Advances in Applied Mechanics

Preface

Mechanics continues to be an enabling science in many endeavors and thereby interfaces with numerous areas in the physical sciences. One of these areas is miniaturization of devices, where, together with materials science, solid mechanics plays a central role in understanding and predicting the processes during manufacturing and use. Treatment of the interfaces between different materials is among the most challenging problems in this subject area. In the present volume, Fried and Gurtin present a comprehensive, unified treatment of interfaces and their evolution. Working strictly within a continuum thermodynamic framework and exploiting the notion of configurational forces, they carefully construct the governing equations for atomic transport and deformation in the bulk and along interfaces, with due connections and improvements to existing approaches. The general theory is worked out in detail for grain boundaries, for strained solid – vapor interfaces as occur in deposition methods and for coherent phase transitions. The paper by Pozrikidis is concerned with interfaces between two fluids of different viscosities. Problems of this kind arise in materials processing, in various engineering devices and processes, and in geophysics. The problems also provide paradigms for the behavior of soft matter in biophysical applications. The role of surfactants is included in the treatment. Taken together, the two papers in this volume provide state-of-the-art perspectives on the role of interfaces in fluids and solids. We hope they will be of interest to a broad cross section of readers in the mechanical sciences. HASSAN AREF

ix

AND

ERIK

VAN DER

GIESSEN

List of Contributors

Numbers in parentheses indicate the pages on which the authors’ contributions begin.

ELIOT FRIED (1), Department of Mechanical Engineering, Washington University in St. Louis, St. Louis, MO 63130-4899, USA MORTON E. GURTIN (1), Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA C. POZRIKIDIS (179), Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA

vii

A Unified Treatment of Evolving Interfaces Accounting for Small Deformations and Atomic Transport with Emphasis on Grain-Boundaries and Epitaxy ELIOT FRIEDa and MORTON E. GURTINb a

Department of Mechanical Engineering, Washington University in St. Louis, St. Louis, MO 63130-4899, USA b

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Some Important Interface Conditions . . . . . . . . . . . . . . B. The Need for a Configurational Force Balance . . . . . . . C. A Format for the Study of Evolving Interfaces in the Presence of Deformation and Atomic Transport . D. The Normal Configurational Force Balance for a Solid– Vapor Interface . . . . . . . . . . . . . . . . . . . . . . . . E. Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

... ... ...

5 5 8

...

9

... ...

10 11

DEFORMATION AND ATOMIC TRANSPORT IN BULK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

II. Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

III. Balance Law for Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

IV. Thermodynamics. The Free-Energy Imbalance . . . . . . . . . . . . A. Chemical Potentials. Balance of Energy. Entropy Imbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Isothermal Conditions. The Free-Energy Imbalance . . . . . .

15

V. Substitutional Alloys . . . . . . . . . . . . . . . . . . . . . . . . A. Lattice Constraint. Vacancies. . . . . . . . . . . . . . . . B. Substitutional Flux Constraint. Relative Chemical Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Elimination of the Lattice Constraint . . . . . . . . . .

15 16

....... .......

17 18

....... .......

18 22

VI. Global Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

VII. Constitutive Theory for Multiple Atomic Species in the Absence of a Lattice Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Basic Constitutive Theory for an Elastic Material with Fickean Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ADVANCES IN APPLIED MECHANICS, VOL. 40 ISSN 0065-2156 DOI: 10.1016/S0065-2156(04)40001-5

1

25 25

q 2004 Elsevier Inc. All rights reserved.

E. Fried and M.E. Gurtin

2

B. C. D. E.

Consequences of the Thermodynamic Restrictions Free Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanically Simple Materials . . . . . . . . . . . . . . Cubic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

27 29 30 32

VIII. Digression: The Gibbs Relation and Gibbs – Duhem Equation at Zero Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

IX. Constitutive Theory for a Substitutional Alloy . A. Larche´ –Cahn Derivatives . . . . . . . . . . . . . B. Constitutive Equations . . . . . . . . . . . . . . . C. Thermodynamic Restrictions . . . . . . . . . . . D. Free Enthalpy. Moduli . . . . . . . . . . . . . . . E. Mechanically Simple Substitutional Alloys. F. Cubic Symmetry . . . . . . . . . . . . . . . . . . . .

.. .. .. .. .. .. ..

. . . . . . .

.. .. .. .. .. .. ..

. . . .

. . . . . . .

. . . .

. . . . . . .

.. .. .. ..

.. .. .. .. .. .. ..

. . . .

. . . . . . .

36 36 39 41 43 45 49

X. Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

CONFIGURATIONAL FORCES IN BULK . . . . . . . . . . . . . .

52

XI. Configurational Forces. Power . . . . . . . . . . . . . . . . . . . . . A. Configurational Force Balance . . . . . . . . . . . . . . . . . . . B. Migrating Control Volumes. Accretion . . . . . . . . . . . . C. Power Expended on a Migrating Control Volume RðtÞ .

. . . . . . .

. . . .

. . . .

. . . . . . .

. . . .

. . . .

XII. Thermodynamical Laws for Migrating Control Volumes. The Eshelby Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Migrational Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . B. The Eshelby Relation as a Consequence of Invariance . . . . C. Consistency of the Migrational Balance Laws with Classical Forms of these Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Isothermal Conditions. The Free-Energy Imbalance . . . . . . E. Generic Free-Energy Imbalance for Migrating Control Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52 52 53 55 56 57 58 61 61 62

XIII. Role and Influence of Constitutive Equations. . . . . . . . . . . . . .

62

INTERFACE KINEMATICS . . . . . . . . . . . . . . . . . . . . . . . . .

64

XIV. Definitions and Basic Results . . . . . . . . . . . . . . . . . . . . . . A. Curvature. Normal Velocity. Normal Time-Derivative . B. Commutator and Transport Identities . . . . . . . . . . . . . . C. Evolving Subcurves CðtÞ of SðtÞ . . . . . . . . . . . . . . . . . D. Transport Theorem for Integrals . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

64 64 67 68 70

XV. Deformation of the Interface . . . . . . . . . . . . . . . . . . . . . . . . . A. Interfacial Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Interfacial-Strain Vector . . . . . . . . . . . . . . . . . . . . . . . . . .

70 71 71

XVI. Interfacial Pillboxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

A Unified Treatment of Evolving Interfaces

3

GRAIN BOUNDARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

XVII. Simple Theory Neglecting Deformation and Atomic Transport. A. Configurational Force Balance . . . . . . . . . . . . . . . . . . . . . B. Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Free-Energy Imbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Evolution Equation for the Grain Boundary. Parabolicity and Backward Parabolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Backward Parabolicity. Facets and Wrinklings . . . . . . . . . . G. Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Digression: General Theory of Interfacial Constitutive Relations with Essentially Linear Dissipative Response . . .

74 75 77 78 81

XVIII. Interfacial Couples. Allowance for an Energetic Dependence on Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Configurational Torque Balance . . . . . . . . . . . . . . . . . . . . B. Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Free-Energy Imbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Evolution Equation for the Grain Boundary . . . . . . . . . . . .

84 86 87 88 91 91 93 94 96 96

XIX. Grain – Vapor Interfaces with Atomic Transport . . . . . . . . . . . . A. Configurational Force Balance . . . . . . . . . . . . . . . . . . . . . . B. Power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Atomic Flows due to Diffusion, Evaporation –Condensation, and Accretion. Atomic Balance . . . . . . . . . . . . . . . . . . . . . D. Free-Energy Imbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Constitutive Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Nearly Flat Interface at Equilibrium . . . . . . . . . . . . . . . . .

99 100 102 103 104

STRAINED SOLID – VAPOR INTERFACES. EPITAXY . . . .

107

XX. Configurational and Standard Forces . . . . . . . . . . . . . . . . . . . . A. Configurational Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Standard Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

108 108 109

XXI. Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. External Power Expenditures . . . . . . . . . . . . . . . . . . . . . . . B. Internal Power Expenditures. Power Balance . . . . . . . . . . .

110 110 111

XXII. Atomic Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Atomic Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Net Atomic Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112 112 114

XXIII. Free-Energy Imbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Energy Flows due to Atomic Transport. Global Imbalance . B. Dissipation Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 115 116

97 98 99

4

E. Fried and M.E. Gurtin XXIV. Normal Configurational Force Balance Revisited . . . . . A. Mechanical Potential F . . . . . . . . . . . . . . . . . . . . . . B. Substitution Alloys. Interfacial Chemical Potentials Terms of the Relative Chemical-Potentials mab . . . .

... ... ma ...

.. .. in ..

118 118

XXV. Constitutive Equations for the Interface . . . . . . . . . . . . . . . . . A. General Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Uncoupled Relations for ha and g . . . . . . . . . . . . . . . . . . .

121 121 123

XXVI. Governing Equations at the Interface . . . . . . . . . . . . . . . . A. Equations with Adatom Densities Included . . . . . . . . . B. Equations when Adatom Densities are Neglected . . . . . C. Addendum: Importance of the Kinetic Term g ¼ 2bv .

. . . .

124 124 129 134

XXVII. Interfacial Couples. Allowance for an Energetic Dependence on Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135

XXVIII. Allowance for Evaporation – Condensation . . . . . . . . . . . . . . .

140

COHERENT PHASE INTERFACES . . . . . . . . . . . . . . . . . . .

141

XXIX. Forces. Power . . . . . . . . . . A. Configurational Forces . B. Standard Forces . . . . . . C. Power . . . . . . . . . . . . .

. . . .

. . . .

.. .. .. ..

. . . .

. . . .

.. .. .. ..

. . . .

. . . .

.. .. .. ..

. . . .

. . . .

.. .. .. ..

. . . .

.. .. .. ..

. . . .

. . . .

.. .. .. ..

. . . .

. . . .

. . . .

119

. . . .

. . . .

142 142 142 143

XXX. Atomic Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

144

XXXI. Free-Energy Imbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Dissipation Inequality for Unconstrained Materials . . . . . . B. Interfacial Flux Constraint and Free-Energy Imbalance for Substitutional Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 146

XXXII. Global Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

148

XXXIII. Normal Configurational Force Balance Revisited . . . . . . . . . . .

150

XXXIV. Constitutive Equations for the Interface . . . . . . . . . . . . . . . . .

152

XXXV. General Equations for the Interface . . . . . . . . . . . . . . . . . . . .

153

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Justification of the Free-Energy Conditions (9.40) at Zero Stress. Gibbs Relation . . . . . . . . . . . . . .

154

147

154

Appendix B: Equivalent Formulations of the Basic Laws. Control-Volume Equivalency Theorem . . . . . . .

157

Appendix C: Status of the Theory as an Approximation of the Finite-Deformation Theory . . . . . . . . . . . . . . . .

160

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

172

A Unified Treatment of Evolving Interfaces

5

I. Introduction This review presents a unified treatment of several topics at the intersection of continuum mechanics and materials science; the thrust concerns processes involving evolving interfaces, focusing on grain-boundaries, solid – vapor interfaces (with emphasis on epitaxy), and coherent phase-transitions. Central to our discussion is the interaction of deformation, atomic transport, and accretion within a dissipative, dynamical framework, but as our interest is crystalline materials, we restrict our attention to small deformations.1 To avoid geometrical complications associated with surfaces in three-dimensional space, we work in two space-dimensions when discussing interfaces but in three spacedimensions when discussing the theory in bulk.

A. Some Important Interface Conditions The past half-century has seen much activity among materials scientists and mechanicians concerning interface problems, a central outcome being the realization that such problems generally result in an interface equation over and above those that follow from the classical balances for forces, moments, and mass. This extra interface condition takes a variety of forms, the most important examples being: (i)

Herring’s (1951) equation. This is an equation ! ›2 c U ¼2 cþ K; ›q2

ð1:1Þ

relating the chemical potential U of a solid – vapor interface to its curvature K: Here cðqÞ . 0 is the free energy (density) of the interface with q ¼ qðxÞ the orientation; that is, the counterclockwise angle to the interface tangent t (Fig. 14.1, page 65). Invoking an assumption of local equilibrium, Herring defines the chemical potential as the variational derivative of the total free energy with respect to variations in the configuration of the interface. Following Herring, Wu (1996), Norris (1998), and Freund (1998), generalizing earlier work of Asaro and Tiller (1972) and Rice and Chuang (1981), compute the chemical 1 Most applications in which deformation, atomic transport, and accretion are present involve small deformations. An abbreviated account of the formal analysis involved in the approximation of small deformations within a finite-deformational framework is provided in Appendix C.

E. Fried and M.E. Gurtin

6

potential of a solid – vapor interface in the presence of deformation, allowing for interfacial stress. Their result, which allows for finite deformations, is, when set within a framework of small deformations, given by U ¼ C 2 Tn·ð7uÞn 2 ðc 2 s
1ÞK 2

›t : ›s

ð1:2Þ

Here C is the bulk free energy (density); T is the bulk stress; u is the displacement; n and t are the interface normal and tangent; s denotes arc length; 1 ¼ t·ð7uÞt is the tensile strain within the interface. The result Eq. (1.2) is derived variationally; consequently, it is based on a constitutive equation c ¼ c^ð1; qÞ for the interfacial free energy, with s
and t defined by

s
¼

›c^ð1; qÞ ; ›1

t¼

›c^ð1; qÞ : ›q

(ii) Mullins’s (1956, 1957) equations. These are geometric equations 

 L c ›2 K V ¼2 2 ; r ›s2

bV ¼ cK;

ð1:3Þ

for the respective motions of an isotropic grain-boundary and an isotropic grain – vapor interface, neglecting evaporation –condensation: Here V is the (scalar) normal velocity of the grain-boundary (or interface) S; while c; b; r; and L are strictly positive constants, with c the interfacial free energy (density), b a kinetic modulus (or, reciprocal mobility), r the atomic density of the solid, and L the mobility for Fickean diffusion within S: Mullins’s argument in support of Eq. (1.3)1 is physical in nature and based on the work of Smoluchowski (1951), Turnbull (1951), and Beck (1952). To derive Eq. (1.3)2, Mullins appeals to balance of mass supplemented by Fick’s law and Herring’s equation (1.1) for the chemical potential. (iii) Kinetic Maxwell equation. This is a condition ""

C2

N X

a¼1

## a

a

r m 2 Tn·ð7uÞn

¼ bV:

ð1:4Þ

A Unified Treatment of Evolving Interfaces

7

for a propagating coherent phase interface between two phases composed of atomic species a ¼ 1; 2; …; N: Here, ma and ra are the chemical potentials and atomic densities in bulk, v f b represents the jump in a field f across the interface, and, as in Eq. (1.3)1, b is a constitutively determined kinetic modulus. The kinetic Maxwell condition was first obtained by Heidug and Lehner (1985), Truskinovsky (1987), and Abeyaratne and Knowles (1990), who ignored atomic diffusion but allowed for both inertia and finite deformations.2 Their derivations are based on determining the energy dissipation, per unit interfacial area, associated with the propagation of the interface and then appealing to the second law. When the kinetic modulus b ¼ 0; Eq. (1.4) reduces to the classical Maxwell equation ""

C2

N X

## a

a

r m 2 Tn·ð7uÞn

¼ 0;

a¼1

first derived variationally by Larche´ and Cahn (1978).3 (iv) Leo– Sekerka (1989) relation. This is a condition for an interface in equilibrium. Relying on a variational framework set forth by Larche´ and Cahn (1978) (cf. also, Alexander and Johnson, 1985; Johnson and Alexander, 1986, Leo and Sekerka consider coherent and incoherent solid – solid interfaces as well as solid –fluid interfaces, and allow for finite deformations. For an interface between a vapor and an alloy composed of atomic species a ¼ 1; 2; …; N; neglecting vapor pressure and thermal influences and assuming small deformations, the Leo – Sekerka relation takes the form N X

a¼1

ðra 2 da KÞma ¼ C 2 Tn·ð7uÞn 2 ðc 2 s
1ÞK 2

›t : ›s

ð1:5Þ

Relation (1.5) is based on a constitutive equation c ¼ c^ð1; q; d~Þ; with d~ ¼ ðd1 ; d2 ; …; dN Þ; for the interfacial free energy, supplemented by 2 Under these circumstances, the kinetic Maxwell condition is Eq. (1.4) with ma ¼ 0; a ¼ 1; 2; …; N; and with T the Piola stress. Although the first derivations ignored atomic diffusion, its inclusion is straightforward. 3 Cf. also Eshelby (1970), Robin (1974), Grinfeld (1981), James (1981), and Gurtin (1983), who neglect compositional effects.

E. Fried and M.E. Gurtin

8

the definitions

s
¼

›c^ð1; q; d~Þ ; ›1

t¼

›c^ð1; q; d~Þ ; ›q

ma ¼

›c^ð1; q; d~Þ ›da

Here da is the interfacial atomic-density of species a:

B. The Need for a Configurational Force Balance One cannot deny the applicability of the interface conditions (1.1) – (1.5); nor can one deny the great physical insight underlying their derivation. But in studying these derivations one is left trying to ascertain the status of the resulting equations (1.1) –(1.5): are they balances, or constitutive equations, or neither?4 This and the disparity between the physical bases underlying their derivation would seem to at least indicate the absence of a basic unifying principle. That additional configurational forces5 may be needed to describe phenomena associated with the material itself is clear from the seminal work of Eshelby (1951, 1956, 1970, 1975), Peach and Koehler (1950), and Herring (1951) on lattice defects. But these studies are based on variational arguments, arguments that, by their very nature, cannot characterize dissipation. Moreover, the introduction of configurational forces through such formalisms is, in each case, based an underlying constitutive framework, and hence, restricted to a particular class of materials.6 A completely different point of view is taken by Gurtin and Struthers (l990),7 who—using an argument based on invariance under observer changes— concluded that a configurational force balance should join the standard (Newtonian) force balance as a basic law of continuum physics. Here the operative word is ‘basic’. Basic laws are by their very nature independent of constitutive assumptions; when placed within a thermodynamic framework such 4 Successful theories of continuum mechanics are typically based on a clear separation of balance laws and constitutive equations, the former describing large classes of materials, the latter, particular materials. 5 We use the term configurational to differentiate these forces from classical Newtonian forces, which we refer to as standard. 6 A vehicle for the discussion of configurational forces within a dynamical, dissipative framework derives configurational force balances by manipulating the standard momentum balance, supplemented by hyperelastic constitutive relations (e.g., Maugin, 1993). But such derived balances, while interesting, are satisfied automatically whenever the momentum balance is satisfied and are, hence, superfluous. 7 This work is rather obtuse; better references for the underlying ideas are Gurtin (1995, 2000).

A Unified Treatment of Evolving Interfaces

9

laws allow one to use the now standard procedures of continuum thermodynamics to develop suitable constitutive theories. C. A Format for the Study of Evolving Interfaces in the Presence of Deformation and Atomic Transport We here develop a complete theory of evolving interfaces in the presence of deformation and atomic transport using a format based on (a) Standard (Newtonian) balance laws for forces and moments that account for standard stresses in bulk and within the interface. (b) An independent balance law for configurational forces that accounts for configurational stresses in bulk and within the interface.8 (c) Atomic balances, one for each atomic species. These balances account for bulk and surface diffusion. (d) A mechanical (isothermal) version of the first two laws of thermodynamics in the form of a free-energy imbalance. This imbalance, which accounts for temporal changes in free energy, energy flows due to atomic transport, and power expended by both standard and configurational forces, may be derived as a consequence of more typical forms of the first two laws under isothermal conditions. (e) Thermodynamically consistent constitutive relations for the interface and for the interaction between the interface and its environment. We show that each of the interface conditions in (i) – (iv) may be derived within this framework without assumptions of local equilibrium. One of the more interesting outcomes of the format we use is an explicit relation for the configurational surface tension s in terms of other interfacial fields; viz.,

s¼c2

N X

da ma 2 s
1:

ð1:6Þ

a¼1

This relation, a direct consequence of the free-energy imbalance applied to the interface, is a basic relation valid for all isothermal interfaces, independent of constitutive assumptions, and hence, of material; it places in perspective the basic difference between the configurational surface tension s and standard surface stress s
(cf. footnote 63). There is much confusion in the literature concerning surface tension s and its relation to surface free energy c: By Eq. (1.6), we see 8

As extended by Davı` and Gurtin (1990), Gurtin (1991), Gurtin and Voorhees (1993), and Fried and Gurtin (1999, 2003) to account for atomic transport.

E. Fried and M.E. Gurtin

10

that these two notions coincide if and only if standard interfacial stress as well as interfacial atomic densities are negligible. D. The Normal Configurational Force Balance for a Solid– Vapor Interface To illustrate the format described above, we consider the special case of a solid –vapor interface discussed in Part E. In this case, neglecting the vapor pressure (and hence, configurational forces exerted by the vapor on the interface), the configurational force balance for the interface takes the simple form

›c þ g ¼ Cn: ›s Here c is the configurational surface stress, g is an internal force associated with the attachment kinetics of vapor atoms at the solid surface, and C is the limit, at the interface, of the configurational stress in the solid. The tangential and normal components of c,

s ¼ c·t;

t ¼ c·n;

are the configurational surface tension and the configurational shear;9 the theory in bulk shows C to be the Eshelby tensor ! N X a a C¼ C2 r m 1 2 ð7uÞT T ð1:7Þ a¼1

(cf. Eq. (12.15)). Of most importance is the component

sK þ

›t þ g ¼ n·Cn; ›s

g ¼ g·n;

of the configurational force balance normal to interface, as this is the component relevant to the motion of the interface. For a solid –vapor interface in the presence of deformation and atomic transport, the normal configurational force balance, when combined with Eqs. (1.6) and (1.7), and the standard force, moment, and atomic balances, yields the interface condition (Fried and Gurtin, 2003) N X

a¼1

9

ðra 2 da KÞma ¼ C 2 Tn·ð7uÞn 2 ðc 2 s
1ÞK 2

›t 2 g; ›s

ð1:8Þ

Thus, in contrast to more classical discussions, the surface tension actually represents a force tangent to the interface, with no a priori relationship to surface energy.

A Unified Treatment of Evolving Interfaces

11

with s
the standard surface-tension. This balance is basic, as its derivation utilizes only basic laws; as such it is independent of material. A consequence of the theory is that, thermodynamically, the force g is conjugate to the normal velocity V of the interface, and dissipative. Within a constitutive framework, thermodynamics renders this force often, but not always, of the form g ¼ 2bV; with b $ 0 a kinetic modulus. The force g represents the sole dissipative force associated with the exchange of atoms between the solid and the vapor at the interface, the corresponding energy dissipated, per unit interfacial area, being 2gV: If we take g ; 0; then the normal configurational force balance (1.8) reduces to the Leo –Sekerka relation (1.5). The Leo –Sekerka relation follows rigorously as an Euler – Lagrange equation associated with the variational problem of minimizing the total free energy of a solid particle surrounded by a vapor. Thus, for solid – vapor interfaces in equilibrium, the format adopted here is completely consistent with results derived variationally. The Leo – Sekerka relation (or similar relations for other type of phase interfaces) is typically applied, as is, to dynamical problems, often with an accompanying appeal to an hypothesis of ‘local equilibrium’, although the precise meaning of this assumption is never spelled out. Within the more general framework leading to the normal configurational force balance (1.8), the question as to when the Leo – Sekerka relation is applicable is equivalent to the question as to when the internal force g is negligible. Our more general framework provides an answer to this question: for sufficiently small length scales the internal force g cannot be neglected, because the term emanating from g in the evolution equations for the interface is of the same order of magnitude as the other kinetic term in these equations, which results from accretion (cf. Section XXVI.C). On the other hand, for sufficiently large length scales the force g is negligible. Quantification of the terms ‘small’ and ‘large’ would require a knowledge of the kinetic modulus b: If we restrict attention to a single atomic species, neglect the adatom density, and take g ¼ 0; then the normal configurational force balance reduces to the Wu –Norris– Freund relation (1.2) with U ¼ rm: The chemical potential U of Wu, Norris, and Freund is, by its very definition, a potential associated with the addition of material at the solid – vapor interface, without regard to the specific composition of that material. As such, U cannot be used to discuss alloys.

E. Scope We begin with a discussion of the theory in bulk, as this allows for a simple presentation of basic ideas. Our discussion of substitutional alloys follows

12

E. Fried and M.E. Gurtin

Larche´ and Cahn (1985),10 who introduce a scalar constant, rsites ; that represents the density of substitutional sites, per unit volume, available for occupation by atoms. The atomic densities ra for a substitutional alloy are then required to satisfy the lattice constraint N X ra ¼ rsites ; a¼1

a constraint that Larche´ and Cahn show to have important consequences, the most important being the result that Fickean diffusion in bulk is driven not by the individual chemical potentials ma ; but instead by the relative chemical potentials mab ¼ ma 2 mb : Larche´ and Cahn arrive at this result using a variational argument. Here following the framework set forth in Section I.C, we show that this result of Larche´ and Cahn is independent of constitutive equations, as it follows directly from the bulk free-energy imbalance and the requirement that the bulk atomic-fluxes a satisfy the substitutional flux constraint N X

a ¼ 0

a¼1

˚ gren, 1982; Cahn and Larche´, 1983). (A To arrive at thermodynamically consistent equations, we follow Coleman and Noll (1963), who use the laws of continuum thermodynamics to suitably restrict constitutive equations. This process involves differentiating the constitutive relation for the bulk free energy with respect to atomic densities. For substitutional alloys any such differentiation must respect the lattice constraint. We overcome this obstacle with the aid of the Larche´ and Cahn (1985) derivative, a procedure that results in constitutive relations for the relative chemical potentials. To our knowledge, ours is the first work to combine the approach of Coleman and Noll with that of Larche´ and Cahn. We consider also unconstrained materials, which are materials whose atomic densities are unencumbered by a lattice constraint. A material whose mobile atoms are interstitial would be unconstrained, as the high density of interstitial vacancies renders a lattice constraint unimportant. More generally, some workers (cf. Mullins and Sekerka, 1985) circumvent the discussion of a lattice constraint by assuming the existence of a defect mechanism that accommodates an excess or deficiency of substitutional atoms; materials treated under such an assumption are, by fiat, unconstrained. Our discussion of interfaces is based on the format presented in Section I.C. We begin with a discussion of grain boundaries (Part D). Anisotropy often renders the 10

Mechanicians seem unaware of this work.

A Unified Treatment of Evolving Interfaces

13

underlying evolution equations backward-parabolic, and hence, unstable, leading to the formation of facets and wrinklings (Section XVII.F); we show that the use of a curvature-dependent energy (along with concomitant configurational moments) may be used to regularize the resulting evolution equations (Section XVIII). We also discuss grain – vapor interactions with atomic diffusion and evaporation – condensation, but within a more or less classical setting (Section XIX). With this as background, we turn to more general grain – vapor interactions, focusing on the derivation of equations of sufficient complexity to characterize phenomena such as molecular-beam epitaxy (Part E). We close with a discussion of coherent solidstate phase-transitions (Part F). Although worthy of discussion, other phenomena, such as incoherent phase transitions, are not included due to limitations of space.

DEFORMATION AND ATOMIC TRANSPORT IN BULK II. Mechanics We consider a homogeneous crystalline body B that occupies a region of three-dimensional space. We work within the framework of ‘small deformations’ as described by a displacement field uðx; tÞ and infinitesimal strain Eðx; tÞ related through the strain –displacement relation E¼

1 2

ð7u þ 7uT Þ:

ð2:1Þ

When we wish to emphasize its time-dependent nature, we will refer to u as a motion; the time-rate u_ of u, which represents the velocity of material points, will be referred to as the motion velocity. For convenience, we neglect inertia as it is generally unimportant in solidstate problems involving the interaction of composition and stress. We associate with each motion of B a system of forces represented by a stress (tensor) Tðx; tÞ: Given any part P of B and letting n denote the outward unit normal to ›P; Tn represents the surface traction (force per unit area) exerted on P across ›P; to simplify the presentation, we neglect external body forces. The balance laws for forces and torques then take the form ð ð Tn da ¼ 0; ðx 2 0Þ £ Tn da ¼ 0; ð2:2Þ ›P

›P

for every part P: These yield the local force and moment balances div T ¼ 0;

T ¼ TT :

ð2:3Þ

E. Fried and M.E. Gurtin

14

Given any part P; WðPÞ ¼

ð

Tn·u_ da

ð2:4Þ

›P

represents the power expended by the tractions on P: Using the moment balance _ and the force balance (2.3)1, we find that (2.3)2, which implies that T·7u_ ¼ T·E; ð _ dv: WðPÞ ¼ T·E ð2:5Þ P

III. Balance Law for Atoms Our treatment of solids is, in some respects, more complicated than descriptions usually encountered in continuum mechanics as the theory, although macroscopic, allows for microstructure by associating with each x [ B a lattice (or network) through which atoms diffuse. We consider N species of atoms, labelled a ¼ 1; 2; …; N; and let ra ðx; tÞ denote the atomic density of species a; which is the density measured in atoms Ð per unit volume. If P is a part of B; then P ra dv represents the number of atoms of a in P: Changes in the number of a-atoms in P are generally brought about by the diffusion of species a across the boundary ›P: This diffusion is characterized by an atomic flux (vector) a ðx; tÞ; measured in atoms per unit area, per unit time, Ð so that 2 ›P a ·n da represents the number of a-atoms entering P across ›P; per unit time. The balance law for atoms, therefore, takes the form ð d ð a r dv ¼ 2 a ·n da; dt P ›P

ð3:1Þ

for all species a and every part P:11 Bringing the time derivative in Eq. (3.1) inside the integral and using the divergence theorem on the integral over ›P; we find that ð

ðr_a þ div a Þdv ¼ 0; P

since P is arbitrary, this leads to a (local) balance law for atoms: for any species a;

r_a ¼ 2div a :

11

ð3:2Þ

If we multiply Eq. (3.1) by the mass of an a-atom, the resulting equation then represents a mass balance for a-atoms.

A Unified Treatment of Evolving Interfaces

15

IV. Thermodynamics. The Free-Energy Imbalance We base the theory on a free-energy imbalance that represents the first two laws of thermodynamics under isothermal conditions. In this section we derive this free-energy imbalance from versions of the first two laws appropriate for continuum mechanics.

A. Chemical Potentials. Balance of Energy. Entropy Imbalance Ð We write e ðx; tÞ for the internal energy, per unit volume, so that P e dv represents the internal energy of a part P:12 Changes in the internal energy of P are balanced by energy carried into P by atomic transport, heat transferred to P; and power expended on P: We view chemical potentials as primitive quantities that enter the theory through the manner in which they appear in the basic law expressing balance of energy. This contrasts sharply with what is done in the materials science literature, where chemical potentials are defined as derivatives of free energy with respect to composition, or introduced variationally—via an assumption of equilibrium—as Lagrange multipliers corresponding to a mass constraint; in either case the chemical potentials require a constitutive structure. To the contrary, the framework we use considers balance of energy as basic, and in a continuum theory that involves a flow of atoms through the material it is necessary to account for energy carried with the flowing atoms.13 To characterize the energy carried into parts P by atomic transport, we introduce the chemical potentials ma ðx; tÞ of the individual species a; specifically, the flow of atoms of species a; as represented by a ; is presumed to carry with it a flux of energy described by ma a ; thus 2

N ð X

a¼1

ma a ·n da

ð4:1Þ

›P

represents the net rate at which energy is carried into P by the flow of atoms across ›P: 12

We use e for internal energy and 1 for interfacial tensile strain. While it is difficult to differentiate between these symbols, it should be clear from the context which is meant. Moreover, our discussion of internal energy is limited to Section IV, where there is no mention of interfacial strain. 13 Eckart (1940), in his discussion of fluid mixtures, notes that chemical potentials should enter balance of energy through terms of the form (4.1). (Jaumann (1911) and Lohr (1917) seem also to have this view, but we are unable to fully comprehend their work.) While Eckart employs constitutive equations, their use is unnecessary. Related works are Meixner and Reik (1959), Mu¨ller (1968), Gurtin and Vargas (1971), Davi and Gurtin (1990), and Gurtin (1991).

16

E. Fried and M.E. Gurtin

The heat transferred to P is characterized by a heat flux (vector) qðx; tÞ; measured per unit area, that represents heat conduction across ›P; precisely, Ð 2 ›P q·n da represents the net heat transferred to P: Thus since the expended power is given by Eq. (2.4), balance of energy has the form N ð ð ð X d ð e dv ¼ Tn·u_ da 2 q·n da 2 ma a ·n da dt P ›P ›P a¼1 ›P

ð4:2Þ

for all parts P of B: The second law of thermodynamics is the requirement that the entropy of a part P change at a rate not less than the entropy flow into P: Parallel to our treatment of internal energy, we write the entropy of an arbitrary part P as an Ð integral P h dv with hðx; tÞ the entropy, per unit volume. We let

uðx; tÞ . 0 denote the absolute temperature and assume that, given any P; the conduction of heat induces a net transfer of entropy to P of amount ð q 2 ·n da: ›P u The second law is, therefore, represented by the entropy imbalance14 ð q d ð h dv $ 2 ·n da; dt P ›P u

ð4:3Þ

to be satisfied for all parts P:

B. Isothermal Conditions. The Free-Energy Imbalance Assume now that isothermal conditions prevail, so that u ; constant; and consider the (Helmholtz) free energy (density) defined by

C ¼ e 2 uh:

ð4:4Þ

Multiplying the entropy imbalance (4.3) by u and subtracting the result from the energy balance (4.2) then yields the free-energy imbalance N ð ð X d ð C dv # Tn·u_ da 2 ma a ·n da: dt P ›P › P a¼1

ð4:5Þ

We, henceforth, restrict our attention to isothermal processes and for that reason base the theory on the free-energy imbalance (4.5). 14

Usually referred to as the Clausius–Duhem inequality (cf. Truesdell and Toupin, 1960).

A Unified Treatment of Evolving Interfaces

17

If, in the free-energy imbalance, we bring the time derivative inside the integral and use the divergence theorem on the integral over ›P together with the expression (2.5) for the expended power, we find that ! N ð X a a _ _ C 2 T·E þ divðm  Þ dv # 0; P

a¼1

so that, since P is arbitrary,

C_ 2 T·E_ þ

N X

divðma a Þ # 0:

a¼1

Thus expanding the divergence and appealing to the atomic balance (3.1), we are led to the inequality N X C_ 2 T·E_ þ ðma r_a 2 a ·7ma Þ # 0: ð4:6Þ a¼1

The quantity def

d ¼2

N X

_ $0 ð a ·7ma 2 ma r_a Þ þ T·E_ 2 C

ð4:7Þ

a¼1

represents the dissipation per unit volume, since its integral over any part P gives the right-hand side of Eq. (4.5) minus the left. For that reason, we refer to local forms of the free-energy imbalance as dissipation inequalities.

V. Substitutional Alloys Our discussion to this point does not distinguish between substitutional and interstitial species. Here, following Larche´ and Cahn (1985, Section 2), we use the terms ‘substitutional’ and ‘interstitial’ in the following sense: “[Lattice] sites that are mostly filled are occupied by what are called substitutional atoms, while sites that are mostly vacant are occupied by interstitial atoms.” The high density of interstitial vacancies renders a corresponding lattice constraint unimportant.15 15 We do not allow for interstitial defects, which are substitutional atoms forced into interstitial positions, and which are, hence, incompatible with the lattice constraint. According to DeHoff (1993, p. 411): “At the same temperature it can be expected that the concentration of interstitial defects is very much smaller (usually by several orders of magnitude) than that of vacancies at equilibrium.” Allowance for interstitial defects may be important when considering neutron irradiation. (DeHoff, 1993, p. 411) or deformation (Shewmon, 1969, p. 47). Finally, DeHoff (1993, p. 406) notes that: “Even in the extreme, near the melting point, defect sites occur at only about one site in 10,000… Nonetheless, this small fraction of defect sites plays a crucial role in materials science.” Some workers (cf. Mullins and Sekerka, 1985) circumvent the discussion of a lattice constraint by assuming the existence of a defect mechanism that accommodates an excess or deficiency of substitutional atoms.

18

E. Fried and M.E. Gurtin

This section is concerned solely with substitutional alloys, neglecting the presence of interstitials.

A. Lattice Constraint. Vacancies We introduce a scalar constant rsites that represents the density of substitutional sites, per unit volume, available for occupation by atoms. We restrict attention to substitutional alloys, so that the atoms are constrained to lie on lattice sites.16 A minor abuse of terminology allows for vacancies (unoccupied substitutional sites) within the same framework as the theory without vacancies: when vacancies are to be considered, we reserve the label ‘v’ of one substitutional species for vacancies,17 so that rv ðx; tÞ represents the vacancy density, v ðx; tÞ the vacancy flux, and mv ðx; tÞ the chemical potential for vacancies. Then, whether or not vacancies are being considered, the substitutional densities must be consistent with the lattice constraint N X ra ¼ rsites : ð5:1Þ a¼1

A consequence of the lattice constraint is conservation of substitutional atoms, N X r_a ¼ 0; ð5:2Þ a¼1

a condition that, by virtue of the local atomic balance (3.2), is equivalent to the diffusional constraint N X div a ¼ 0: ð5:3Þ a¼1

B. Substitutional Flux Constraint. Relative Chemical Potentials 1. Importance of Relative Chemical Potentials A restriction stronger than the diffusional constraint (5.3) is the substitutional flux constraint N X

a ¼ 0

a¼1

16 17

Our discussion of the lattice constraint follows Larche´ and Cahn (1985, Section II). Thus ‘all atoms’ means ‘all atoms and vacancies,’ and so forth.

ð5:4Þ

A Unified Treatment of Evolving Interfaces

19

Fig. 5.2.1. Schematic of an atom–vacancy exchange.

˚ gren (1982) and Cahn and Larche´ (1983), who argue that Eq. (5.4) discussed by A is a consequence of the requirement that diffusion, as represented by atomic fluxes, arises, microscopically, from exchanges of atoms or exchanges of atoms with vacancies (Fig. 5.2.1). Flux Hypothesis for Substitutional Alloys: we assume, henceforth, that the substitutional flux constraint is satisfied. Essential to the treatment of substitutional alloys are the relative chemical potentials defined by

maz ¼ ma 2 mz :

ð5:5Þ

Direct consequences of this definition are the identities

maa ¼ 0;

mab ¼ 2mba ;

mab ¼ maz 2 mbz :

ð5:6Þ

The next result is fundamental to the discussion of substitutional alloys. Theorem (relative chemical potentials) Given any choice of reference species z, we may, without loss in generality, replace the free-energy imbalance (45) with that obtained by replacing each chemical potential ma by the corresponding relative chemical potential maz : N ð ð X d ð C dv # Tn·u_ da 2 maz a ·n da: dt P ›P a¼1 ›P

ð5:7Þ

To establish this result, we first show that the free-energy imbalance (4.5) is invariant under all transformations of the form

ma ðx; tÞ ! ma ðx; tÞ þ lðx; tÞ

for all species a;

ð5:8Þ

with lðx; tÞ independent of a: In view of the substitutional flux constraint, given any such field lðx; tÞ; N X

a¼1

ðma þ lÞ a ¼

N X

a¼1

ma a þ l

N X

a¼1

a ¼

N X

a¼1

ma a ;

ð5:9Þ

E. Fried and M.E. Gurtin

20

and hence, N ð ð X d ð C dn # Tn·u_ da 2 ðma þ lÞ a ·n da: dt P ›P › P a¼1

ð5:10Þ

Thus the free-energy imbalance is invariant under the transformation (5.8). The specific choice l ¼ 2mz in Eq. (5.10) yields the desired result (5.7). This completes the proof of the theorem. The free-energy imbalance (5.7), when localized, yields the dissipation inequality18

C_ 2 T·E_ 2

N X

ðmaz r_a 2 a ·7maz Þ # 0;

ð5:11Þ

a¼1

which is to hold for any given choice of z: This inequality will be useful in developing a suitable constitutive theory for substitutional alloys.

2. Remarks (1) Of the basic laws, it is only the free-energy imbalance that involves chemical potentials. We may, therefore, conclude from the theorem on relative chemical potentials that the individual chemical potentials are irrelevant to the theory in bulk. At external or internal boundaries, however, it is often the individual chemical potentials that are needed, a specific example being a solid – vapor interface (cf. Section XXIV.B as well as Larche´ and Cahn (1985)). (2) The free-energy imbalance (5.7) and the dissipation inequality (5.11) may be written with the chemical potentials expressed relative to that of any arbitrarily chosen species z; in which case both Eqs. (5.7) and (5.11) are independent of rz and z : (3) Larche´ and Cahn (1973, 1985) were apparently the first to emphasize the importance of the relative chemical potentials when discussing substitutional alloys. Specifically, Larche´ and Cahn (1973) consider a variational problem that, within our framework, consists in minimizing a body’s free energy under a mass constraint for each atomic species. Larche´ and Cahn define the chemical potentials ma ; a ¼ 1; 2; …; N; to be the Lagrange multipliers associated with the mass constraints; they show that only the 18

When dealing with relative chemical potentials, we will often encounter expressions such as Eq. (5.11), in which z appears as a free-index.

