email protected] S.P.A. Gill University of Leicester United Kingdom [email protected] Isaac Goldhirsch Tel-Aviv University Israel [email protected]

List of Participants

List of Participants

Michael Griebel University of Bonn Germany [email protected] P. Hansson Lund Institute of Technology Sweden [email protected] E. Hernandez Institut de Ciencia de Materials de Barcelona Spain [email protected] James Hill University of Wollongong Australia [email protected] Josef Janˇcáˇr Brno University of Technology Czech Republic [email protected] L.R. Jensen Aalborg University Denmark [email protected] Rajiv K. Kalia University of Southern California USA [email protected] T.E. Karakasidis University of Thessaly Greece [email protected] Bhushan L. Karihaloo Cardiff University United Kingdom [email protected]

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L. Kolmorgen Aalborg University Denmark [email protected] Tomasz Kowalewski Polish Academy of Sciences Poland [email protected] C.T. Lim National University of Singapore Singapore [email protected] Martin Müser University of Western Ontario Canada [email protected] Oleg Naimark Russian Academy of Sciences Russia [email protected] G.M. Odegard Michigan Technological University USA [email protected] N. Olhoff Aalborg University Denmark [email protected] P. Olsson Lund Institute of Technology Sweden [email protected] Bogdan Palosz Institute of High Pressure Physics PAN Poland [email protected]

List of Participants

List of Participants

K. Pedersen Aalborg University Denmark [email protected] Ch. Persson Lund Institute of Technology Sweden [email protected] S. Melin Petersson Lund Institute of Technology Sweden [email protected] R. Pyrz Aalborg University Denmark [email protected] J.Chr. Rauhe Aalborg University Denmark [email protected] Ioannis N. Remediakis University of Crete Greece [email protected] Jörg Rottler University of British Columbia Canada [email protected] A. Rozanski Polish Academy of Sciences Poland [email protected] J. Schjødt-Thomsen Aalborg University Denmark [email protected]

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Siegfried Schmauder University of Stuttgart Germany [email protected] Ole Sigmund Technical University of Denmark Denmark [email protected] Ramesh Talreja Texas A&M University USA [email protected] Vincent B.C. Tan National University of Singapore Singapore [email protected] K.Y. Volokh Technion – Israel Institute of Technology Israel [email protected] Daniel Wagner The Weizmann Institute of Science Israel [email protected] Jianxiang Wang Peking University China [email protected] Grethe Winther Technical University of Denmark – Risø Denmark [email protected] Robert J. Young The University of Manchester United Kingdom [email protected]

List of Participants

List of Participants

J. Yvonnet Université Paris-Est France [email protected] Liangchi Zhang The University of Sydney Australia [email protected] Tong-Yi Zhang Hong Kong University of Science and Technology China [email protected] Ya-Pu Zhao Chinese Academy of Sciences China [email protected]

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Crack Initiation, Kinkind and Nanoscale Damage in Silica Glass: MultimillionAtom Molecular Dynamics Simulation Y.C. Chen, K. Nomura, Z. Lu, R. Kalia, A. Nakano and P. Vashishta, pp. 13–17.

Fig. 1 Nucleation of nanocavities and crack nanocolumns in a tensile-stress region around the pre-crack tip. The inset shows the setup of the simulation. The white rectangular plate is a rigid indenter, the light blue parallelepiped is the silica glass of dimensions 120 × 120 × 15 nm3 , and the dark blue region denotes a pre-crack of length 40 nm and width 15 nm. Dotted line indicates the direction along which the nanocavities nucleate. Damage nanocavities (red, orange, yellow, green) and nanocolumns (blue) in the tensile stress region. The impact loading speed is 0.05vR , where vR is the Rayleigh wave speed.

R. Pyrz and J.C. Rauhe (eds.), IUTAM Symposium on Modelling Nanomaterials and Nanosystems, 317–338. © Springer Science+Business Media B.V. 2009

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Fig. 2 Formation of a wing crack via growth and coalescence of nano-columns and nanocavities at an impact loading speed of 0.05vR . (a) A snapshot taken after 19 ps shows cavities (red, orange, yellow, green and dark blue) around nanocolumns (blue). (b) In the next couple of picoseconds, nanocolumns merge and coalesce with nanocavities to form a wing crack.

Fig. 3 Second healing of the wing crack at the loading speed of 0.05vR . (Right) A snapshot of the wing and primary cracks (blue) just after healing begins. The wing-crack tip is split up into two nanocolumns and there are a few damage cavities (green and red) near the tip. (Middle) In 4 ps the wing crack has receded considerably and left several cavities (red, yellow, green and blue). (Left) A snapshot of the wing crack and cavities after the crack stops healing. The residual length of the wing crack is slightly less than half of the maximum length.

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Multiscale Modelling of Layered-Silicate/PET Nanocomposites during SolidState Processing Ł. Figiel, F.P.E. Dunne and C.P. Buckley, pp. 19–26.

Fig. 1 Simulated stress-strain curves – effect of silicate loading; T = 95◦ C.