A Unified Treatment of Evolving Interfaces

21

relative chemical potentials ma 2 mb enter the corresponding equilibrium conditions. (4) Like the pressure in an incompressible body, the individual chemical potentials are indeterminate in bulk. (5) One might refer to invariance of the free-energy imbalance under all transformations of the form

ma ðx; tÞ ! ma ðx; tÞ þ lðx; tÞ

for all species a

as invariance of the lattice chemistry. As is clear from the proof of the theorem on relative chemical potentials, invariance of the lattice chemistry is equivalent to the conclusions of that theorem. Moreover, as we shall show, invariance of the lattice chemistry is equivalent to the substitutional flux constraint, so that we could equally well have taken—as our starting hypothesis—invariance of the lattice chemistry rather than the flux hypothesis for substitutional alloys. In view of Eqs. (5.8) – (5.10), to prove our assertion of equivalence we have only to show that invariance of the lattice chemistry implies the substitutional flux constraint. Indeed, if Eq. (5.10) holds for all fields l and all parts P: Then N ð X

a¼1

l a ·n da ¼ 0;

ð5:12Þ

›P

for otherwise we could choose l to violate Eq. (5.10). Thus ð

lz·n da ¼ 0; ›P

z¼

N X

a

ð5:13Þ

a¼1

for all fields l and all parts P: A standard argument in the calculus of variations then implies that z ; 0: (6) Invariance of the lattice chemistry has the following physical interpretation. Roughly speaking, the chemical potential of a given species at a point x represents the energy the body would gain, per unit time, were we to add one atom, per unit time, of that species at x. Because of the lattice constraint, adding an atom A of a given species involves removing an atom B of that or another species. Thus, increasing the chemical potential of each species by the same amount should not affect the free-energy imbalance, because the marginal increase in energy associated with the addition of A would be balanced by the marginal decrease associated with the removal of B.

E. Fried and M.E. Gurtin

22

C. Elimination of the Lattice Constraint Because of the lattice constraint (5.1), we may omit the atomic balance for the substitutional species z; say, and simply define N X rz ¼ rsites 2 ra : ð5:14Þ a¼1 a–z

Thus and by the substitutional flux constraint (5.4), N N X X r_z ¼ 2 r_a ; z ¼ 2 a ; a¼1 a–z

a¼1 a–z

so that the atomic balance for species z is satisfied automatically provided the atomic balances for each remaining species a – z are satisfied. In view of this discussion, we may, without loss in generality, use the following normalization in which a given species z is used as reference: † We consider the atomic density rz and the atomic flux z defined by the lattice constraint and the substitutional flux constraint, respectively. † We omit the atomic balance law for the species z: † We use as chemical potentials for the species a the relative chemical potentials maz : † We use the free-energy imbalance and dissipation inequality (5.11) with z as reference (since these are independent of rz and z Þ: As we shall see, for solid – vapor interfaces with interfacial atomic transport, the absence of a lattice constraint at the interface renders this normalization of little use (cf. Section XXIV.B).

VI. Global Theorems Granted appropriate boundary conditions, the atomic balance ð d ð a r dv ¼ 2 a ·n da dt B ›B

ð6:1Þ

(cf. Eq. (2.4)) and the free-energy imbalance ! N ð X d ð a a C dv # Tn·u_ 2 m  ·n da dt B ›B a¼1

ð6:2Þ

A Unified Treatment of Evolving Interfaces

23

(cf. Eqs. (3.1) and (4.5)) applied to the body itself yield important global conservation and decay relations. Such relations are important for two reasons: they suggest variational principles appropriate to a discussion of equilibrium; and they are useful for establishing a priori estimates and, hence, results concerning the existence and qualitative properties of solutions to initial-boundary-value problems. With a view toward establishing such global relations, we introduce the following definitions. Let A be a subsurface of ›B: We say that (i)

A is fixed if u_ ¼ 0

on A;

(ii) A is subject to dead loads if there is a constant symmetric (stress) tensor Tp such that Tn ¼ Tp n

on A;

(iii) A is impermeable if, for each atomic species a; a ·n ¼ 0

on A;

(iv) (for unconstrained materials) A is in chemical equilibrium if, for each atomic species a; there is a constant chemical potential map such that

ma ¼ map

on A;

(iv0 ) (for substitutional alloys) A is in chemical equilibrium if, for some fixed choice of species z and any other species a; there is a constant relative chemical potential maz p such that

maz ¼ maz p

on A:

(When A separates the solid from a vapor, the boundary values maz p would be given by the corresponding difference in vapor potentials: a z maz p ¼ mv 2 mv :) A direct consequence of (i) and (iii) with A ¼ ›B; Eqs. (6.1) and (6.2) is the Theorem for an isolated body If the body is isolated, that is if ›B is fixed and impermeable, then the total number of atoms of each species remains fixed, while the total free energy is nonincreasing: d ð a r dv ¼ 0; dt B

a ¼ 1; 2; …; N;

d ð C dv # 0: dt B

E. Fried and M.E. Gurtin

24

For a nonisolated body under sufficiently simple boundary conditions one can still prove a global decay relation for a physically meaningful integral. Global decay theorem Assume that a portion A of ›B is fixed and the remainder, ›B w A; subject to dead loads. (a) If ›B is impermeable, then d ð a r dv ¼ 0; a ¼ 1; 2; …; N; dt B d ð ðC 2 Tp ·EÞdv # 0: dt B (b) If a portion E of ›B is impermeable and the remainder, ›B w E; in chemical equilibrium, then ! N X d ð a a C 2 Tp ·E 2 mp r dv # 0; dt B a¼1 if the material is unconstrained, while ! N X d ð az a C 2 Tp ·E 2 mp r dv # 0 dt B a¼1 if the material is a substitutional alloy. To prove this theorem, note first that, by hypothesis, ð

Tn·u_ da ¼ ›B

ð

Tp n·u_ da ¼

›B

¼

d ð T ·E dv; dt B p

d ð d ð Tp n·u da ¼ T ·7u dv dt ›B dt B p ð6:3Þ

hence, assertion (a) follows from Eqs. (6.1) and (6.2), and the stipulated boundary condition a ·n ¼ 0 on ›B for all a: To establish part (b) of the theorem, consider an unconstrained material. Since, by (iv), for each a; ma on A ¼ ›B w E has the constant value map ; while a ·n ¼ 0 on ›B; we may use the atomic balance (6.1) to conclude that ( N ) 

ð N ð N ð X X d X a a a d a a a m  ·n da ¼ mp r dv ¼ m r dv : ð6:4Þ 2 dt B dt a¼1 B p a¼1 ›B a¼1

A Unified Treatment of Evolving Interfaces

25

Similarly, for a substitutional alloy we may use the substitutional flux constraint (5.4) and (iv0 ) to conclude that  N ð N ð N

X X X d ð az a 2 ma a ·n da ¼ 2 maz a ·n da ¼ mp r dv : ð6:5Þ dt B a¼1 ›B a¼1 ›B a¼1 Assertion (b) follows from Eqs. (6.2) – (6.5).

VII. Constitutive Theory for Multiple Atomic Species in the Absence of a Lattice Constraint The force and moment balances, the balance law for atoms, and the freeenergy imbalance are basic laws, common to large classes of materials; we keep such laws distinct from specific constitutive equations, which differentiate between particular materials. We view the dissipation inequality as a guide in the development of suitable constitutive theories. In this regard we do not seek general constitutive equations consistent with the dissipation inequality, but instead we begin with constitutive equations close to those upon which more classical theories are based.

A. Basic Constitutive Theory for an Elastic Material with Fickean Diffusion Guided by the dissipation inequality (4.6) and by standard theories of elasticity and diffusion, we assume that the free energy, stress, and chemical potential are prescribed functions of the strain and the list def

r~ ¼ ðr1 ; r2 ; …; rN Þ of atomic densities,

C ¼ C^ ðE; r~Þ;

^ T ¼ TðE; r~Þ;

ma ¼ m^a ðE; r~Þ;

ð7:1Þ

and that the atomic flux is given by Fick’s law a ¼ 2

N X

Mab ðE; r~Þ7mb ;

ð7:2Þ

b¼1

with Mab ðE; r~Þ the mobility tensor for species a with respect to species b: Such a ‘mixed’ description with ma as independent variables in Eq. (7.1) and 7ma as dependent variables in Eq. (7.2) is widely used by materials scientists

E. Fried and M.E. Gurtin

26

^ m^a ; and Mab (Larche´ and Cahn, 1985, Section VIII.A).19 The functions C^ ; T; represent constitutive response functions for the material. The constitutive equation (7.2) is simple in form but complicated in nature, as each of the mobilities Mab ðE; r~Þ is a second-order tensor reflecting the underlying symmetry of the material. The mobilities can be arranged in a matrix array 2

M11

6 6 21 6M 6 6 . 6 . 6 . 4 MN1

M12

···

M1N

M22 .. .

··· .. .

M2N .. .

MN2

···

MNN

3 7 7 7 7 7 7 7 5

ð7:3Þ

with tensor entries, but it should be kept in mind that, since each mobility tensor has nine components, Eq. (7.3) represents 9N 2 scalar constitutive moduli. We refer to Eq. (7.3) as the mobility matrix. Following the procedure of Coleman and Noll (1963), we require that the dissipation inequality Eq. (5.11) hold in all ‘processes’ related through the constitutive equations (7.1) and (7.2); equivalently, (

) ( ) N ^ ðE; r~Þ X ›C^ ðE; r~Þ › C a ^ 2 TðE; r~Þ ·E_ þ 2 m^ ðE; r~Þ r_a a ›E › r a¼1 2

N X

7ma ·Mab ðE; r~Þ7mb # 0;

ð7:4Þ

a;b¼1

_ ra ; with ð›C^ =›EÞij ¼ ›C^ =›Eij : We can always find fields u and r~ such that E; E; a a r_ ; and 7m (for each aÞ have arbitrarily prescribed values at some ðx; tÞ: Thus, since r_a and E_ appear linearly in Eq. (7.4), their ‘coefficients’ must vanish, for _ may be chosen to violate Eq. (7.4). This leaves the inequality otherwise r_a and E PN a ab b a;b¼1 7m ·M ðE; r~Þ7m $ 0: Therefore, as thermodynamic restrictions, the free energy must determine the stress and the chemical potentials through 19 We, therefore, do not adhere to the principle of equipresence, as discussed by Truesdell and Toupin (1960) and Truesdell and Noll (1965), which asserts that “a quantity present as an independent variable in one constitutive equation should be so present in all, unless…its presence contradicts some law of physics or rule of invariance.” According to Truesdell and Noll (1965, Section 96), “This principle forbids us to eliminate any of the ‘causes’ present from interacting with any other as regards a particular ‘effect.’ It reflects on the scale of gross phenomena the fact that all observed effects result from a common structure such as the motions of molecules.” A general treatment consistent with equipresence is provided by Fried and Gurtin (1999). Our results concur with theirs and, hence, equipresence provided the relation between the chemical potentials and atomic densities is invertible.

A Unified Treatment of Evolving Interfaces

27

the ‘state relations’

›C^ ðE; r~Þ ^ ; r~Þ ¼ TðE; ›E

m^a ðE; r~Þ ¼

›C^ ðE; r~Þ ; ›r a

ð7:5Þ

and the mobility matrix (7.3) must be positive semi-definite.20 N X

aa ·Mab ðE; r~Þab $ 0;

ð7:6Þ

a;b¼1

for all vector-lists a~ ¼ ða1 ; a2 ; …; aN Þ: Reversing this argument we see that the restrictions (7.5) and (7.6) are sufficient that all process related through the constitutive equations (7.1) and (7.2) obey the dissipation inequality (5.11) (irrespective of whether or not the condition specified in footnote 20 is satisfied). Thus, for a single species, ·7m # 0; asserting that atoms flow down a gradient in chemical-potential. More generally, note that the dissipation (4.7), which is the negative of the left-hand side of Eq. (7.4), is given by

d¼

N X

7ma ·Mab ðE; r~Þ7mb $ 0:

a;b¼1

B. Consequences of the Thermodynamic Restrictions Immediate consequences of Eq. (7.5) are the Maxwell relations

›T^ ›m^a ¼ a ›r ›E

ð7:7Þ

C_ ¼ T·E_ þ ma r_a :

ð7:8Þ

and the Gibbs relation21

It is convenient to define scalar and tensor moduli 9 ^ ›TðE; r~Þ ›2 C^ ðE; r~Þ > ¼ ;> CðE; r~Þ ¼ > = ›E ›E 2 ^ > ›TðE; r~Þ ›m^a ðE; r~Þ ›2 C^ ðE; r~Þ > ; Aa ðE; r~Þ ¼ ¼ ¼ :> a ›r ›E ›E › ra

ð7:9Þ

At least when the set of ðE; r~Þ at which the matrix with entries ›m^a ðE; r~Þ=›rb is invertible is dense in the space of all ðE; r~Þ: 21 In the materials literature (cf., e.g., Caroli et al. (1984)) the Gibbs relation is generally a postulate rather than a consequence of the underlying thermodynamical development. 20

E. Fried and M.E. Gurtin

28

We refer to C as the elasticity tensor and to Aa as stress-composition (or chemistry-strain) tensors for a: The elasticity tensor C is a symmetric linear transformation of symmetric tensors into symmetric tensors; that is, C associates with each symmetric tensor U a symmetric tensor H ¼ C½U (or, more precisely, H ¼ CðE; r~Þ½UÞ: For each atomic species a; Aa is a symmetric tensor that represents the marginal increase in stress due to an incremental increase in the atomic density ra ; holding the other densities and the strain fixed, or equivalently, the marginal increase in ma due to an incremental increase in the strain holding the densities fixed. The elasticity tensor has components Cijkl ¼

›2 C^ ›Eij ›Ekl

and for symmetric tensors H and U; H ¼ C½U has the component form Hij ¼ Cijkl Ukl ; with components that satisfy Cijkl ¼ Cklij ¼ Cijlk :

ð7:10Þ

Because of these symmetries, there are at most 21 independent elastic moduli. ^ For E ¼ Eðx; tÞ; r~ ¼ r~ðx; tÞ; and T ¼ TðEðx; tÞ; r~ðx; tÞÞ; the definitions (7.9) of the elasticity and stress-composition tensors are consistent with the chain-rule calculation _ ¼ CðE; r~Þ½E _ þ T

N X

Aa ðE; r~Þr_a :

ð7:11Þ

a¼1

Note that, by Eq. (7.9)2, Fick’s law becomes 0 1 N N 2 ^ X X › C ðE; r ~ Þ g b A Mab ðE; r~Þ@ a ¼ 2 b ›rg 7r þ A ðE; r~Þ7E ; › r b¼1 g¼1

ð7:12Þ

where, using Cartesian components, def

ðAb 7EÞj ¼ Abkl

›Ekl ; ›x j

ð7:13Þ

so that jai

¼

N X

0

N X 2 Mijab ðE; r~Þ@ b¼1 g¼1

1 ›2 C^ ðE; r~Þ ›rg ›Ekl A b þ Akl ðE; r~Þ : ›x j ›rb ›rg ›x j

ð7:14Þ

Thus both density gradients and strain gradients may drive atomic diffusion.

A Unified Treatment of Evolving Interfaces

29

C. Free Enthalpy It is often more convenient to use stress rather than strain as an independent variable. As is reasonable within the context of small elastic deformations, we ^ assume that TðE; r~Þ is a smoothly invertible function of E with inverse ~ E ¼ EðT; r~Þ; we may then define new functions for the free energy and the chemical potential through ~ C ¼ C~ ðT; r~Þ ¼ C^ ðEðT; r~Þ; r~Þ; ~ ma ¼ m~a ðT; r~Þ ¼ m^a ðEðT; r~Þ; r~Þ: With stress as independent variable, it is most convenient to work with the (Gibbs) free-enthalpy (density) defined by the Legendre transformation

F ¼ C 2 T·E

ð7:15Þ

and given by the constitutive response function ~ F~ ðT; r~Þ ¼ C~ ðT; r~Þ 2 T·EðT; r~Þ:

ð7:16Þ

(We consistently use a ‘tilde’ to denote a function of ðT; r~Þ; retaining the ‘hat’ for a function of ðE; r~Þ:Þ Then, using the chain-rule and the restrictions (7.5), a straightforward calculation shows that

›F~ ðT; r~Þ ~ EðT; ; r~Þ ¼ 2 ›T

m~a ðT; r~Þ ¼

›F~ ðT; r~Þ ; ›ra

ð7:17Þ

a direct consequence of which is the Maxwell relation

›E~ ›m~a : ¼2 a ›r ›T We can also define moduli analogous to those of Eq. (7.9): 9 ~ ›EðT; r~Þ ›2 C~ ðT; r~Þ > > KðT; r~Þ ¼ ¼2 ;> = ›T ›T2 ~ ›EðT; r~Þ ›m~a ðT; r~Þ > > ; ;> Na ðT; r~Þ ¼ ¼2 a ›r ›T

ð7:18Þ

ð7:19Þ

with K the compliance tensor and to Na the strain-composition (or chemistrystrain) tensor for a: The tensor Na represents the marginal increase in stress due to an incremental increase in the atomic density ra ; holding the other densities and the stress fixed, or equivalently, the marginal increase in ma due to an incremental increase in the stress holding the densities fixed.

E. Fried and M.E. Gurtin

30

~ For T ¼ Tðx; tÞ; r~ ¼ r~ðx; tÞ; and E ¼ EðTðx; tÞ; r~ðx; tÞÞ; the definitions (7.19) of the compliance and chemistry-strain tensors are consistent with the chain-rule calculation _ þ E_ ¼ KðT; r~Þ½T

N X

Na ðT; r~Þr_a :

ð7:20Þ

~ for E ¼ EðT; r~Þ

ð7:21Þ

a¼1

The compliance tensor K obeys KðT; r~Þ ¼ CðE; r~Þ21

(i.e., Hij ¼ Cijkl Ukl if and only if Uij ¼ Kijkl Hkl Þ; K; therefore, has symmetries analogous to those displayed in Eq. (7.10). Differentiating the identity ^ EðT; ~ Tð r~Þ; r~Þ ¼ T with respect to ra ; we arrive at the important relation Na ¼ 2K½Aa 

ð7:22Þ

in which, for convenience, we have omitted arguments. Thus, 9 C; Aa ða ¼ 1; 2; …; NÞ are independent of strain and composition > > = if and only if > > ; K; Na ða ¼ 1; 2; …; NÞ are independent of stress and composition:

ð7:23Þ

Finally, we note that the free energy and the chemical potentials at zero stress,

C0 ðr~Þ ¼ C~ ðT; r~ÞlT¼0 ;

ma0 ðr~Þ ¼ m~a ðT; r~ÞlT¼0 ;

ð7:24Þ

are, by Eqs. (7.16) and (7.17), related through

ma0 ðr~Þ ¼

›C0 ðr~Þ : › ra

ð7:25Þ

D. Mechanically Simple Materials We refer to a material as being mechanically simple if: the elasticity tensor C and stress-composition tensors Aa are independent of strain and composition (cf. Eq. (7.23)); (ii) the mobilities Mab are independent of strain.

(i)

Assumption (i) has strong consequences. Since C and Aa are independent of E ^ ›ra ; from an arbitrary reference list r~0 to r~ and r~; we may integrate Aa ¼ ›T= ^ and then use the relation C ¼ ›T=›E; the result is an equation for the stress of

A Unified Treatment of Evolving Interfaces

31

the form T ¼ C½E þ

N X

Aa ðra 2 ra0 Þ:

ð7:26Þ

a¼1

Next, to obtain the free energy, we integrate the relation ›C^ =›E ¼ T using Eq. (7.26); the result is the relation

C¼

N X 1 E·C½E þ ðra 2 ra0 ÞAa ·E þ Fðr~Þ; 2 a¼1

ð7:27Þ

which, when differentiated with respect to the density ra ; yields an expression

ma ¼

›Fðr~Þ þ Aa ·E ›ra

ð7:28Þ

for the chemical potential ma : Next, using Eq. (7.22), we may explicitly invert Eq. (7.26) to obtain a relation E ¼ K½T þ

N X

ðra 2 ra0 ÞNa

ð7:29Þ

a¼1

for E in terms of T and r~: Then, by Eqs. (7.26), (7.27) and (7.29), ! N N X 1 1 X a a a C 2 Fðr~Þ ¼ E· C½E þ ðr 2 r0 ÞA þ ðra 2 ra0 ÞAa ·E 2 2 a¼1 a¼1 ! N N X 1 1 X a a a K½T þ ðr 2 r0 ÞN ·T þ ðra 2 ra0 ÞAa ·E ¼ 2 2 a¼1 a¼1 ¼

N 1 1 X T·K½T þ ðra 2 ra0 Þðrb 2 rb0 ÞAa ·Nb 2 2 a;b¼1

ð7:30Þ

and it follows that

C¼

1 T·K½T þ C0 ðr~Þ 2

ð7:31Þ

and, hence, that Fðr~Þ is related to C0 ðr~Þ; the free energy at zero stress, through Fðr~Þ ¼ C0 ðr~Þ 2

N 1 X ðra 2 ra0 Þðrb 2 rb0 ÞAa ·Nb : 2 a;b¼1

ð7:32Þ

32

E. Fried and M.E. Gurtin Next, using Eqs. (7.29) and (7.31) in Eq. (7.16), we find that N X 1 F ¼ 2 T·K½T 2 ðra 2 ra0 ÞNa ·T þ C0 ðr~Þ; 2 a¼1

ð7:33Þ

thus, by Eqs. (7.17)2 and (7.25), the chemical potentials may be expressed alternatively as

ma ¼ ma0 ðr~Þ 2 Na ·T:

ð7:34Þ

Turning to Fick’s law (7.12), since the mobility is also independent of strain, we see that, by Eq. (7.28), 0 1 N N 2 X X › Fð r ~ Þ a ab g b M ðr~Þ@ 7r þ A 7EA: ð7:35Þ  ¼2 b g b¼1 g¼1 ›r ›r Alternatively, appealing to Eqs. (7.2) and (7.34), 0 1 N N 2 X X › C ð r ~ Þ 0 a ¼ 2 Mab ðr~Þ@ 7rg 2 Nb 7TA: b g b¼1 g¼1 ›r ›r

ð7:36Þ

Thus, spatial variations of either strain or stress may drive atomic diffusion.

E. Cubic Symmetry A special but important class of materials consists of those with cubic symmetry. Here, we consider the ramifications of cubic symmetry for mechanically simple materials. The elasticity tensor for a cubic material involves only three independent elasticities, as discussed in Gurtin (1972, page 88). The compliance tensor admits a similar representation. Moreover, the tensors Ma ; Aa ; Na are isotropic: Ma ¼ ma 1;

Aa ¼ aa 1;

Na ¼ ha 1:

ð7:37Þ

Further, because C½1 is a (second-order) tensor, it must be isotropic, and hence, of the form C½1 ¼ 3k1;

ð7:38Þ

with k; the compressibility. By Eq. (7.22), Aa ¼ 2C½Na ; the moduli aa and ha are, therefore, related through the compressibility k via aa ¼ 23kha :

ð7:39Þ

A Unified Treatment of Evolving Interfaces

33

In view of Eqs. (7.37)2,3 and (7.38), the free energy (Eq. (7.27)) and free enthalpy (Eq. (7.33)) specialize to 9 N > X 1 > C ¼ E·C½E þ aa ðra 2 ra0 Þtr E þ Fðr~Þ; > > > = 2 a¼1 > N X > 1 > F ¼ 2 T·K½T 2 ha ðra 2 ra0 Þtr T þ C0 ðr~Þ; > > ; 2 a¼1

ð7:40Þ

where, by Eqs. (7.32) and (7.39), Fðr~Þ ¼ C0 ðr~Þ þ

N 9 X kha hb ðra 2 ra0 Þðrb 2 rb0 Þ: 2 a;b¼1

ð7:41Þ

We, therefore, have the equivalent sets of relations: T ¼ C½E þ

N X

a

a

a ðr 2

›Fðr~Þ m ¼ þ aa tr E; ›r a

ra0 Þ1;

a

a¼1

E ¼ K½T þ

N X

9 > > > > > =

> > > ma ¼ ma0 ðr~Þ 2 ha tr T; > > ;

ha ðra 2 ra0 Þ1;

a¼1

ð7:42Þ

with ma0 given by Eq. (7.25). Note that we may write the stress as T ¼ C½E 2 Ecom ;

def

Ecom ¼

N X

ha ðra 2 ra0 Þ1:

ð7:43Þ

a¼1

We refer to Ecom as the compositional strain and to ha as the solute-expansion modulus for species a: Since we would generally expect the body to expand when atoms are added, we should have

ha . 0;

aa , 0;

ð7:44Þ

where the second inequality follows from Eq. (7.39), assuming that k . 0: Granted Eq. (7.44), if the body is, instead, constrained to have vanishing strain, then, by Eq. (7.42)2, the resulting compositional stress would be aa ðra 2 ra0 Þ1 and compressive when atoms are added.

E. Fried and M.E. Gurtin

34

By Eq. (7.37)1, the alternative expressions (7.35) and (7.36) of Fick’s law become 0 1 9 N N 2 > X X > › Fð r ~ Þ > a ¼ 2 mab ðr~Þ@ 7rg þ ab 7 tr EA; > > b g > = b¼1 g¼1 ›r ›r ð7:45Þ 0 1 > N N > 2 X X > › C ð r ~ Þ 0 > a ¼ 2 mab ðr~Þ@ 7rg 2 hb 7 tr TA: > > b ›rg ; › r b¼1 g¼1

VIII. Digression: The Gibbs Relation and Gibbs– Duhem Equation at Zero Stress Consider the free energy C0 ðr~Þ and the chemical potentials ma0 ðr~Þ; at zero stress, as defined in Eq. (7.24). Our derivation of the Gibbs– Duhem equation at zero stress utilizes the atomic density, atomic volume, and concentrations: def

r¼

N X

ra ;

def

V¼

a¼1

1 ; r

def

ca ¼ Vra :

The mass density rm is related to the atomic masses ma of the individual species P P through rm ¼ Na¼1 ra ma ¼ r Na¼1 ca ma : Thus, for vm ¼ 1=rm the specific volume, vm ¼

V N X

:

ð8:1Þ

ca ma

a¼1

The free energy C0 is measured per unit volume, so that vm C0 represents the specific-free energy. We derive the Gibbs relation by noting that for p the thermodynamic pressure, 2p is the derivative, with respect to vm ; of the specific free energy at fixed composition ~c: Thus, by Eq. (8.1), 2p is the derivative, with respect to V; of VC0 at fixed composition: " !# N X › c1 c2 cN ›C0 ðr~Þ ca VC0 ¼ C0 ðr~Þ 2 ; ; …; ›V V V V ›ra V a¼1 ¼ C0 ðr~Þ 2

N X

ra ma0 ðr~Þ:

ð8:2Þ

a¼1

On the other hand, if we identify this pressure with the actual pressure, then, since T ; 0; Eq. (8.2) must vanish; thus we have the Gibbs relation (Gibbs, 1878,

A Unified Treatment of Evolving Interfaces

35

Eq. (12)):

C0 ðr~Þ ¼

N X

ra ma0 ðr~Þ:

ð8:3Þ

a¼1

The relation (8.3) is well defined for all r~ and provides a method of determining the free energy from a knowledge of the chemical potentials. But the latter cannot be arbitrary, but instead must be consistent with the Gibbs –Duhem equation (Kirkwood and Oppenheim, 1961, Eq. (6.61)) N X ›ma ðr~Þ ra 0 b ¼ 0 ð8:4Þ ›r a¼1 for all species b: This set of N relations follows upon differentiating Eq. (8.3) with respect to rb and appealing to Eq. (7.25). The Gibbs – Duhem equation (8.4) represents a condition that is both necessary and sufficient that Eq. (8.3) hold. A simple set of constitutive relations at zero stress is based on the assumption that the chemical potentials ma0 ðr~Þ depend on r1 ; r2 ; …; rN through the concentrations c1 ; c2 ; …; cN : Granted this, we may use the identity ›c a ¼ Vðdab 2 ca Þ; dab ¼ Kronecker delta; ð8:5Þ ›rb to write the Gibbs –Duhem equation (8.4) in the form N N X X ›ma ›ma def ca b0 ¼ ca g0 cg ¼ L: ›c ›c a¼1 a¼1

ð8:6Þ

If, in addition, we assume that ma is a function of ca0 (only), then L is independent of composition and Eq. (8.6) yields ›ma ca a0 ¼ L; ð8:7Þ ›c which has the explicit solution ma0 ¼ U a þ L ln ca ; ð8:8Þ so that22

C0 ðr~Þ ¼

N X

ra ðU a þ L ln ca Þ:

ð8:9Þ

a¼1

22 Materials scientists typically choose L ¼ Ru; with R the gas constant and u the absolute temperature, and rewrite Eq. (8.9) in the form N X C0 ðr~Þ ¼ ra ma0 ð~cÞ; ma0 ð~cÞ ¼ ðUpa þ Ru lnðga ca ÞÞ;

a¼1

where ga is an activity coefficient and Upa is the chemical potential when ra ¼ 1=ga ; and where ga and Upa generally depend on u:

E. Fried and M.E. Gurtin

36

IX. Constitutive Theory for a Substitutional Alloy The lattice constraint N X

ra ¼ rsites

a¼1

renders the constitutive theory for a substitutional alloy more difficult than that for an unconstrained material. In many respects the substitutional theory mirrors that for unconstrained materials; in particular, the theory is based on constitutive equations in which the density-list r~ ¼ ðr1 ; r2 ; …; rN Þ appears as an independent variable. Difficulties arise because each such list r~ must be admissible; that is, must satisfy the lattice constraint and must have 0 # ra , 1 for all atomic species a: Thus, since varying one of the densities while holding the others fixed, violates the lattice constraint, standard partial differentiation of the constitutive response functions with respect to the atomic densities is not well-defined. Section IX.A comes to grips with this problem.

A. Larche´ –Cahn Derivatives Let f ðr~Þ be defined on the set of admissible density lists. As noted above, the standard partial derivatives ›f =›ra are not defined. To free f of the lattice constraint, choose a species z as reference, use the lattice constraint to express rz as a function

rz ¼ rsites 2

N X

ra

a¼1 a–z

of the list ðr1 ; r2 ; …; rz21 ; rzþ1 ; …; rN Þ of remaining densities, and consider f as a function f ðzÞ of the remaining densities by defining N f ðzÞ ðr1 ; r2 ; …; rz21 ; rzþ1 ; …; rN Þ ¼ f ðr~Þl z sites X ra |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} r ¼r 2

rz

missing

ð9:1Þ

a¼1 a–z

The domain of f ðzÞ is then open, since the arguments of f ðzÞ may be varied slightly without violating the lattice constraint; thus the partial derivatives

› f ð zÞ ; ›ra

› 2 f ðz Þ ›ra ›rb

are well defined. Note that when a; say, is equal to z; the left-hand side of

A Unified Treatment of Evolving Interfaces

37

Eq. (9.1) is independent of rz ; so that, trivially,

› f ð zÞ ¼ 0; ›rz

ð9:2Þ

We refer to f ðzÞ as the description of f relative to z: An alternative treatment of differentiation that respects the lattice constraint may be developed as follows. Choose species a and z: If the list r~ ¼ ðr1 ; r2 ; …; rN Þ is consistent with the lattice constraint, then so also is the list ðr1 ; …; ra þ 1; …; rz 2 1; …; rN Þ obtained by increasing the atomic density of species a by an amount 1 and decreasing the density of z by an equal amount (while holding the remaining densities fixed). Bearing this in mind, we define the Larche´ –Cahn derivative ›ðzÞ =›ra by

›ðzÞ f ðr~Þ d f ðr1 ; …; ra þ 1; …; rz 2 1; …; rN Þl1¼0 ; ¼ d1 ›ra

ð9:3Þ

›ðzÞ f ðr~Þ=›ra represents the change in f ðr~Þ due to a unit increase in the density of a-atoms and an equal decrease in the density of z-atoms.23 Second Larche´ –Cahn derivatives are defined similarly: ›2ðzÞ f ðr~Þ d2 ¼ f ðr1 ;…; ra þ 1;…; rb þ l;…; rz 2 1 2 l;…; rN Þl1¼l¼0 : a b d1 dl › r ›r

ð9:4Þ

For convenience, we define

› ð zÞ f ¼ 0: ›rz

ð9:5Þ

A direct consequence of Eq. (9.3) is then the skew-symmetry relation

› ð zÞ f › ða Þ f ¼2 z ; a ›r ›r valid for all species a and z: Thus, N N N X X X › ð zÞ f › ða Þ f › ðz Þ f ¼ 2 ¼ 2 z ›ra ›ra a;z¼1 a;z¼1 ›r a;z¼1

23

Larche´ and Cahn (1985, Eq. (3.7)) use the notation ›=›raz rather than ›ðzÞ =›ra :

ð9:6Þ

38

E. Fried and M.E. Gurtin

and we have N X › ð zÞ f ¼ 0: ›ra a;z¼1

ð9:7Þ

Using the description f ðzÞ of f relative to z; the Larche´ – Cahn derivative may be given an alternative representation which is convenient in calculations. Increasing an argument ra by an amount 1 (while holding the other arguments of f ðzÞ fixed) corresponds, via the definition (9.1), to decreasing the argument rz in f by 1: Therefore, as a consequence of Eq. (9.3), the Larche´ – Cahn derivative ›f ðzÞ =›ra is simply the derivative of f with respect to ra taken with the density rz eliminated via the lattice constraint; thus and by Eqs. (9.2) and (9.5),

› ð zÞ f › f ð zÞ ¼ ; ›ra ›ra

ð9:8Þ

and similarly for second derivatives,

›2ðzÞ f › 2 f ð zÞ ¼ : a b ›r ›r ›ra ›rb

ð9:9Þ

Note that Eqs. (9.8) and (9.9) are meaningful even though their left-hand sides are functions of the complete list r~ ¼ ðr1 ; r2 ; …; rz ; …; rN Þ; while their right-hand sides are functions of the list ðr1 ; r2 ; …; rz21 ; rzþ1 ; …; rN Þ ; |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} rz missing

indeed, the left-hand sides are defined only for those arguments r~ consistent with the lattice constraint, a constraint that renders rz known when the other densities are known (cf. Eq. (9.1)). It may happen that f ðr~Þ may be extended smoothly to an open region of RN :24 In that case the Larche´ –Cahn derivative may be computed as the difference

› ð zÞ f ›f ›f ¼ 2 z; a a ›r ›r ›r

ð9:10Þ

e.g., for the function defined on the set of admissible density lists by f ðr~Þ ¼ la ra 24

For example, when f is the free energy described in Footnote 22, a free energy used by materials scientists also for substitutional alloys (cf. Larche´ and Cahn, 1985, Section IV.B).

A Unified Treatment of Evolving Interfaces

39

with each of the ls constant,

› ð zÞ f ¼ la 2 lz : ›ra

ð9:11Þ

Next, choose a species z and bear in mind that ›f ðzÞ =›ra is a standard partial derivative. Then, for r~ðtÞ an admissible, time-dependent density list and wðtÞ ¼ f ðr~ðtÞÞ;

w_ ¼

N N X X ›f ðzÞ ðr~Þ a ›ðzÞ f ðr~Þ a r ¼ r_ ; _ ›ra ›ra a¼1 a¼1 |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} chain-rule

ð9:12Þ

by Eq: ð9:8Þ

which is the chain-rule for Larche´ – Cahn derivatives.

B. Constitutive Equations 1. General Relations Given a fixed choice of the reference species z and guided by the dissipation inequality (5.11), viz.,

C_ 2 T·E_ 2

N X

ðmaz r_a 2 a ·7maz Þ # 0;

ð9:13Þ

a¼1

and the requirement that the bulk theory for a substitutional alloy depend only on the relative chemical potentials (cf. the theorem containing Eq. (5.7)), we base the theory on constitutive equations

C ¼ C^ ðE; r~Þ;

^ T ¼ TðE; r~Þ;

ð9:14Þ

for the free energy and stress, constitutive equations

mab ¼ m^ab ðE; r~Þ

ð9:15Þ

for the relative chemical potentials, and Fick’s law a ¼ 2

N X

b¼1

for the atomic fluxes.

Mab ðE; r~Þ7mbz

ð9:16Þ

40

E. Fried and M.E. Gurtin 2. Constraints on m^ab

The constitutive relations (9.15), which are prescribed for all relative chemical potentials, are presumed to be consistent with the identities (5.6); more pragmatically, we need only assume that the response functions m^az are prescribed for all a and some fixed choice of reference species z; for then the response functions relative to any other species b may be defined by

m^ab ¼ m^az 2 m^bz ;

ð9:17Þ

and, granted this, the skew symmetry relation

m^ab ¼ 2m^ba

ð9:18Þ

is satisfied for each pair of species, so that, in particular, m^aa ¼ 0 (no sum).

3. Mobility Constraints We require that the mobility tensors Mab ðE; r~Þ : (a) be consistent with the substitutional flux constraint; (b) render Fick’s law (9.16) independent of the choice of reference species z: To discuss the implications of these constraints, we suppress the arguments E and r~; which are irrelevant to the following discussion. For (b) to hold it is sufficient that N X

Mab 7mbz ¼

b¼1

N X

Mab 7mbg

ð9:19Þ

b¼1

for all choices of z and g and all a: By Eq. (9.17), the relative chemical potentials necessarily satisfy

mbz ¼ mbg 2 mzg for all choices of z and g and all b; therefore, Eq. (9.19) will be satisfied provided N X

b¼1

Mab 7mzg ¼ 0

A Unified Treatment of Evolving Interfaces

41

for all choices of z and g and for all a; and, hence, provided N X

Mab ¼ 0

ð9:20Þ

b¼1

for all a: Next, consider requirement (a). The substitutional flux constraint (5.4) applied to Fick’s law is the requirement that ! N N N N X X X X a ab bz ab 7mbz ¼ 0;  ¼2 M 7m ¼ 2 M a¼1

a;b¼1

b¼1

a¼1

an equation that will be satisfied for each choice of z provided the term in parenthesis vanishes. Thus, recalling Eq. (9.20), we have the mobility constraints of Larche´ and Cahn (1985, Eqs. (8.2) and (8.3)): N X

Mab ðE; r~Þ ¼ 0;

a¼1

N X

Mab ðE; r~Þ ¼ 0:

ð9:21Þ

b¼1

C. Thermodynamic Restrictions Our next step is to determine restrictions on the constitutive equations that ensure satisfaction of the dissipation inequality (9.13). Because of the lattice constraint, thermodynamic arguments involving arbitrary variations of the atomic densities are delicate. In this regard, the following lemma is useful: Lemma Given any admissible density list n~ ; any scalar a, any two atomic species a – b; and any time t; there is a time-dependent, admissible density-list r~ðtÞ such that, at t;

r~ðtÞ ¼ n~;

r_a ðtÞ ¼ 2r_b ðtÞ ¼ a;

r_g ðtÞ ¼ 0 for g – a; b:

ð9:22Þ

To prove this lemma note first that a simple choice r~ðtÞ consistent with the lattice constraint and with Eq. (9.22) is given by

ra ðtÞ ¼ na þ ðt 2 tÞa;

rb ðtÞ ¼ nb 2 ðt 2 tÞa;

and

rg ðtÞ ¼ ng

for g – a; b:

ð9:23Þ

But this choice does not furnish a solution of our problem, since the densities ra ðtÞ and rb ðtÞ may be negative. This is easily remedied: given any a . 0; we can _ tÞ ¼ 1; and lTðtÞl , a: always find a scalar function TðtÞ such that TðtÞ ¼ 0; Tð

E. Fried and M.E. Gurtin

42

The density list r~ðtÞ defined by

ra ðtÞ ¼ na þ TðtÞa;

rb ðtÞ ¼ nb 2 TðtÞa;

supplemented by Eq. (9.23), satisfies Eq. (9.22) and will be admissible for all t provided we choose a small enough. This completes the proof of the lemma. Recall that, by Eqs. (9.2) and (9.5), and the sentence containing Eq. (9.18),

›ðzÞ C^ ›C^ ðzÞ ¼ ¼ m^zz ¼ 0: ›rz ›rz Fix a species z and choose an arbitrary process consistent with the constitutive equations. Then, by Eq. (9.12), N ›C^ ðE; r~Þ _ X ›ðzÞ C^ ðE; r~Þ a ·E þ C_ ¼ r_ : ›E ›ra a¼1

ð9:24Þ

The requirement that the dissipation inequality (9.13) holds in all such processes leads to the inequality ( ) ( ) N X ›C^ ðE; r~Þ ›C^ ðzÞ ðE; r~Þ az ^ _ 2 TðE; r~Þ ·E þ 2 m^ ðE; r~Þ r_a ›E ›ra a¼1 2

N X

7maz ·Mab ðE; r~Þ7mbz # 0;

ð9:25Þ

a;b¼1

for each choice of the free-index z: If, for the moment, we restrict attention to spatially constant processes, then this inequality reduces to ( ) ( ) N ð zÞ ^ X ›C^ ðE; r~Þ › C ðE; r ~ Þ az ^ 2 TðE; r~Þ ·E_ þ 2 m^ ðE; r~Þ r_a # 0: ð9:26Þ a ›E › r a¼1 This inequality must hold for all EðtÞ and all admissible density lists r~ðtÞ: Assuming that the atomic densities are independent of time leads to the requirement that T^ ¼ ›C^ =›E and Eq. (9.26) reduces to an inequality involving only the density-rates. As noted in the density-variation lemma (cf. Eq. (9.22)), given any species a – z; we can always find an admissible density list r~ðtÞ such that, at some time t; the values r~ðtÞ and r_a ðtÞ ¼ 2r_z ðtÞ are arbitrary, while the remaining rates r_a ðtÞ vanish. For this choice of r~ðtÞ; Eq. (9.26), at time t; becomes ( ) ›ðzÞ C^ ðE; r~Þ az 2 m^ ðE; r~Þ r_a # 0: ›ra Thus, since E, r~; and r_a are arbitrary at t; we must have m^az ¼ ›ðzÞ C^ =›ra for all

A Unified Treatment of Evolving Interfaces

43

admissible r~; a result reduces Eq. (9.25) to the inequality N X

7maz ·Mab 7mbz $ 0;

a;b¼1

for each choice of the free-index z: Summarizing, the second law in the form of the dissipation in equality requires that 9 ^ > ^TðE; r~Þ ¼ ›CðE; r~Þ ; > > = ›E ð9:27Þ > ›ðzÞ C^ ðE; r~Þ > az > m^ ðE; r~Þ ¼ ;; ›ra for all atomic species a and z; and that matrix (7.3) of mobilities must be positive semi-definite (cf. the discussion associated with Eq. (7.6)). The dissipation Eq. (4.7), which is the negative of the left-hand side of Eq. (9.25), is given by

d¼

N X

7ma ·Mab ðE; r~Þ7mb :

a;b¼1

Immediate consequences of Eq. (9.27)1 and (9.27)2 are the Maxwell relations

›ðzÞ T^ ›m^az ¼ a ›r ›E

ð9:28Þ

C_ ¼ T·E_ þ maz r_a ;

ð9:29Þ

and the Gibbs relation

which hold for each choice of the free-index z:

D. Free Enthalpy. Moduli Further, assuming an invertible stress – strain relation as discussed in Section VII.C, the free enthalpy defined by ~ F~ ðT; r~Þ ¼ C~ ðT; r~Þ 2 T·EðT; r~Þ

ð9:30Þ

yields the relations

›F~ ðT; r~Þ ~ ; EðT; r~Þ ¼ 2 ›T

m~az ðT; r~Þ ¼

›F~ ðzÞ ðT; r~Þ ; ›ra

ð9:31Þ

E. Fried and M.E. Gurtin

44

and these in turn yield the Maxwell relations

›ðzÞ E~ ›m~az ¼ a ›r ›T

ð9:32Þ

as well as direct counterparts of the other results of Section VII.C. As before, we define tensor moduli 9 ^ ›TðE; r~Þ ›2 C^ ðE; r~Þ > > CðE; r~Þ ¼ ¼ ;> > > ›E ›E2 > > > > ð b Þ ab ^ > › ðE; r ~ Þ › m ^ ðE; r ~ Þ T > ab > ; A ðE; r~Þ ¼ ¼ > a = ›r ›E ~ > ›EðT; r~Þ ›2 C~ ðT; r~Þ > > KðT; r~Þ ¼ ¼2 ;> > 2 > ›T ›T > > > > ðb Þ ab ~ ›E ðT; r~Þ ›m~ ðT; r~Þ > > ab > : N ðT; r~Þ ¼ ¼ 2 ; ›ra ›T

ð9:33Þ

Aab and Nab ; respectively, are the stress- and strain-composition tensors for a relative to b: Aab represents the marginal increase in stress due to both an incremental increase in ra and an incremental decrease of the same amount in rb ; holding the other densities and the strain fixed. An analogous meaning applies to Nab : In view of the skew-symmetry relations (9.6) and (9.18), the tensors Aab and Nab also satisfy skew-symmetry relations: Aab ¼ 2Aba ;

Nab ¼ 2Nba ;

ð9:34Þ

consequences of which are the identities N X

Aab ¼ 0;

a;b¼1

N X

Nab ¼ 0:

ð9:35Þ

a;b¼1

It is also useful to note that, as in the unconstrained theory, K ¼ C21 ;

Nab ¼ 2K½Aab ;

ð9:36Þ

and 9 C; Aab ða; b ¼ 1; 2; …; NÞ are independent of strain and composition > > = if and only if > > ; K; Nab ða; b ¼ 1; 2; …; NÞ are independent of stress and composition: ð9:37Þ

A Unified Treatment of Evolving Interfaces

45

Also, as in the unconstrained theory, we note that the free energy and the relative chemical potentials at zero stress,

C0 ðr~Þ ¼ C~ ðT; r~ÞlT¼0 ;

ab mab 0 ðr~Þ ¼ m~ ðT; r~ÞlT¼0 ;

ð9:38Þ

are, by Eqs. (9.30) and (9.31), related through

mab 0 ðr~Þ ¼

›ðbÞ C0 ðr~Þ : ›ra

ð9:39Þ

E. Mechanically Simple Substitutional Alloys Consistent with our discussion of the unconstrained theory in Section VII.D, we refer to a substitional alloy as being mechanically simple if: the elasticity tensor C and stress-composition tensors Aab are independent of strain and composition (cf. Eq. (9.37)); (ii) the mobilities Mab are independent of strain. In addition, for a substitutional alloy without vacancies, we stipulate that (iii) there exist functions ma0 ðr~Þ such that

(i)

C0 ðr~Þ ¼ ra ma0 ðr~Þ;

b a mab 0 ðr~Þ ¼ m0 ðr~Þ 2 m0 ðr~Þ:

ð9:40Þ

We refer to Eq. (9.40) as the free-energy conditions at zero stress, to ma0 ðr~Þ as the species-a chemical potential at zero stress.25 We now show that, interestingly, the free energy and stress of a mechanically simple material within this constrained theory have a form identical to that of a mechanically simple material in the theory without a lattice constraint (cf. Eqs. (7.27) and (7.28)1); only the expression for the chemical potential is different. We now verify this result, which is based on the constancy of the tensors C and Aab : Fix the species b and consider the relations

›T^ ðbÞ ¼ Aab ; ›ra

a ¼ 1; 2; …; b 2 1; b þ 1; …; N : |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

ð9:41Þ

b missing

The derivative on the left may be viewed as the ordinary partial derivative of T^ with respect to ra taken with the density rb eliminated via the lattice constraint 25

An argument in support of the free-energy conditions at zero stress is given in Appendix A. These conditions are often used by materials scientists (cf., e.g., Spencer et al. (2001)). The condition (9.40)1 is used only in our discussion of an interface between a vapor and a substitutional alloy without vacancies (Sections XXVI.A and XXVI.B).

E. Fried and M.E. Gurtin

46

(cf. the discussion following Eq. (9.8)). We may, therefore, integrate Eq. (9.41) from an arbitrary reference list r~0 to r~; the result is an equation for the stress of the form T¼

N X

Aab ðra 2 ra0 Þ þ SðEÞ;

ð9:42Þ

a¼1

with S(E) an arbitrary function of strain. Even though Abb ¼ 0 (no sum), we may view Eq. (9.42) as a constitutive equation giving the stress as a function of the strain and all densities, with the interpretation that rb ; which does not appear on the right-hand side, has been eliminated via the lattice constraint. The species b was fixed in the foregoing argument, but we are at liberty to view Eq. (9.42) as N equations, each of which delivers the stress T. If we sum Eq. (9.42) over all b and divide the result by N; we find that T¼

N 1 X Aab ðra 2 ra0 Þ þ SðEÞ; N a;b¼1

ð9:43Þ

and, therefore, for def

Aa ¼

N 1 X Aab ; N b¼1

ð9:44Þ

Eq. (9.43) becomes T¼

N X

Aa ðra 2 ra0 Þ þ SðEÞ:

ð9:45Þ

a¼1

Note that, by Eqs. (9.35) and (9.44), N X

Aa ¼ 0;

ð9:46Þ

a¼1

which would seem natural in lieu of the lattice constraint. Next by Eq. (9.33)1,

›SðEÞ ¼ C; ›E therefore, S½E ¼ CðEÞ þ S0 ; with S0 a constant tensor. Thus Eq. (9.45) becomes T ¼ C½E þ

N X

Aa ðra 2 ra0 Þ þ S0 ;

a¼1

and if we assume that reference list r~0 is chosen so that T ¼ 0 when E ¼ 0 and

A Unified Treatment of Evolving Interfaces

47

r~ ¼ r~0 ; then S0 ¼ 0 and we arrive at a constitutive relation for the stress: T ¼ C½E þ

N X

Aa ðra 2 ra0 Þ;

ð9:47Þ

a¼1

Eq. (9.47) is identical to Eq. (9.41) of the unconstrained theory. Finally, to obtain the free energy we integrate the relation ›C^ =›E ¼ T using Eq. (9.47); the result is the relation (7.27) for the free energy of the unconstrained theory. In Eq. (7.27) P the function Na¼1 ðra 2 ra0 ÞAa ·E is well defined for all density lists r~; irrespective of whether the list is consistent with the lattice constraint. Thus, by Eq. (9.10) its Larche´ – Cahn derivative ›ðbÞ =›ra applied to this function gives ðAa 2 Ab Þ·E: Thus, in view of the state relation m^ab ¼ ›ðbÞ C^ =›ra ; we arrive at the following expression for the relative chemical potentials:

mab ¼

›ðbÞ Fðr~Þ þ ðAa 2 Ab Þ·E: ›ra

ð9:48Þ

Summarizing, the constitutive equations for the free energy, the stress, and the chemical potentials of a mechanically simple, subsitutional alloy must have the specific form 9 N X 1 > a a a > C ¼ E·C½E þ ðr 2 r0 ÞA ·E þ Fðr~Þ; > > > 2 > a¼1 > > > = N X a a a ð9:49Þ A ðr 2 r0 Þ; > T ¼ C½E þ > > a¼1 > > > > ðb Þ > › Fð r ~ Þ ab a b > ; m ¼ þ ðA 2 A Þ·E: a ›r By Eq. (9.33)2, the derivative of the right-hand side of Eq. (9.49)3 with respect to E should be Aab ; thus Aab ¼ Aa 2 Ab : The remainder of the proof closely follows the argument for the unconstrained theory ensuing from Eq. (7.29). Solving Eq. (9.49)2 for E yields E ¼ K½T 2

N X

K½Aa ðra 2 ra0 Þ

ð9:50Þ

a21

and, by Eq. (9.33), the Larche´ – Cahn derivative of the right-hand side of the equation should be Nab : Nab ¼ 2K½Aa 2 Ab :

ð9:51Þ

E. Fried and M.E. Gurtin

48

If we define Na through the obvious analog of Eq. (9.44), we find, upon summing Eq. (9.51) over b; that Na ¼ 2K½Aa ;

ð9:52Þ

Nab ¼ Na 2 Nb :

ð9:53Þ

and hence, that

Equation (9.52) is identical to its counterpart in the unconstrained theory. Because of this, the calculation (7.30) remains valid, so that

C¼

1 2

T·K½T þ C0 ðr~Þ

ð9:54Þ

and Fðr~Þ is related to C0 ðr~Þ; the free energy at zero stress, through Fðr~Þ ¼ C0 ðr~Þ 2

N 1 X ðra 2 ra0 Þðrb 2 rb0 ÞAa ·Nb : 2 a;b¼1

ð9:55Þ

Further, by Eq. (9.35), N X

Na ¼ 0:

ð9:56Þ

a¼1

Finally, using Eqs. (9.50) and (9.54) in Eq. (9.30), we find that N X 1 F ¼ 2 T·K½T 2 ðra 2 ra0 ÞNa ·T þ C0 ðr~Þ; 2 a¼1

ð9:57Þ

and, therefore, by Eq. (9.31)2, a b mab ¼ mab 0 ðr~Þ 2 ðN 2 N Þ·T:

ð9:58Þ

At any given point, the term 2ðNa 2 Nb Þ·T ¼ 2Nab ·T

ð9:59Þ

represents energy the body would gain, per unit time, were we to replace a b-atom by and a-atom in the presence of the stress T. An analogous interpretation applies to the term ðAa 2 Ab Þ·E:

A Unified Treatment of Evolving Interfaces

49

The results concerning stress as independent variable are summarized as follows: 9 1 > C ¼ T·K½T þ C0 ðr~Þ; > > > 2 > > = n X a a a ð9:60Þ ðr 2 r0 ÞN ; > E ¼ K½T þ > > a¼1 > > > ; ab ab a b m ¼ m0 ðr~Þ 2 ðN 2 N Þ·T: Next, since the mobilities are independent of the strain, it follows that, for any z; a ¼ 2

N X

Mab ðr~Þ7mbz

b¼1

1 N 2ðzÞ X › Fð r ~ Þ g b z A ¼2 Mab ðr~Þ@ b g 7r þ ðA 2 A Þ7E : b¼1 g¼1 ›r ›r N X

0

Similarly, we have the alternative expression 0 1 N N 2ðzÞ X X › C ð r ~ Þ 0 g b z A Mab ðr~Þ@ a ¼ 2 b ›rg 7r 2 ðN 2 N Þ7T : › r b¼1 g¼1

ð9:61Þ

ð9:62Þ

In view of the mobility constraint (9.21), we may replace Ab 2 Az in Eq. (9.61) by Ab and Nb 2 Nz in Eq. (9.62) by Nb :

F. Cubic Symmetry The cubic specializations of the foregoing are straightforward and give results analogous to those presented in the unconstrained case. Because of their importance, we present these here. First, the tensors Mab ; Aab ; Nab ; Aa ; and Na are isotropic, so that Mab ¼ mab 1;

Aab ¼ aab 1;

Nab ¼ hab 1;

Na ¼ ha 1;

Aa ¼ aa 1; ð9:63Þ

with aab ¼ aa 2 ab and hab ¼ ha 2 hb : Further, the moduli aa and ha are related through the compressibility k via aa ¼ 23kha :

ð9:64Þ

E. Fried and M.E. Gurtin

50

and, by Eqs. (9.46) and (9.56), satisfy N X

aa ¼ 0;

a¼1

N X

ha ¼ 0:

a¼1

In addition, the relations (9.49)1 and (9.57) determining the free energy and free enthalpy specialize to 9 N > X 1 > a a a > C ¼ E·C½E þ a ðr 2 r0 Þtr E þ Fðr~Þ; > > = 2 a¼1 ð9:65Þ > N X > 1 > a a a > F ¼ 2 T·K½T 2 h ðr 2 r0 Þtr T þ C0 ðr~Þ; > ; 2 a¼1 with C0 ðr~Þ given by Eq. (7.41). Further, as consequences of Eqs. (9.49)2,3, (9.60)2,3, and (9.63)4,5 we have the equivalent sets of relations: 9 N > X ›ðbÞ Fðr~Þ > a a a ab a b a ðr 2 r0 Þ1; m ¼ þ ða 2 a Þtr E; > T ¼ C½E þ > a > = › r a¼1 ð9:66Þ > N X > > ab a a a ab a b E ¼ K½T þ h ðr 2 r0 Þ1; m ¼ m0 ðr~Þ 2 ðh 2 h Þtr T; > > ; a¼1

mab 0

given by Eq. (7.25). Finally, we have alternative expressions of with Fick’s law: 0 1 9 N N 2ðzÞ > X X > › Fð r ~ Þ > a ¼ 2 mab ðr~Þ@ 7rg þ ab 7 tr EA; > > b g > = b¼1 g¼1 ›r ›r ð9:67Þ 0 1 > N N > 2ðzÞ X X > › C ð r ~ Þ 0 g b A > > a ¼ 2 mab ðr~Þ@ b ›rg 7r 2 h 7 tr T ; > ; › r b¼1 g¼1 in which the choice of species z is arbitrary.

X. Governing Equations The local balance laws for forces and atomic densities, div T ¼ 0;

r_a ¼ 2div a ;

ð10:1Þ

supplemented by the constitutive equations form the governing equations of the theory, which are coupled partial differential equations for the displacement and the atomic densities. For the general theory the resulting equations, while not

A Unified Treatment of Evolving Interfaces

51

difficult to write, are complicated and afford little insight. For that reason, we display only the governing equations applicable to mechanically simple unconstrained materials and mechanically simple substitutional alloys without vacancies. Note that, because of the strain displacement relation E ¼ 12 ð7u þ 7uT Þ and the symmetry Cijkl ¼ Cijlk ; we may write the components of C½E in the form ðC½EÞij ¼ Cijkl

›uk : ›xl

For an unconstrained material the basic equations take the form 9 N a X > ›2 uk a ›r > Cijkl þ Aij ¼ 0; > > > ›xl ›xj a¼1 ›x j = 8 0 19 N N <X = > X > ›2 Fðr~Þ g b A ;> > r_a ¼ div Mab @ 7 r þ A 7E b g ; :b¼1 ; > g¼1 ›r ›r

ð10:2Þ

where, for L and P tensor fields, the vector field L7P has the component form ðL7PÞk ¼ Lij

›Pij ›x k

The atomic balance (10.2)2 may also be written in terms of stress and, by Eq. (7.36), has the form 8 0 19 N N 2 <X = X › C ð r ~ Þ 0 g b A : ð10:3Þ r_a ¼ div Mab @ 7 r 2 N 7T b g :b¼1 ; g¼1 ›r ›r In writing these equations we have suppressed the dependencies of Mab ; F; and C0 on r~: For a substitutional alloy, the only change in the basic equations is that

›2ðzÞ Fðr~Þ ›2 Fðr~Þ should replace ›rb ›rg ›rb ›rg and

›2ðzÞ C0 ðr~Þ ›2 C0 ðr~Þ should replace b g ›r ›r ›rb ›rg In using the atomic balances for substitutional materials, one should bear in mind that, for their validity, the mobility tensors Mab must be consistent with the mobility constraints (9.21).

52

E. Fried and M.E. Gurtin

CONFIGURATIONAL FORCES IN BULK In dynamical problems, defect structures, such as phase interfaces and dislocation lines, may move relative to the material. In variational treatments of related equilibrium problems, independent kinematical quantities may be independently varied, and each such variation yields a corresponding Euler – Lagrange balance. In dynamics with general forms of dissipation there is no encompassing variational principle, but experience has demonstrated the need for an additional balance associated with the kinematics of the defect. An additional balance of this sort is the relation that ensues when one formally sets the variationally derived expression for the driving force on a defect equal to a linear function of the velocity of that defect.26 Here, guided by variational treatments in which such a balance is a consequence of the assumption of equilibrium, we follow Gurtin and Struthers (1990) and Gurtin (1995, 2000) and introduce, as primitive objects, configurational forces together with an independent configurational force balance. Roughly speaking, configurational forces are related to the integrity of the body’s material structure and expend power in the transfer of material and in the evolution of defects. In this part we discuss configurational forces within a context that neglects defect structures. Within that context such forces are extraneous to the solution of actual boundary-value problems. But in general situations knowledge of the structure of configurational forces in bulk away from defects is central to the understanding of their localized behavior at defects.

XI. Configurational Forces. Power A. Configurational Force Balance The configurational force system we envisage has two ingredients: a stress Cðx; tÞ and a body force fðx; tÞ; both distributed continuously over the body. This system is required to satisfy the configurational force balance ð

Cn da þ ›P

26

ð

f dv ¼ 0

ð11:1Þ

P

Classical examples of driving forces are those on dislocations (Peach and Koehler, 1950); triple junctions (Herring, 1951); vacancies, interstitial atoms, and inclusions (Eshelby, 1951); interfaces (Eshelby, 1956, 1970); crack tips (Eshelby, 1956; Atkinson and Eshelby, 1968; Rice, 1968). Other examples of additional balances are discussed in Section I.A.

A Unified Treatment of Evolving Interfaces

53

Fig. 11.2.1. The migrating control volume R ¼ RðtÞ; v›R is viewed as the velocity with which an external agency adds material to ›R; the normal component V›R of v›R must coincide with the normal velocity of ›R:

for all parts P; a requirement equivalent to the local force balance27 div C þ f ¼ 0:

ð11:2Þ

B. Migrating Control Volumes. Accretion To characterize the manner in which configurational forces perform work, a means of capturing the kinematics associated with the transfer of material is needed. Following Gurtin (1995, 2000), we accomplish this with the aid of control volumes RðtÞ that migrate through B (Fig. 11.2.1) and thereby result in the transfer of material to—and the removal of material from—RðtÞ at ›RðtÞ; a process we refer to as accretion. Here it is essential that parts not be confused with migrating control volumes: † a part P is a fixed subregion of B; † a migrating control volume RðtÞ migrates through B. Let R ¼ RðtÞ be a migrating control volume with V›R ðx; tÞ the (scalar) normal velocity of ›RðtÞ in the direction of the outward unit normal n: To describe power expenditures associated with the migration of RðtÞ; we introduce a field v›R ðx; tÞ 27

Configurational force balances may be derived as consequences of invariance of the power under Galilean changes of referential observer (Gurtin, 2000; Podio-Guidugli, 2002).

54

E. Fried and M.E. Gurtin

defined over ›RðtÞ for all t; which we view as the velocity with which an external agency adds material at ›RðtÞ: Compatibility then requires that v›R have V›R as its normal component, v›R ·n ¼ V›R ;

ð11:3Þ

but v›R is otherwise arbitrary. We refer to any such field v›R as a velocity field for ›R: Non-normal velocity fields, while not intrinsic, are important. For example, given an arbitrary time-dependent parametrization x ¼ rðp1 ; p2 ; tÞ of ›R; the field defined by v›R ¼ ›r=›t (holding ðp1 ; p2 Þ fixed) generally represents a nonnormal velocity field for ›R: But while it is important that we allow for the use of non-normal velocity fields, it is essential that the theory itself not depend on the particular velocity field used to describe a given migrating control volume. As we shall see (Section XII.B), this observation has important consequences. Let R ¼ RðtÞ be a migrating control volume and v›R a velocity field for ›R: One might picture v›R as a velocity field for particles (not material points) evolving on the migrating surface ›R; in which case the trajectory zðtÞ of the particle that passes through position x on ›RðtÞ at time t would be the unique solution of (Fig. 11.2.2) dzðtÞ ¼ v›R ðzðtÞ; tÞ; dt

zðtÞ ¼ x:

ð11:4Þ

Given a field f ðx; tÞ; its rate of change following ›R; as described by v›R ; is the derivative with respect to time along such trajectories: % % tÞ ¼ d f ðzðtÞ; tÞ% fðx; ð11:5Þ % t¼t dt

Fig. 11.2.2. The trajectory zðtÞ of a particle that passes through x on ›RðtÞ at time t.

A Unified Treatment of Evolving Interfaces

55

In particular, using the chain-rule, we find, for the displacement field u;

tÞ ¼ uðx;

% % d _ tÞ þ 7uðx; tÞv›R ðx; tÞ: uðzðtÞ; tÞ%% ¼ uðx; dt t¼t

ð11:6Þ

We refer to u ¼ u_ þ ð7uÞv›R as the motion velocity following (the migration of) ›R as described by v›R :

C. Power Expended on a Migrating Control Volume RðtÞ The notion of power is basic to all of mechanics. In its essence, this notion involves the pairing, via an inner product, of velocities with associated forces. In such a pairing the velocity is said to be power conjugate to its associated force. In our development of the free-energy imbalance in Section IV, the power expended on an arbitrary part P by tractions acting on ›P was expressed by Eq. (2.4), viz.

WðPÞ ¼

ð

Tn·u_ da:

ð11:7Þ

›P

_ represents the power expended by the traction Tn Here the integrand Tn·u; over the motion velocity u_ of material points on ›P; so that u_ is the power conjugate velocity for Tn: Consider now a migrating control volume RðtÞ: The migration of RðtÞ induces a transfer of material across ›RðtÞ; and one would expect that transfer to be accompanied by an expenditure of power over and above the standard expenditure represented by Eq. (11.7). A central tenet of our treatment of configurational forces is the presumption that (Gurtin, 1995, 2000): configurational forces expend power in consort with transfers of material. In particular, for the migrating control volume RðtÞ; we view the traction Cn as a force associated with the transfer of material across ›RðtÞ; since any given velocity field v›R for ›RðtÞ represents a velocity field with which material is transferred across ›RðtÞ; we take v›R to be an appropriate power conjugate velocity for Cn: We, therefore, assume that the migration of ›R is accompanied

56

E. Fried and M.E. Gurtin

by an expenditure of power equal to ð ›RðtÞ

Cn·v›R da:

Classically, the standard traction Tn on ›R would be power-conjugate to _ as in Eq. (11.7), but ›R when migrating has no the motion velocity u; intrinsic material description, as material is continually being added and removed, and it would seem appropriate to use as power-conjugate velocity for Tn the motion velocity u following ›R as described by v›R ; granted this, ð Tn·u da ð11:8Þ ›RðtÞ

represents the associated expenditure of power. Material is added to R only along its boundary ›R; there is no transfer of material to the interior of P; nor is there any change in the material structure. For that reason the configurational body force f performs no work. We, therefore, write the power expenditure of the standard and configurational force systems in the form ð WðRðtÞÞ ¼ ðCn·v›R þ Tn·uÞda: ð11:9Þ ›RðtÞ

Writing the expended power in the form (11.9) might, at first sight, appear artificial, since the tangential component of the velocity v›R is not uniquely determined by the migration of ›R; this issue is addressed in Section XII.B, where we determine conditions both necessary and sufficient for WðRðtÞÞ to be independent of the specific choice of v›R : If v›R ; 0; then RðtÞ is independent of t; so that RðtÞ ; P; with P a part. In this case, by Eq. (11.6), u reduces to u_ and the expended power (11.9) assumes the classical form (11.7).

XII. Thermodynamical Laws for Migrating Control Volumes. The Eshelby Relation We now present extensions of the atomic balance, balance of energy, and the entropy imbalance appropriate to a treatment of configurational forces.28 28

Due to their absence of temporal derivatives, the standard force and moment balances (2.3) and the configurational force balance (11.1) remain valid as is for migrating control volumes.

A Unified Treatment of Evolving Interfaces

57

Our extensions are based on the use of migrating control volumes to characterize the manner in which configurational forces expend power.

A. Migrational Balance Laws We begin rewriting the atomic balance (3.1), balance of energy (4.2), and the entropy imbalance (4.3) for an arbitrary part P of B : 9 ð d ð a > r dv ¼ 2 a ·n da; > > > dt P ›P > > > > N = ð ð ð ð X d a a 1 dv ¼ Tn·u_ da 2 q·n da 2 m  ·n da; > dt P ›P ›P > a¼1 ›P > > > > ð ð > d q ; h dv $ 2 ·n da: > dt P ›P u

ð12:1Þ

Generalizations, to migrating control volumes RðtÞ; of Eq. (12.1)1,3, the atomic balance and the entropy inequality are straightforward, but the generalization of the energy balance (12.1)2 is not at all obvious. Indeed, the standard generalization of this balance, namely ð d ð 1 dv 2 1V›R da dt RðtÞ ›RðtÞ N ð ð ð X ¼ Tn·u_ da 2 q·n da 2 ›RðtÞ

›RðtÞ

a¼1

ma a ·n da; ›RðtÞ

ð12:2Þ

is inapplicable because it does not account explicitly for the rate at which work is performed by the configurational-force system. Here, following Gurtin (1995, 2000, Section 6c), we consider a more general development based on † an atomic balance for each species a in the standard form ð ð d ð ra dv ¼ 2 a ·n da þ ra V›R da ; dt RðtÞ ›RðtÞ ›RðtÞ |fflfflfflfflffl{zfflfflfflfflffl} accretive flow of atoms

ð12:3Þ

58

E. Fried and M.E. Gurtin

† balance of energy in the form WðRðtÞÞ

zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ ð ð d ð 1 dv ¼ ðCn·v›R þ Tn·uÞda 2 q·n da dt RðtÞ ›RðtÞ ›RðtÞ N ð N ð ð X X 2 ma a ·n da þ QV›R da þ ma ra V›R da ; ›RðtÞ ›RðtÞ a¼1 ›RðtÞ a ¼1 |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} accretive heating

accretive flow of chemical energy

ð12:4Þ with Q a field defined over the body for all time; † an entropy imbalance in the form ð ð d ð q QV›R h dv $ 2 ·n da þ da : dt ›RðtÞ u ›RðtÞ u ›RðtÞ |fflfflfflfflffl{zfflfflfflfflffl}

ð12:5Þ

accretive flow of entropy

B. The Eshelby Relation as a Consequence of Invariance We require that the migrational laws (12.3) –(12.5) be satisfied for any choice of the migrating control volume RðtÞ and—in view of our discussion in the paragraph following Eq. (11.3)—that these laws be independent of the choice of velocity field v›R ðx; tÞ used to describe the migration of ›RðtÞ: Among the migrational laws it is only the energy balance (12.4) that is influenced by changes in v›R ðx; tÞ and this influence is felt only through the expended power WðRðtÞÞ as described by Eq. (11.9). Thus we are led to the Intrinsicality hypothesis Given any migrating control volume RðtÞ; the expended power Eq. (11.9) is independent of the choice of velocity field v›R ðx; tÞ used to describe the migration of ›RðtÞ: This hypothesis has strong consequences. Choose a migrating control volume RðtÞ and an arbitrary tangential vector field on ›R: Then, by Eq. (11.3) v› R ¼ V› R n þ a

ð12:6Þ

is a velocity field for ›RðtÞ; and the requirement that the expended power be invariant under changes in v›R is equivalent to the requirement that this power be

A Unified Treatment of Evolving Interfaces

59

independent of the choice of tangential field a: By Eqs. (11.6) and (12.6),

u ¼ u_ þ V›R ð7uÞn þ ð7uÞa; thus if we let G ¼ C þ ð7uÞT T; then Cn·v›R þ Tn·u ¼ Tn·u_ þ V›R n·Gn; þ a·Gn; and the expended power (11.9) becomes

WðRðtÞÞ ¼

ð

Tn·u_ da þ ›RðtÞ

ð ›RðtÞ

V›R n·Gn da þ

ð

a·Gn da:

ð12:7Þ

›RðtÞ

Since a appears linearly in Eq. (12.7), and not elsewhere in Eq. (11.9), the invariance of Eq. (11.9) under changes in a is equivalent to the requirement Ð that ›RðtÞ a·Gn da ¼ 0 for all RðtÞ and all fields aðx; tÞ tangential to ›RðtÞ: Since both R and a are arbitrary, Gn·a ¼ 0 for all orthogonal vectors n and a; thus Gn must be parallel to n for all n and every vector must be an eigenvector of G: Thus there is a scalar field p such that G ¼ p1; hence, (Gurtin, 2000, p. 37)

C ¼ p1 2 ð7uÞT T

ð12:8Þ

and the expended power lias the intrinsic form

WðRðtÞÞ ¼

ð

Tn·u_ da þ ›RðtÞ

ð ›RðtÞ

pV›R da:

ð12:9Þ

The field p represents a configurational bulk tension that performs work in conjunction with the addition of material at the boundary of a migrating control volume. Conversely, the relation (12.8) renders the theory consistent with the hypothesis of intrinsicality. Summarizing, we have shown that the intrinsicality hypothesis is equivalent to the requirement that the configurational stress and expended power have the respective forms Eqs. (12.8) and (12.9). But more can be said if we account for the full set of migrational laws. A standard transport theorem asserts that, for RðtÞ a migrating control volume and

E. Fried and M.E. Gurtin

60

Qðx; tÞ a field on the body for all time, ð ð d ð Q dv ¼ Q_ dv þ QV›R da: dt RðtÞ RðtÞ ›RðtÞ

ð12:10Þ

Thus we may rewrite Eq. (12.5) as ð

h_ dv þ

ð

RðtÞ

›RðtÞ

hV›R da $ 2

ð

ð q QV›R ·n da þ da: u u ›RðtÞ

›RðtÞ

ð12:11Þ

On the other hand, we may use Eqs. (12.9) and (12.10) to write balance of energy (12.4) in the form ð

1_ dv þ RðtÞ

2

ð ›RðtÞ

ð

1V›R da ¼

q·n da 2 ›RðtÞ

ð

N ð X

a¼1

Tn·u_ da þ ›RðtÞ

ð ›RðtÞ

a a

m  ·n da þ ›RðtÞ

ð

pV›R da

Qþ ›RðtÞ

N X

! a a

m r

V›R da:

a¼1

ð12:12Þ Given a time t; it is possible to find a second migrating control volume R0 ðtÞ with R0 ðtÞ ¼ RðtÞ; but with V 0›R ; the normal velocity of ›R0 at t ¼ t; an arbitrary scalar field on ›R0 ; satisfaction of Eqs. (12.11) and (12.12) for all such V 0›R implies that p ¼ 1 2 hq 2

Q ¼ hq;

N X

ma ra ;

ð12:13Þ

a¼1

so that p¼C2

N X

m a ra ;

ð12:14Þ

a¼1

with C ¼ 1 2 uh the free energy. Thus Eq. (12.8) yields the Eshelby relation C¼ C2

N X

! a

a

r m

1 2 ð7uÞ` T:

ð12:15Þ

a¼1

This relation, which applies to both unconstrained materials and substitutional alloys, plays a fundamental role in our discussion of evolving interfaces.

A Unified Treatment of Evolving Interfaces

61

C. Consistency of the Migrational Balance Laws with Classical Forms of these Laws By Eq. (12.13), we may rewrite the migrational laws for energy and entropy in the form

9 ð ð ð d ð > 1 dv 2 1V›R da ¼ ðCn·v›R þ Tn·uÞda 2 q·n da > > > dt RðtÞ ›RðtÞ ›RðtÞ ›RðtÞ > > ! > N ð N > ð X X = a a a a 2 m  ·n da 2 C2 r m V›R da; ›RðtÞ > a¼1 ›RðtÞ a¼1 > > > > > ð ð ð > d q ; h dv 2 hV›R da $ 2 ·n da: > dt RðtÞ ›RðtÞ ›RðtÞ u ð12:16Þ

The second of these and the atomic balance (12.3) are standard. On the other hand, balance of energy in the form (12.16)2 is nonstandard, as it explicitly accounts for power expended by configurational forces, but we may use Eqs. (12.9) and (12.14) to reduce Eq. (12.16)2 to its classical form Eq. (12.2). The migrational laws are, therefore, fully consistent with classical versions of these laws.

D. Isothermal Conditions. The Free-Energy Imbalance Assume now that isothermal conditions prevail: u ; constant: Multiplying the entropy imbalance (12.16)3 by u and subtracting the result from the energy balance (12.16)2 then yields the (migrational) free-energy imbalance

N ð ð X d ð C dv # ðCn·v›R þ Tn·uÞda 2 ma a ·n da dt RðtÞ ›RðtÞ a¼1 ›RðtÞ accretive flow of chemical energy

zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ N ð X þ ma ra V›R da : a¼1

›RðtÞ

ð12:17Þ

E. Fried and M.E. Gurtin

62

This free-energy imbalance should be compared to its classical counterpart ð d ð C dv 2 CV›R da dt RðtÞ ›RðtÞ |fflfflfflfflffl{zfflfflfflfflffl} accretive flow of free energy

#

ð ›RðtÞ

Tn·u_ da 2

N ð X

a¼1

ma a ·n da; ›RðtÞ

ð12:18Þ

which accounts explicitly for the accretive flow of free energy, but accounts neither for power expended by the configurational force system nor for the accretive flow of chemical energy. Finally, in view of the lattice constraint Eq. (5.1), C transforms according to C ! C 2 lrsites 1 under the transformations ma ! ma þ l and, consequently, the free-energy imbalance (12.17) is invariant under such transformations.

E. Generic Free-Energy Imbalance for Migrating Control Volumes The free-energy imbalance (12.17) has the generic form d {free energy of RðtÞ} dt # {power expended on RðtÞ by configurational and standard forces} þ {energy flow into RðtÞ by atomic diffusion} þ {accretive flow of chemical energy into RðtÞ}:

ð12:19Þ

With the exception of Section XIII, the remainder of this work focuses on evolving interfaces neglecting thermal influences. Our discussion there is based on free-energy imbalances of the generic structure (12.19) with the role of a migrating control volume replaced by that of an interfacial pillbox (Section XVI).

XIII. Role and Influence of Constitutive Equations Until this point in our discussion of configurational forces, no use has been made of constitutive theory. Hence, the results are completely independent of constitutive equations and apply to broad classes of materials, allowing, for example, for plasticity, viscoelasticity, and other more complicated forms of

A Unified Treatment of Evolving Interfaces

63

dissipative material response that couple the mechanical and chemical degrees of freedom. We now consider the implications of assuming that the free energy, stress, and chemical potentials are given by thermocompatible constitutive equations depending upon the strain and the atomic densities. First note that, since ðð7uÞT TÞij ¼ Tkj ›uk =›xi ; computing div C using the Eshelby relation (12.15) gives ! N X ›Cij ›Tkj ›uk › ›2 uk a a ¼ C2 r m 2 Tkj 2 ; ›x i ›x j › x › x ›x j ›x i j i a¼1 and, therefore, using the balance div T ¼ 0; the symmetry of T; and the strain – displacement relation (2.1), we find that the configurational body force f in the balance div C þ f ¼ 0 has the decomposition fi ¼ 2

N N X X ›Ekj ›C › ra ›m a þ Tkj þ ma þ ra : ›x i ›x i ›xi ›x i a¼1 a¼1

ð13:1Þ

By Eq. (7.5), N X ›Ekj ›C ›ra ¼ Tkj þ ma ð13:2Þ ›x i ›x i ›x i a¼1 P for an unconstrained material and, therefore, f ¼ Na¼1 ra 7ma : For a substitutional alloy the computation is similar, but a bit more complicated; the crucial step is noting that, by virtue of the lattice constraint Eq. (5.1),  a X  a N N X a ›r az ›r m m ¼ ›xi ›xi a¼1 a¼1

for any reference species z; so that, by Eq. (9.27), N N X X ›Ekj ›Ekj ›C ›ra ›ra ¼ Tkj þ maz ¼ Tkj þ ma : ›x i ›x i ›x i ›x i ›xi a¼1 a¼1

ð13:3Þ

Thus, for unconstrained materials and for substitutional alloys, f¼

N X

ra 7ma ;

ð13:4Þ

a¼1

showing that configurational body forces arise in response to spatial variations in the chemical potentials. We review f as an internal body force. If we had, from the outset, included an external body force b in the standard force system, so that div T þ b ¼ 0; then we would get a concomitant external body force 2ð7uÞT b in the configurational system (cf. Eshelby, 1956; Maugin, 1993).

64

E. Fried and M.E. Gurtin

A major difference between the standard and configurational force systems is the presence of internal configurational forces such as f: These forces are connected with the material structure of the body B; corresponding to each configuration of B there is a distribution of material and internal configurational forces that act to hold the material in place in that configuration. Such forces characterize the resistance of the material to structural changes and are basic when discussing the kinetics of defects. Finally, as is clear from Eqs. (12.15) and (13.4), the configurational fields are completely determined by the fields u; T; C; m~; and r~; there is no need for additional constitutive assumptions. As we shall see, this will not be so when we discuss evolving interfaces. INTERFACE KINEMATICS We consider an interface SðtÞ separating bulk phases or grains; in the latter case we will often, but not always, refer to S as a grain boundary. To avoid cumbersome mathematical formality associated with surfaces, we restrict attention to two space-dimensions; the interface is then presumed to be a smooth curve SðtÞ that evolves smoothly with t: XIV. Definitions and Basic Results This section contains mathematical results of a preliminary nature concerning the evolution of curves.29 A. Curvature. Normal Velocity. Normal Time-Derivative We use the following notation for quantities associated with S: tðx; tÞ and nðx; tÞ; respectively, denote the tangent and normal fields t ¼ ðcos q; sin qÞ;

n ¼ ð2sin q; cos qÞ;

ð14:1Þ

with n directed outward from the region occupied by the (2)-phase, and with q the counterclockwise angle from the ð1; 0Þ-axis to t (Fig. 14.1). Then Eqs. (14.1) 29

These results are taken from Angenent and Gurtin (1989). Cf. also Gurtin (1993, Sections I, II), where a detailed presentation may be found.