Fig. 2 Simulated deformation and contour plots representing onset of crystallization at applied strains: (A) 0.5, (B) 1; temperature: 95◦ C; applied strain rate: 1 s−1 ; legend: 1 – lock-up of viscous flow due to crystallization at all integration points of a finite element, 0 – no lock-up.

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Fig. 3 Strain amplification factor at different volume fractions; T = 95◦ C; applied strain rate: 1 s−1 .

Fig. 4 Effect of processing temperature on the nanocomposite morphology; (A) T = 100◦ C, (B) T = 110◦ C; strain rate: 1 s−1 .

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Multiscale Modeling of Amorphous Materials with Adaptivity V.B.C. Tan, M. Deng, T.E. Tay and K.M. Lim, pp. 37–42.

Fig. 1 Strain contours from multiscale simulation just before crack propagation.

Fig. 2 Strain contours at the end of multiscale simulation.

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Thermodynamically-Consistent Multiscale Constitutive Modeling of Glassy Polymer Materials P.K. Valavala and G.M. Odegard, pp. 43–51.

Fig. 1 Molecular RVEs of two polymer systems.

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Discrete-Continuum Transition in Modelling Nanomaterials R. Pyrz and B. Bochenek, pp. 63–74.

Fig. 4 Non-homogeneity measure for three shear deformation levels of the diamond sheet.

Fig. 5 Non-homogeneity measure for the real structure and the structure deformed in an affine manner (a); colour code slip vector module that indicates displacement difference between nonaffine (real) and affine displacement field (b).

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Fig. 6 Colour code atomic shear strain components for three shear deformation levels.

Fig. 8 Non-homogeneity measure at different strain levels.

Fig. 9 Colour code atomic tensile strain components at different deformation levels.

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Multiscale Modelling of Mechanical Anisotropy of Metals G. Winther, pp. 89–98.

Fig. 4 Predicted GNB planes based on two models (see text) and experimental data for grain the orientation space of rolling.

Micromechanical Modeling of the Elastic Behavior of Multilayer Thin Films; Comparison with In Situ Data from X-Ray Diffraction G. Geandier, L. Gélébart, O. Castelnau, E. Le Bourhis, P.-O. Renault, Ph. Goudeau and D. Thiaudière, pp. 99–108.

Fig. 3 Typical bilayer periodic microstructure and Finite Element mesh used for generating fullfield reference solutions. The top and thick layer is soft and made of anisotropic grains (“Cu-like” behaviour). The thin bottom layer is made of isotropic W grains.

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Fig. 4 Distribution of equivalent elastic strain in (top) the soft “Cu-like” layer, and (bottom) the stiff W layer. The applied macroscopic axial stress is 100 MPa. Results generated for a = 100. Note the different color scales for both figures.

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Monitoring of Molecule Adsorption and Stress Evolutions by In-Situ Microcantilever Systems H.L. Duan, Y. Wang and X. Yi, pp. 133–140.

Fig. 4 Variation of curvature κ with island coverage q and island size L (ϕ = 10◦ , tf = 1.4 nm, ts = 0.5 µm).

Using Thermal Gradients for Actuation in the Nanoscale E.R. Hernández, R. Rurali, A. Barreiro, A. Bachtold, T. Takahashi, T. Yamamoto and K. Watanabe, pp. 141–150.

Fig. 1 Schematic picture of the mobile element of the nanofabricated device. The long inner nanotube (shown in yellow) is suspended between the two electrodes; the gold platelet is attached to the outer nanotube (shown in red), which can slide down and rotate around the inner nanotube due to the low friction contact between nanotube walls.

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Fig. 2 Displacements of the centre of mass of the outer (17,14) nanotube along the inner (12,9) tube, under three different thermal gradients.

Fig. 3 Displacements of the centre of mass of a C60 cluster encapsulated inside a (10,10) nanotube subject to different thermal gradients.

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Fig. 4 The left panel shows the dispersion relation for the low energy phonon bands of a (10,10) SWCNT calculated with the Brenner [12] potential. The lowest optical band, shown in red, was used to construct the phonon wave packet shown on the right panel.

Systematic Design of Metamaterials by Topology Optimization O. Sigmund, pp. 151–159.

Fig. 1 The inverse homogenization problem. White arrows indicate the conventional forward homogenization approach, and black arrowe indicate the inverse homogenization approach.

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Fig. 3 Left: Topology optimized negative Poisson’s ratio materials (from [12]). Right: Topology optimized negative thermal expansion material (from [21, 22]).

Fig. 4 Top: Irreducible Brillouin zone and band diagram for periodic wave propagation problems. Bottom: Composite pictures showing topology optimized maximum band gap materials as well as the geometrically optimized partitions (from [19]).

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Thermo-Elastic Size-Dependent Properties of Nano-Composites with Imperfect Interfaces H.L. Duan, B.L. Karihaloo and J. Wang, pp. 201–209.

Fig. 1 Normalized αeT /αm as a function of C and D.