A Unified Treatment of Evolving Interfaces

65

Fig. 14.1. The interface SðtÞ: Our convention is such that K . 0 on concave upward portions of SðtÞ:

yield the Frenet formulas

›t ¼ Kn; ›s

›n ¼ 2Kt; ›s

with s the arclength and K¼

ð14:2Þ

›q ›s

ð14:3Þ the curvature. We consistently use the term interfacial field for a scalar, vector, or tensor field wðx; tÞ defined on SðtÞ for all t: Let x ¼ rðs; tÞ denote an arc length parametrization of SðtÞ with s increasing in the direction of t: Then ›r V¼ ·n ð14:4Þ ›t and v ¼ Vn

ð14:5Þ

denote the scalar and vector normal velocities of S; while v ¼ 2t·ð›r=›tÞ is termed the arc velocity. Note that, since t ¼ ›r=›s and t·›t=›t ¼ 0; we may use Eqs. (14.2)1 and (14.4) to conclude that   ›v › ›r ›r ›t ›r ¼ 2t· · ¼ 2 ·ðKnÞ ¼ 2KV: ð14:6Þ 2 › t ›s ›s ›t ›s ›t Given a point x0 on the interface at some time t0 ; the normal trajectory through x0 at t0 is the smooth curve zðtÞ defined as the solution of the initial-value problem dzðtÞ ¼ vðzðtÞ; tÞ; dt

zðt0 Þ ¼ x0 :

ð14:7Þ

The normal trajectory zðtÞ may also be described uniquely in terms of arc length; indeed since the mapping x ¼ rðs; tÞ represents a one-to-one correspondence between s and x;30 there is a function s ¼ SðtÞ such that zðtÞ ¼ rðSðtÞ; tÞ: Then, since the trajectory zðtÞ is normal, tðSðtÞ; tÞ· 30

drðSðtÞ; tÞ ¼0 dt

The curve SðtÞ is presumed to be nonintersecting.

ð14:8Þ

E. Fried and M.E. Gurtin

66

and hence, SðtÞ is a solution of the initial-value problem dSðtÞ ¼ vðSðtÞ; tÞ; dt

Sðt0 Þ ¼ s0 ;

ð14:9Þ

where x0 ¼ rðs0 ; tÞ: The normal time-derivative w of an interfacial field wðx; tÞ is the derivative of w following the normal trajectories of SðtÞ:31

w ðx0 ; t0 Þ ¼

% dwðzðtÞ; tÞ %% %t¼t : dt 0

ð14:10Þ

The field wðx; tÞ may equally well be described as a function wðs; tÞ of arc length and time. We refer to wðs; tÞ as the arc length description of w: In the arc length description the normal time derivative has the equivalent form

w ðs0 ; t0 Þ ¼

% dwðSðtÞ; tÞ %% %t¼t ; dt 0

so that, by Eq. (14.9),

w ¼

›w ›w þv ; ›t ›s

ð14:11Þ

where here and in what follows

›w=›t denotes the partial derivative of w holding s fixed. Basic to our discussion of constitutive equations is the chain-rule for the normal time-derivative: given a function w ¼ w^ðl~Þ; with l~ ¼ ðl1 ; l2 ; …; lN Þ and each ln an interfacial field,

w ¼

31

N X ›w^ðl~Þ l n: ›ln n¼1

ð14:12Þ

The normal-time derivative is a counterpart, for the interface S; of the time-derivative following the motion of the surface ›R of a migrating control volume R (cf. Section XI.B).

A Unified Treatment of Evolving Interfaces

67

The verification of Eq. (14.12) is based on the definition of the normal timederivative as specified in Eq. (14.10). Indeed,

w ðx0 ; t0 Þ ¼

¼

& ' % N X dwðzðtÞ; tÞ %% ›w^ðl~Þ dln ðzðtÞ; tÞ ¼ %t¼t dt dt ›ln t¼t0 0 n¼1 N X ›w^ðl~ðx0 ; t0 ÞÞ l n ðx0 ; t0 Þ: ›ln n¼1

ð14:13Þ

B. Commutator and Transport Identities Often in what follows it becomes necessary to interchange the differential operators ›=›s and ð· · ·Þ: We now establish the commutator associated with such an interchange. Let w be an interfacial field. Then, by Eq. (14.11),   ›w › ›w ›w ›2 w ›2 w ›v ›w ¼ þv þv 2 þ ; ¼ ›s ›t ›s ›s ›t ›s ›s ›s ›s so that, by Eq. (14.6), we have the commutator relation

›w ¼ ›s



 ›w ›w : 2KV ›s ›s

ð14:14Þ

Also important are the transport identities:

q ¼

›V ; ›s

K ¼

›2 V þ K 2 V; ›s 2

t ¼ q n;

n ¼ 2 q t:

ð14:15Þ

The identities (14.15)3,4, follow from Eq. (14.1). Next, applying Eq. (14.11) with w ¼ r; r ¼

›r þ vt; ›t

ð14:16Þ

so that, by Eqs. (14.4) and (14.8),   ›r r ¼ ðn· r Þn ¼ n· n ¼ Vn: ›t Therefore, by Eq. (14.2)2, n·

›r ›V ¼ : ›s ›s

ð14:17Þ

E. Fried and M.E. Gurtin

68

On the other hand, the commutator relation with w ¼ r yields n·

›r ¼ n· t ¼ q : ›s

The last two relations imply Eq. (14.15)1. Finally, since K ¼ ›q=›s; the commutator relation with w ¼ q yields

›q ¼ K 2 K2V ›s

ð14:18Þ

and Eq. (14.15)2 follows from Eq. (14.15)1.

C. Evolving Subcurves CðtÞ of SðtÞ Throughout, CðtÞ denotes an arbitrary evolving subcurve of SðtÞ: We consistently use the following notation: xa ðtÞ and xb ðtÞ; respectively, denote the initial and terminal points (in the sense of arc length) of the curve CðtÞ; for any interfacial field wðx; tÞ;

wa ðtÞ ; wla ðtÞ ¼ wðxa ðtÞ; tÞ;

wb ðtÞ ; wlb ðtÞ ¼ wðxb ðtÞ; tÞ:

ð14:19Þ

The endpoints of CðtÞ may also be marked by their arc length values Sa ðtÞ and Sb ðtÞ; where xa ðtÞ ¼ rðSa ðtÞ; tÞ;

xb ðtÞ ¼ rðSb ðtÞ; tÞ;

ð14:20Þ

so that, using the arc length description of w;

wa ðtÞ ¼ wðSa ðtÞ; tÞ;

wb ðtÞ ¼ wðSb ðtÞ; tÞ;

ð14:21Þ

hence,

wlba ; wb 2 wa ¼

ð ›w ds: C ›s

The functions Wa ðtÞ and Wb ðtÞ defined by Wa ¼ ta ·

dxa ; dt

Wb ¼ tb ·

dxb dt

are the tangential endpoint velocities of CðtÞ: Since the normal velocities of the endpoints must coincide with the normal velocity of SðtÞ; V a ¼ na ·

dxa ; dt

V b ¼ nb ·

dxb ; dt

A Unified Treatment of Evolving Interfaces

69

Fig. 14.3.1. The subcurve C of S and the velocity of the endpoint xb : Wb ðtÞ is the tangential endpoint velocity of xb :

hence (Fig. 14.3.1) dxa ¼ Wa ta þ Va na ; dt

dxb ¼ W b t b þ Vb n b : dt

ð14:22Þ

On the other hand, by Eq. (14.20), % dxa ›r %% dSa ¼ ; þt dt ›t %a a dt so that, by Eq. (14.21),   dxa dSa ¼ r la þ ta 2 va ; dt dt

ð14:23Þ

and an analogous relation holds for the other endpoint. Thus, since t· r ¼ 0; Wa ¼

dSa 2 va ; dt

Wb ¼

dSb 2 vb : dt

ð14:24Þ

Given an interfacial field w; argument leading to Eq. (14.23) yields  %  d wa ›w %% dSa ¼ w la þ 2 va dt ›s %a dt and similarly for the other endpoint. Therefore, by Eq. (14.24), % dw a ›w %% ¼ w la þ W ; dt ›s % a a

% d wb ›w %% ¼ w lb þ W ; dt ›s % b b

ð14:25Þ

E. Fried and M.E. Gurtin

70

and applying these relations with w ¼ q yields, since ›q=›s ¼ K; dq a ¼ q la þ Ka Wa ; dt

dq b ¼ q lb þ Kb Wb : dt

ð14:26Þ

D. Transport Theorem for Integrals Let w be an interfacial field. Then ð

w ds ¼ CðtÞ

ðS2 ðtÞ

wðs; tÞds:

S1 ðtÞ

Thus, writing _ ba ¼ wb S_ b 2 wa S_ a ½wS

ð14:27Þ

and suppressing the argument t where convenient, we may use Eq. (14.11) to conclude that  ð ›w ð  d ð ›w _ ba ¼ _ ba : ds þ ½wS w ds ¼ w 2v ds þ ½wS dt CðtÞ ›s C ›t C Thus if we integrate the term v ›w=›s by parts and use Eqs. (14.6) and (14.24), we arrive at the following theorem (Angenent and Gurtin, 1989): Transport theorem for integrals. For CðtÞ a smoothly evolving subcurve of SðtÞ and wðx; tÞ a smooth interfacial field, ð d ð w ds ¼ ðw 2 wKVÞds þ ½wWba ; dt CðtÞ CðtÞ

ð14:28Þ

½wWba ¼ wb Wb 2 wa Wa :

ð14:29Þ

with

Notation analogous to Eq. (14.29) (for endpoint differences) will be used repeatedly throughout what follows.

XV. Deformation of the Interface We now turn to a discussion of deformation. We consider both solid –solid and solid –vapor phase transitions; in the former case we restrict attention to coherent interfaces.

A Unified Treatment of Evolving Interfaces

71

A. Interfacial Limits We assume that the interface SðtÞ separates phases labeled (þ) and (2), with the normal n pointing into the (þ)-phase. When discussing solid – vapor interfaces, the (þ)-phase always denotes the vapor. Consider an arbitrary field f ðx; tÞ that is continuous up to SðtÞ from either side. Let v f b and Rf S designate the jump and average of f across the interface, while f ^ denote the limiting values of f ; specifically, for x on SðtÞ; 9 (( )) f ðx; tÞ ¼ f þ ðx; tÞ 2 f 2 ðx; tÞ; > > > > > 2 1 þ Rf Sðx; tÞ ¼ 2 ðf ðx; tÞ þ f ðx; tÞÞ; = ð15:1Þ > > f ^ ðx; tÞ ¼ lim f ðx ^ 1nðx; tÞ; tÞ: > > > 1!0 ; 1.0

Our discussion of solid – vapor interfaces is limited to situations in which the vapor may, in essence, be represented by the limiting values of its basic fields at the interface; bulk values of these fields away from the interface play no role. In this instance, the field f þ simply represents the value of f in the vapor at the interface. B. Interfacial-Strain Vector 1. Solid –Vapor Interface We assume that the displacement uðx; tÞ is smooth up to the interface from the solid phase, so that the interfacial strain e¼

›u ¼ ð7uÞt ›s

ð15:2Þ

is well defined on S; as is the temporal derivative u ¼ u_ þ ð7uÞv:

ð15:3Þ

2. Coherent Solid-Solid Interface In our discussion of solid – solid phase transitions we restrict attention to interfaces S that are coherent in the sense that u is continuous across the interface:32 ½½u ¼ 0: 32

Cf. Cermelli and Gurtin (1994a,b) for discussions of incoherent interfaces.

ð15:4Þ

E. Fried and M.E. Gurtin

72

We do not require that the derivatives of u be continuous across the interface, but we do require that all such derivatives be continuous up to the interface from either side, an assumption that allows us to compute the interfacial strain e¼

›u ›s

ð15:5Þ

using the limiting values of 7u on each side of the interface: e ¼ ð7uÞþ t ¼ ð7uÞ2 t ¼ R7uSt:

ð15:6Þ

Similarly, for x ¼ rn ðp; tÞ a normal parametrization of SðtÞ;33 we may use Eq. (14.10) and the chain-rule to compute u as follows: _ þ R7uSv; u ¼ u_ þ þ ð7uÞþ v ¼ u_ 2 þ ð7uÞ2 v ¼ RuS

ð15:7Þ

this yields the classical compatibility condition ½½u_  þ V ½½7un ¼ 0:

ð15:8Þ

3. Identities Involving the Interfacial Strain It is convenient to introduce the interfacial tensile and shear strains, 1 and g; defined by e ¼ 1t þ g n:

ð15:9Þ

Then 1 ¼ t·

›u ; ›s

g ¼ n·

›u ; ›s

so that, using the Frenet formulas (14.2),     ›1 ›2 u ›u ›g ›2 u ›u ¼ t· 2 þ n· ¼ n· 2 2 t· K; K: ›s ›s ›s ›s ›s ›s

ð15:10Þ

ð15:11Þ

Next, the fields u; u ; and e ¼ ›u=›s are, for each of the two types of phase transitions under consideration, well defined on the interface. We may therefore, use the commutator relation (14.14) to conclude that

›u ¼ e 2 KVe: ›s 33

That is, ›rn =›t ¼ Vn ¼ v; with ›=›t the derivative holding p fixed.

ð15:12Þ

A Unified Treatment of Evolving Interfaces

73

By Eq. (14.15)3,4, the interfacial strain-rate (following the motion of the interface) is given by e ¼ ð1 2 g q Þt þ ðg þ 1 q Þn:

ð15:13Þ

Thus, interestingly, while the strain 1 represents stretching of the interface, the stretch rate, as defined by t· e ; is given by t· e ¼ 1 2 g q

ð15:14Þ

and hence, involves the shear strain g via a term arising from temporal changes in the orientation of the interface. Let CðtÞ; with endpoints xa ðtÞ and xb ðtÞ; be an arbitrary evolving subcurve of SðtÞ; and let ua ðtÞ ¼ uðxa ðtÞ; tÞ;

ub ðtÞ ¼ uðxb ðtÞ; tÞ

denote the corresponding endpoint displacements (see Fig. 14.3.1), so that dua ; dt

dub dt

ð15:15Þ

represent motion velocities following these endpoints. Then, by the vectorial counterpart of Eq. (14.25), dua ¼ u la þ ea Wa ; dt

dub ¼ u lb þ eb Wb : dt

ð15:16Þ

(These relations may also be derived using the chain-rule and, say, Eq. (C.4): dua dx _ a þ ð7uÞa a ¼ ðuÞ _ a þ ð7uÞa ðWa ta þ Va na Þ ¼ u la þ ea Wa ; etc:Þ ¼ ðuÞ dt dt XVI. Interfacial Pillboxes We will discuss interfaces separating grains, solid phases, and solid and vapor phases. For the purpose of this discussion, assume that the interface SðtÞ separates phases labelled (þ) and (2), with the normal n pointing into the (þ)-phase. Our discussion of configurational forces in bulk was based on the use of control volumes that migrate through the body. The counterpart of this notion for the interface S ¼ SðtÞ is an interfacial pillbox, which we now define. Consider an arbitrary evolving subcurve CðtÞ of SðtÞ: Our discussion of basic laws views C as a interfacial pillbox of infinitesimal thickness containing a portion of S; a view that allows us to isolate the physical processes in the individual phases that interact with S: The geometric boundary of C consists of its endpoints xa and xb : But C viewed as pillbox has a pillbox boundary

E. Fried and M.E. Gurtin

74

Fig. 16.1. An interfacial pillbox separating (þ) and (2) phases.

consisting of (Fig. 16.1): † a surface Cþ with unit normal þn that lies in the (þ)-phase; † a surface C2 with unit normal 2n that lies in the (2)-phase; † end faces represented by the endpoints xa and xb of C: The interactions of C with the bulk phases are then represented by tractions exerted on—and flows of atoms and energy across—the surfaces Cþ and C2 ; the interaction of C with the remainder of S is represented by forces on—and flows of atoms and energy across—the endpoints of C:

GRAIN BOUNDARIES We here consider a class of theories central to materials science. These theories involve only configurational forces; deformation and standard forces are neglected.34

XVII. Simple Theory Neglecting Deformation and Atomic Transport We here discuss an evolving grain boundary S ¼ SðtÞ neglecting deformation and atomic transport.35 34

Some workers who discuss configurational forces do not accept the notion of an independent configurational force balance, but consider that balance to depend in some way on the standard force balance. The theories discussed in this part show that the configurational force balance is, in general, independent of the standard force balance, since in these theories there are no standard forces; there are only configurational forces. 35 The results of this section are due to Angenent and Gurtin (1989); cf. also Gurtin (1993, 2000).

A Unified Treatment of Evolving Interfaces

75

A. Configurational Force Balance The configurational force system for the grain boundary consists of an interfacial stress (vector) c; an internal force density g; distributed continuously over S; and contributions associated with the interaction of the bulk phases with the grain boundary (Fig. 17.1.1). The stress c characterizes forces such as surface tension that act within S; g represents internal forces associated with the exchange of atoms between grains at S: Let C ¼ CðtÞ be an arbitrary interfacial pillbox. The portion of S external to C then exerts forces 2ca and cb at xa and xb :36 Further, the bulk material in the (þ )-phase exerts a configurational traction Cþ n on Cþ ; while that in the (2 )-phase exerts a configurational traction 2C2 n on C2 ; so that the net configurational traction exerted at each point of C by the bulk phases is Cþ n 2 C2 n ¼ vCbn: The configurational force balance for C; therefore, takes the form37 clba þ

ð C

g ds þ

ð

½½Cn ds ¼ 0;

C

clba ¼ cb 2 ca ;

ð17:1Þ

and has the immediate consequence that c must be a continuous function of arc length: ð17:2Þ Ð Further, since cb 2 ca ¼ C ›c=›s ds; Eq. (17.1) yields the local balance

›c þ g þ ½½Cn ¼ 0: ›s

ð17:3Þ

Fig. 17.1.1. Configurational forces on an interfacial pillbox C: 36

i.e., e.g., ca ðtÞ ¼ cðxa ; tÞ: The balance (17.1) is an interfacial force-balance (11.1) for Ð counterpart of the bulk configurational Ð b a part P: The role of the net Ð traction ›P Cn daÐin Eq. (11.1) is played by cla þ C vCbn ds; while that of the net internal force P f da is played by C g ds: 37

76

E. Fried and M.E. Gurtin

We define the configurational surface tension s and shear stress t through the decomposition c ¼ s t þ t n:

ð17:4Þ

Then, appealing to the Frenet formulas ›t=›s ¼ Kn and ›n=›s ¼ 2Kt;

›c ¼ ›s



   ›s ›t 2 tK t þ þ sK n; ›s ›s

ð17:5Þ

whereby the normal component of Eq. (17.3), the normal configurational force balance, is given by

sK þ

›t þ g þ n·½½Cn ¼ 0; ›s

ð17:6Þ

with g ¼ g·n

ð17:7Þ

the normal internal force. On the other hand, we shall find that g·t is indeterminate, thereby rendering the tangential component of Eq. (17.3), namely

›s 2 tK þ g·t þ t·½½Cn ¼ 0; ›s

ð17:8Þ

inconsequential to the theory (cf. the discussion following Eq. (17.30)). The bulk configurational stress C is determined by the Eshelby relation (12.15), which, since we neglect deformation and atomic transport, has the form C ¼ C 1; with C the bulk free energy; thus ½½C ¼ ½½C1;

ð17:9Þ

which we assume to be constant. The normal and tangential components of the configurational force balance, therefore, reduce to

sK þ

›t þ g þ ½½C ¼ 0 ›s

ð17:10Þ

and

›s 2 tK þ g·t ¼ 0: ›s

ð17:11Þ

A Unified Treatment of Evolving Interfaces

77

B. Power 1. External Expenditure of Power The configurational forces on an interfacial pillbox C are assumed to expend power in conjunction with the migration of the pillbox. The stress c exerts force at the endpoints xa and xb of C; and we, therefore, take dxa =dt and dxb =dt as the corresponding power-conjugate velocities for c. Analogously, the configuration tractions Cþ n and 2C2 n represent forces exert on Cþ and C2 and are, therefore, taken to be power-conjugate to the normal velocity v ¼ Vn of Cþ and C2 : The (net) power expended on C; therefore, has the form &

dx c· dt

'b

þ

ð

a

½½Cn·v ds;

ð17:12Þ

C

with &

dx c· dt

'b a

¼ cb ·

dxb dx 2 ca · a : dt dt

2. Internal Expenditure of Power. Power Balance By Eqs. (14.22) and (17.4), & ' dx b c· ¼ ½sW þ tVba ; dt a

ð17:13Þ

while, by Eqs. (14.5) and (17.9), vCbn·v ¼ vCbV; hence, the power expended on C has the form & ' ð dx b ð c· ð17:14Þ þ ½½Cn·v ds ¼ ½sW þ tVba þ ½½CV ds: dt a C C Consider the term ½tVba : Using the identity ›V=›s ¼ q (cf. Eq. (14.15)1) and the normal configurational force balance (17.10), it follows that  ð ð  ›t V ds ¼ ½tV ba ¼ tq þ ½t q 2 ðsK þ g þ FÞV ds: ð17:15Þ ›s C C Combining Eq. (17.15) with Eq. (17.14) then yields the power balance & ' ð dx b ð c· þ ½½Cn·v ds ¼ ½sWba þ ½t q 2 ðsK þ gÞVÞ ds : ð17:16Þ dt a C C |fflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflffl ffl} power expended on C

power expended within C

78

E. Fried and M.E. Gurtin

Fig. 17.2.1. Power expenditures within an evolving grain boundary.

As noted by Gurtin (2000, pp. 71, 105), the individual terms comprising the internal power have the following physical interpretations (Fig. 17.2.1): † The term ½sWba represents power expended internally by the surface tension as material is added to C at its endpoints. † The term t q represents an expenditure of power associated with changes in the orientation of the grain boundary. † The term 2sKV represents an expenditure of power associated with changes in interfacial length due to the curvature of the grain boundary. † The term 2gV represents power expended in the exchange of material at the grain boundary; the negative sign signifies that g expends (positive) power when and only when it opposes motion of the grain boundary. A complete catalog of internal power expenditures is possible only with the introduction of configurational forces within a structure that embodies the notion of power. Because they bypass such fundamental notions, ad hoc methods such as gradient-flow arguments obscure the underlying physics.

C. Free-Energy Imbalance 1. Global Imbalance Guided by the arguments given in Section XII.D in support of the free-energy imbalance (12.17) for a migrating control volume in bulk, and bearing in mind that the present theory does not allow for atomic transport, we posit a free-energy imbalance for each interfacial pillbox CðtÞ in the general form d {free energy of CðtÞ} # {power expended on CðtÞ}: dt

ð17:17Þ

As in our treatment of power expended on a migrating control volume, the right side does not account for flows of bulk and interfacial free energy into the pillbox CðtÞ across its boundaries due to its migration, but it is meant to include power expended on CðtÞ by configurational forces.

A Unified Treatment of Evolving Interfaces

79

Let c denote the free energy of the grain boundary, measured per unit length, so that ð cðx; tÞds CðtÞ

represents the net free energy of the interfacial pillbox CðtÞ at any time t: Thus, bearing in mind the expression (17.12) for the power expended by the configurational forces, we write the free-energy imbalance for C in the form & ' d ð dx b ð c ds # c· þ ½½Cn·v ds: ð17:18Þ dt C dt a C 2. Growth and Decay of an Isolated Grain The global free-energy imbalance has interesting consequences. Consider a grain isolated from all other grains, with S the closed curve that represents the grain boundary. Identify the grain with the (2 )-phase, so that the normal n points into the surrounding matrix. Then, since vCbn·v ¼ vCbV and vCb is (assumed) constant, ð _ ½½CV ds ¼ ½½CAðtÞ; SðtÞ

where AðtÞ is the area enclosed by SðtÞ (that is, the area of the grain). Therefore, 

d ð ð17:19Þ c ds 2 ½½CAðtÞ # 0; dt SðtÞ and the quantity in braces decreases with time. If, for example, the bulk free energy of the matrix is strictly larger than that of the grain, then vCbdA=dt . 0; so that, were interfacial energy negligible, the area of the grain would increase with time, thereby lowering the net free energy of the bulk material. On the other hand, assuming that c . 0; for vCbdA=dt # 0 the area of the grain would decrease with time. If the grain and the surrounding matrix have equal free-energy densities, so that vCb ¼ 0; then the total free energy of the grain boundary must decrease with time. If, in addition, the grain-boundary energy c . 0 is constant, then d {length of SðtÞ} # 0 dt and the grain boundary must shorten with time.

E. Fried and M.E. Gurtin

80

3. Indeterminacy of the Tangential Component of the Internal Force g Note that there is no expenditure of power associated with ‘tangential motion’ of the grain boundary S (which is to be expected, since only the normal motion of S is intrinsic). Consistent with a ‘constraint’ of this type, we leave as indeterminate the tangential component g·t of the internal force. This assumption renders the tangential balance (17.8) irrelevant and allows us to restrict attention to the normal configurational force balance (17.10). This will be the case throughout what follows; for that reason, we will often leave the tangential configurational force balance unmentioned (cf. Remark 6 in Section XVII.D.2).6

4. Equality of Surface Tension and Interfacial Free Energy In view of the transport theorem (14.28) and the power balance (17.16), the free-energy imbalance for C; namely Eq. (17.18), becomes ð C

ð#Þ

zfflfflfflffl}|fflfflfflffl{ ð ðc 2 cKVÞds þ ½ðc 2 sÞWba # ½t q 2 ðsK þ gÞVds:

ð17:20Þ

C

Since C is arbitrary, so also are the tangential velocities Wa and Wb of the endpoints of C; since the only term in Ineq. (17.20) dependent on these velocities is the term (#), it follows that

s ¼ c:

ð17:21Þ

There are many misconceptions concerning the relation between surface tension and interfacial free energy. Here surface tension enters the theory via a force balance, whereas free energy enters via an energy imbalance; the fact that they coalesce is a consequence of the theory. As we shall see, in more general theories allowing for interfacial torques, standard interfacial stress, or atomic transport accounting for atomic densities within the interface, the relation s ¼ c is no longer valid. Note that, by Eq. (17.21), we may rewrite the normal configurational force balance (17.10) as

cK þ

›t þ g þ ½½C ¼ 0: ›s

ð17:22Þ

A Unified Treatment of Evolving Interfaces

81

5. Dissipation Inequality By Eq. (17.21), the free-energy inequality (17.20) reduces to ð

ðc 2 t q þ gVÞds # 0; C

since C is arbitrarily, we have the dissipation inequality

c 2 t q þ gV # 0:

ð17:23Þ

D. Constitutive Equations As in our discussion of bulk behavior, we use the dissipation inequality (17.23) as a guide in the development of constitutive equations for the grain boundary. Moreover, we require that the local dissipation inequality hold in all ‘processes’ related through the constitutive equations.

1. Basic Constitutive Equations. Restrictions It would seem clear from the dissipation inequality (17.23) that, at bottom: † c and t should depend on q; an assumption common to more classical theories (cf., e.g., Herring (1951)); † g should depend on V and, since classical theories display linear kinetics (cf., e.g., Mullins (1956)), this dependence might be linear; anisotropy would require a dependence of g also on q: We, therefore, begin with constitutive equations

c ¼ c^ðqÞ;

t ¼ t^ðqÞ;

g ¼ 2bðqÞV:

ð17:24Þ

Here bðqÞ is a kinetic modulus associated with the attachment kinetics of atoms at the grain boundary. By the chain rule (14.12), c ¼ ð›c^=›qÞ q ; and since the constitutive relations (17.24) are independent of V; we may conclude, upon taking V ¼ 0 in the

82

E. Fried and M.E. Gurtin

dissipation inequality (17.23), that c 2 t q # 0: Thus compatibility with Eq. (17.23) requires that ( ) dc^ðqÞ 2 t^ðqÞ q # 0 ð17:25Þ dq for all choices of the orientation field q: We may, therefore, restrict attention to spatially uniform functions qðtÞ; so that q ¼ q_ : Given any choice of time t0 ; we can always find a choice of qðtÞ such that qðt0 Þ and q_ ðt0 Þ take on arbitrary prescribed values. Thus, since q appears linearly in Eq. (17.25), its coefficient must vanish: t^ ¼ ›c^=›q: The free energy must, therefore, determine the configurational shear through the relation

t¼

dc^ðqÞ : dq

ð17:26Þ

Finally, by Eq. (17.26), the dissipation inequality reduces to gV # 0; which renders bðqÞ $ 0:

ð17:27Þ

2. Remarks (1) Anisotropy of the interface manifests itself in a nontrivial dependence of c^ðqÞ on q: An interesting and important consequence of Eq. (17.26) is that, for a grain boundary with anisotropic free energy, the surface shear cannot vanish.38 (2) By Eqs. (17.4), (17.21), and (17.26), we may consider the configurational stress c as a function c ¼ cðqÞ; with cðqÞ ¼ c^ðqÞtðqÞ þ

dc^ðqÞ nðqÞ: dq

ð17:28Þ

(3) The normal internal force g is a dissipative force associated with the rearrangement of atoms at the grain boundary. The term def

D ¼ 2 gV ¼ bðqÞV 2 38

Cf. Angenent and Gurtin (1989, Eq. 4.2), although this result is clear from the work of Herring (1951) in his discussion of the equilibrium theory within a variational framework.

A Unified Treatment of Evolving Interfaces

83

represents the dissipation per unit length of the interface, as its integral over C represents the right-hand side of the free-energy imbalance (17.17) (the expended power) minus its left-hand side (the net rate of change of free energy); this dissipation is characterized by the kinetic modulus bðqÞ: (4) Were we to begin with constitutive relations of the form

c ¼ c^ðq; VÞ;

t ¼ t^ðq; VÞ;

g ¼ g^ ðq; VÞ;

ð17:29Þ

thereby satisfying equipresence, the dissipation inequality would render c and t independent of V and consistent with Eq. (17.26), and would yield the reduced dissipation inequality g^ ðq; VÞ # 0; then, as in the discussion leading to Eq. (17.24)3, linearity of g ¼ g^ ðq; VÞ in V would require an additional hypothesis. (5) More general constitutive equations are also possible. For example, a constitutive relation of the form

t¼

dc^ðqÞ 2 CðqÞ q ; dq

CðqÞ . 0;

would be consistent with the dissipation inequality. (6) Since, by Eqs. (17.21) and (17.26), s ¼ c and t ¼ dc^ðqÞ=dq;

›s ›q ¼t ¼ tK; ›s ›s and we may conclude from the tangential configurational balance (17.8) that g·t ¼ 0:

ð17:30Þ

Thus the tangential component of the internal force g vanishes. This result is a consequence of the special theory under consideration; for theories that account for the density of atoms within the interface, the tangential force g·t is needed to balance spatial inhomogeneities on the interface induced by variations in chemical potential (cf. Eq. (13.4) and the discussion following Eq. (17.8)). For the theories discussed in Sections XVIII and XIX, which account, respectively, for configurational torques and the diffusion of a single atomic species, it is also the case that g·t ¼ 0:

84

E. Fried and M.E. Gurtin E. Evolution Equation for the Grain Boundary. Parabolicity and Backward Parabolicity

We now return to our discussion of grain boundaries as described by the constitutive equations

t¼

dc^ðqÞ ; dq

g ¼ 2bðqÞV;

assuming that bðqÞ . 0: For any function f ðqÞ we write f 0 ðqÞ ¼

df ðqÞ ; dq

and, when there is no danger of confusion, we write

cðqÞ ¼ c^ðqÞ: Then, since K ¼ ›q=›s; Eq. (17.26) yields ›t=›s ¼ c 00 ðqÞK; thus, by Eq. (17.24)3, the normal configurational force balance reduces to the curvature-flow equation39 bðqÞV ¼ ½cðqÞ þ c 00 ðqÞK þ ½½C:

ð17:31Þ

Note that the larger the dissipation, as characterized by the modulus bðqÞ; the slower the motion of the interface. For an isotropic grain boundary both b and c are positive constants. Then, for vCb ¼ 0 and, with an appropriate rescaling of space and time, Eq. (17.31) has the simple (and beautiful) form V ¼ K; a parabolic partial differential equation with a large literature.40 This equation, which was the forerunner of Eq. (17.31), was introduced by Burke and Turnbull (1952) and Mullins (1956) to study the motion of grain boundaries. 39

Proposed by Uhuwa (1987, Eq. 2) and independently by Gurtin (1988, Eq. 8.3). Cf. also Angenent and Gurtin (1989). Evolution according to Eq. (17.31) is studied by Angenent (1991), Chen et al. (1991), Barles et al. (1993), and Soner (1993). A formulation of Eq. (17.31) using a variational definition of the curvature term is given by Taylor et al. (1992), who give extensive references. The term cðqÞ þ c 00 ðqÞ appears first in the study of Herring (1951), who shows that it represents the variational derivative of the net interfacial free-energy with respect to variations in the position of the interface. 40 Cf. Gurtin (2000) for references.

A Unified Treatment of Evolving Interfaces

85

Fig. 17.5.1. Sign conventions when the interface is a graph y ¼ hðx; tÞ:

Locally the interface may be represented as the graph of a function y ¼ hðx; tÞ; provided the x- and y-axes are chosen appropriately. Consider the choice indicated in Fig. 17.5.1 (with orientation such that arc length increases with increasing x) and let q¼

›h : ›x

Then, for ð2p=2Þ , q , ðp=2Þ; q ¼ tan q;

K¼

1 ›2 h ; ð1 þ q2 Þ3=2 ›x2

1 ›h ; V ¼ pffiffiffiffiffiffiffiffi 2 1 þ q ›t

ð17:32Þ

and, assuming that bðqÞ . 0 for all such q; we may use these relations to rewrite the evolution equation (17.31) in the form

›h ›2 h ¼ AðqÞ 2 þ PðqÞ; ›t ›x

ð17:33Þ

with AðqÞ ¼

cðqÞ þ c 00 ðqÞ ; bðqÞ cos qð1 þ tan2 qÞ3=2

PðqÞ ¼

F ; bðqÞ cos q

q ¼ tan q:

For the angle range 2ðp=2Þ , q , ðp=2Þ under consideration, sgn AðqÞ ¼ sgn½cðqÞ þ c 00 ðqÞ: Thus the evolution equation ((17.33) and hence) (17.31) is (Angenent and Gurtin, 1989) (i) parabolic on any angle interval over which cðqÞ þ c 00 ðqÞ . 0; (ii) backward parabolic (and hence, unstable as a partial differential equation) on any angle interval over which cðqÞ þ c 00 ðqÞ , 0:

E. Fried and M.E. Gurtin

86

Let def

fðqÞ ¼ cðqÞ þ c 00 ðqÞ:

ð17:34Þ

Materials scientists often refer to fðqÞ as the interfacial stiffness. When fðqÞ . 0 for all q; Eq. (17.31), being parabolic, exhibits behavior that is not much different than that for V ¼ K; and is well understood.41 What makes Eq. (17.31) nonstandard is the possibility of backward parabolicity for interfacial free-energy densities that satisfy fðqÞ , 0 for certain angle-intervals. Such energy densities are not mathematical curiosities: materials scientists (Gjostein, 1963; Cahn and Hoffman, 1974) give strong arguments in support of free-energy densities with fðqÞ , 0 for some angles q:

F. Backward Parabolicity. Facets and Wrinklings In analyzing energies with backward-parabolic angle-intervals an important concept is the Frank diagram F (Frank, 1963), which is the graph in polar coordinates of the function r ¼ 1=cðqÞ: F is locally strictly convex where fðqÞ . 0 and locally strictly concave where fðqÞ , 0: One method of dealing with the backward-parabolic intervals is to allow the interface to contain corners (jumps in tangent angle) that exclude the backward-parabolic ranges of q: In the presence of a corner, the evolution equation (17.31) does not by itself characterize the motion of the interface; there is an additional condition (17.2) requiring that the configurational stress cðqÞ ¼ cðqÞtðqÞ þ

dcðqÞ nðqÞ be a continous function of arc length dq

(cf. Eq. (17.28)). Thus for a corner corresponding to an angle jump from q1 to q2 we must have cðq1 Þ ¼ cðq2 Þ; a condition that has important consequences. One can show that: (i) the tangent line to F at q1 must also be a tangent line to F at q2 (that is, q1 and q2 must be angles of bitangency for F Þ; and (ii) there is exactly one (maximal) angle interval between q1 and q2 on which fðqÞ , 0: Thus, by restricting attention to an interface with corners such that there is one corner for each backward parabolic interval and such that across each corner the tangent angle jumps between bitangency angles of the Frank diagram, one arrives at an interface whose 41

Cf. Angenent (1991), Chen et al. (1991), Barles et al. (1993), and Soner (1993).

A Unified Treatment of Evolving Interfaces

87

Fig. 17.6.1. A wrinkled portion C of the grain boundary S: The angles q1 and q2 associated with the facet-tangents t1 and t2 must be bitangency angles of the Frank diagram.

evolution is governed by a parabolic equation (Angenent and Gurtin, 1989). This procedure leads to a free-boundary problem, since the positions of the corners vary with time. The presence of unstable angle-intervals allows for facets (flat sections) and wrinklings, which are interfacial sections consisting of facets whose tangent angle is q1 alternating with facets whose tangent angle is q2 (Fig. 17.6.1). Because K ; 0 on each facet, facets with tangent angle q1 must, by Eq. (17.31), have constant normal velocity V ; V1 ¼ F=bðq1 Þ; while V ; V2 ¼ F=bðq2 Þ for those with angle q2 : Then, by compatibility, the wrinkling must evolve as a rigid body with velocity w defined by w·nðqi Þ ¼ F=bðqi Þ; i ¼ 1; 2:42

G. Junctions In the two-dimensional theory under consideration, grain boundaries meet at junctions such as the one shown in Fig. 17.7.1. This figure shows the pillbox used to determine the configurational force balance for the junction; this figure omits the forces exerted by the bulk material on the pillbox as well as the internal forces exerted on the grain boundaries; we assume that these forces approach zero as the pillbox collapses to the junction. For the case in which N grain boundaries, labelled n ¼ 1; 2; …; N; meet at a junction, and for which the individual grain boundaries are oriented so that the unit tangent field of each is directed away from the junction, the configurational force balance for the junction has the form N X

cn þ {net force exerted on the pillbox by the bulk material}

n¼1

þ {net internal force exerted on the pillbox} ¼ 0;

ð17:35Þ

where cn denotes the force exerted on the pillbox by the portion of grainboundary n that lies outside the pillbox. Assuming that the two terms written {· · ·} tend to zero as the pillbox tends to the junction, we arrive at 42

Cf. Gurtin (1993, Section XI) for a thorough discussion of wrinklings and related phenomena.

E. Fried and M.E. Gurtin

88

Fig. 17.7.1. A triple-junction with a corresponding junction pillbox. Only the interfacial tractions on the pillbox are shown.

Herring’s (1951) junction balance:43 N X

cn ¼ 0:

ð17:36Þ

n¼1

or, equivalently, N X

ðsn tn þ tn nn Þ ¼ 0:

n¼1

A classical consequence of the junction balance is that, for a triple junction, if the free energies of the grain boundaries are constant and equal, then the angles between adjacent boundaries are equal, with 2p=3 the common angle.

H. Digression: General Theory of Interfacial Constitutive Relations with Essentially Linear Dissipative Response In our development of constitutive equations for grain-boundaries, we used the dissipation inequality c 2 t q þ gV # 0 (and experience with classical theories) to motivate constitutive relations giving c and t as functions of q 43 In describing the forces on the junction, we neglected the internal configurational force gp on the junction, which if included would appear on the left-hand side of Eq. (17.36). If we neglect junction energy, then the appropriate free-energy imbalance for the junction would lead to the inequality gp ·vp # 0; where vp is the junction velocity. In this manner we would be led to a constitutive equation gp ¼ 2Bvp ; with B a positive semi-definite tensorial modulus that may depend on the junction angles of the grain boundaries as well as the mismatch angles of the grains. In this manner, we would arrive at P the balance n cn ¼ Bvp : Cf. Simha and Bhattacharya (1998) and Gurtin (2000, Part H); cf. also Suo (1997, Eq. (2.17)), whose analysis is restricted to isotropic surface energies and to a junction between two grains and a vapor, a situation of great interest in discussing grain-boundary grooving (cf. Mullins (1957)).

A Unified Treatment of Evolving Interfaces

89

together with a relation giving g as a function of q and V: This procedure will be used repeatedly in this article, where each of the individual theories is based on a dissipation inequality of the form

c2

N X

tn q n þ

m¼1

N X

gm Vm # 0:

ð17:37Þ

m¼1

In these theories we use the terms

c2

N X

tn q n ;

m¼1

N X

gm Vm ;

m¼1

respectively, to motivate constitutive relations giving44

c; t~ as functions of q~;

~ g~ as a function of ðq~; VÞ;

ð17:38Þ

where

t~ ¼ ðt1 ; t2 ; …; tN Þ; g~ ¼ ðg1 ; g2 ; …; gM Þ;

q~ ¼ ðq1 ; q2 ; …; qN Þ; V~ ¼ ðV1 ; V2 ; …; VM Þ:

and where neither of the fields q~ and V~ involves temporal derivatives of the other. In view of the chain-rule (14.12), thermodynamic compatibility requires that, ~ for any choice of the fields q~ and V; N X n¼1

(

) M X ›c^ðq~Þ ~ ~ m # 0: 2 t^n ðqÞ q n þ g^ m ðq~; VÞV ›qn m¼1

ð17:39Þ

It is always possible to find an interfacial field q~ whose values and whose (normal) time-derivative-values at any given time are arbitrary. Thus, since the inequality (17.39) is linear in the variables q n ; we must have t^n ¼ ›c^=›qn (for all n). We, therefore, have the following thermodynamic restrictions:

44 This structure cannot include theories involving temporal derivatives (or past histories) of the independent constitutive variables, nor can it include theories whose constitutive equations involve variables not present in the dissipation inequality. But it does include constitutive theories that we believe to be appropriate for the class of applications under consideration, without encumbering the presentation with lengthy arguments involving thermodynamical reductions of general constitutive theories. Moreover, as is clear from Remark 4 of Section XVII.C.2 above, the a priori splitting ~ may, in some cases, be presumed in Eq. (17.38), in which c and t~ are taken to be independent of V; unnecessary (e.g., in Section XVIII). But in discussions involving surface diffusion, spatial derivatives ~ while the chemical potentials themselves enter the list q~: of chemical potentials enter the list V;

E. Fried and M.E. Gurtin

90

† c must determine t~ through the relations

›c^ðq~Þ ; ›qn

tn ¼

ð17:40Þ

† the constitutive response function for g~ must be consistent with the reduced dissipation inequality M X

~ m # 0: g^ m ðq~; VÞV

ð17:41Þ

m¼1

In the special cases we shall consider, def

D¼ 2

M X

~ m g^ m ðq~; VÞV

m¼1

represents the dissipation. Of special interest here are constitutive equations giving g~ as a linear function of V~ for each fixed value of q~ : gm ¼ 2

M X

Amj ðq~ÞVj ;

ð17:42Þ

j¼1

in this case we refer to the constitutive equations as having essentially linear dissipative response. (The term ‘essential’ refers to the fact that Eq. (17.42) is ~ but the dependence on q~ is unrestricted). In linear in the primary variable V; addition, if Amj ¼ 0

for m – j;

ð17:43Þ

so that, writing Am ¼ Amm ; gm ¼ Am ðq~ÞVm ;

m ¼ 1; 2; …; M;

ð17:44Þ

we refer to the constitutive equations as having uncoupled essentially linear dissipative response. The dissipation corresponding to the uncoupled constitutive relations (17.42) has the form D¼

M X

Amj ðq~ÞVm Vj

ð17:45Þ

m; j¼1

and, by virtue of the reduced dissipation inequality (17.41), the coefficients Amj ðq~Þ form a positive semi-definite matrix.

A Unified Treatment of Evolving Interfaces

91

XVIII. Interfacial Couples. Allowance for an Energetic Dependence on Curvature Within the theory for grain boundaries presented above, facets are modeled as sharp corners. That crystalline solids may exhibit departures from this idealization was recognized by Herring (1951),45 who argued that, when the radius of curvature of a crystal surface is sufficiently small, the free energy of that surface should depend not only on orientation but also on curvature. A theory that incorporates a dependence on curvature has numerous benefits. The procedure discussed in Section XVII.F cannot, by itself, characterize the nucleation of facets and wrinklings, nor can it be used for an initial-value problem in which the initial interface has angle intervals for which the interfacial stiffness cðqÞ þ c 00 ðqÞ , 0: To analyze behavior within such angle-intervals, a regularization of the evolution equation (17.31) is needed. Such a regularization, proposed by Angenent and Gurtin (1989) and developed by DiCarlo et al. (1992),46 entails a curvature-dependent free energy. Within the present framework, a curvaturedependent free energy requires (configurational) interfacial couples together with a configurational balance for torques. A theory including these ingredients provides a physically based regularization of the evolution equation (17.31), in contrast to the pragmatic alternative of simply adding supplemental terms involving higher-order derivatives.

A. Configurational Torque Balance We now expand the configurational force system discussed in Section XVII.A to include a (scalar) interfacial couple-stress M and a (scalar) internal couple m distributed continuously over S (Fig. 18.1.1). Let C ¼ CðtÞ be an arbitrary interfacial pillbox S: The portion of S external to C then exerts torques 2Ma 2 ðxa 2 0Þ £ ca and Mb ðxb 2 0Þ £ cb at xa and xb :47 In addition to the interfacial couple m distributed uniformly over C; there is the torque ðx 2 0Þ £ g associated 45

Cf. also, Gjostein (1963) and Cahn and Hoffman (1974). Cf. Stewart and Goldenfeld (1992), who define the chemical potential as the variational derivative of the interfacial energy with respect to changes in the location of the surface and, starting with a mass balance that includes surface diffusion and evaporation terms, derive an evolution equation which, after linearization, yields conditions for the onset of instabilities (and the subsequent formation of facets along the surface). Cf. also Golovin et al. (1998, 1999), who use variational arguments to obtain an evolution equation for an interface z ¼ hðx; tÞ (respectively, z ¼ hðx; y; tÞÞ with a curvaturedependent surface tension, assuming that dh=dx is small (respectively, ›h=›x and ›h=›y are small). 47 Here ‘£’ denotes the (scalar) two-dimensional cross-product; in components p £ k ¼ p1 k2 2 p2 k1 : 46

E. Fried and M.E. Gurtin

92

Fig. 18.1.1. Configurational couples on an interfacial pillbox C: In the configurational torque balance these couples are suppllemented by torques exerted by the configurational force system.

with the internal force g. Further, the bulk material in the (þ)-phase exerts a torque ðx 2 0Þ £ Cþ n on Cþ ; while that in the (2)-phase exerts a torque 2ðx 2 0Þ £ C2 n on C2 ; so that the net configurational torque exerted at each point of C by the bulk phases is ðx 2 0Þ £ ðCþ n 2 C2 nÞ ¼ ðx 2 0Þ £ vCbn: The configurational torque balance for C; therefore, takes the form ½M þ ðx 2 0Þ £ cba þ

ð

m ds þ C

ð

½ðx 2 0Þ £ ðg þ ½½CnÞds ¼ 0:

ð18:1Þ

C

This balance supplements the configurational force balance (17.1). Consider the term ½M þ ðx 2 0Þ £ cba : Since ›x=›s ¼ t; and since, by Eq. (17.4), t £ c ¼ t; ½M þ ðx 2 0Þ £ cba ¼

 ð  ›M ›c þ t þ ðx 2 0Þ £ ds: ›s ›s C

Thus, by the local consequence (Eq. (17.3)), namely ›c=›s þ g þ vCbn ¼ 0; of the configurational force balance (17.1), the torque balance (18.1) reads  ð  ›M þ m þ t ds ¼ 0; ›s C since C is the arbitrary, this yields the local torque balance

›M þ m þ t ¼ 0: ›s

ð18:2Þ

Differentiating each term of Eq. (18.2) with respect to s and using the normal configurational force balance (17.6) to give an expression for ›t=›s; it follows that

›2 M ›m 2 sK 2 g 2 n·½½Cn ¼ 0; þ 2 ›s ›s

ð18:3Þ

A Unified Treatment of Evolving Interfaces

93

or equivalently, granted C ¼ C1 as in the theory without couples, that

›2 M ›m 2 sK 2 g 2 ½½C ¼ 0: þ 2 ›s ›s

ð18:4Þ

This balance is basic to what follows.

B. Power Let C ¼ CðtÞ be an interfacial pillbox. In addition to the power expended by configurational forces, we must now account for power expended by the couple stress M: This stress acts at the endpoints of C and the corresponding torques Ma and Mb should be power-conjugate to the angle-rates dqa =dt and dqb =dt at these endpoints (cf. Eq. (14.21)). Thus & ' dq b dq dq M ; Mb b 2 Ma a dt a dt dt represents the power expended on C by the couple stress, and the (net) power expended on C has the form & ' dx dq b ð c· þM þ ½½Cn·v ds: ð18:5Þ dt dt a C By Eq. (14.26), &

dq M dt

'b a

¼ ½Mðq þ KWÞba ;

ð18:6Þ

so that, using Eq. (14.18), the left-hand side of Eq. (18.6) equals 0 1 ð › q › M ½MKWba þ @M þ q Ads ›s ›s C ¼ ½MKWba þ

ð  C

MðK 2 K 2 VÞ þ

 ›M q ds: ›s

On the other hand, Eq. (17.16) (which holds here also) and Eq. (18.2) yield & ' ð dx b ð c· þ ½½Cn·v ds ¼ ½sWba þ ½t q 2 ðsK þ gÞVds dt a C CðtÞ    ð  ›M b ¼ ½sWa 2 þ m q ds: ðsK þ gÞV þ ›s C

E. Fried and M.E. Gurtin

94

Adding the last two relations, we arrive at the power balance &

dx dq þM c· dt dt

'b ð þ ½½Cn·v ds a

C

¼ ½ðs þ MKÞWba þ

ð

½M K 2 m q 2 ððs þ MKÞK þ gÞVds;

ð18:7Þ

C

which bears comparison to Eq. (17.16). Note that here: † The term ½ðs þ MKÞWba (rather than ½sWba Þ represents power expended internally as material is added to C at its endpoints. † The term M K represents an expenditure of power associated with changes in the curvature of the grain boundary. † The term 2m q (rather than t q Þ represents the expenditure of power associated with changes in the orientation of the grain boundary. † The term 2ðs þ MKÞKV (rather than 2sKVÞ represents the expenditure of power associated with changes in interfacial area due to the curvature of the grain boundary. As before, the tangential component of g expends no power internally, but now, interestingly, neither does the shear t; as its role in the power balance is replaced by the internal torque m: This is consistent with the theory without couples, since, by Eq. (18.2), t ¼ 2m when M ¼ 0: Classically, forces that expend no power internally are presumed to be indeterminate; for that reason, t joins g·t as an indeterminate field, an assumption that allows us to consider the torque balance (18.2) as a defining relation for t: Note that, granted t is defined by the torque balance, the balance (18.4) is equivalent to the normal configurational force balance (17.10) of the theory without couples.

C. Free-Energy Imbalance 1. Global Imbalance As before, we let c denote the free energy of the grain boundary; the freeenergy imbalance for C then takes the form & ' d ð dx dq b ð þM c ds # c· þ ½½Cn·v ds: dt CðtÞ dt dt a CðtÞ

ð18:8Þ

A Unified Treatment of Evolving Interfaces

95

Note that, since vCbn·v ¼ vCbV; with vCb (assumed) constant, the decay relation (17.19) and its consequences remain valid within this more general theory.

2. Dissipation Inequality In view of the transport identity (14.28) and the power balance (18.7), the freeenergy inequality (18.8) becomes ð#Þ

ð C

zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ ðc 2 cKVÞds þ ½ðc 2 ðs þ MKÞÞWba

#

ð

½M K 2 m q 2 ððs þ MKÞK þ gÞVds:

ð18:9Þ

C

Since C is arbitrary, so also are the tangential velocities Wa and Wb of the endpoints of C; thus, since the only term in Eq. (18.9) dependent on these velocities is the term (#), we have the relation

s ¼ c 2 MK;

ð18:10Þ

thus, in contrast to classical theories, the present theory requires that the correspondence between surface tension and interfacial free energy be generalized to account for the influence of the interfacial couple-stress M: Using Eq. (18.10), we may rewrite the normal configurational force balance (17.10) as ðc 2 MKÞK þ

›t þ g þ ½½C ¼ 0: ›s

ð18:11Þ

Next, by Eq. (18.10), the free-energy inequality (18.9) reduces to ð

ðc þ m q 2 M K þ gVÞds # 0;

ð18:12Þ

C

and this yields the dissipation inequality

c þ m q 2 M K þ gV # 0:

ð18:13Þ

96

E. Fried and M.E. Gurtin D. Constitutive Equations

Guided by Eq. (18.13) and the discussion of Section XVII.D for the theory without configuration moments, we consider constitutive equations giving ) c; m; M as functions of ðq; KÞ; ð18:14Þ g as a function of ðq; K; VÞ: Then, appealing to the general constitutive theory discussed in Section XVII.H, we find, as a consequence of thermocompatibility and an assumption of essentially linear dissipative response, that: † the free energy determines the internal couple and the interfacial couplestress through the relations m¼2

›c^ðq; KÞ ; ›q

M¼

›c^ðq; KÞ : ›K

ð18:15Þ

† the normal internal force is given by the linear relation g ¼ 2bðq; KÞV;

ð18:16Þ

with kinetic modulus bðq; KÞ $ 0:

E. Evolution Equation for the Grain Boundary The basic equations of the theory consist of the balance (18.4) with s ¼ c 2 MK; viz.,

›2 M ›m 2 ðc 2 MKÞK 2 g 2 ½½C ¼ 0; þ ›s ›s 2

ð18:17Þ

supplemented by the constitutive relations (18.15) and (18.16). In general, the coupling between orientation and curvature induced by the constitutive relation for the free energy renders the resulting evolution complicated. But, for small curvatures, a quadratic dependence on curvature should provide a reasonable approximation and, as one shall see, such an energy provides a parabolic regularization to the evolution equation (17.31). We, therefore, consider the simple constitutive equation

c^ðq; KÞ ¼ c0 ðqÞ þ 12 lK 2 ;

ð18:18Þ

A Unified Treatment of Evolving Interfaces

97

with l . 0 constant. The restrictions (18.15) then yield the specific relations m ¼ 2c 00 ðqÞ;

M ¼ lK;

thus, assuming that b ¼ bðqÞ . 0; Eqs. (18.16) and (18.17) yield the evolution equation ! ›2 K 1 3 00 bðqÞV ¼ ½c0 ðqÞ þ c 0 ðqÞK 2 l ð18:19Þ þ K þ ½½C: 2 ›s 2 For the interface a graph y ¼ hðx; tÞ; we can use Eq. (17.32) to rewrite the evolution equation (18.19) in a form analogous to Eq. (17.33). The right-hand side of the resulting equation then contains a term of the form 2ð· · ·Þ

›4 h ; ›x 4

and all other terms as well as the coefficient ð· · ·Þ depend on partial derivatives of h with respect to x of order strictly less than four. Moreover, the coefficient ð· · ·Þ is strictly positive. Thus the evolution equation (18.19) is equivalent to a fourthorder parabolic partial differential equation. In that sense, Eq. (18.19) with l small represents a parabolic regularization of the evolution equation (17.31). This regularization should be useful in analyzing situations for which the interfacial stiffness is negative and Eq. (17.31) is backward parabolic for certain angle intervals. In a sense, the regularized equation (18.19) represents a counterpart for interfaces of the classical Cahn – Hilliard equation (Cahn and Hilliard, 1958, 1959, 1971), as discussed by DiCarlo et al. (1992).48

XIX. Grain –Vapor Interfaces with Atomic Transport In this section we consider an interface S ¼ SðtÞ that separates a grain from a vapor environment. We include atomic transport in bulk, on the interface, and from the vapor, but neglect deformation of the grain and flow of the vapor.49 For simplicity, we restrict attention to a single atomic species. We model the vapor as 48

Cf. Watson et al. (2003) and Watson (2003). Cf. Spencer et al. (1991, 1993), Spencer and Meiron (1994), Guyer and Voorhees (1996, 1998), and Spencer et al. (2001), who investigate the morphological stability of epitaxially strained singlecomponent and alloy thin films in the presence of surface diffusion and an isotropic surface freeenergy. See also Zhang and Bower (1999, 2001), who examine shape changes of strained islands on lattice-mismatched substrates due to surface diffusion in the presence of both isotropic and anisotropic surface energies. 49

98

E. Fried and M.E. Gurtin

a reservoir whose sole interaction with the grain is through the evaporation of atoms from—and, the condensation of atoms on—the grain boundary.

A. Configurational Force Balance We assume that the configurational stress in the vapor vanishes:50 considering the vapor to be the (þ)-phase, the grain, the (2)-phase, it follows that Cþ ¼ 0: Apart from this restriction, the discussion of configurational forces is identical to that presented for a grain – grain interface. Thus, writing C ¼ C2 for the interfacial limit of the configurational stress in the grain, the configurational force balance (17.1) takes the form clba þ

ð

g ds 2 C

ð

Cn ds ¼ 0;

ð19:1Þ

C

and this yields the normal configurational force balance

sK þ

›t þ g 2 n·Cn ¼ 0; ›s

ð19:2Þ

with s and t defined by Eq. (17.4) and g ¼ g·n; as before. The bulk configurational force C is determined by the Eshelby relation (12.15), which, since deformation is neglected here, has the form C ¼ ðC 2 rmÞ1;

ð19:3Þ

with m; the bulk chemical-potential, assumed continuous up to the grain boundary. The normal configurational force balance (19.1), therefore, takes the form

rm ¼ C 2 sK 2

›t 2 g; ›s

ð19:4Þ

the normal configurational force balance thus provides a relation for the chemical potential of the grain boundary. Finally, the tangential component of the configurational force balance reduces to Eq. (17.11). 50

Tacit are the following assumptions: the free energy of the grain is reckoned relative to that of the vapor; the atomic density of the vapor and the vapor pressure are negligible.

A Unified Treatment of Evolving Interfaces

99

B. Power Consistent with the requirement that the configurational stress vanishes in the vapor and the notational convention C ¼ C2 ; the (net) power expended on a migrating pillbox C has the form & ' dx b ð c· 2 Cn·v ds: ð19:5Þ dt a C By Eqs. (14.5) and (17.13), and the Eselby relation (19.3), Cn·v ¼ ðC 2 rmÞV; hence, the power expended on C has the form & ' ð dx b ð c· 2 Cn·v ds ¼ ½sW þ tVba 2 ðC 2 rmÞV ds ð19:6Þ dt a C C which, by Eq. (17.15) and the normal configurational force balance (19.4), yields the power balance & ' ð dx b ð c· 2 Cn·v ds ¼ ½sWba þ ½t q 2 ðsK þ gÞV ds: ð19:7Þ dt a C C

C. Atomic Flows due to Diffusion, Evaporation – Condensation, and Accretion. Atomic Balance In addition to the bulk atomic density r and bulk atomic flux ; we account for an atomic supply r from the vapor and for a (scalar) interfacial atomic flux h, the associated vectorial flux being ht: Without loss of generality, we write r ¼ r2 and  ¼ 2 for the interfacial limits of the density and atomic flux in the grain. Finally, we account for the transport of adatoms along the interface through the flux h but assume that the adatom density is negligibly small. Let C ¼ CðtÞ be an interfacial pillbox. Then surface diffusion in the portion of S exterior to C results in fluxes ha and 2hb of atoms into C across xa and xb ; (since n points into the vapor) bulk diffusion results in a flow ·n of atoms into C across C2 from the solid: r represents the rate at which atoms are supplied from the vapor (Fig. 19.3.1). Hence, the net rate at which atoms added to C by diffusive transport and by evaporation – condensation is 2hlba þ

ð

ð ·n þ rÞds:

ð19:8Þ

C

Atoms are also carried into C as it migrates. Since we neglect the adatom density, the only such accretive flow is 2rV across C2 into C (Fig. 19.3.1), so that

100

E. Fried and M.E. Gurtin

Fig. 19.3.1. Atomic fluxes and supplies to an interfacial pillbox C:

the net rate at which atoms are added to C by accretion is 2

ð

rV ds:

ð19:9Þ

C

Thus, the atomic balance for C takes the form 2hlba þ

ð

ð ·n 2 rV þ rÞds ¼ 0;

ð19:10Þ

C

since C is arbitrary, this yields the local balance

rV ¼ 2

›h þ ·n þ r: ›s

ð19:11Þ

D. Free-Energy Imbalance The general free-energy imbalance (17.17) is now modified to include energy flow into the pillbox CðtÞ by atomic transport: d {free energy of CðtÞ} # {power expended on CðtÞ} dt þ {energy flow into CðtÞ by atomic transport}: ð19:12Þ To discuss this inequality, we let mv denote the chemical potential of the vapor. We assume that the chemical potential m of the solid at the surface is the limiting value m ¼ m2 of the bulk chemical potential; m; therefore, represents the chemical potential for surface diffusion. Since we attribute no specific structure to the vapor, mv represents the chemical potential of the vapor at the grain boundary. In general, we admit the possibility that m differs from mv :

A Unified Treatment of Evolving Interfaces

101

1. Energy Flows due to Diffusion, Evaporation –Condensation, and Accretion Atomic transport induces energy flows associated with diffusion, evaporation –condensation, and accretion. The diffusion of atoms within the interface results in energy fluxes 2ma ha and mb hb across xa and xb ; the diffusion of atoms within the solid results in an energy flow m ·n across C2 ; the supply r of atoms from the vapor results in an energy flow mv r: Hence, the net rate at which energy is added to C by diffusion and evaporation – condensation is ð 2½mhba þ ðm ·n þ mv rÞds: ð19:13Þ C

The motion of the interface results in an energy flow 2rmV associated with accretive transport of atoms from the grain across C2 ; so that the rate at which energy is added to C by the accretive transport of atoms is ð 2 rmV ds: ð19:14Þ C

2. Global Imbalance As before, we let c denote the free energy of the grain boundary; in view of Eqs. (19.13) and (19.14), the free-energy imbalance for an arbitrary interfacial pillbox then takes the form (Davı` and Gurtin, 1990)

& ' dx b ð d ð c ds # c· 2 Cn·v ds dt CðtÞ dt a CðtÞ |fflffl{zfflffl} |fflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflffl} free energy

power expended by configurational forces

þ

ð

ðmð ·n 2 rVÞ þ mv rÞds 2 ½mhba : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð19:15Þ

CðtÞ

energy flow by atomic transport

3. Dissipation Inequality By Eq. (19.7) and the transport theorem (14.28), the free-energy imbalance (19.15) becomes ð#Þ

ð C

zfflfflfflffl}|fflfflfflffl{ ðc 2 cKVÞds þ ½ðc 2 sÞWba

#

ð C

ðt q 2 ðsK þ gÞV þ mð ·n 2 rVÞ þ mv rÞds 2 ½mhba

ð19:16Þ

E. Fried and M.E. Gurtin

102

We must, therefore, have s ¼ c; as before, and hence, ð ð c ds # ðt q 2 gV þ mð ·n 2 rVÞ þ mv rÞds 2 ½mhba : C

ð19:17Þ

C

Next, using the atomic balance (19.11),   ð  ›h ð  ›m ›m ½mhba ¼ þh m mð ·n 2 rV þ rÞ þ h ds ¼ ds: ›s ›s ›s C C Thus, Eq. (19.17) reduces to  ð  ›m þ ðm 2 mv Þr þ gV ds # 0; c 2 tq þ h ›s C since C is arbitrary, we have the dissipation inequality

c 2 tq þ h

›m þ ðm 2 mv Þr þ gV # 0: ›s

ð19:18Þ

E. Constitutive Equations Our discussion of constitutive equations follows the format set out in Section XVII.H. We consider constitutive equations giving 9 c; t as functions of q; > =   ð19:19Þ ›m ; m 2 mv ; V : > h; r; g as functions of q; ; ›s Then, appealing to the discussion of Section XVII.H we find, as a consequence of thermocompatibility and an assumption of essentially linear dissipative response, that the free energy determines the shear through the relation

t ¼ c 0 ðqÞ; that the constitutive equations for h; r; and g have the specific form 9 ›m

> 2 lðqÞðm 2 mv Þ 2 lðqÞV; > h ¼ 2LðqÞ > > ›s > > = ›m

qÞðm 2 mv Þ 2 kðqÞV; 2 kð r ¼ 2KðqÞ > ›s > > > > ›m >

2 bðqÞðm 2 mv Þ 2 bðqÞV; ; g ¼ 2BðqÞ ›s 51

ð19:20Þ 51

ð19:21Þ

Cf. Davı` and Gurtin (1990), who neglect the internal force g and hence do not include dependences on V:

A Unified Treatment of Evolving Interfaces

103

and that, for each q; the coefficients that define the linear relations (19.21) form a positive semi-definite matrix. It is the purpose of this section of discuss more classical theories in which the linear relations (19.21) are uncoupled. We, therefore, assume that † Surface diffusion is given by Fick’s law (Herring, 1951; Mullins, 1957), h ¼ 2LðqÞ

›m ; ›s

ð19:22Þ

with LðqÞ $ 0 a modulus that describes the mobility of the atoms on the interface; † evaporation –condensation is described by the relation r ¼ 2kðqÞðm 2 mv Þ;

ð19:23Þ

with kðqÞ $ 0 an evaporation modulus; † kinetics is defined by the relation g ¼ 2bðqÞV;

ð19:24Þ

with bðqÞ $ 0: The dissipation then has the form

,

›m

-2

D ¼ LðqÞ ›s þ |fflfflfflffl{zfflfflfflffl} dissipation induced by surface diffusion

kðqÞðm 2 mv Þ2 þ bðqÞV 2 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

ð19:25Þ

dissipation accompanying the attachment of vapor atoms

and results in two terms, kðqÞðm 2 mv Þ2 and bðqÞV 2 ; associated with the attachment of vapor atoms to the lattice. One might expect the dynamical term bðqÞV 2 to be negligible for standard evaporation – condensation, and, in fact, theories discussed by materials scientists typically do not include the kinetic term g ¼ 2bðqÞV:52

F. Basic Equations The basic equations for the interface consist of the normal configurational force-balance (19.4) (with s ¼ c) and the atomic balance

rm ¼ C 2 cK 2 52

›t 2 g; ›s

rV ¼ 2

›h þ ·n þ r; ›s

Cf. Section XXVI.C for a detailed discussion of this kinetic term.

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E. Fried and M.E. Gurtin

supplemented by the constitutive equations c ¼ c^ðqÞ; Eqs. (19.20), (19.22) and (19.23): 9 rm ¼ C 2 ½cðqÞ þ c 00 ðqÞK þ bðqÞV; > =   ð19:26Þ › ›m LðqÞ rV ¼ þ ·n þ kðqÞðmv 2 mÞ: > ; ›s ›s These equations are coupled to the bulk atomic balance

r_ ¼ 2div 

ð19:27Þ

and the bulk constitutive equations

m ¼ m^ðrÞ ¼

dC^ ðrÞ ; dr

 ¼ 2MðrÞ7m

ð19:28Þ

(cf. Eqs. (7.2) and (7.5)2), with Eq. (19.28)2 equivalent to  ¼ 2DðrÞ7r;

DðrÞ ¼

dm^ðrÞ MðrÞ: dr

ð19:29Þ

G. Nearly Flat Interface at Equilibrium For a flat interface at equilibrium, K ¼ V ¼ 0 and the normal configurational force balance (19.26) implies that

C 2 rm ¼ 0;

ð19:30Þ

which is the familiar assertion that, in equilibrium, the grand canonical potential C 2 rm of the solid must coincide with that of the vapor, here normalized to be zero (cf. Larche´ and Cahn, 1985). Let the constants C0 ; r0 ; and m0 denote the values of the bulk free energy, bulk atomic-density, and chemical potential when the grain and vapor are in equilibrium, with the interface flat; then

C0 ¼ C^ ðr0 Þ;

% dC^ %% m0 ¼ m^ðr0 Þ ¼ % ; dr % 0

where the subscript zero denotes evaluation at r ¼ r0 : Then, by Eq. (19.30), C0 2 m0 r0 ¼ 0: Thus C 2 rm; considered as a function of r; has the following

A Unified Treatment of Evolving Interfaces

105

expansion for r close to r0 : % % dC^ %% dm^ %% C 2 rm ¼ ðr 2 r0 Þ þ oðlr 2 r0 lÞ % ðr 2 r0 Þ 2 m0 ðr 2 r0 Þ 2 r0 dr % 0 dr %0 ¼ 2r0

% dm^ %% ðr 2 r0 Þ þ oðlr 2 r0 lÞ: d r %0

On the other hand,

m ¼ m0 þ

% dm^ %% ðr 2 r0 Þ þ oðlr 2 r0 lÞ; dr % 0

so that

C 2 rm ¼ 2r0 ðm 2 m0 Þ þ oðlr 2 r0 lÞ:

ð19:31Þ

If, in addition, we assume that the vapor is not supersaturated, then its chemical potential must coincide with the equilibrium potential of the grain: mv ¼ m0 : Thus, neglecting the term of oðlr 2 r0 lÞ in Eq. (19.31), we may write the normal configurational balance and the interfacial atomic balance, Eq. (19.26), with the approximation rV ¼ r0 V; in the form53 9 r0 ðm 2 m0 Þ ¼ 2fðqÞK þ bðqÞV; > =   ð19:32Þ › ›m LðqÞ r0 V ¼ þ ·n 2 kðqÞðm 2 m0 Þ; > ; ›s ›s with f the interfacial stiffness as defined in Eq. (17.34). These equations, when combined, form a single equation for the interface   › › LðqÞ 2 kðqÞ ð2fðqÞK þ bðqÞVÞ þ r0 ·n: r20 V ¼ ð19:33Þ ›s ›s Similarly, to within a term of oðlr 2 r0 lÞ; we may rewrite Fick’s law (19.29) as  ¼ 2D0 7r ;

ð19:34Þ

and the bulk atomic balance has the approximate form

r_ ¼ divðD0 7rÞ; 53

ð19:35Þ

In statical situations the configurational balance (19.32)1 reduces to the classical relation

r0 ðm 2 m0 Þ ¼ 2fðqÞK due to Herring (1951), whose derivation utilizes virtual variations of the position of the interface (cf. (i) of Section I.A).

106

E. Fried and M.E. Gurtin

or, for Dij the components of D0 ;

r_ ¼ Dij

›2 r : ›x i ›x j

Note the presence of two terms in the interface system (19.32) involving the velocity V: † bðqÞV; a dissipative term associated with the attachment kinetics of atoms at the free surface; † r0 V; a nondissipative term associated with the accretive transport of atoms to and from the bulk material at the interface. Equations (19.32)– (19.35) govern the motion of the interface assuming small departures from equilibrium and a non-supersaturated vapor. These equations are complicated; two important special cases are described below.54 For the remainder of this section we restrict our attention to behavior close to equilibrium with vapor not supersaturated, and therefore, work with the approximations (19.32) and (19.35).

1. Interface Motion by Surface Diffusion The equations simplify considerably when bulk diffusion, evaporation – condensation, and kinetics are neglected. In this instance we omit the bulk diffusion equation (19.35), take ·n ¼ 0 at the interface, and take bðqÞ ¼ kðqÞ ¼ 0: Then Eq. (19.33) reduces to   › › 2 LðqÞ ðfðqÞKÞ : r0 V ¼ 2 ð19:36Þ ›s ›s If we neglect the dependences of the coefficients on q (that is, if we assume that the material response of the interface is well-approximated as isotropic), then this equation has the simple form V ¼ 2A

›2 K ; ›s2

A¼

Lc ; r20

ð19:37Þ

and represents a fourth-order parabolic equation for the evolution of the interface. 54 For an isotropic material, linearized versions of Eqs. (19.37) and (19.38) (with b ¼ 0) are discussed by Mullins (1957) in his discussion of thermal grooving. Cf. Davı` and Gurtin (1990), who work within the present framework.

A Unified Treatment of Evolving Interfaces

107

2. Interface Motion by Evaporation– Condensation and Kinetics If we neglect surface and bulk diffusion, and hence, take LðqÞ ¼ 0; drop the bulk diffusion equation (19.35), and take ·n ¼ 0 at the interface, then Eq. (19.32) combine to give BðqÞV ¼ ½cðqÞ þ c 00 ðqÞK;

BðqÞ ¼ bðqÞ þ

r20 : kðqÞ

ð19:38Þ

Thus the equations reduce to the curvature-flow equation (17.31) (with F ¼ 0), but now the kinetic term consists of two parts as represented by the constants bðqÞ and r20 =kðqÞ: Note that the limit k ! 0 yields B ! 1 and hence, V ! 0; which is consistent with the underlying assumptions and the balances (19.32); in this limit the first of Eq. (19.32) yields r0 ðm 2 m0 Þ ¼ 2fðqÞK:

STRAINED SOLID –VAPOR INTERFACES. EPITAXY We now broadly generalize certain aspects of the discussion of solid – vapor interfaces presented in Section XIX to account for deformation, standard interfacial stress, multiple species of atoms, and adatoms. The inclusion of deformation is of particular importance in discussions of epitaxy (Fig. 20.1), as differences in lattice parameters between film and substrate can induce large

Fig. 20.1. Schematic describing epitaxy. Undulations of the film– vapor interface typically result from instabilities induced, for example, by stresses arising from a mismatch in lattice parameters between film and substrate.

108

E. Fried and M.E. Gurtin

stresses in the film.55 Further, as discussed by Shchukin and Bimberg (1999), standard interfacial stress may strongly influence the formation of surface patterns.56 We begin by characterizing the solid –vapor interaction through prescribed supplies of vapor atoms to the solid surface, a characterization that would seem appropriate to molecular beam epitaxy; later sections discuss more general constitutive relations for the solid – vapor interface.57

XX. Configurational and Standard Forces When considering the configurational forces that act on an isolated portion P of a body, it is generally necessary to account not only for forces that describe interactions between P and the remainder of the body but also for internal forces within P; as such internal forces are of importance in the generation and evolution of defects. The situation for standard forces is quite different. There, internal forces are typically absent.

A. ConFIGURATIONAL Forces For an interface separating solid and vapor phases, the configurational forces acting on a pillbox are identical to those arising in our discussion of grain – vapor interfaces. Thus, the configurational force balance (19.1) and its local consequences remain valid. In particular, we have the normal configurational force balance

sK þ

›t þ g 2 n·Cn ¼ 0: ›s

ð20:1Þ

55 See, for example, the reviews of Stringfellow (1982), Cammarata and Sieradzki (1994), Ibach (1997), Politi et al. (2000), and Spaepen (2000). For sufficiently thick films, stresses may be induced by dislocations and other defects—even in the absence of a lattice mismatch between film and substrate. See Freund (1994) and Gao and Nix (1999) studies of defects in films. 56 Andreussi and Gurtin (1977) show that a decreasing dependence of standard surface stress on tensile surface strain leads to the wrinkling of a free surface. See, also, Shenoy and Freund (2002), who demonstrate the existence of an instability induced solely by surface strain, more precisely, by a compressive-strain induced nonconvexity in the dependence of surface energy on orientation. 57 The results of Part E are small-deformation counterparts of those obtained by Fried and Gurtin (2003).

E. Fried and M.E. Gurtin

110

s
represents the standard scalar interfacial stress. Since ›t=›s ¼ Kn; we can rewrite the interface condition (20.3)1 as s
Kn þ

›s
t ¼ Tn; ›s

ð20:5Þ

or equivalently, as

›s
¼ t·Tn: ›s

s
K ¼ n·Tn;

ð20:6Þ

The first of Eq. (20.6) represents a counterpart, for a solid –vapor interface, of the classical Laplace – Young relation for a liquid – vapor interface.

XXI. Power A. External Power Expenditures Our discussion of power follows the discussions leading to Eqs. (11.9) and (17.14). Consider an arbitrary interfacial pillbox C ¼ CðtÞ: The configurational and standard stresses c and s act at the endpoints xa ðtÞ and xb ðtÞ of CðtÞ: As in our discussion of grain boundaries (cf. Section XVII.B), we take dxa =dt and dxb =dt as the power-conjugate velocities for c. For s, we reason by analogy to our treatment of the power expended by standard tractions on a migrating control volume and take as power-conjugate velocities the motion velocities dua =dt and dub =dt following the evolution of the endpoints xa ðtÞ and xb ðtÞ (cf. Eq. (15.16)). The portion C2 of the pillbox boundary adjacent to the solid is acted on by the tractions 2Cn and 2Tn: As in our discussion of grain – vapor interfaces (cf. Section XIX), we assume that the configurational traction 2Cn is power-conjugate to the normal velocity v. Further, consistent with our treatment of the power expended by standard tractions on a migrating control volume, we use as a power-conjugate velocity for 2Tn the motion velocity u following SðtÞ (cf. Eq. (11.8)). Finally, the configurational force g; being internal, expends no external power. The (net) external power expended on CðtÞ; therefore, has the form (cf. Eq. (19.5)) & c·

dx du þ s· dt dt

'b 2 a

ð

ðCn·v þ Tn· u Þds: C

ð21:1Þ

A Unified Treatment of Evolving Interfaces

111

B. Internal Power Expenditures. Power Balance First of all, using the identity ›V=›s ¼ q and the normal configurational force balance (20.1), it follows that (cf. Eqs. (17.13) and (17.15)) & ' ð dx b c· ¼ ½sW þ tVba ¼ ½sWba þ ½t q 2 ðsK þ g 2 n·CnÞVds; ð21:2Þ dt a C thus, since v ¼ Vn; & ' ð dx b ð 2 Cn·v ds ¼ ½sWba þ ½t q 2 ðsK þ gÞVds: c· dt a C C

ð21:3Þ

Further, since s ¼ s
t and 1 ¼ t·e (cf. Eq. (15.13)), we may use Eq. (15.16) to obtain & ' du b s· ¼ ½s· u þ s
1Wba : ð21:4Þ dt a Thus, since › u =›s ¼ e 2 KVe (cf. Eq. (15.12))  & ' ð  ›s du b s· · u þ s·ðe 2 KVeÞ ds: ¼ ½s
1Wba þ dt a C ›s

ð21:5Þ

On the other hand, since s ¼ s
t and t· e ¼ 1 2 g q (cf. Eq. (15.14)), s· e ¼ s
ð1 2 g q Þ; therefore,  & ' ð  ›s du b s· · u þ s
ð1 2 g q Þ 2 s
1KV ds ¼ ½s
1Wba þ dt a C ›s and, using the standard force balance (20.3)1, & ' ð du b ð s· 2 Tn· u ds ¼ ½s
1Wba þ ½s
ð1 2 g q Þ 2 s
1KVds: dt a C C Combining Eqs. (21.8) and (29.8) then yields the power balance & ' dx du b ð þ s· c· 2 ðCn·v þ Tn· u Þds dt dt a C ð ¼ ½ðs þ s
1ÞWba þ ½s
1 þ t
q 2 ððs þ s
1ÞK þ gÞVds;

ð21:6Þ

ð21:7Þ

ð21:8Þ

ð21:9Þ

C

where we have introduced the reduced configurational shear

t
¼ t 2 sg
:

ð21:10Þ

112

E. Fried and M.E. Gurtin

The contributions to the internal power-expenditure, which should be compared with those (cf. Section XVII.B) for a grain boundary, therefore, have the form: † The term ½ðs þ s
1ÞWba (rather than ½sWba Þ represents power expended internally as material is added to C at its endpoints. † The term s
1 represents an expenditure of power associated with interfacial stretching. † The term t
q (rather than t q Þ represents an expenditure of power associated with changes in the orientation of the interface.59 † The term 2ðs þ s
1ÞKV (rather than 2sKVÞ represents the expenditure of power associated with changes in interfacial length due to the curvature of the interface. The expenditure 2gV is as discussed following Eq. (17.16).

XXII. Atomic Transport A. Atomic Balance As in our treatment of bulk atomic transport, we consider N species of atoms, labelled a ¼ 1; 2; …; N: In addition to atomic densities ra and fluxes a distributed over the solid, we account for interfacial atomic densities da ; (scalar) interfacial atomic fluxes ha ; and a prescribed supply r a of atoms from the vapor to the solid surface. As before, we write ra ¼ ra2 and a ¼ a2 for the appropriate interfacial limits. Let C ¼ CðtÞ be an arbitrary interfacial pillbox. Then surface diffusion in the portion of S exterior to C results in fluxes haa and 2hab of a-atoms across xa and xb ; bulk diffusion results in a flow a ·n of a-atoms from the solid into C across C2 ; and there is a flow r a of a-atoms into C across Cþ from the vapor (Fig. 22.1.1). Hence, the net rate at which atoms are added to C by diffusive transport and 59 One might, at first sight, be surprised at the term sg
in the power expenditure t
q ¼ ðt 2 sg
Þq: In the finite-strain theory the interfacial stress s is tangent to the deformed interface; i.e., s ¼ s

t with
t the unit tangent to the deformed interface. If in that theory we let u denote the displacement relative to the reference configuration and consider 7 and ›=›s as ‘material’ operators and ð· · ·Þ as the timederivative following the interface as described materially, then, defining e ¼ ›u=›s; it follows, as before, that e ¼ ð1 2 g q Þt þ ðg þ 1 q Þn: Thus in the finite-strain theory the interfacial stress power has the form s· e ¼ s
ð1 2 g q Þð
t·tÞ þ ðg
t þ 1 q Þð
t·nÞ and hence reduces to that used in the ‘smalldeformation theory’ in the small-strain limit, where
t ! t and n
! n: This lends further credence to the form of the power expenditures described here.