Modeling the Stress Transfer between Carbon Nanotubes and a Polymer Matrix during Cyclic Deformation C.C. Kao and R.J. Young, pp. 211–220.

Fig. 2 Shift of G’ band position with axial strain for each loading cycle.

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Finite Element Modelling Clay Nanocomposites and Interface Effects on Mechanical Properties J.Y.H. Chia, pp. 241–248.

Fig. 1 The position vectors and position system of two obliquely positioned clay particles.

Fig. 2 The position vectors and position system of two coplanar clay particles.

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Fig. 4 The predicted hydrostatic stress-strain curve of a clay/epoxy nanocomposite; the corresponding damage state in a single clay particle compared with the damage state of the entire nanocomposite; the contour plots of the growing plastic zones in the epoxy matrix and damage evolving in the clay particles.

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Multiscale Molecular Modelling of Dispersion of Nanoparticles in Polymer Systems of Industrial Interest M. Fermeglia and S. Pricl, pp. 261–270.

Fig. 2 Coarse grained modeling from atomistic model.

Fig. 3 From mesoscale morphology to FEM analysis.

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Fig. 4 Distribution of a (or B) covered nanoparticles in diblock copolymers: nanoparticles are located in the center of each domain; left: distribution of the nanoparticles with respect to the copolymer domains; right: position of the center of mass of the nanoparticles.

Fig. 5 Distribution of A and B equal coverage nanoparticles in diblock copolymers: nanoparticles are located at the interface of the domains; left: distribution of the nanoparticles with respect to the copolymer domains; right: 3D representation of the nanoparticles at the interface.

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Modeling Electrospinning of Nanofibers T.A. Kowalewski, S. Barral and T. Kowalczyk, pp. 279–292.

Fig. 2 Idealized electrostatic configuration: the potential is prescribed between the tip of the needle and a grounded infinite plane. The needle electrode is modeled as a point electrode, the charge of which is computed such as to always maintain the prescribed potential at the needle tip. Image charges are used to implement the potential condition φ = 0 on the infinite plane.

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Fig. 3 Illustration of the clustering method used in our treecode implementation: neighbor charges are recursively grouped two-by-two and the smallest enclosing spheres are calculated at each clustering level, until all charges are contained in a binary tree which root is a sphere that contains all charges.

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Fig. 4 Simulated jet paths at Ug = 10 kV (upper row) and 15 kV (bottom row), for two charge densities 10 C/m3 (left) and 20 C/m3 (right).

Author Index

Altenbach, H., 123 Bachtold, A., 141 Baowan, D., 109 Barral, S., 279 Barreiro, A., 141 Bochenek, B., 63 Buckley, C.P., 19 Bull, S.J., 161 Burczy´nski, T., 231 Castelnau, O., 99 Chen, J., 161 Chen, Y.C., 13 Chia, J.Y.H., 241 Chu, H.J., 181 Cox, B.J., 109 Deng, M., 37 Duan, H.L., 133, 181, 201 Dunne, F.P.E., 19 Dziatkiewicz, G., 231 Eremeyev, V.A., 123 Fermeglia, M., 261 Figiel, Ł., 19 Gélébart, L., 99 Górski, R., 231 Geandier, G., 99 Gierlotka, S., 75 Gill, S.P.A., 27 Goldenberg, C., 249 Goldhirsch, I., 249 Goudeau, Ph., 99 Grzanka, E., 75

He, Q.-C., 191 Hernández, E.R., 141 Hill, J.M., 109 Jancar, J., 293 Jolley, K., 27 Kalia, R., 13 Kao, C.C., 211 Karihaloo, B.L., 181, 201 Kelires, P.C., 171 Kopidakis, G., 171 Kowalczyk, T., 279 Kowalewski, T.A., 279 Ku´s, W., 231 Le Bourhis, E., 99 Le Quang, H., 191 Lim, K.M., 37 Lu, Z., 13 Melin, S., 221 Monteiro, E., 191 Mrozek, A., 231 Naimark, O.B., 271 Nakano, A., 13 Nomura, K., 13 Odegard, G.M., 43 Olsson, P.A.T., 221 Palosz, B., 75 Palosz, W., 75 Plekhov, O.A., 271 Pricl, S., 261 Proffen, T., 75 Pyrz, R., 63

339

340

Remediakis, I.N., 171 Renault, P.-O., 99 Rich, R., 75 Rurali, R., 141 Sigmund, O., 151 Stelmakh, S., 75 Takahashi, T., 141 Tan, V.B.C., 37 Tay, T.E., 37 Thamwattana, N., 109 Thiaudière, D., 99 Valavala, P.K., 43 Vashishta, P., 13

Author Index

Volokh, K.Y., 1 Wang, C.Y., 53 Wang, J., 181, 201 Wang, Y., 133 Watanabe, K., 141 Winther, G., 89 Wojdyr, M., 75 Yamamoto, T., 141 Yi, X., 133 Young, R.J., 211 Yvonnet, J., 191 Zhang, L.C., 53