A Unified Treatment of Evolving Interfaces

113

Fig. 22.1.1. Transport of a-atoms into an interfacial pillbox C:

through the vapor supply is 2ha lba þ

ð

ð a ·n þ r a Þds:

ð22:1Þ

C

Accretion of the portion of the interface exterior to C results in fluxes 2daa Wa and dab Wb of a-atoms across xa and xb ;60 while accretion of C results in a flow 2ra V of a-atoms into C across C2 (Fig. 22.1.1). Hence, the net rate at which atoms are added to C by accretive transport is ½da Wba 2

ð

ra V ds:

ð22:2Þ

C

In view of Eqs. (22.1) and (22.2), the atomic balance for C takes the form ð d ð da ds ¼ 2½ha 2 da Wba þ ð a ·n 2 ra V þ r a Þds; dt CðtÞ CðtÞ

ð22:3Þ

for each species a; and by virtue of the integral transport theorem (14.28), we have the local balance61

d a þ ðra 2 da KÞV ¼ 2

›ha þ a ·n þ r a ›s

ð22:4Þ

on the interface for each species a: 60

i.e., e.g., daa ðtÞWa ðtÞ ¼ da ðxa ðtÞ; tÞWa ðtÞ: Comparison of the interfacial atomic balance (22.4) with its bulk counterpart (3.2) reveals several formal similarities and differences. The adatom densities and fluxes enter Eq. (22.4) in a manner completely analogous to that in which the bulk atomic densities and fluxes enter Eq. (3.2). Apart from the curvature term and the term accounting for the supply of atoms from the vapor, what most distinguishes Eq. (22.4) from Eq. (3.2) is the presence of terms that account for bulk diffusion and accretion. 61

E. Fried and M.E. Gurtin

114

B. Net Atomic Balance Defining the net bulk and adatom densities and the atomic volume V through

r¼

N X

rb ;

d¼

b¼1

N X

db ;

V¼

b¼1

1 ; r

ð22:5Þ

and the net fluxes and supply through ¼

N X

b¼1

b ;

h¼

N X

hb ;

N X

r¼

b¼1

rb ;

ð22:6Þ

b¼1

we see that Eq. (22.4), when summed over all atomic species, yields the net atomic balance

d þ ðr 2 dKÞV ¼ 2

›h þ ·n þ r: ›s

ð22:7Þ

If adatom densities are neglected, then this net balance reduces to an equation for the normal velocity:   ›h V¼V 2 þ ·n þ r : ›s

ð22:8Þ

For a substitutional alloy the lattice constraint (5.1) and the substitutional flux constraint (5.4) imply that r ¼ rsites and  ¼ 0; hence, the net atomic balance (22.7) takes the form

d 2 dKV þ rsites V ¼ 2

›h þr ›s

ð22:9Þ

whose sole coupling with the remaining field equations is through the kinematical terms V and K and, possibly, through the net flux h via additional constitutive information. If adatom densities are neglected, then Eq. (22.9) reduces to   ›h V¼V 2 þr ; ›s

V¼

1 : rsites

ð22:10Þ

Thus, when both adatom densities and surface diffusion are neglected, the evolution of an interface between a substitutional alloy and a vapor is governed solely by the atomic supply r:

A Unified Treatment of Evolving Interfaces

115

XXIII. Free-Energy Imbalance A. Energy Flows due to Atomic Transport. Global Imbalance We assume that, if the material is unconstrained, then, for each species a; the chemical potential ma of the solid at its surface is the limiting value ma2 of the bulk chemical potential. For a substitutional alloy, the individual chemical potentials are not well defined in bulk; the relative chemical potentials must be used. We assume that, for any two species a and b; the limiting value mab2 of the relative chemical potential is equal to the difference ma 2 mb of chemical potentials of the solid at its surface. Thus we have the chemical interfaceconditions: ) ma ¼ ma2 for an unconstrained material; ð23:1Þ ma 2 mb ¼ mab2 for a substitutional alloy: Motivated by the desire to describe processes such as molecular beam epitaxy, where atoms are supplied directly at the film surface and where the vapor cannot simply be modeled as a reservoir, we introduce for each species a an external supply r a of a-atoms with concomitant supply ma r a of energy. Therefore, in contrast to the discussion of more classical evaporation – condensation in Section XIX, we do not find it necessary to endow the vapor with a chemical potential.62 However, in Section XXVIII we will discuss a different class of vapor-interaction equations in which the vapor is considered as a reservoir for atoms endowed with chemical potentials mav ; and for which r a is no longer arbitrarily prescribed, but is, instead, a constitutive variable. The energy flow into a migrating pillbox C due to atomic transport includes contributions associated with diffusion, accretion, and the supply of atoms from the vapor. The diffusion of a-atoms within the interface results in energy fluxes maa haa and 2mab hab across xa and xb ; the diffusion of a-atoms in the solid results in an energy flow ma a ·n across C2 ; the supply r a of a-atoms from the vapor results in an energy flow ma r a : The net rate at which energy is supplied to C by diffusion and evaporation –condensation is, hence, given by (cf. Eq. (19.13)) 2

N X

a¼1

62

½ma ha ba þ

N ð X

a¼1

ma ð a ·n þ ra Þds:

ð23:2Þ

C

The discussion of evaporation–condensation (Section XIX, for a single species) was based on the tacit assumption that there exist an infinitesimally thin transition-layer across which there is flow r of atoms from a quiescent reservoir at chemical potential mv to the solid at potential m; a flow proportional to mv 2 m:

E. Fried and M.E. Gurtin

116

The motion of the interface results in energy fluxes 2maa daa Wa and mab dab Wb associated with the accretive transport of a-atoms across the endpoints xa and xb of C: In addition, the motion of the interface results in an energy flow 2ma ra2 V associated with the accretive transport of a-atoms from the solid across C2 : Hence, the net rate at which energy is added to C by the accretive transport of atoms is N X

½ma da Wba 2

a¼1

N ð X

a¼1

ma ra V ds:

ð23:3Þ

C

Letting c denote the free energy of the interface and bearing in mind Eqs. (21.1), (23.2) and (23.3), the free-energy imbalance for an arbitrary interfacial pillbox C ¼ CðtÞ takes the form & ' & ' d ð dx b ð du b ð c ds # c· 2 Cn·v ds þ s· 2 Tn·u ds dt C dt a dt a C C |ffl{zffl} |fflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflffl ffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} free energy

power expended by configurational forces

þ

N X

power expended by standard forces

½ma ð2ha þ da WÞba þ

N ð X

ma ða ·n 2 ra V þ ra Þds : ð23:4Þ C

a¼1

a¼1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} energy flow by atomic transport

B. Dissipation Inequality In view of Eqs. (21.9) and (22.4), and the integral transport theorem (14.28), the free-energy imbalance (23.4) for C becomes ð#Þ

zfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl ffl{ " ! # b N ð X a a ðc 2 cKVÞds þ c 2 m d 2 s
1 2 s W C

a¼1

#

ð C

2

s
1 þ t
q 2 s þ

N X

a¼1

N X

!

a

!

a a

m d þ s
1 KV 2 gV ds

a¼1

½ma ha ba

þ

  ›ha a m d þ ds: ›s C

N ð X

a¼1

a

ð23:5Þ

Since C is arbitrary, so also are the tangential velocities Wa and Wb of the endpoints of C; thus, since the only term in Eq. (23.5) dependent on these

A Unified Treatment of Evolving Interfaces

117

velocities is the term (#), we have the interfacial Eshelby relation63

s¼c2

N X

da ma 2 s
1;

ð23:6Þ

a¼1

which is an analog of the bulk Eshelby relation (12.15). Since N X

a¼1

½ma ha ba ¼

 N ð  X ›ha ›ma þ ha ma ds; ›s ›s a¼1 C

ð23:7Þ

we may use Eq. (23.6) to rewrite Eq. (23.5) as ð C

! N  a X a a a ›m c 2 s
1 2 t
q 2 m d 2h þ gV ds # 0; ›s a¼1

ð23:8Þ

63

In the absence of standard interfacial stress, the configurational stress within the interface has the P form c ¼ s t þ t n; with s ¼ c 2 Na¼1 da ma : This result, which, like the relation s ¼ c arising in our discussion of grain boundaries (cf. Eq. (17.21)), has a purely energetic structure. In this case it is legitimate to view s as an interfacial tension. But when standard stress is taken into account this interpretation is no longer valid. To explain this, consider the three-dimensional theory. There the interface is a surface S and the interfacial stresses are tensorial. The standard interfacial stress T is a tangential tensor field on S: in terms of Cartesian coordinates with subscripts 1 and 2 associated with the tangent plane at a point x on S; the component matrix of TðxÞ has the form 2 3 T11 T12 0 6 7 6 T21 T22 0 7 4 5 0

0

0

with T12 ¼ T21 (cf., e.g., Gurtin and Murdoch (1975) and the review of Shchukin and Bimberg (1999)). On the other hand, the configurational surface stress is a tensor field with both normal and tangential parts; the tangential part, which is the part relevant to a discussion of surface tension and which might be compared to the vector field c of the two-dimensional theory, has the form ! N X da ma P 2 ð7S uÞ` T; Ctan ¼ c 2 a¼1

where 7S is the surface gradient on S; and P; the projector onto S; has diagonal component matrix at x with diagonal ð1; 1; 0Þ: Only in the absence of standard surface stress does Ctan take the form Ctan ¼ P ðc 2 Na¼1 da ma ÞP of a pure tension; otherwise, Ctan is a generic tangential tensor field. One could, of course, think of c 2 da ma as a surface tension, and this is often done (cf. Shchukin and Bimberg, 1999), but what is more cogent is the form of Ctan ; which should be compared with the form P C ¼ ðc 2 Na¼1 ra ma Þ1 2 ð7uÞ` T of the bulk Eshelby tensor established in Section XXII, a comparison that identifies Ctan as an Eshelby tensor for the surface S (cf. Gurtin, 1995, 2000).

E. Fried and M.E. Gurtin

118

since C is arbitrary, this yields the interfacial dissipation inequality  N  X ›m a c 2 s
1 2 t
q 2 ma d a 2 ha þ gV # 0: ›s a¼1

ð23:9Þ

When adatom densities are neglected, the ‘constraint’ d~ ¼ 0~ renders Eq. (23.9) independent of the chemical potentials ma ; in this case, as in classical theories of continua, the chemical potentials ma on S should be considered as indeterminate (not constitutively determinate).

XXIV. Normal Configurational Force Balance Revisited A. Mechanical Potential F Since we account for deformation and atomic transport, the bulk configurational stress C is determined by the Eshelby relation (12.15): ! N X a a C¼ C2 r m 1 2 ð7uÞT T: a¼1

Thus, in view of the interfacial Eshelby relation (23.6), we may write the normal configurational force balance (20.1) in a form, N X

ðra 2 da KÞma ¼ C 2 Tn·ð7uÞn 2 ðc 2 s
1ÞK 2

a¼1

›t 2 g; ›s

ð24:1Þ

which equates terms of a chemical nature to those that are purely mechanical. Thus, for def

F ¼ C 2 Tn·ð7uÞn 2 ðc 2 s
1ÞK 2

›t 2 g; ›s

ð24:2Þ

we may write the normal configurational force balance in the form N X

ðra 2 da KÞma ¼ F ;

ð24:3Þ

a¼1

we refer to F as the mechanical potential. When g ¼ 0; F coincides with the variational derivative of the total free energy with respect to variations of the position of the interface, holding the composition fixed. This field appears first in the works of Wu (1996), Freund (1998), and Norris (1998), who, working within a framework that does not explicitly account for chemistry, follow Herring (1951) (cf. our discussion of

A Unified Treatment of Evolving Interfaces

119

Eq. (1.1)) in viewing the surface gradient of this potential as the driving force for surface diffusion.64 That Eq. (24.3) is compatible with this point of view follows upon neglecting adatom densities and restricting attention to a single species. A useful alternative form for F follows upon noting that, since t ¼ sg
þ t
(cf. Eq. (21.10)) and ›s
=›s ¼ t·Tn ¼ n·Tt (cf. Eq. (20.6)),

›t ›s
›g ›t
›g ›t
¼ þ ¼ g n·Tt þ s
þ ; g þ s
›s ›s ›s ›s ›s ›s

ð24:4Þ

hence, F ¼ C 2 Tn·ð7uÞn 2 g n·Tt 2 ðc 2 s
1ÞK 2 s


›g ›t
2 2 g: ›s ›s

ð24:5Þ

B. Substitution Alloys. Interfacial Chemical Potentials ma in Terms of the Relative Chemical-Potentials mab For a substitutional alloy the bulk densities ra are subject to the lattice constraint (5.1). As a consequence of this constraint, the individual chemical potentials ma in bulk away from the interface are not well defined; only the relative chemical potentials mab have meaning. At the interface the limiting values of these relative potentials are related to the chemical potentials of the solid at the interface through the chemical interface-conditions (23.1), ma 2 mb ¼ mab2 : Thus, limiting our discussion to the interface and writing mab ¼ mab2 ; we have the interface condition

ma 2 mb ¼ mab :

ð24:6Þ

On the other hand, there is no lattice constraint for the diffusion of adatoms and, as is clear from Eq. (24.1), it is the individual chemical potentials ma that enter the basic equation on S: In accord with this, we now show that the individual chemical potentials on S are uniquely determined—by what is essentially the normal configurational force balance—when the relative chemical potentials are known in the film. 64

Related studies are those of Asaro and Tiller (1972), Rice and Chuang (1981), and Spencer et al. (1991), who restrict attention to a surface free-energy that is constant. See, also, Freund and Jonsdottir (1993), Grilhe (1993), Gao (1994), Freund (1995), Spencer and Meiron (1994), Yang and Srolovitz (1994), Suo and Wang (1997), Wang and Suo (1997) and Xia et al. (1997), Le´onard and Desai (1998), Gao and Nix (1999), Shchukin and Bimberg (1999), Danescu (2001), Spencer et al. (2001), and Xiang and E (2002).

E. Fried and M.E. Gurtin

120

If we define effective densities

raef ¼ ra 2 da K;

ð24:7Þ

raef ma ¼ F :

ð24:8Þ

then Eq. (24.3) becomes

It is convenient to define an effective net density ref , effective concentrations caef ; and an effective net atomic volume Vef through

ref ¼

N X

raef ;

a¼1

caef ¼

raef ; ref

Vef ¼

1 : ref

ð24:9Þ

Then, choosing an atomic species a arbitrarily, we may use an argument of Larche´ and Cahn (1985, Appendix 1) to express ma in terms of the N 2 1 relative chemical-potentials mab and the sum rbef mba : Since mb ¼ ma 2 mab ; N X

rbef mb ¼

b¼1

N X

rbef ðma 2 mab Þ ¼ ref ma 2

b¼1

N X

rbef mab ;

b¼1

and the normal configurational force balance (24.8) yields an identity,

ma ¼

N X

cbef mab þ Vef F

b¼1

¼



›t 2g ; cbef mab þ Vef C 2 Tn·ð7uÞn 2 ðc 2 s
1ÞK 2 ›s b¼1 N X

ð24:10Þ

which, for a substitutional alloy, gives the individual chemical potential ma of each species a in terms of the chemical-potentials maz of a relative to all other species z: We refer to Eq. (24.10) as the configurational-chemistry relations. Conversely, if Eq. (24.10) holds for all a; then mab ¼ ma 2 mb and Eq. (24.8) is satisfied. Thus we have the following result:

Equivalency theorem for substitutional alloys. The configurational-chemistry relations,

ma ¼

N X

b¼1

cbef mab þ Vef F ;

a ¼ 1; 2; …; N;

ð24:11Þ

A Unified Treatment of Evolving Interfaces

121

are satisfied if and only if both the normal configurational force balance, N X

ðra 2 da KÞma ¼ F ;

a¼1

and the relative chemical-potential relations,

mab ¼ ma 2 mb ;

a; b ¼ 1; 2; …; N;

are satisfied. Thus, when discussing substitutional alloys we may equally well use the normal configurational force balance or the configurational-chemistry relations, provided that in the former case we account also for the relative chemicalpotential relations. This result is central to what follows.

XXV. Constitutive Equations for the Interface Since there is no lattice constraint for the diffusion of adatoms along the interface, the discussion of this section is valid whether or not the associated bulk material is subject to a lattice constraint. A. General Relations Our discussion of constitutive equations is guided by the interfacial dissipation inequality (23.9), viz.,  N  X ›m a c 2 s
1 2 t
q 2 ma d a 2 ha þ gV # 0; ›s a¼1 and follows the format set out in Section XVII.H. Let

d~ ¼ ðd1 ; d2 ; …; dN Þ;

m~ ¼ ðm1 ; m2 ; …; mN Þ:

Granted essentially linear dissipative response, we consider constitutive equations giving65

c; s
; t
; m~ as functions of ð1; q; d~Þ 65 Since 1 ¼ t·ð7uÞt is invariant for transformations of the form 7u ! 7u þ W; with W an arbitrary skew-symmetric tensor, the constitutive equations considered here have the requisite invariance under infinitesimal changes of observer.

E. Fried and M.E. Gurtin

122

in conjunction with constitutive equations for ha and g of the specific form ha ¼ 2

N X

Lab

b¼1

›m b 2 la V; ›s

N X

g¼2

a¼1

Ba

›m a 2 bV; ›s

ð25:1Þ

with coefficients possibly dependent on ð1; q; d~Þ:66 Then, appealing to the discussion of Section XVII.H, we find that the free energy determines the standard scalar interfacial stress, the reduced shear, and the chemical potentials through the relations

s
¼

›c^ð1; q; d~Þ ; ›1

t
¼

›c^ð1; q; d~Þ ; ›q

ma ¼

›c^ð1; q; d~Þ ; ›da

ð25:2Þ

and that the coefficient matrix of Eq. (25.1) must be positive semi-definite. In view of Eq. (21.10), t ¼ sg
þ t
and Eq. (25.2)1,2 yield an auxiliary constitutive relation showing that the configurational shear t depends also on the interfacial shear –strain g: Further, since 1 ¼ e·tðqÞ; we may define a function c~ðe; q; d~Þ through c~ðe; q; d~Þ ¼ c^ð1; q; d~Þ; differentiating this function with respect to q holding e fixed yields, by virtue of the identitites n ¼ ›t=›q (cf. Eq. (14.1)) and g ¼ n·e (cf. Eq. (15.10)2), the relation

›c~ðe; q; d~Þ ›c^ð1; q; d~Þ ›c^ð1; q; d~Þ þ ¼g ; ›q ›1 ›q

ð25:3Þ

which, by Eq. (25.2)1,2 and the relation t ¼ sg
þ t
gives

t¼

›c~ðe; q; d~Þ : ›q

ð25:4Þ

As the theory is limited to small deformations,67 it is reasonable to restrict attention to free energies that are quadratic in the tensile strain 1 and, thus, have the form

c^ð1; q; d~Þ ¼ c0 ðq; d~Þ þ wð1; q; d~Þ;

ð25:5Þ

with strain energy wð1; q; d~Þ ¼ s
0 ðq; d~Þ1 þ

1 2

kðq; d~Þ12 :

ð25:6Þ

66 The experiments of Barvosa-Carter and Aziz (2001) and first-principles calculations of Van de Walle et al. (2002) suggest that strain dependence of the mobilities may be important. 67 In Appendix C we develop the theory of small deformations as an approximation within a theory that allows for finite deformations. The resulting asymptotic analysis, which is delicate, leads to a theory that differs slightly from the one presented here, as the torque balance (20.3)2 is no longer satisfied and the constitutive equation for the strain energy contains an extra term 12 s
0 1g2 : Appendix C.2.5 gives a detailed discussion of the differences between the two theories.

A Unified Treatment of Evolving Interfaces

123

Here, c0 ðq; d~Þ is the strain-free surface energy, s
0 ðq; d~Þ is the residual surface stress, and kðq; d~Þ is the surface elasticity. Because atoms on the free surface are not bonded to the maximum number of nearest neighboring atoms, a residual surface stress is to be expected (Shchukin and Bimberg, 1999). The expressions for s
; t
; and ma determined by Eq. (25.2) under this specialization are straightforward. However, the final field equations are cumbersome. For this reason, we work with the generic expression c ¼ c^ð1; q; d~Þ:

B. Uncoupled Relations for ha and g A simplified form of Eq. (25.1) would take the classical Fickean form N X ›mb ha ¼ 2 ð25:7Þ Lab ›s b¼1 in conjunction with the simple kinetic relation g ¼ 2bV

ð25:8Þ

that formed the basis of our discussion of grain boundaries. Here the coefficients Lab ; which represent the mobility of the atoms on the interface, are presumed to form a positive semi-definite matrix, while b $ 0: The dissipation then has the form D¼

N X

a

b

›m ›m Lab ð1; q; d~Þ þ bð1; q; d~ÞV 2 |fflfflfflffl{zfflfflfflffl} › s › s : a;b¼1 |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} dissipation accompanying the

dissipation induced by surface diffusion

ð25:9Þ

attachment of vapor atoms

We, henceforth, work with the uncoupled relations (25.7) and (25.8). This choice is made for convenience only; the generalization of the resulting equations to situations involving the more general coupled relations (25.1) is straightforward. An argument in support of the uncoupled relations is that Eq. (25.7) represents the flow of atoms within the interface, while Eq. (25.8) represents an interaction of the solid surface with the vapor environment. But we do believe that there might be situations in which there is coupling between the diffusive adatom-flow described by h~ and the solid – vapor interaction as described by g: Such coupling might be especially important when studying film growth, since the kinetics of the deposition process and the small length scales involved render this process of a nature far different than the processes leading, for example, to relations that describe classical surface diffusion.

E. Fried and M.E. Gurtin

124

XXVI. Governing Equations at the Interface A. Equations with Adatom Densities Included While generally considered negligible, adatom densities would seem important in describing the segregation of atomic species at a solid – vapor interface.68 Moreover, the parabolic nature of the atomic balances when adatom densities are included might serve to regularize the overall system of partial differential equations, which is typically unstable. Regardless of whether adatom densities are included or neglected, our theory accounts for the diffusion of adatoms via the interfacial atomic balances.

1. General Relations The basic interfacial balances are the standard force-balance, the atomic balance, and the normal configurational force balance; for the case of a substitutional alloy, the configurational balance with the relations for the relative chemical potentials together are equivalent to the configurational-chemistry relations. The standard force and atomic balances are (cf. Sections XX.B and XXII) 9 ›s
> Tn ¼ s
Kn þ t; > ›s = ð26:1Þ a > ›h > þ a ·n þ r a ; ; d a þ ðra 2 da KÞV ¼ 2 ›s while the normal configurational force balance and the configurational-chemistry relations for substitutional alloys take the respective forms N X

ðra 2 da KÞma ¼ F ;

a¼1

ma ¼

N X

cbef mab þ Vef F

ð26:2Þ

b¼1

(cf. Eqs. (24.3), (24.5) and (24.11)), with F ¼ C 2 Tn·ð7uÞn 2 g n·Tt 2 ðc 2 s
1ÞK 2 s


›g ›t
2 2 g: ›s ›s

ð26:3Þ

As asserted in the equivalency theorem for substituted alloys (Section XXIV.B), 68

Spencer et al. (2001) assert that “surface segregation can make the surface composition differ from the bulk and the overall surface density of components could be nonuniform.” See also Lu and Suo (2001, 2002), who uses a Cahn– Hilliard type theory to study segregation of a planar two-phase monolayer on a strained substrate.

A Unified Treatment of Evolving Interfaces

125

for such materials imposing Eq. (26.2)2 ensures satisfaction of both Eq. (26.2)1 and the relations mab ¼ ma 2 mb : The balances are coupled to the constitutive relations

s
¼

›c^ð1; q; d~Þ ; ›1

›c^ð1; q; d~Þ ; ›q

t
¼

ma ¼

›c^ð1; q; d~Þ ›da

ð26:4Þ

(cf. Section XXV.B) and ha ¼ 2

N X

Lab

b¼1

›mb ; ›s

g ¼ 2bV:

ð26:5Þ

In Eq. (26.5) the constitutive moduli Lab and g are possibly dependent on ð1; q; d~Þ: By Eq. (26.4), writing cð1; q; d~Þ ¼ c^ð1; q; d~Þ and omitting the arguments ð1; q; d~Þ; we find that ! ›g ›t
›2 c ›c þ ¼ cþ K ðc 2 s
1ÞK þ s
21 ›s ›s ›1 ›q 2 þ

N X ›2 c ›1 ›c ›g ›2 c ›db þ þ : b ›1 ›q ›s ›1 ›s ›s b¼1 ›d ›q

ð26:6Þ Thus, combining the interfacial balances and the constitutive relations, we arrive at the standard force and atomic balances 0 1 9 N 2 2 2 b > X ›c › c › 1 › c › c › d > At; > > Tn ¼ Kn þ @ 2 þ Kþ > b ›s > ›1 ›1 ›q ›1 ›s = b¼1 ›1 ›d ð26:7Þ 0 1 > N > b X > › › m > @ A þ a ·n þ ga ; > d a þ ðra 2 da KÞV ¼ Lab > ; ›s ›s b¼1

and find that the mechanical potential F used in describing the normal configurational force balance and configurational-chemistry relations (26.2) has the form ! ›2 c ›c K 21 F ¼ C 2 Tn·ð7uÞn 2 g n·Tt 2 c þ ›1 ›q2 2

N X ›2 c ›1 ›c ›g ›2 c ›db 2 2 þ bV: b ›1 ›q ›s ›1 ›s ›s b¼1 ›d ›q

ð26:8Þ

E. Fried and M.E. Gurtin

126

Consider the atomic balance (26.7)2. If we define moduli

kab ad ¼

›2 c ; ›da ›db

Dab ad ¼

N X

Lag kgb ad ;

ð26:9Þ

g¼1

with Dab ad the surface diffusivity of a relative to b; then the interfacial-diffusion term in Eq. (26.7)2 has the form 0 1 N b › @X ab ›m A L ›s b¼1 ›s 8 !9 N N b 2 2 = X › <X › d › c › 1 › c › q ab þ þ : Dab L ¼ ›s :b¼1 ad ›s ›db ›1 ›s ›db ›q ›s ; b¼1

ð26:10Þ

Thus, bearing in mind the term ðda Þ in Eq. (26.7)2, we note that if the matrix with entries Dab ad is positive definite, then Eq. (26.7)2, as a system of partial differential equations for the adatoms da (considering the remaining fields as fixed), is parabolic and as such should have a regularizing effect on the evolution of the interface.

2. Chemical Potentials at the Surface. Chemical Compatibility For an unconstrained material, the chemical potentials ma of the surface, needed in the normal configurational force balance (26.2)1, may be determined directly through the limit relations

ma ¼ ma2 :

ð26:11Þ

On the other hand, these chemical potentials are given by the relations ma ¼ ›c=›da and by the limit relations (26.11) supplemented by the bulk constitutive relations (7.17):

ma ¼

›c^ð1; q; d~Þ ›F~ ðT; r~Þ ¼ : a ›d ›ra

ð26:12Þ

Similarly, for a substitutional alloy, the relative chemical potentials mab on the solid surface may be determined through the limit relations

mab ¼ mab2 ;

ð26:13Þ

A Unified Treatment of Evolving Interfaces

127

and, in addition, Eq. (9.31) yields

mab ¼

›c^ð1; q; d~Þ ›c^ð1; q; d~Þ ›F~ ðbÞ ðT; r~Þ 2 ¼ ; a ›d ›ra ›db

ð26:14Þ

Granted a knowledge of these relative chemical potentials, the surface values of the chemical potentials ma are given by the configurational-chemistry relations (26.2)2, a procedure that ensures satisfaction of the normal configurational force balance. Equations (26.12) and (26.14) might be termed chemical-compatibility relations as they represent compatibility between the composition of the interface and that of the bulk material. When ! ›c^ð1; q; d~Þ ›c^ð1; q; d~Þ ›c^ð1; q; d~Þ ; ; …; ›dN ›d1 ›d2 is an invertible function of d~; Eq. (26.12) furnishes explicit relations giving the adatom densities as functions of the bulk densities (as well as 1; q; and T).69 A similar assertion cannot be made for Eq. (26.14). Note that, for mechanically simple materials, Eqs. (26.12) and (26.14) have the respective forms (cf. Eqs. (7.34), (9.58), and (9.59))

ma ¼

›c^ð1; q; d~Þ ¼ ma0 ðr~Þ 2 Na ·T ›da

ð26:15Þ

and

mab ¼

›c^ð1; q; d~Þ ›c^ð1; q; d~Þ 2 ¼ ma0 ðr~Þ 2 mb0 ðr~Þ 2 Nab ·T: a ›d ›db

ð26:16Þ

3. Equations Neglecting Standard Surface Stress ðs ; 0Þ In this case c ¼ cðq; d~Þ and the standard force-balance (26.7)1 has the simple form Tn ¼ 0;

ð26:17Þ

asserting that the solid –vapor interface be (standard) traction-free, and the 69

Spencer et al. (2001) do not consider a dependence of interfacial free-energy on adatom densities, but instead posit constitutive relations of the form da ¼ lra ; with l; constant. For a nondeformable, unconstrained material, relations of this form would follow from Eq. (26.12), granted interfacial and P P bulk free-energies of the form c ¼ 12 L Na¼1 ðda Þ2 and C ¼ 12 lL Na¼1 ðra Þ2 ; with L; constant.

E. Fried and M.E. Gurtin

128

atomic balance (26.7)2 becomes 0 1 N b › @X ab ›m A d þ ðr 2 d KÞV ¼ L þ a ·n þ r a : ›s b¼1 ›s a

a

a

ð26:18Þ

Further, the normal configurational force balance (26.8) for unconstrained materials and the configurational-chemistry relations for substitutional alloys have the respective forms ! N N X X ›2 c ›2 c ›db a a a þ bV ð26:19Þ K2 ðr 2 d KÞm ¼ C 2 c þ 2 b ›s ›q a¼1 b¼1 ›d ›q and a

m ¼

N X

cbef mab

b¼1

8 9 ! N < = X ›2 c ›2 c ›db þ bV : K2 þ Vef C 2 c þ 2 b : ; ›s ›q b¼1 ›d ›q ð26:20Þ

We now specialize the normal configurational force balance and the configurational-chemistry relations to mechanically simple materials. For such materials the relation (7.31) for the free energy and the Gibbs relation (8.3) yield

C¼

1 2

T·K½T þ

N X

ra ma0 ðr~Þ;

ð26:21Þ

a¼1

hence, the normal configurational force balance (26.19) for an unconstrained, mechanically simple material takes the form ! N N 2 X X › c a a a a a K ðr 2 d KÞm ¼ r m0 þ 12 T·K½T 2 c 2 ›q 2 b¼1 a¼1 2

N X

b¼1

›2 c ›db þ bV: ›db ›q ›s

ð26:22Þ

The analogous result for a substitutional alloy is not as simple to derive. To begin with, we assume there are no vacancies; the relations (9.53) and (9.60)1,3 for the free energy and chemical potentials and the free-energy conditions at zero-stress (Eq. (9.40)) then imply that

C¼

1 2

T·K½T þ

N X

a¼1

ra ma0 ðr~Þ;

mab ¼ ma0 ðr~Þ 2 mb0 ðr~Þ 2 Nab ·T: ð26:23Þ

A Unified Treatment of Evolving Interfaces Thus, writing

129

! N X ›2 c ›2 c ›db þ bV; R¼2 cþ K2 2 b ›s ›q b¼1 ›d ›q

we may use Eqs. (24.7), (24.9), and (26.20) to show that

ref ma ¼

N X

rbef mab þ C þ R

b¼1

¼ ref ma0 2

N X

rbef mb0 2

b¼1

¼ ref ma0 þ

N X

N X

rbef Nab ·T þ 12 T·K½T þ

b¼1

db mb0 K 2

b¼1

N X

rb mb0 þ R

b¼1

N X

rbef Nab ·T þ 12 T·K½T þ R:

b¼1

The configurational-chemistry relations (26.20) for a mechanically simple, substitutional alloy without vacancies, therefore, takes the form 8 <

1 N 2 X › c ma 2 ma0 ¼ cbef Nba ·T þ Vef 12 T·K½T 2 @c þ 2 db mb0 AK 2 : › q b¼1 b¼1 9 N = X ›2 c ›db ð26:24Þ 2 þ bV : b ; ›s b¼1 ›d ›q N X

0

B. Equations when Adatom Densities are Neglected 1. Balances The basic interfacial balances are Eqs. (26.1) and (26.2) with the terms involving adatoms deleted. The balances for standard forces and atoms, therefore, take the form Tn ¼ s
Kn þ

›s
t; ›s

ra V ¼ 2

›ha þ a ·n þ r a ; ›s

ð26:25Þ

while the normal configurational force balance and the configurational-chemistry relations (for substitutional alloys) take the respective forms N X

a¼1

ra m a ¼ F ;

ma ¼

N X

b¼1

cb mab þ VF ;

ð26:26Þ

E. Fried and M.E. Gurtin

130

with mechanical potential F ; as before, given by Eq. (26.3). In writing Eq. (26.26) we have used the fact that, for d~ ; 0; the defining relations for Vef in Eq. (24.7) and cbef in Eq. (24.9) take the simple forms

Vef ¼ V ¼

1

r

sites

cbef ¼ cb ¼

;

rb ; rsites

ð26:27Þ

with V the atomic volume and cb the concentration of species b:

2. Constitutive Relations. Indeterminacy of the Chemical Potentials The treatment of chemical potentials in the absence of an accounting of adatom densities is somewhat delicate: since ma da ¼ 0 for each a; the interfacial dissipation inequality (23.9) reduces to

c 2 s
1 2 t
q þ

N X

ha

a¼1

›ma þ gV # 0; ›s

ð26:28Þ

the absence of a field conjugate to the surface chemical potentials ma renders them indeterminate on S; that is, not viable as independent variables in the constitutive relations for the interface. Thus, in view of the dissipation inequality (26.28), arguing as for Eqs. (26.4) and (26.5), we are led to the constitutive relations

s
¼

›c^ð1; qÞ ; ›1

t
¼

›c^ð1; qÞ ; ›q

ð26:29Þ

g ¼ 2bV;

ð26:30Þ

and ha ¼ 2

N X

b¼1

Lab

›mb ; ›s

with moduli Lab and g allowed to depend on ð1; qÞ: By Eq. (26.29), writing cð1; qÞ ¼ c^ð1; qÞ; we find that ! ›g ›t
›2 c ›c ›2 c ›1 ›c › g ðc 2 s
1ÞK þ s
þ ¼ cþ K þ þ : 2 1 2 ›s ›s › 1 › 1 › q › s ›1 ›s ›q ð26:31Þ Thus, if we combine the interfacial balances and the constitutive relations, we

A Unified Treatment of Evolving Interfaces

131

arrive at the standard force and atomic balances ! 9 > ›c ›2 c ›1 ›2 c > Kn þ þ Tn ¼ K t; > > > ›1 ›1 ›q ›12 ›s = 0 1 > N > › @X ›m b A > ra V ¼ Lab þ a ·n þ r a : > > ; ›s b¼1 ›s

ð26:32Þ

Further, the normal configurational force balance and the configurationalchemistry relations for substitutional alloys are given by Eq. (26.26) with mechanical potential ! ›2 c ›c K 21 F ¼ C 2 Tn·ð7uÞn 2 g n·Tt 2 c þ ›1 ›q2 2

›2 c ›1 ›c ›g 2 þ bV; ›1 ›q ›s ›1 ›s

ð26:33Þ

which is simply Eq. (26.8) with the term involving adatoms dropped.

3. Equations Based on a Quadratic Strain-Energy A theory that might be useful in accessing the effects of surface stress might be based on a free energy of the form67

c ¼ c0 ðqÞ þ wð1; qÞ;

ð26:34Þ

with quadratic strain-energy wð1; qÞ ¼ s
0 ðqÞ1 þ

1 2

kðqÞ12 :

ð26:35Þ

In this case, the scalar surface stress and reduced configurational shear are

s
¼ s
0 ðqÞ þ kðqÞ1;

t
¼ c 00 ðqÞ þ s
00 ðqÞ1 þ 12 k 0 ðqÞ12

ð26:36Þ

(cf. Eq. (26.4)1,2). Further, by Eqs. (15.10) and (15.11), the normal and tangential components (20.6)1 and (20.6)2 of the standard force balance reduce to   ›u n·Tn ¼ s
0 ðqÞ þ kðqÞt· K ð26:37Þ ›s and t·Tn ¼



s
00 ðqÞ

 ›u ›u ›2 u 0 þ k ðqÞt· þ kðqÞn· K þ kðqÞt· 2 ; ›s ›s ›s

ð26:38Þ

E. Fried and M.E. Gurtin

132

while the mechanical potential becomes F ¼ C 2 Tn·ð7uÞn 2 AK 2 a·

›2 u þ bV; ›s 2

ð26:39Þ

with   ›u ›u ›u 2 þ 2s
00 ðqÞn· þ kðqÞ n· A ¼ c0 ðqÞ þ c 000 ðqÞ 2 ðs
0 ðqÞ 2 s
000 ðqÞÞt· ›s ›s ›s  2    › u › u › u 2 12 ð3kðqÞ 2 k 00 ðqÞÞ t· þ2k 0 ðqÞ t· n· ð26:40Þ ›s ›s ›s and a ¼ s
0 ðqÞn þ s
00 ðqÞt þ kðqÞ



     ›u ›u ›u 0 t· n þ n· t þ k ðqÞ t· t: ›s ›s ›s

ð26:41Þ

A simplifying assumption of potential value for assessing the importance of surface stress might be to take the residual stress and elasticity for the surface to be constant. Then, Eqs. (26.37) and (26.38) simplifying slightly to   ›u n·Tn ¼ s
0 þ kt· K ð26:42Þ ›s and   ›u ›2 u t·Tn ¼ kn· K þ kt· 2 ; ›s ›s while Eqs. (26.40) and (26.41) reduce to 9     3 ›u 2 ›u 2 ›u > þk n· 2s
0 t· ; > A ¼ c0 ðqÞ þ c 000 ðqÞ 2 k t· > 2 ›s ›s ›s = &   ' > > ›u ›u ›u > ; a ¼ s
0 n þ k t· n þ n· t · ; ›s ›s ›s

ð26:43Þ

ð26:44Þ

so that even within this drastically simplified theory the standard force balance, the normal configurational force balance, and, for a substitutional alloy, the configurational-chemistry relations are quite complicated. Alternatively, consistent with the view of Shchukin and Bimberg (1999), we might allow the free energy to be anisotropic and ignore interfacial elasticity, so that

c ¼ c0 ðqÞ þ s
0 ðqÞ1;

s
¼ s
0 ðqÞ;

t
¼ c 00 ðqÞ þ s
00 ðqÞ1:

ð26:45Þ

Then, the appropriately simplified versions of Eqs. (26.37) and (26.38) combine

A Unified Treatment of Evolving Interfaces

133

to yield Tn ¼ s
0 ðqÞKn þ s
00 ðqÞKt;

ð26:46Þ

thus, writing ›u=›n ¼ ð7uÞn; it follows that Tn·ð7uÞn ¼ s
0 ðqÞKn·

›u ›u þ s
00 ðqÞKt· ›n ›n

ð26:47Þ

and we may replace Eq. (26.44) by A ¼ c0 ðqÞ þ c 00 ðqÞ þ ðs
0 ðqÞ þ ð2s
00 ðqÞÞnÞ· 2s
000 ðqÞÞt·

›u ; ›s

a ¼ ðs
0 ðqÞn þ s
00 ðqÞtÞ·

9 ›u > 2 ðs
0 ðqÞ 2 s
00 ðqÞ > > > ›n > > > = > > > > > > > ;

2

› u þ bV: ›s 2

ð26:48Þ Finally, if s
0 ¼ constant and k ¼ 0; then Eqs. (26.39) and (26.46) specialize to yield Tn ¼ s
0 Kn;

ð26:49Þ

and ( F ¼C2

c0 ðqÞ þ c 000 ðqÞ þ s
0



›u ›u n· 2 t· ›n ›s

)

K 2 s
0 n·

›2 u þ bV: ›s 2

ð26:50Þ

As discussed in Section XVII.E, the local stability of the evolution equation (17.31) for a grain boundary is determined by the sign of the coefficient of the curvature—the interfacial stiffness. In Eq. (26.50), the coefficient of the curvature is the sum of the interfacial stiffness and the underlined term. Thus, notwithstanding the term s
0 n·›2 u=›s2 ; it appears that a tensile normal strain (for which n·›u=›n . 0Þ and a compressive tangential strain (for which 1 ¼ t·›u=›s , 0Þ should be stabilizing.

4. Equations Neglecting Standard Surface Stress Here the interface remains traction-free and the atomic balance becomes 0 1 N b › @X a ab ›m A r V¼ L ð26:51Þ þ a ·n þ r a ; ›s b¼1 ›s

E. Fried and M.E. Gurtin

134

the normal configurational force balance has the form ! N X ›2 c a a K þ bV; r m ¼C2 cþ ›q2 a¼1

ð26:52Þ

and, for a substitutional alloy, the configurational chemistry relations becomes ( ! ) N X ›2 c a b ab K þ bV : ð26:53Þ m ¼ c m þV C2 cþ ›q2 b¼1 Finally, restricting attention to mechanically simple materials, we find that Eqs. (26.52) and (26.53), respectively, reduce to ! N X ›2 c a a a 1 K þ bV ð26:54Þ r ðm 2 m0 Þ ¼ 2 T·K½T 2 c þ ›q2 a¼1 and a

m 2

ma0

¼

N X

b¼1

( b

ba

c N ·T þ V

1 2

! ) ›2 c K þ bV : T·K½T 2 c þ ›q 2

ð26:55Þ

Granted cubic symmetry, so that Nab ¼ hab 1 (cf. Eq. (9.63)), and assuming that b ; 0; Eq. (26.55) reduces to the following result of Spencer et al. (2001, Eqs. (2.10), (2.11)):70 ( ! ) N X ›2 c a a b ba 1 K : m 2 m0 ¼ c h tr T þ V 2 T·K½T 2 c þ ›q2 b¼1

C. Addendum: Importance of the Kinetic Term g ¼ 2bV In each of the cases discussed in Section XXVI.B.3 and XXVI.B.4, the normal configurational force balance contains the kinetic term bV resulting from the constitutive equation g ¼ 2bV for g; a dissipative force associated with the attachment of vapor atoms to the solid surface. The dissipation associated with g;71 measured per unit length of the interfacial curve, is bV 2 ; without this term the attachment process is nondissipative. Interface conditions that play the role of the normal configurational force balance are typically derived using a chemical potential defined to be the variational derivative of the total free energy with respect to variations in the 70 71

Note that our hba is V times their modulus hba : Cf. Eq. (25.9) and the remark following Eq. (23.9).

A Unified Treatment of Evolving Interfaces

135

configuration of the interface; thus the possibility of having a dynamical interface condition involving V are ruled out from the start by the use of such variational paradigms, which, by their very nature, cannot involve the normal velocity V. To assess the importance of the kinetic term, consider a single atomic species and neglect bulk diffusion, so that  ; 0 and the atomic volume V ¼ 1=r is constant. Assume further that the mobility L for surface diffusion and the kinetic modulus b are constant, so that the atomic balance (26.51) and the normal configurational force balance (26.52) take the form ( ! ) ›2 m ›2 c V ¼ VL 2 þ r; K þ bV : ð26:56Þ m¼V C2 cþ ›s ›q2 The field

(

! ) ›2 c K m ¼V C2 cþ ›q 2 eq

represents the equilibrium chemical potential (the chemical potential when V ¼ 0); using this field we may write the normal configurational force balance in the form m ¼ meq þ VbV: This relation and the atomic balance (26.56)1 yield the evolution equation V 2x2 with

›2 V ›2 meq ¼ V L þr ›s 2 ›s 2 pffiffiffiffi x ¼ V bL

ð26:57Þ

ð26:58Þ

a material length-scale; the kinetic term g ¼ 2bV is, therefore, important at length scales of order x and smaller.72 Thus, whether or not the kinetic term is important depends on the magnitude of the product bL; consequently, when the Fickean mobility L is sufficiently large, the term g ¼ 2bV may be important even when the modulus b is small.

XXVII. Interfacial Couples. Allowance for an Energetic Dependence on Curvature The notorious instability of strained solid –vapor interfaces results in a wide variety of surface patterns and morphologies, an example being the faceted 72

This conclusion arose from conservations with Peter Voorhees.

136

E. Fried and M.E. Gurtin

Fig. 27.1. (a) STM image ð108 £ 108 nm2 Þ of a faceted island obtained by depositing Si0:6 Ge0:4 on Si(001). (b) Cross-section of the region near the top of the island, from an average of 30 line-scans taken from left to right close to the middle of the island. Scale in nm. (Images from Tersoff et al. (2002).)

islands (Fig. 27.1) observed by Tersoff et al. (2002). Such instabilities are reflected by the underlying evolution equations73 and by the resulting difficulty of performing reliable simulations, which motivates the need for physically based regularized theories.74 Regularizations that account for an energetic dependence 73 Asaro and Tiller (1972), Grinfeld (1986), and Srolovitz (1989) showed that a planar layer under stress may be unstable. Subsequent studies of related instabilities include Freund and Jonsdottir (1993), Grilhe (1993), Gao (1994), Freund (1995), Spencer and Meiron (1994), Yang and Srolovitz (1994), Suo and Wang (1997), Wang and Suo (1997), Xia et al. (1997), Le´onard and Desai (1998), Gao and Nix (1999), Shchukin and Bimberg (1999), Phan et al. (2001), Danescu (2001), Spencer et al. (2001), and Xiang and E (2002). 74 Our perspective here is identical to that in Section XVIII. As opposed to a pragmatic approach in which terms involving higher-order derivatives are added to stabilize an equation, we seek regularizations that reflect relevant physical mechanisms.

A Unified Treatment of Evolving Interfaces

137

on curvature (DiCarlo et al., 1992) but neglect surface stress and adatom densities, are used by Tersoff et al. (2002) to describe the initial stages of island formation and by Siegel et al. (2003) to study the formation of wrinklings on a void surface.75 To arrive at theory that allows for an energetic dependence on curvature, one may follow the steps taken in our extension of the classical theory of grain boundaries to allow for the study of facets and wrinklings (cf. Section XVIII). Specifically, we may introduce an interfacial couple stress and an interfacial internal couple together with a configurational torque balance. Taking into account the torques exerted by the configurational force system, the configurational torque balance for an interfacial pillbox C is (cf. Eq. (18.1)) ½M þ ðx 2 0Þ £ cba þ

ð

m ds þ

C

ð

ðx 2 0Þ £ ðg 2 CnÞ ds ¼ 0:

ð27:1Þ

C

The foregoing considerations leave intact the basic balances considered in our previous treatment of strained solid – vapor interfaces. Thus, using the configurational force balance and proceeding as in the derivation of Eq. (18.3), we obtain the local form of the configurational torque balance:

›2 M ›m 2 sK 2 g þ n·Cn ¼ 0: þ 2 ›s ›s

ð27:2Þ

In the presence of interfacial couples, the net power expended on an interfacial pillbox C must account for the action of the couple stress M at the endpoints of C; we are thus led to a free-energy imbalance of the form ' & ' & ' & d ð dq b dx b ð du b ð c ds # M þ c· 2 Cn·v ds þ s· 2 Tn· u ds dt C dt a dt a dt a C C N N ð X X ½ma ð2ha þ da WÞba þ ma ð a ·n 2 ra V þ ra Þds: þ a¼1

a¼1

C

ð27:3Þ Arguing as in Sections XVIII.C and XXIII.B, we find that, when interfacial configurational torques are taken into consideration, the interfacial Eshelby relation (23.6) must be modified to read

s¼c2

N X

da ma 2 s
1 2 MK

ð27:4Þ

a¼1

75 As discussed following Eq. (26.10), the inclusion of adatoms may provide at least a partial regularization. However, we believe that constraining the curvature of the interface via an energetic dependence on curvature should provide a more effective regularization.

E. Fried and M.E. Gurtin

138

and the interfacial dissipation inequality is of the form  N  X ›m a c 2 s
1 þ m ma d a 2 ha þ gV # 0;
q 2MK 2 ›s a¼1

ð27:5Þ

with ð27:6Þ

m
¼ m þ sg
; the reduced internal configurational couple. Using the bulk Eshelby relation (12.15) and the identity

›m ›m ›g ›s
›m ›g

¼ 2 s
2 2 g n·Tt 2 s
; g¼ ›s ›s ›s ›s ›s ›s

ð27:7Þ

which follows from Eq. (27.6) and ›s
=›s ¼ t·Tn (cf. Eq. (20.6)), we find that the balance (27.2) can be expressed in the form N X

ðra 2 da KÞma ¼ C 2 Tn·ð7uÞn 2 g n·Tt 2 ðc 2 s
1 2 MKÞK

a¼1

2 s


›g ›m ›2 M
þ þ 2 g: ›s ›s ›s 2

ð27:8Þ

For unconstrained materials, Eq. (27.8) supersedes the configurational balance (26.2). For substitutional alloys, Eq. (27.8) is replaced by

N X ma ¼ cbef mab þ Vef C 2 Tn·ð7uÞn 2 g n·Tt 2 ðc 2 s
1 2 MKÞK a¼1

) ›g ›m ›2 M
2s
þ þ 2g : ›s ›s ›s 2

ð27:9Þ

which generalize the configurational-chemistry relations (26.3). Guided by Eq. (27.5), we consider constitutive equations giving

c; s
; m;
M; m~ as functions of ð1; q; K; d~Þ in conjunction with constitutive equations for ha and g of the form Eq. (25.1) with coefficients now possibly dependent on ð1; q; K; d~Þ: Appealing to the discussion of Section XVII.H, we then find that the free energy determines the standard interfacial stress, reduced configurational couple, couple stress, and chemical potentials through the relations

›c^ ; ð27:10Þ ›da where, for convenience, we have omitted the argument ð1; q; K; d~Þ: For simplicity,

s
¼

›c^ ; ›1

m
¼2

›c^ ; ›q

M¼

›c^ ; ›K

ma ¼

here—after we work only with the uncoupled relations (26.5) for ha and g:

A Unified Treatment of Evolving Interfaces

139

When interfacial couples and adatom densities are taken into account, the general relations are, therefore, Eqs. (26.1), (26.11), and (27.8) for an unconstrained material, Eqs. (26.13) and (27.9) for a substitutional alloy, along with the constitutive relations (26.5) and (27.10). If we neglect adatom densities and, following our treatment of grain boundaries, assume that the free energy has the form76

cð1; qÞ þ 12 lK 2 ;

ð27:11Þ

with l . 0 constant, it then follows from Eq. (27.10)1–3 that

s
¼

›c^ð1; qÞ ; ›1

m
¼2

›c^ð1; qÞ ; ›q

M ¼ lK:

ð27:12Þ

! 9 > ›c ›2 c ›1 ›2 c > Kn þ þ Tn ¼ K t; > > > ›1 ›1 ›q ›12 ›s = 0 1 > N > › @X ›m b A > ra V ¼ Lab þ a ·n þ r a ; > > ; ›s b¼1 ›s

ð27:13Þ

In this case, the basic interfacial balances are

the normal configurational force balance77 ! ›2 c ›c K r m ¼ C 2 Tn·ð7uÞn 2 g n·Tt 2 c þ 21 ›1 ›q 2 a¼1 ! ›2 c ›1 ›c ›g ›2 K 1 3 2 2 þ bV þ l þ 2K ›1 ›q ›s ›1 ›s ›s 2 N X

a

a

ð27:14Þ

for unconstrained materials, and the configurational chemistry relations ! ›2 c ›c K m ¼ c m þ V C 2 Tn·ð7uÞn 2 g n·Tt 2 c þ 21 ›1 ›q2 a¼1 !) › 2 c ›1 ›c ›g ›2 K 1 3 2 þ bV þ l ð27:15Þ þ 2K 2 ›1 ›q ›s › 1 ›s ›s 2 a

N X

(

b

ab

for substitutional alloys. When the interface is the graph of a function y ¼ hðx; tÞ; the term l ›2 K=›s2 in Eqs. (27.14) and (27.15) would have the form A ›4 h=›x4 ; 76

Cf. DiCarlo et al. (1992), Stewart and Goldenfeld (1992), Liu and Metiu (1993), and Golovin et al. (1998, 1999). 77 Cf. Gurtin and Jabbour (2002), who consider a single atomic species in three space-dimensions, neglecting surface stress.

E. Fried and M.E. Gurtin

140

with A . 0 a function of ›h=›x; ›2 h=›x2 ; and ›3 h=›x3 ; (cf. Section XVII.E). Therefore (considering the remaining fields as fixed), Eqs. (27.14) and (27.15) would represent elliptic partial differential equations and as such should have a regularizing effect on the evolution of the interface.

XXVIII. Allowance for Evaporation – Condensation To model processes in which evaporation – condensation is of importance, we now follow the approach taken in our treatment of grain –vapor interfaces (cf. Section XIX) and endow the vapor with a chemical potential mav for each atomic species a: In addition, we reinterpret the role of the supplies r a in the theory: we view r a as the rate at which a-atoms of chemical potential mav are supplied from the vapor to the solid at the interface. Thus, while this change in perspective leaves unaltered the atomic balances (22.3), it is necessary to modify the freeÐ energy imbalance (23.4) for a pillbox C to account for the net rate C mav r a ds at which energy is added to C by evaporation – condensation; granted this net rate is accounted for, the free-energy imbalance for an interfacial pillbox C ¼ CðtÞ reads (cf. Eq. (19.15)) & ' & ' d ð dx b ð du b ð c ds # c· 2 Cn·v ds þ T· 2 Tn· u ds dt C dt a dt a C C N N ð X X ½ma ð2ha þ da WÞba þ ðma ð a ·n 2 ra VÞ þ mav r a Þds: ð28:1Þ þ a¼1

a¼1

C

Arguing as in Section XXIII.B, we find that our consideration of evaporation – condensation leaves the interfacial Eshelby relation (23.6) unchanged and results in an interfacial dissipation inequality of the form (cf. Eq. (19.18))  N  X ›m a 2 ðma 2 mav Þr a þ gV # 0: ð28:2Þ c 2 s
1 2 t
q 2 ma d a 2 ha ›s a¼1 The inequality (28.2) differs from that, Eq. (23.9), valid in the absence of evaporation – condensation only by the presence of the term ðma 2 mav Þr a : Thus the state relations (25.2) determining s
; t
; and ma remain valid, leaving the residual dissipation inequality  N  X ›ma þ ðma 2 mav Þr a þ gV # 0: ha ð28:3Þ ›s a¼1 Granted essentially linear dissipative response, Eq. (28.3) leads to constitutive equations for ha ; r a and g of a (multi-species) form strictly analogous to

A Unified Treatment of Evolving Interfaces

141

Eq. (19.21), but with coefficients possibly dependent on ð1; q; d~Þ and coefficient matrix positive semi-definite. If the constitutive equations for ha ; r a ; and g are uncoupled, so that ha ¼ 2

N X

Lab

b¼1

›m b ; ›s

ra ¼ 2

N X

k
ab ðmb 2 mbv Þ;

g ¼ 2bV;

ð28:4Þ

b¼1

then the basic equations of the theory remain exactly as described in Sections XXVI.A and XXVI.B, but now the atomic supplies r a are not arbitrarily prescribable, but are instead given by Eq. (28.4)2. The consecutive equations for r a , and g describe the solid –vapor interaction. A somewhat more robust description of this interaction is given by the coupled relations (Fried and Gurtin, 2003) ra ¼ 2

N X

k
ab ðmb 2 mbv Þ 2 ka V;

b¼1

g¼2

N X

b
a ðma 2 mav Þ 2 bV: ð28:5Þ

a¼1

Under this assumption the resulting equations, while complicated, are easily obtained from Eqs. (26.1) –(26.4), (28.4)1, and (28.5). For example, if we neglect adatom densities and surface stress, then the resulting equations for a substitutional alloy without vacancies are Tn ¼ 0; 0 1 N N b X X › › m @ Lab A þ a ·n 2 ðra þ ka ÞV ¼ k
ab ðmb 2 mbv Þ: ›s b¼1 ›s b¼1 and 0 @1 þ

N X

b¼1

1 b
b Ama ¼

N X

b¼1

ðcb þ b
b Þmab

! ) N X ›2 c a a
b mv þ bV : K2 þV C2 cþ ›q2 a¼1 (

ð28:6Þ

COHERENT PHASE INTERFACES We now consider a class of theories for a coherent phase interface. That bulk deformation and stress have a pronounced effect on the shape of coherent precipitates is well understood.78 Further, as argued by Cahn (1989), interfacial 78

Cf. the reviews of Johnson and Voorhees (1992) and Voorhees (1992) and the references therein.

142

E. Fried and M.E. Gurtin

stress may also be of importance. Hence, in addition to allowing for a multiplicity of atomic species, we account for stress in the bulk phases and on the interface. However, we neglect interfacial atomic densities.

XXIX. Forces. Power A. Configurational Forces For a coherent interface separating two solid phases, the configurational forces acting a pillbox are identical to those arising in our discussion of grain boundaries. Thus, the configurational force balance (17.6) and its local consequences remain valid. In particular, we have the normal configurational force balance

sK þ

›t þ n·½½Cn þ g ¼ 0: ›s

ð29:1Þ

B. Standard Forces Let C ¼ CðtÞ be an arbitrary interfacial pillbox. In addition to the standard forces 2sa and sb exerted at xa and xb by the portion of S exterior to C; the solid in the (þ)-phase exerts a traction Tþ n on Cþ and the solid in the (2)-phase exerts a traction 2T2 n on C2 ; the net traction exerted at each point of C by the bulk phases is then Tþ n 2 T2 n ¼ vTbn (cf. Section XX.B). The standard torque acting on a pillbox is determined analogously. In view of the proceeding discussion, the standard force and torque balances for C take the form 9 ð > b sla þ ½½Tn ds ¼ 0; > > = C ð29:2Þ ð > > b > ½ðx 2 0Þ £ sa þ ðx 2 0Þ £ ½½Tn ds ¼ 0; ; C

since C is arbitrary, these yield the interfacial balances

›s þ ½½Tn ¼ 0; ›s

t £ s ¼ 0:

ð29:3Þ

Thus, as in the case of a solid – vapor interface, the stress vector s is tangent to the interface; hence, s ¼ s
t; with s
the standard scalar interfacial stress.

A Unified Treatment of Evolving Interfaces

143

Since ›t=›s ¼ Kn; we can rewrite the interface condition (29.3)1 as

s
Kn þ

›s
t þ ½½Tn ¼ 0; ›s

ð29:4Þ

or equivalently, as n·½½Tn ¼ 2s
K;

t·½½Tn ¼ 2

›s
: ›s

ð29:5Þ

the first of Eq. (29.5) represents a counterpart, for a coherent interface, of the classical Laplace – Young relation for a liquid – vapor interface.

C. Power Consider a migrating pillbox C ¼ CðtÞ: Arguing as for a solid –vapor interface (Section XXI.A), we view (i) the endpoint velocities dxa =dt and dxb =dt as the power-conjugate velocities for c; (ii) the motion velocities dua =dt and dub =dt following xa ðtÞ and xb ðtÞ as the power-conjugate velocities for s; (iii) the normal velocity v of C as the power-conjugate velocity for the configurational tractions Cþ n and 2C2 n acting on Cþ and C2 ; and (iv) the motion velocity u following SðtÞ as the power-conjugate velocity for the standard tractions Tþ n and 2T2 n acting on Cþ and C2 : The (net) external power expended on CðtÞ is, therefore, presumed to have the form & ' dx du b ð þ s· c· þ ð½½Cn·v þ ½½Tn· u Þds: dt dt a C

ð29:6Þ

Using the identity ›V=›s ¼ q and the normal configurational force balance (29.1), it follows that (cf. Eqs. (17.13) and (17.15)) & ' dx b c· ¼ ½sW þ tVba dt a ð ¼ ½sWba þ ½t q 2 ðsK þ g þ n·½½CnÞVds;

ð29:7Þ

C

thus, since v ¼ Vn; & c·

dx dt

'b ð ð þ ½½Cn·v ds ¼ ½sWba þ ½t q 2 ðsK þ gÞVds: a

C

C

ð29:8Þ

A Unified Treatment of Evolving Interfaces

145

for each species a; and this yields the local balance (( a )) ›ha (( a )) þ  ·n: r V¼ ›s

ð30:4Þ

XXXI. Free-Energy Imbalance Basic to the theory is the assumption that the chemical potentials ma are continuous across the interface: (( a )) m ¼ 0:

ð31:1Þ

This requirement, often referred to as an assumption of local chemical equilibrium, allows us to consider the bulk fields ma ; when evaluated on S; as appropriate interfacial chemical-potentials. Restrict attention to a single atomic species a: As in the case of a solid – vapor interface, the energy flow into a migrating pillbox C due to atomic transport includes contributions associated with diffusion and accretion. In addition to the diffusive energy fluxes maa haa and 2maa haa associated with the flow of a-atoms across xa and xb ; the diffusion of a-atoms in the (þ)-phase results in an energy flow 2ma aþ ·n across Cþ while diffusion of a-atoms in the (2)-phase results in an energy flow ma a2 ·n across C2 ; so that the diffusive energy flow of a-atoms to each point of C from the bulk material is 2ma v a b·n: Hence, the net rate at which energy is added to C by diffusive transport is 2

N X

½ma ha ba 2

a¼1

N ð X

a¼1

(( )) ma a ·n ds:

ð31:2Þ

C

The motion of the interface results in accretive energy flows ma raþ V and 2ma ra2 V of a-atoms from the solid across Cþ and C2 ; so that the net accretive energy flow of a-atoms to each point of C from the bulk material is ma vra bV: Hence, the net rate at which energy is added to C by the accretive transport of atoms is N ð X

a¼1

(( )) ma ra V ds:

ð31:3Þ

C

Letting c denote the free energy of the interface and bearing in mind Eqs. (21.1), (31.2), and (31.3), the free-energy imbalance for an arbitrary

E. Fried and M.E. Gurtin

146

interfacial pillbox C takes the form d ð c ds # dt C |fflffl{zfflffl} free energy

&

' dx b ð c· þ ½½Cn·v ds dt a C |fflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflffl ffl}

power expended by configurational forces

2

N X

½ma ha ba þ

a¼1

N ð X

a¼1

& ' du b ð þ s· þ ½½Tn· u ds dt a C |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

power expended by standard forces

(( )) (( )) ma ð2 a ·n þ ra VÞds :

ð31:4Þ

C

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} energy flow by atomic transport

A. Dissipation Inequality for Unconstrained Materials In view of Eqs. (29.10) and (30.3), the free-energy imbalance (31.4) for C becomes ð#Þ

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ ðc 2 cKVÞds þ ½ðc 2 s
1 2 sÞWba

ð C

#

ð C

2

½s
1 þ t
q 2 ðs þ s
1ÞKV 2 gVds

N X

½ma ha ba þ

a¼1

N ð X

a¼1

ma C

› ha ds: ›s

ð31:5Þ

Since C is arbitrary, so also are the tangential velocities Wa and Wb of the endpoints of C; thus, since the only term in Eq. (31.5) dependent on these velocities is (#), we have the superficial Eshelby relation

s ¼ c 2 s
1;

ð31:6Þ

which differs from the Eshelby relation (23.6) for a strained solid – vapor interface only because of our present neglect of interfacial atomic densities. By Eq. (31.6) and N X

½ma ha ba ¼

a¼1

 N ð  X ›ha ›ma þ ha ma ds; ›s ›s a¼1 C

ð31:7Þ

Equation (31.5) becomes ð C

! ›ma þ gV ds # 0; c 2 s
1 2 t
q þ h ›s a¼1 N X

a

ð31:8Þ

A Unified Treatment of Evolving Interfaces

147

since C is arbitrary, this yields the interfacial dissipation inequality N X

c 2 s
1 2 t
q þ

a¼1

ha

›ma þ gV # 0: ›s

ð31:9Þ

B. Interfacial Flux Constraint and Free-Energy Imbalance for Substitutional Alloys Our treatment of substitutional alloys in bulk was based on the substitutional flux constraint N X

a ¼ 0

ð31:10Þ

a¼1

˚ gren (1982) and Cahn and Larche´ (1983), who argue that discussed by A Eq. (31.10) is a consequence of the requirement that diffusion, as represented by atomic fluxes, arises from exchanges of atoms or exchanges of atoms with vacancies. A consequence of coherency is that (cf. Eq. (15.6)) ½½7ut ¼ 0; so that, roughly speaking, infinitesimal pieces of the lattices on the two sides of ˚ gren and Cahn and Larche´ should, the interface fit together.79 The argument of A therefore, apply at the interface, and noting that, by continuity, the substitutional flux constraint (31.10) is satisfied at the interface, it would seem reasonable to require that N X

ha ¼ 0;

ð31:11Þ

a¼1

and this we shall do. We refer to Eq. (31.11) as the substitutional flux constraint for the interface. By the lattice constraint and the substitutional flux constraint in bulk, N (( )) X ra ¼ 0;

N (( X

a¼1

a¼1

)) a ¼ 0;

ð31:12Þ

therefore, Eq. (30.4) when summed over all species is satisfied identically. The interfacial flux constraint allows us to establish a counterpart of the theorem on relative chemical potentials (Section V.B), and the proof is not much different. We first show that the free-energy imbalance (31.4) is invariant under 79

Cf. the discussion of Cermelli and Gurtin (1994a) following their Eq. (3.7).

E. Fried and M.E. Gurtin

148

all transformations of the form

ma ðx; tÞ ! ma ðx; tÞ þ lðx; tÞ

for all species a;

ð31:13Þ

with lðx; tÞ independent of a: Given any such field lðx; tÞ; Eqs. (31.11) and P (31.12) imply that, for pa equal to ha ; vra b; or v a b; Na¼1 ðma þ lÞpa ¼ 0; and hence, by Eq. (31.4), & ' & ' d ð dx b ð du b ð c ds # c· þ ½½Cn·v ds þ s· þ ½½Tn· u ds dt C dt a dt a C C 2

N X

½ðma þ lÞha ba þ

a¼1

N ð X

a¼1

(( )) ðma þ lÞð2 a ·n C

(( )) þ ra VÞds;

ð31:14Þ

thus, since the field l is arbitrary, the free-energy imbalance (31.4) is invariant under all transformations of the form (31.13).80 Choosing a species z arbitrarily and taking l ¼ 2mz in Eq. (31.14), so that ma þ l reduces to the relative chemical potential maz ; we arrive at the free-energy imbalance for substitutional alloys: & ' & ' d ð dx b ð du b ð c ds # c· þ ½½Cn·v ds þ s· þ ½½Tn· u ds dt C dt a dt a C C 2

N X

½ðmaz ha ba þ

a¼1

N ð X

a¼1

(( )) (( )) maz ð2 a ·n þ ra VÞds:

ð31:15Þ

C

Steps identical to those used to establish the interfacial dissipation inequality (31.9) for unconstrained materials, but with the chemical potentials ma replaced by the relative chemical potentials maz ; yield the interfacial dissipation inequality

c 2 s
1 2 t
q þ

N X

a¼1

ha

›maz þ gV # 0: ›s

ð31:16Þ

XXXII. Global Theorems Following Gurtin and Voorhees (1993), we now generalize the global theorems presented in Section VI for a two-phase body that contains a single 80 Conversly, granted the (bulk) substitutional flux constraint, invariance of Eq. (31.4) under all transformations of the form (31.13) yields the substitutional flux constraint for the interface. The proof is identical to that of the remark containing Eq. (5.13).

A Unified Treatment of Evolving Interfaces

149

closed interface S disjoint from ›B and across which the conditions (15.4) and (31.1) of coherency and local chemical equilibrium are met. Consider first an unconstrained material. As in Section VI, the global conservation and decay relations are based upon expressions of the atomic balance and the free-energy imbalance for the body itself. To obtain these statements, we utilize the control-volume equivalency theorem of Appendix B to state the atomic balance and free-energy imbalance for a general migrating control-volume R containing a portion C ¼ S > R of the interface, giving ð d ð a r da ¼ 2 ð a ·n 2 ra V›R Þds 2 ½ha ba ; dt R ›R

ð32:1Þ

and (cf. Eq. (B.1)) 

ð d ð C da þ c ds dt R C & ' ð dx du b # þ s· ðCn·v›R þ Tn·uÞds þ c· dt dt a ›R N ð N X X þ ma ð2 a ·n þ ra V›R Þds 2 ½ma ha ba : a¼1

›R

ð32:2Þ

a¼1

Thus, choosing R ¼ B and bearing in mind that S is closed and disjoint from ›B; we find that Eqs. (32.1) and (32.2) specialize to yield N ð X d ð a r da ¼ 2 a ·n ds dt B › B a¼1

ð32:3Þ

and  ð

ð ð d ð C da þ c ds # Tn·u_ ds 2 ma a ·n ds: dt B S ›B ›B

ð32:4Þ

For a substitutional alloy, Eq. (32.4) remains valid, but with the chemical potentials ma replaced by the relative chemical potentials maz : By Eqs. (32.3) and (32.4) with u_ ¼ 0 and a ·n ¼ 0 on ›B; we have the Global theorem for an isolated two-phase body. Consider a two-phase body B containing a single closed interface S which is disjoint from ›B: If the body is isolated, that is if ›B is fixed and impermeable, then the total number of atoms of

150

E. Fried and M.E. Gurtin

each species remains fixed, while the total free energy is nonincreasing: d ð a r da ¼ 0; a ¼ 1; 2; …; N; dt B 

ð d ð C da þ c ds # 0: dt B S Global theorem for a two-phase body. Assume that a portion of ›B is fixed and the remainder is subject to the dead loads. (a) If ›B is impermeable, then d ð a r da ¼ 0; a ¼ 1; 2; …; N; dt B 

ð d ð ðC 2 Tp ·EÞda þ c ds # 0: dt B S (b) If a portion E of ›B is impermeable and the remainder, ›B w E; in chemical equilibrium, then ! ( ) N ð X d ð a a C 2 Tp ·E 2 mp r da þ c ds # 0 dt B S a¼1 if the material is unconstrained, while ! ( ) N ð X d ð az a C 2 Tp ·E 2 mp r da þ c ds # 0 dt B S a¼1 if the material is a substitutional alloy. Assertion (a) of this theorem follows on using Eq. (6.3) and the boundary condition a ·n ¼ 0 in Eqs. (32.3) and (32.4); similarly, assertion (b) follows on using Eqs. (6.3) – (6.5) in Eq. (32.4).

XXXIII. Normal Configurational Force Balance Revisited Since we account for deformation and atomic transport, the bulk configurational stress C is determined by the full Eshelby relation (12.15), namely ! N X a a C¼ C2 r m 1 2 ð7uÞT T; a¼1

A Unified Treatment of Evolving Interfaces

151

while the interfacial configurational tension s is determined by the interfacial Eshelby relation (23.6). Thus, we may write the normal configurational force balance (29.1) in the form N (( )) X ›t þ g; ra ma ¼ ½½C 2 Tn·ð7uÞn þ ðc 2 s
1ÞK þ ›s a¼1

ð33:1Þ

which, like the statement (24.1) arising in our treatment of strained solid – vapor interfaces, equates terms associated with atomic transport to terms that are purely mechanical. Thus, trivially, for def

G ¼ ½½C 2 Tn·ð7uÞn þ ðc 2 s
1ÞK þ

›t þ g; ›s

ð33:2Þ

we may write the normal configurational force balance succinctly as N (( )) X ra ma ¼ G:

ð33:3Þ

a¼1

When g ¼ 0; G coincides with the variational derivative of the total free energy with respect to variations of the position of the interface. If, in addition to taking g ¼ 0; we neglect interfacial structure, so that c ¼ 0; s ¼ 0; t ¼ 0; and s
¼ 0; then G reduces to the interfacial driving traction vC 2 Tn·ð7uÞnb: The requirement that the driving traction vanish is the classical Maxwell equation for the equilibrium of a coherent interface (Eshelby, 1970; Robin, 1974; Larche´ and Cahn, 1978; Grinfeld, 1981; James, 1981; Gurtin, 1983). For a substitutional alloy, the lattice constraint (5.1) and the identity mb ¼ a m 2 mab reduce Eq. (33.1) to N X hh ii ba rb m ¼ G;

ð33:4Þ

b¼1

which, for each species a; involves the chemical potentials relative to all other species b: Hence, in contrast to the situation at a strained interface separating a substitutional alloy from vapor, it is not possible to determine the individual chemical potentials on a coherent interface in a substitutional alloy. Because of Eq. (33.4), the remaining results are essentially the same for substitutional alloys as they are for unconstrained materials. For that reason, we limit the ensuing discussion to unconstrained materials with the proviso that for substitutional alloys the chemical potentials ma be interpreted as chemical potentials maz relative to a fixed choice of species z:

E. Fried and M.E. Gurtin

152

We next obtain counterparts of relations obtained in Section XXIV.A for a strained solid – vapor interface. The first of these follows on noting that, by the equation t ¼ sg
þ t
used to define t
(cf. Eq. (21.10)) and the tangential component ›s
=›s ¼ 2t·vTbn ¼ 2n·vTbt of the standard interfacial force balance (cf. Eq. (29.5)),

›t ›s
›g ›t
›g ›t
¼ þ ¼ 2g n·½½Tt þ s
þ ; g þ s
›s ›s ›s ›s ›s ›s

ð33:5Þ

which yields the identity (( )) ›g ›t
þ þ g: G ¼ C 2 Tn·ð7uÞn 2 g n·Tt þ ðc 2 s
1ÞK þ s
›s ›s

ð33:6Þ

If surface stress is neglected, so that s
¼ 0; vTbn ¼ 0; and t
¼ t; then, since T ¼ T` ; 1 ¼ n ^ n þ t ^ t; and v7ubt ¼ 0;81 ½½Tn·ð7uÞn ¼ RTSn·½½7un þ ½½Tn·R7uSn ¼ RTS·½½7un ^ n ¼ RTS·½½7uð1 2 t ^ tÞ ¼ RTS·½½7u 2 RTSt·½½7ut ¼ RTS·½½E;

ð33:7Þ

thus, in this case, G ¼ ½½C 2 RTS·½½E 2 cK þ

›t þ g: ›s

ð33:8Þ

XXXIV. Constitutive Equations for the Interface Our approach to the constitutive theory is guided by the interfacial dissipation inequality (31.16), which is almost identical to the inequality (23.9) that arises in the theory for strained solid – vapor interfaces. Thus, reasoning as in Section XXV, we arrive at constitutive equations

s
¼

›c^ð1; qÞ ; ›1

t
¼

›c^ð1; qÞ ; ›q

ð34:1Þ

determining the standard scalar interfacial stress and the reduced shear in conjunction with constitutive equations82 ha ¼ 2Lab 81

›mb ; ›s

g ¼ 2bV;

ð34:2Þ

Equation (4.15) of Gurtin and Voorhees (1993) erroneously asserts that vTn·ð7uÞnb ¼ vT·Eb ð†Þ; which differs from Eq. (33.7). Because of this, results ensuing from (†), such as their Eq. (8.3), are incorrect. 82 For convenience, we neglect coupling from the outset.

A Unified Treatment of Evolving Interfaces

153

for the interfacial atomic fluxes and the normal internal force. Here the matrix with coefficients Lab is positive semi-definite, while b $ 0; these moduli may depend on ð1; qÞ:

XXXV. General Equations for the Interface The basic interfacial balances are the standard force-balance, the atomic balance, and the normal configurational force balance (cf. Sections XXIX.B and XXX) 9 (( a )) ›s
›ha (( a )) > t; þ  ·n; > r V¼ ½½Tn ¼ 2s
Kn 2 > = ›s ›s N (( )) X > (( )) ›g ›t
> þ þ g: > ra ma ¼ C 2 Tn·ð7uÞn 2 g t·Tn þ ðc 2 s
1ÞK þ s
; › s › s a¼1

ð35:1Þ These interfacial balances are coupled to the interfacial constitutive relations (34.1) and (34.2). Thus, combining the interfacial balances and the constitutive relations and using the identity (26.6), which applies here also, we arrive at the standard force and atomic balances ! 9 > ›c ›2 c ›1 ›2 c > Kn 2 þ K t; > ½½Tn ¼ 2 > 2 > ›1 ›1 ›q ›1 ›s = 0 1 ð35:2Þ N b > (( a )) (( a )) > › @X > ab ›m A L  ·n ¼ þ r V; > > ; ›s ›s b¼1

and the normal configurational force balance. ! N (( )) X (( )) ›2 c ›c a a K r m ¼ C 2 Tn·ð7uÞn 2 g n·Tt þ c þ 21 ›1 ›q2 a¼1 þ

›2 c ›1 › c ›g þ 2 bV: ›1 ›q ›s ›1 ›s

ð35:3Þ

If we neglect surface stress, then c ¼ cðqÞ and the standard force-balance (35.2)1 has the simple form ½½Tn ¼ 0

ð35:4Þ

asserting that the (standard) traction be continuous across the interface, the

E. Fried and M.E. Gurtin

154

atomic balance (35.2)2 reads ((

0 1 N b X ) ) (( )) › › m @ A þ ra V; Lab a ·n ¼ ›s b¼1 ›s

and the normal configurational force balance (35.3) becomes ! N (( )) X ›2 c a a K 2 bV: r m ¼ ½½C 2 RTS·½½E þ c þ ›q2 a¼1

ð35:5Þ

ð35:6Þ

The internal force g ¼ 2bV; a dissipative force associated with the rearrangement of atoms at the interface, is of a physical nature akin to that encountered in our discussion of grain boundaries. Such a force is generally not included in discussions of migrating coherent interfaces, typically because the possibility of having such a force is precluded from the outset by an appeal to local equilibrium. The other sources of kinetics for such problems are bulk and interfacial diffusion; because the time-scales associative with such processes are typically very long, the force g ¼ 2bV could possibly be relevant. We know of no investigations as to the relative importance of these disparate measures of kinetics.

Acknowledgments We greatly acknowledge instrumental discussions with John Cahn, Don Carlson, Ben Freund, Vladimir Korchagin, Brian Spencer, Bob Svendsen, Erik Van der Giessen, and Peter Voorhees. This work was supported by the Department of Energy and by the National Science Foundation. Appendix A: Justification of the Free-Energy Conditions (9.40) at Zero Stress. Gibbs Relation Consider a substitutional alloy without vacancies and, as before, let C0 ðr~Þ denote the free energy at zero stress. In addition, consider a second material identical to the substitutional alloy in question except that vacancies with density rv are present, and let C 0 ðr~; rv Þ denote the corresponding free energy at zero stress. We now show that † if C 0 ðr~; rv Þ ! C0 ðr~Þ as rv ! 0; and † if, as rv ! 0; the corresponding chemical potentials ma0 v ðr~; rv Þ relative to vacancies have limiting values,

A Unified Treatment of Evolving Interfaces

155

then, writing ma0 ðr~Þ for the limiting values of ma0 v ðr~; rv Þ as rv approaches zero, the free-energy conditions at zero stress are satisfied. We consider ma0 ðr~Þ to be the species-a chemical-potential for C0 ðr~Þ; ma0 ðr~Þ should be interpreted as a chemical potential, relative to vacancies, in the limit of vanishing vacancies. Note that ma0 is defined solely as a limiting value. This limiting value cannot be equal to ›C0 =›ra ; which, in light of the lattice constraint, is not meaningful. On the other hand, by Eq. (9.40), the difference ma0 ðr~Þ 2 mb0 ðr~Þ represents, for C0 ðr~Þ; the chemical potential of a relative to b: To establish the foregoing assertions, we assume that (i)

C 0 ðr~; rv Þ is continuous with continuous Larche´ –Cahn derivatives on the set ( ) N X def v a sites v b v Dvac ¼ ðr~; r Þ : r ¼ r þ r ; r . 0 ðb ¼ 1; 2; …; NÞ; r $ 0 a¼1

ðA:1Þ (so that the smoothness described above is up to rv ¼ 0Þ; (ii) C0 ðr~Þ is the zero-vacancy limit of C 0 ðr~; rv Þ;

C0 ðr~Þ ¼ C 0 ðr~; 0Þ:

ðA:2Þ

By (i) and (ii), C0 ðr~Þ is continuous with continuous Larche´ – Cahn derivatives on the constraint set ( ) N X def b sites a Dcon ¼ r~ : r ¼ r ; r . 0 ðb ¼ 1; 2; …; NÞ : ðA:3Þ a¼1

Further, by Eq. (9.39) applied to C 0 ;

ma0 v ðr~; rv Þ ¼

›ðvÞ C 0 ðr~; rv Þ ; ›ra

v mab 0 ðr~; r Þ ¼

›ðbÞ C 0 ðr~; rv Þ ›ra

ðA:4Þ

Moreover, Eq. (5.6)3 yields bv v av v v mab 0 ðr~; r Þ ¼ m0 ðr~; r Þ 2 m0 ðr~; r Þ:

ðA:5Þ

By (i), the Larche´ – Cahn derivatives of C 0 exist and are continuous up to rv ¼ 0: The limit rv ! 0; therefore, yields

›ðbÞ C 0 ðr~; rv Þ ›ðbÞ C 0 ðr~; 0Þ ›ðbÞ C0 ðr~Þ ! ¼ a a ›r ›r ›ra

ðA:6Þ

E. Fried and M.E. Gurtin

156

and

›ðvÞ C 0 ðr~; rv Þ ›ðvÞ C 0 ðr~; 0Þ def a ! ¼ m0 ðr~Þ: ›ra ›ra

ðA:7Þ

Thus, by Eqs. (A.4)–(A.7), b v a mab 0 ðr~; r Þ ! m0 ðr~Þ 2 m0 ðr~Þ;

ðA:8Þ

and by Eqs. (A.4)2 and (A.6), v mab 0 ðr~; r Þ ¼

›ðbÞ C 0 ðr~Þ ›ðbÞ C0 ðr~Þ ! ¼ mab ðr~Þ; a ›r ›ra

therefore, appealing to Eq. (9.39),

mab 0 ðr~Þ ¼

›ðbÞ C0 ðr~Þ ¼ ma0 ðr~Þ 2 mb0 ðr~Þ; ›ra

ðA:9Þ

which is Eq. (9.40)2. We are now in a position to establish the Gibbs relation (9.40)1. Let C0ðvÞ ðr~Þ denote C 0 ðr~; rv Þ with rv eliminated via the lattice constraint: % % C0ðvÞ ðr~Þ ¼ C 0 ðr~; rv Þ% N X rv ¼rsites 2 ra a¼1

The domain of

C0ðvÞ ðr~Þ

ðvÞ def

D

(

¼ r~ :

is the set

N X

) a

r #r

sites

b

; r . 0 ðb ¼ 1; 2; …; NÞ :

ðA:10Þ

a¼1

and the boundary of DðvÞ contains the constraint set Dcon : By Eq. (9.8), % ›C0ðvÞ ðr~Þ ›ðvÞ C 0 ðr~; rv Þ %% ¼ % N X % ›ra ›ra rv ¼rsites 2 ra a¼1

and, in view of (i), Define

C0ðvÞ ðr~Þ

is continuously differentiable on DðvÞ ; up to Dcon :

mðvÞa ðr~Þ ¼

›C0ðvÞ ðr~Þ : ›ra

ðA:11Þ

Then, by Eqs. (A.2) and (A.7), since Dcon corresponds to rv ¼ 0;

C0 ðr~Þ ¼ C0ðvÞ ðr~Þ;

ma0 ðr~Þ ¼ mðvÞa ðr~Þ

for all r~ [ Dcon :

ðA:12Þ

A Unified Treatment of Evolving Interfaces

157

Finally, since C0ðvÞ ðr~Þ and mðvÞa ðr~Þ are consistent with Eq. (A.11) and are unencumbered by the lattice constraint in the interior of Dcon ; we may argue as in Section VIII to show that C0ðvÞ ðr~Þ ¼ ra mðvÞa ðr~Þ: Thus, by Eq. (A.12), passing to the limit rv ¼ 0; we have the Gibbs relation (9.40)1.

Appendix B: Equivalent Formulations of the Basic Laws. Control-Volume Equivalency Theorem In our development of the equations governing the bulk material, we formulated the basic laws—the configurational force, standard force, and atomic balances, and the imbalance for free energy—first for fixed parts P of the body and then, to account for the role of configurational forces, for migrating control volumes RðtÞ: On the other hand, our discussion of interfaces is based almost exclusively on the use of interfacial pillboxes. Alternatively we could base this discussion on the use of migrating control volumes that contain a portion of the interface. The purpose of this section is to show that the two methods of formulating basic laws are equivalent. With this in mind, let RðtÞ denote a migrating control volume (in the sense of Section XI.B). We then refer to RðtÞ: † as a migrating bulk control-volume if RðtÞ lies solely in the bulk material; † as a migrating interfacial control-volume if RðtÞ contains a portion of the interface in its interior; † as a general migrating control-volume if RðtÞ is either a migrating bulk control-volume or a migrating interfacial control-volume. Control-volume equivalency theorem. For each of the basic laws L under consideration, L is satisfied for all general migrating control-volumes if and only if: (i) L is satisfied for all migrating bulk control-volumes; (ii) L is satisfied for all interfacial pillboxes. Since the family of all general migrating control-volumes includes the family of migrating bulk control-volumes, to establish the theorem it suffices to show that, granted L is satisfied for all migrating bulk control-volumes, L is satisfied for all migrating interfacial control-volumes if and only if L is satisfied for all interfacial pillboxes. The basic laws under consideration are the balance laws for configurational forces, standard forces, and atoms, and the imbalance for free

E. Fried and M.E. Gurtin

158

energy. Here we shall establish equivalence for L ¼ {free-energy imbalance}; but not for the three balance laws, as their proof is similar and not as difficult. Thus consider a migrating interfacial control-volume RðtÞ; and let CðtÞ; with xa ðtÞ and xb ðtÞ the endpoints of CðtÞ; denote the portion of the interface in RðtÞ: For such a control volume the appropriate generalization of the free-energy imbalance (12.17) has the form  ð

& ' ð d ð dx du b þ s· C da þ c ds # ðCn·v›R þ Tn·uÞds þ c· dt RðtÞ dt dt a CðtÞ ›RðtÞ N ð N X X þ ma ð2 a ·n þ rV›R Þds þ ½ma ð2ha þ da WÞba : ðB:1Þ a¼1

›RðtÞ

a¼1

On the other hand, the free-energy imbalance for a interfacial pillbox CðtÞ has the form & ' ð d ð dx du b þ s· c ds # ð½½Cn·v þ ½½Tn· u Þds þ c· dt CðtÞ dt dt a CðtÞ N ð N X X ( ( ) ) ( ( ) ) þ ma ð2 a ·n þ ra VÞds þ ½ma ð2ha þ da WÞba : a¼1

CðtÞ

a¼1

ðB:2Þ The interface is assumed to be coherent and the (bulk) chemical potential m; assumed continuous across the interface, represents the chemical potential of S: The formulation of Eq. (B.2) is discussed at length in Part E, and one may use that discussion and the bulk imbalance (12.17) to infer the imbalance (B.1). Specifically, we will show that, granted (cf. Eq. (12.17)) N ð ð X d ð C da # ðCn·v›R þ Tn·uÞds 2 ma ð a ·n 2 ra V›R Þds dt RðtÞ ›RðtÞ › RðtÞ a¼1

ðB:3Þ for all migrating bulk control-volumes RðtÞ; the imbalance (B.1) is satisfied for all migrating interfacial control-volumes if and only if the imbalance (B.2) is satisfied for all interfacial pillboxes. To establish this assertion, let RðtÞ be a migrating interfacial control volume and let Rþ ðtÞ and R2 ðtÞ denote the bulk control volumes represented by the portions of RðtÞ that lie in the (þ) and (2) phases, so that the portion of ›Rþ ðtÞ that coincides with CðtÞ is viewed as lying in the (þ )-phase at the interface, and similarly for ›R2 ðtÞ: Dropping the argument t when convenient, let n›R^ denote the outward unit normal on ›R^ : Then, by

A Unified Treatment of Evolving Interfaces

159

Eq. (B.3) with R ¼ R^ ; N ð ð X d ð C da # ðCn·v›R^ þ Tn·uÞds 2 ma ð a ·n 2 ra V›R^ Þds; ^ dt R^ › R^ › P a¼1

ðB:4Þ Moreover, on C;

n›Rþ ¼ 2n; V›Rþ ¼ 2V; n›R2 ¼ n;

v›Rþ ¼ v ¼ Vn;

)

V›R2 ¼ V; v›R2 ¼ v ¼ Vn;

ðB:5Þ

and on both ›Rþ > C and ›R2 > C:

u ¼ u

ðB:6Þ

Therefore, adding the two equations represented by Eq. (B.4), ð ð d ð C da # ðCn·v›R þ Tn·uÞds þ ma ð2 a ·n þ ra V›R Þds dt R ›R ›R |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} I1 ðRÞ

2

I2 ðRÞ

ð

I3 ðRÞ

ð½½Cn·v þ ½½Tn· u Þds CðtÞ

2

N ð X

a¼1

(( )) (( )) ma ð2 a ·n þ ra VÞds:

ðB:7Þ

CðtÞ

Assume that the free-energy imbalance (B.2) for a interfacial pillbox is satisfied. Then adding Eqs. (B.2) and (B.7) we arrive at the free-energy imbalance (B.1) for a migrating interfacial control-volume. To prove the converse assertion, assume that Eq. (B.1) holds for all migrating interfacial control-volumes. Choose an arbitrary evolving subcurve CðtÞ and let Rz ðtÞ denote the migrating interfacial control-volume (of thickness 2z ) defined by Rz ðtÞ ¼ {x : x ¼ z ^ zmðz; tÞ; z [ CðtÞ; 0 # z p 1}; so that Rz ! C as z ! 0: In this limit, using limiting relations for the upper and lower faces of the approximate pillbox Rz analogous to the identities (B.5) and (B.6), we find that ð I1 ðRz Þ ! 0; I2 ðRz Þ ! ð½½Cn·v þ ½½Tn· u Þds; C

I3 ðRz Þ !

N ð X

a¼1

(( )) (( )) ma ð2 a ·n þ ra VÞds: C

160

E. Fried and M.E. Gurtin

Thus passing to the limit z ! 0 in Eq. (B.1) (with R ¼ Rz Þ; we are led to the freeenergy imbalance (B.2) for the interfacial pillbox C: This completes the proof of the portion of the Control-Volume Equivalency Theorem relevant to free-energy imbalances.

Appendix C: Status of the Theory as an Approximation of the Finite-Deformation Theory The theory developed in the body of this study is restricted to small deformations. Here, we give an abbreviated account of the finite-deformation theory and of the formal analysis involved in the approximation of small deformations within that theory.

C.1. Theory for Finite Deformations C.1.a. Kinematics We now use the symbol x to denote an arbitrary material point as labeled in a fixed reference configuration. The interface S ¼ SðtÞ and all pillboxes C ¼ CðtÞ are assumed to lie in the reference configuration. In a theory of finite deformations the deformation y is related to the displacement u through yðx; tÞ ¼ x þ uðx; tÞ; yðx; tÞ represents the point of space occupied by the material point x at time t: By Eq. (C.4), the interfacial deformation-derivative def

f¼

›y ›s

ðC:1Þ

is related to the interfacial surface-strain e ¼ ›u=›s through f ¼ t þ e:

ðC:2Þ

Consistent with the small-deformation theory, we refer to 1 ¼ t·e;

g ¼ n·e ð¼ n·fÞ

as the interfacial tensile and shear strains. The field

l ¼ lfl

ðC:3Þ

A Unified Treatment of Evolving Interfaces

161

represents the interfacial stretch; clearly,

l2 ¼ 1 þ 21 þ lel2 :

ðC:4Þ


t ¼ l21 f

ðC:5Þ

The vector field

represents a (unit) tangent to the deformed interface. Trivially, f ¼ l
t; hence, Eq. (C.2) and the identity t
·
t ¼ 0 imply that l ¼
t· f : Thus, since t ¼ q n Eq. (14.15), the stretch rate satisfies

l ¼
t· e · q
t·n:

ðC:6Þ

C.1.b. Standard and Configurational Forces. Power The treatment of configurational forces follows Section XIX.A and, as before, leads to the normal configurational force balance Eq. (20.1), viz.

sK þ

›t þ g 2 n·Cn ¼ 0: ›s

ðC:7Þ

The standard force and torque balances for a pillbox C take the form slba 2

ð

Tn ds ¼ 0; C

½ðy 2 0Þ £ sba 2

ð

ðy 2 0Þ £ Tn ds ¼ 0;

ðC:8Þ

C

with T the Piola stress, and comparing these balances to their small-deformation counterparts (20.2), we see that the force balance is unchanged, but the torque balance reflects the fact that we are working in a theory of finite deformations. The integral balances (C.8) are equivalent to the local balances83

›s ¼ Tn; ›s


t £ s ¼ 0:

ðC:9Þ

The second of Eq. (C.9) renders s tangent to the deformed interface; hence, there is a scalar field s
; the (scalar) standard surface stress, such that s ¼ s

t:

83

ðC:10Þ

Cf. Gurtin and Murdoch (1975) who derive three-dimensional force and moment balances in a finite-strain setting. See also Fried and Gurtin (2003).

E. Fried and M.E. Gurtin

162

Our discussion of power follows Section XXI.A. The net external power expended on a pillbox CðtÞ has the form &

dx du c· þ s· dt dt

'b 2 a

ð

ðCn·v þ Tn· u Þds: C

The configurational portion of this expenditure is given by Eq. (29.8). To determine the contribution of the standard forces, we use Eq. (15.16) and the identity › u =›s ¼ e 2 KVe (Eq. (15.12)) to obtain  & ' ð  ›s du b · u þ s· e 2 ðs·eÞKV ds s· ¼ ½s· u þ ðs·eÞWba ¼ ½ðs·eÞWba þ dt a C ›s (cf. Eq. (21.1)), and, using the standard force balance (C.9)1, & s·

du dt

'b a

¼

ð C

Tn· u ds ¼ ½ðs·eÞWba þ

ð

ðs· e 2 ðs·eÞKVÞds:

ðC:11Þ

C

Combining Eqs. (29.8) and (C.11) yields the power balance & c·

dx du þ s· dt dt

'b 2 a

ð

ðCn·v þ Tn· u Þds C

¼ ½ðs þ s·eÞWba þ

ð (

) s· e þ t q 2 ððs þ s·eÞK þ gÞV ds:

ðC:12Þ

C

C.1.c. Free-Energy Imbalance Our discussion of atomic transport is as in Section XXII and leads to the atomic balance (22.4). The free-energy imbalance for an arbitrary interfacial pillbox C ¼ CðtÞ takes the form & ' & ' d ð dx b ð du b ð c ds # c· 2 Cn·v ds þ s· 2 Tn· u ds dt C dt a dt a C C N N ð X X ½ma ð2ha þ da WÞba þ ma ð a ·n 2 ra V þ ra Þds þ a¼1

a¼1

C

(cf. Eq. (23.4)), and, in view of Eqs. (C.12) and (22.4), and the integral transport

A Unified Treatment of Evolving Interfaces

163

theorem (14.28), yields the inequality ð#Þ

zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl ffl{ " ! # b N ð X ðc 2 cKVÞds þ c 2 s·e 2 ma da 2 s W C

a¼1

#

ð

s· e þ t q 2 s þ s·e þ C

2

N X

a¼1

N X

a

! a a

m d

!

KV 2 gV ds

a¼1

½ma ha ba þ

N ð X

a¼1

  ›ha ma d a þ ds: ›s C

ðC:13Þ

Since C is arbitrary, so also are the tangential velocities Wa and Wb of the endpoints of C; thus, since the only term in Eq. (C.13) dependent on these velocities is the term (#), we have the interfacial Eshelby relation

s ¼ c 2 s·e 2

N X

da ma ;

ðC:14Þ

a¼1

which should be compared with its counterpart (23.6) of the small-deformation theory. Since  N N ð  X X ›ha ›ma þ ha ½ma ha ba ¼ ma ds; ›s ›s a¼1 a¼1 C we may use Eq. (C.14) to rewrite Eq. (C.13) as ð C

! N  a X a a a ›m c 2 s· e 2 t q 2 m d 2h þ gV ds # 0; ›s a¼1

since C is arbitrary, this yields the interfacial dissipation inequality

c 2 s· e 2 t q 2

N  X

a¼1

ma d a 2 ha

›m a ›s



þ gV # 0:

ðC:15Þ

C.1.d. Normal Configurational Force Balance Revisited As before, we have the bulk Eshelby relation (12.15), but now the interfacial Eshelby relation has the form (C.14); we may, therefore, write the normal configurational force balance and (for a substitutional alloy) the

E. Fried and M.E. Gurtin

164

chemistry-composition relations in the respective forms (24.3) and (24.11), viz. N X

ðra 2 da KÞma ¼ F ;

ma ¼

a¼1

N X

cbef mab þ Vef F ;

ðC:16Þ

b¼1

but with the mechanical force now given by (cf. Eq. (24.2)) F ¼ C 2 Tn·ð7uÞn 2 ðc 2 s·eÞK 2

›t 2 g: ›s

ðC:17Þ

C.1.e. Constitutive Equations Without Regard to Torque Balance Our discussion of constitutive equations closely follows Section XXV.A. Guided by the interfacial dissipation inequality (C.20), we consider constitutive equations giving

c; s; t; m~ as functions of ðe; q; d~Þ

ðC:18Þ

in conjunction with constitutive equations for ha and g of the form (25.1), with coefficients possibly dependent on ðe; q; d~Þ; rather than ð1; q; d~Þ: (The discussion of constitutive equations for ha and g, follows Section XXV.) Appealing to the discussion of Section XVII.H, we find that s¼

›c~ðe; q; d~Þ ; ›e

t¼

›c~ðe; q; d~Þ ; ›q

ma ¼

›c~ðe; q; d~Þ : ›da

ðC:19Þ

The difference between Eq. (C.19) and the constitutive relations (25.2) of the small-deformation theory is reflected by the use of s, t; and e as variables instead of s
; t
; and 1: Remark The theory as developed thus far is valid whether or not the local torque balance
t £ s ¼ 0 is satisfied. This observation is central to what follows. As we shall see, the theory we develop as a small-deformation approximation of the finite-deformation theory satisfies the torque balance only approximately, but otherwise falls within the thermomechanical structure of the finite-deformation theory. We refer to the finite-deformation theory based on the constitutive equations discussed above, but with the torque balance omitted, as the finite-deformation theory without torque balance.

A Unified Treatment of Evolving Interfaces

165

C.1.f. Constitutive Equations Consistent with Torque Balance To establish consequences of the torque balance, we return to the dissipation inequality (C.20), which, by virtue of Eqs. (C.6) and (C.10), we may rewrite as N  a X a a a ›m c 2 s
l 2 t
q 2 m d 2h þ gV # 0; ›s a¼1

ðC:20Þ

where

t
¼ t 2 ð
t·nÞs
is the reduced shear. Consider once again constitutive relations in the form (C.18). By Eqs. (C.2) and (C.5), a dependence on ðe; q; d~Þ is equivalent to a dependence on ðl;
t; q; d~Þ; thus, since s ¼ s

t is a consequence of the torque balance (cf. Eq. (C.10)), Eq. (C.18) is equivalent to relations giving

c; s
; t
; m~ as functions of ðl;
t; q; d~Þ:

ðC:21Þ

Therefore, arguing as before, we conclude, as a consequence of the dissipation inequality, that the free energy must be independent of the deformed tangent
t;

c ¼ c^ðl; q; d~Þ;

ðC:22Þ

and must generate s
; t
; and m~ through the relations (Fried and Gurtin, 2003)

s
¼

›c^ðl; q; d~Þ ; ›l

t
¼

›c^ðl; q; d~Þ ; ›q

ma ¼

›c^ðl; q; d~Þ : ›d a

ðC:23Þ

The finite-deformation theory is discussed at great length by Fried and Gurtin (2003) and the reader is referred there for a thorough discussion of the basic equations as well as of the partial differential equations resulting from various simplifying assumptions.84 84 In the finite-deformation theory of Fried and Gurtin (2003), the standard stresses T and s enter the bulk and interfacial Eshelby relations through the terms of the form FT T and s·f; with F ¼ 1 þ 7u the deformation gradient and f ¼ ›y=›s: Because F and f are not suitable for a discussion of small deformations, we here work with Eshelby relations where the stresses enter through terms of the form ð7uÞT T and s·e ¼ ›u=›s: Gurtin (2000, Section XIII) shows that the bulk theory is independent of which of the two configurational stress tensors one uses and a straightforward modification of his argument shows that the this is also true of the interfacial theory.

166

E. Fried and M.E. Gurtin C.2. Theory for Small Deformations as an Approximation of the Finite-Deformation Theory

The chief differences between the small-deformation theory developed in the body of this study and the theory discussed in this section involves the constitutive interactions between stress, strain, and orientation. So as to not obscure these differences we simplify the theory by neglecting adatoms. The inclusion of adatoms involves only minor modifications.

C.2.a. Small-Strain Estimates We are interested in a theory appropriate to situations in which e is small; thus, using the symbol oðeÞ to denote terms that, in a precise sense, are smaller than e in the limit e ! 0;85 we have the estimates

l ¼ 1 þ 1 þ oðeÞ;


t ¼ t þ g n þ oðeÞ;

ðC:24Þ

where Eq. (C.24)2 follows from Eqs. (C.2) and (C.24)1. C.2.b. Constitutive Relations Appropriate to Small Deformations In developing an approximate theory appropriate to small deformations, the relevant constitutive relation is that for the stress; since s ¼ s

t; the first of Eq. (C.23) yields the relation s ¼ s
ðlÞ
t;

ðC:25Þ

where, for convenience, we have suppressed the argument q and have written s
¼ ›c^=›l: Granted smoothness, we may use Eq. (C.24)1 to conclude that % ›s
%% s
ðlÞ ¼ s
0 þ 1 þ oðeÞ; ›l % 0 where the subscript 0 denotes evaluation at l ¼ 1: Thus, by Eqs. (C.24)2 and (C.25), we have the estimate   % ›s
%% s ¼ s
0 þ 1 þ oðe ðt þ e 2 1t þ oðeÞÞ; ðC:26Þ ›l %0 and, introducing the surface elasticity k¼

85

% ›s
%% ›l % 0

A function f ðzÞ is oðzn Þ if lzl2n f ðzÞ ! 0 as z ! 0; f ðzÞ is Oðzn Þ if lzl2n f ðzÞ is bounded as z ! 0:

A Unified Treatment of Evolving Interfaces

167

and appealing to Eq. (C.3), we have the equivalent estimates86 s ¼ s
0 t þ ðk 2 s
0 Þ1t þ s
0 e þ oðeÞ ¼ ðs
0 þ k1Þt þ s
0 g n þ oðeÞ:

ðC:27Þ

Consider now the second of these estimates with terms of oðeÞ neglected: s ¼ ðs
0 þ k1Þt þ s
0 g n:

ðC:28Þ

(Without the underlined term the stress s is consistent with the torque balance t £ s ¼ 0 (Eq. (29.3)2) of the theory as presented in the main body of the chapter.) Writing 9 w ¼ s
0 1 þ 12 k12 þ 12 s
0 g2 ; = ðC:29Þ w ¼ s
0 ðe·tÞ þ 12 kðe·tÞ2 þ 12 s
0 ðe·nÞ2 ; ; we see that s ¼ ›w=›e; and hence, that w ¼ wðeÞ represents an interfacial strain energy. Thus far the orientation q has been irrelevant to our discussion. We now include orientational dependences and, therefore, write s
0 ¼ s
0 ðqÞ and k ¼ kðqÞ: Then, if c~ðe; qÞ represents the resulting free energy, we must have

c~ðe; qÞ ¼ c0 ðqÞ þ wðe; qÞ;

s¼

›c~ðe; qÞ ; ›e

ðC:30Þ

s
0 ðqÞðe·nÞ2 ;

ðC:31Þ

with wðe; qÞ ¼ s
0 ðqÞðe·tÞ þ

1 2

kðqÞðe·tÞ2 þ

1 2

bearing in mind that t ¼ tðqÞ and n ¼ nðqÞ: For consistency with both Eqs. (25.4) and (C.19)2, we define the configurational shear through

t¼

›c~ðe; qÞ : ›q

ðC:32Þ

We supplement these constitutive relations by Fick’s law (25.7) and the kinetic relation (25.8), allowing for anisotropy, viz. ha ¼ 2

N X

b¼1

Lab ðqÞ

›mb ; ›s

g ¼ 2bðqÞV:

ðC:33Þ

To ensure satisfaction of the dissipation inequality (C.20), we assume that bðqÞ $ 0 and that the matrix with entries Lab ðqÞ is positive semi-definite. The constitutive theory as defined by Eqs. (C.30) – (C.33) is then a special case of 86

These represent a two-dimensional version of (L) of the Addenda to Gurtin and Murdoch (1975).

E. Fried and M.E. Gurtin

168

the constitutive relations (C.19) of the finite-deformation theory without torque balance. C.2.c. Basic Equations of the Theory Bearing in mind that we are neglecting adatom densities, the balances for standard forces and atoms, namely Eqs. (C.9) and (22.4), take the form Tn ¼

›s ; ›s

ra V ¼ 2

›h a þ a ·n þ r a ; ›s

ðC:34Þ

while the normal configurational force balance and the configurational-chemistry relations (for substitutional alloys) take the respective forms N X

a¼1

ra m a ¼ F ;

ma ¼

N X

cb mab þ VF

ðC:35Þ

b¼1

(cf. Eq. (26.26)), with mechanical potential F given by Eq. (C.17). Since the constitutive relations (C.30) – (C.33) render the theory consistent with the dissipation inequality (C.20), the balances (C.34) and (C.35) supplemented by Eqs. (C.30) – (C.33) represents an exact system of equations within the framework of the finite-deformation theory without torque balance. The single basic relation not satisfied is the torque balance; as we shall see, this balance is satisfied to within an error of oðeÞ: The constitutive relations (C.30)–(C.32) have the specific form 9 c ¼ c0 ðqÞ þ s
0 ðqÞ1 þ 12 kðqÞ12 þ 12 s
0 ðqÞg2 ; > > > = s ¼ ðs
0 ðqÞ þ kðqÞ1Þt þ s
0 ðqÞg n; > > > t ¼ c 00 ðqÞ þ s
00 ðqÞ1 þ 12 k0 ðqÞ12 þ s
0 ðqÞg þ kðqÞ1g þ 12 s
00 ðqÞg2 2 s
0 ðqÞ1g; ; ðC:36Þ where the underlined terms in Eq. (C.36) represent those terms emanating from the underlined term in Eq. (C.28). With the exception of the atomic balance, which with Eq. (C.33)2 has the form 0 1 N › @X ›mb A a ab r V¼ L ðqÞ þ a ·n þ r a ›s b¼1 ›s (cf. Eq. (26.32)2), the partial differential equations resulting from this general theory are excessively complicated and nothing is to be gained by writing them explicitly.

A Unified Treatment of Evolving Interfaces

169

C.2.d. Basic Equations with Isotropic Strain Energy The ensuing calculations make repeated use of the kinematical relations Eqs. (15.10) and (15.11). Isotropy of the strain energy renders both s
and k constant; consequently, the normal and tangential components of the standard force balance have the respective forms 9   ›u ›2 u > n·Tn ¼ s
0 þ ðk 2 s
0 Þt· K þ s
0 n· 2 ; > > ›s ›s = ðC:37Þ   > ›u ›2 u > > t·Tn ¼ ðk 2 s
0 Þn· K þ kt· 2 ; ; ›s ›s and the mechanical potential becomes F ¼ C 2 Tn·ð7uÞn 2 AK 2 a·

›2 u þ bðqÞV: ›s 2

ðC:38Þ

where, in contrast to the theory developed in the main body of this chapter, A and a now have the forms     3 ›u 2 ›u 2 ›u A ¼ c0 ðqÞ þ c 000 ðqÞ 2 k t· þk n· 2s
0 t· 2 ›s ›s ›s " 2  2 # ›u 3 ›u ðC:39Þ þ s
0 t· 2 n· 2 ›s ›s and a ¼ s
0 n þ ðk 2 s
0 Þ

&   ' ›u ›u t· n þ n· t : ›s ›s

ðC:40Þ

In the theory developed in the main body of this chapter, the underlined terms in Eq. (C.36) are absent. These terms lead to the presence of additional terms     ›u ›2 u ›u 2s
0 t· K þ s
0 n· 2 ; and 2 s
0 n· K ›s ›s ›s in Eq. (C.37)1 and (C.37)2, respectively, and to the additional terms "    # &   ' 2 ›u 2 3 ›u 2 ›u ›u › u K; and s
0 t· s
0 t· 2 t· n þ n· t · 2 2 ›s ›n ›s ›s ›s in Eq. (C.38).

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170

These equations simplify further when the interface has negligible elasticity in the sense that k ¼ 0; for then the standard force balance has the form ( )  ›u ›2 u ›u n·Tn ¼ s
0 1 2 t· ðC:41Þ t·Tn ¼ 2s
0 n· K; K þ n· 2 ; ›s ›s ›s and Eqs. (C.39) and (C.40) reduce to      ›u ›u ›u ›u 00 2 t· A ¼ c0 ðqÞ þ c 0 ðqÞ þ s
0 n· 2 s
0 n· t· ›n ›s ›n ›s " #        ›u ›u ›u 2 3 ›u 2 2 n· 2 s
0 n· t· þ s
0 t· 2 ›s ›n ›s ›s

ðC:42Þ

and a ¼ s
0

&   ' ›u ›u ›u 2 t· 1 þ n· n 2 n· t : ›n ›s ›s

ðC:43Þ

C.2.e. Comparison of Small-Deformation Theories The general finite-deformation theory discussed in Section C.1 satisfies the torque balance ð ðC:44Þ ½ðy 2 0Þ £ sba 2 ðy 2 0Þ £ Tn ds ¼ 0 C

(cf. Eq. (C.8)2), which has the local form
t £ s ¼ 0;

ðC:45Þ

but the theory developed in this section (Section C.2)—which we refer to as the first-order strain theory—satisfies this torque balance only approximately. Specifically, the first-order strain theory is based on the stress – strain relation s ¼ ðs
0 þ k1Þt þ s
0 g n;

ðC:46Þ

so that, by Eqs. (C.2), (C.3), and (C.5), l
t £ sl ¼ l21 lðt þ 1t þ g nÞ £ ððs
0 þ k1Þt þ s0 g nÞÞl ¼ l21 l1gðs
0 2 kÞl; giving the estimate l
t £ sl ¼ Oðe2 Þ:

ðC:47Þ

Thus, while the local torque balance (C.45) is not satisfied exactly, it is satisfied to within an error of Oðe2 Þ:

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171

Consider, next, the theory developed in the main body of the chapter, but with the specific free energy defined by Eqs. (26.34) and (26.35), a theory that we refer to as the exact infinitesimal strain theory. This theory is based on the torque balance ð ½ðx 2 0Þ £ sba 2 ðx 2 0Þ £ Tn ds ¼ 0 ðC:48Þ C

(cf. Eq. (20.2)), whose localization, t £ s ¼ 0; yields s ¼ s
t (Eq. (20.4)). Thus, in view of Eq. (20.4) and the consecutive relations Eqs. (25.2), (26.34), and (26.35), s ¼ ðs
0 þ k1Þt; hence, the argument leading to Eq. (C.47) yields ½
t £ s ¼ l21 lgðs
0 þ k1Þl and results in the estimate l
t £ sl ¼ OðeÞ

ðC:49Þ

The first-order strain theory, therefore, provide a better approximation to the torque balance (C.45) of the finite-deformation theory than does the exact infinitesimal strain theory. On the other hand, the exact infinitesimal strain theory is compatible with the theory in bulk as developed in Part A, but the first-order strain theory is not. Indeed, the torque balance (C.48) of the exact infinitesimal strain theory is compatible with the bulk torque balance ð ðx 2 0Þ £ Tn da ¼ 0 ðC:50Þ ›P

(cf. Eq. (2.2)2), and hence, the two balances may be combined to form a torque balance for a control volume that contains both bulk and interfacial material. But the stress s of the first-order strain theory does not satisfy the torque balance (C.48), since by Eq. (C.46), lt £ sl ¼ ls
0 gl: Further, the exact infinitesimal strain theory and the theory in bulk are consistent with the form of material frameindifference appropriate to small deformations; namely invariance under transformations of the form 7u ! 7u þ W; with W an arbitrary skew tensor. But the first-order strain theory does not have this invariance; in fact, in that theory, an infinitesimal rigid rotation induces stress. Pragmatically, one might except that the exact infinitesimal strain theory might be suitable to situations in which s
0 is of the same order as k1 (and 1 is small), for then the term s
0 g in the first-order strain theory would be negligible. On the other hand, for s
0 large we would expect the first-order strain theory to be more appropriate. In considering this issue one should bear in mind that, for s
0 large and k negligibly small, the tangential part of the

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standard force balance has the form

  ›u t·Tn ¼ 2s
0 n· K ›s

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Instability of Multi-Layer Channel and Film Flows C. POZRIKIDIS Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Unidirectional Two-Layer Flow . . . . . . . . . . . . . . . . . . B. Unidirectional Multi-Layer Flow . . . . . . . . . . . . . . . . . C. Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Lubrication-Flow Model for Long Waves . . . . . . . . . . . E. Numerical Simulations for Stokes Flow . . . . . . . . . . . . F. Numerical Simulations for Navier –Stokes Flow . . . . . . G. Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Oscillatory, Non-Isothermal, and Non-Newtonian Flow .

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KEYWORDS: liquid layers, liquid films, flow instability, interfacial flow, surfactants, boundary element methods, free-surface flow. I. Introduction Interfaces between adjacent viscous liquid layers are susceptible to various kinds of hydrodynamic instability with different physical origin and diverse mechanisms of growth. The Kelvin– Helmholtz instability is associated with the self-induced motion of a vortex layer established along an interface at moderate and high Reynolds numbers, the capillary instability is associated with variations in the jump of the normal stress caused by the deformation of a curved ADVANCES IN APPLIED MECHANICS, VOL. 40 ISSN 0065-2156 DOI: 10.1016/S0065-2156(04)40002-7

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three-dimensional interface, and the Marangoni instability is associated with variations in surface tension induced by a temperature field or by the uneven distribution of a surfactant. One of the most subtle and least understood types of interfacial instability is associated with rapid viscosity variations in a homogeneous fluid, or differences in the viscosity between two adjacent fluid layers in parallel flow. At high Reynolds numbers, the growth of perturbations may be associated with the discontinuity in the vorticity or slope of the unperturbed velocity across the interface, which is established for the shear stress to remain continuous across the interface. Accordingly, viscosity is important insofar as to establish the discontinuity, and plays a secondary role in determining the growth of perturbations. The growth or decay of interfacial waves may then be studied under the formalism of vorticity dynamics and under the framework of inviscid but not irrotational flow. At low and moderate Reynolds numbers, the physical mechanism by which the instability develops is not entirely clear. However, even when viscous forces are significant or dominant, the instability may still be associated with the discontinuity in the slope of the unperturbed velocity across the interface. Indeed, linear stability analysis of Poiseuille flow of two superposed layers in a channel reveals that, when the slope of the unperturbed velocity profile is continuous across the interface, the flow is either neutrally stable or stable at any Reynolds number (e.g., Yiantsios and Higgins, 1988). At first sight, it might appear that convective transport is necessary for the onset of an instability due to viscosity stratification, and that a slightly perturbed interface will either be neutrally stable or return to its unperturbed position under conditions of Stokes flow. A counter-example is provided by the flow of a liquid film down an inclined plane, with another film of a lubricating fluid of same density lining the wall. Loewenherz and Lawrence (1989) showed that when the viscosity of the lubricant is lower than the viscosity of the upper film and the surface tension is negligible, the flow is unstable regardless of the relative thickness of the two layers. This behavior has been characteristically described by Chen (1993) as ‘anti-lubrication’. A similar instability is observed in the case of three-layer film flow through a channel (Li, 1969) or down an inclined plane wall (Weinstein and Kurz, 1991). An analogy can be made between the instability due to viscosity stratification and the fingering Saffman –Taylor instability of the interface of two viscous fluids in a porous medium or the Hele – Shaw cell. When the surface tension is sufficiently low, a low-viscosity fluid pushing a high-viscosity fluid penetrates the latter by developing elongated fingers. In the case of the fingering instability, the motion can be modeled and analyzed in the context of irrotational flow for the depth-averaged velocity using a well-known formalism of vortex dynamics

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(e.g., Pozrikidis, 1997), and the instability may be attributed to the self-induced motion of an interfacial vortex sheet. An analogous formalism that would provide a physical reason for the interfacial instability is not possible in the case of channel flow. Motivation for studying the instability of multi-layer flow is provided by the long-standing desire to identify the physical mechanisms of mixing and describe the morphology of interfacial patterns generated by stirring and agitation. Intricate interfacial patterns developing on passive interfaces separating fluids with identical physical properties in the absence of surface tension have been described by experimental and numerical methods for various types of flow (e.g., Jana et al., 1994). Less progress has been made in the more interesting case of active interfaces separating fluids with different physical properties and in the presence of surface tension or more complex interfacial rheology. In these cases, the velocity field may no longer be prescribed at the outset and must be computed instead as part of the solution. Previous work has considered predominantly mixing in highReynolds-number and turbulent shear flows, including shear layer and jet flows. Instability due to viscosity stratification has been studied from two viewpoints. The fundamental viewpoint seeks to document and analyze the physical factors that influence the stability of a sharp interface or gradual transition layer in a general shear flow. The practical viewpoint seeks to characterize the behavior of liquid layers in channel and film flow with particular reference to applications involving two-phase flow through pipes, channels, micro-channels and porous media, multi-layer coating, and co-extrusion of polymeric sheets. The dual purpose of this article is to provide a tutorial and to review classical and recent work with particular emphasis on linear stability analysis and numerical simulation. The material updates and extends a previous review by the present author (Pozrikidis, 2000) in several ways, including a discussion of the effect of surfactants causing variations in surface tension. Channel flow is discussed in Section II, film flow is discussed in Section III, and the equations describing surfactant transport and the relation between the surface tension and the surfactant concentration are summarized in the form of a primer in Appendix A.

II. Channel Flow In the first part of this review, we consider several aspects of the multi-layer channel flow in two-dimensional channels. We begin in Sections II.A and II.B by describing the structure of the two-layer unidirectional flow, and outline an algorithm for computing the velocity profile across an arbitrary number of layers. In Section II.C, we discuss the linear stability of the two- and multi-layer flow in

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the absence and presence of surfactants. To describe the non-linear motion for long interfacial waves, in Section II.D we formulate the problem in the context of the lubrication approximation and discuss linear stability analysis and numerical solutions. In Section II.E, we develop a boundary-integral formulation that allows us to simulate the evolution of periodic waves in the limit of Stokes flow. In Section II.F, we review numerical computations for flow at non-zero Reynolds numbers conducted by volume-of-fluid (VOF) and immersed-boundary methods. In Sections II.G and II.H, we present a brief overview of experimental observations and generalizations to unsteady and non-Newtonian flow.

A. Unidirectional Two-Layer Flow Consider the combined unidirectional Couette – Poiseuille-gravity-driven flow of two superposed layers through a two-dimensional channel of width 2h confined between two parallel plane walls located at y ¼ ^h; with a flat interface located at y ¼ yI ; as illustrated in Fig. 2.1.1. The lower and upper walls translate parallel to themselves with respective velocities equal to U1 and U2 ; and the channel is inclined at an angle u0 with respect to the horizontal. A constant streamwise pressure gradient is allowed to act in the direction of the flow; by definition, ›p=›x ¼ 2x; where x is a specified constant. The viscosity and density of the lower fluid are denoted by m1 and r1 ; and the viscosity and density of the upper fluid are denoted by m2 and r2 : The viscosity and density ratios are defined as

l¼

m2 ; m1

d¼

r2 : r1

ð2:1:1Þ

Fig. 2.1.1. Schematic illustration of unidirectional Couette –Poiseuille-gravity-driven two-layer flow in an inclined channel.

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In terms of the position of the interface, the lower- and upper-layer thicknesses are given by h1 ¼ h þ yI and h2 ¼ h 2 yI : The layer thickness ratio is defined as r ¼ h2 =h1 ; accordingly, h1 ¼

2h ; rþ1

h2 ¼

2hr : rþ1

ð2:1:2Þ

Elementary analysis shows that the x and y-components of the Navier –Stokes equation in the two layers satisfy the second-order differential equations 0 ¼ x þ m1

›2 uð1Þ x þ r 1 gx ; ›y2

0¼2

›2 uð2Þ x þ r 2 gx ; 0 ¼ x þ m2 ›y2 where p is the pressure, gx ¼ g sin u0 ;

›pð1Þ þ r1 gy ; ›y

ð2:1:3Þ

›pð2Þ þ r2 gy ; 0¼2 ›y gy ¼ 2g cos u0 ;

ð2:1:4Þ

are the components of the acceleration of gravity vector, and g is the magnitude of the acceleration of gravity. The no-slip wall boundary condition requires ð2Þ uð1Þ x ¼ U1 at y ¼ 2h and ux ¼ U2 at y ¼ h: The kinematic and dynamic conditions expressing continuity of velocity and shear stress across the interface require ð2Þ uð1Þ x ¼ ux ; uI ;

m1

›uð1Þ ›uð2Þ x ¼ m2 x ; ›y ›y

ð2:1:5Þ

both evaluated at y ¼ yI ; where uI is the interfacial velocity. Solving the governing equations by elementary methods, we obtain the piecewise parabolic profiles 1 uð1Þ ðx þ r1 gx Þ þ j1 ðy 2 yI Þ þ uI ; x ¼ 2 2m 1 ð2:1:6Þ 1 ð2Þ ux ¼ 2 ðx þ r2 gx Þ þ j2 ðy 2 yI Þ þ uI ; 2m 2 where DU l h l 2 dr 2 2 j1 ¼ x þ r1 gx h1 l þ r m1 l 2 r2

!

l 2 r2 ; ð1 þ rÞðl þ rÞ

j2 ¼

j1 ; ð2:1:7Þ l

are the shear rates on either side of the interface, and DU ¼ U2 2 U1 is the difference in the wall velocities. The interfacial velocity is given by   rU1 þ lU2 h2 1 þ dr 2r : ð2:1:8Þ uI ¼ þ x þ r1 gx 1 þ r ð1 þ rÞðl þ rÞ rþl m1

184

C. Pozrikidis

Note that, when the densities of the two liquids are matched, d ¼ 1; the two terms enclosed by the large parentheses on the right-hand side of Eqs. (2.1.7) and (2.1.8) combine into the effective negative pressure gradient x þ r1 gx : The corresponding pressure fields are given by pð1Þ ¼ 2xx þ r1 gy ðy 2 yI Þ þ P0 ;

pð2Þ ¼ 2xx þ r2 gy ðy 2 yI Þ þ P0 ; ð2:1:9Þ

where P0 is an indeterminate constant. Consistent with our earlier definition, ›p=›x ¼ 2x in both fluids.

B. Unidirectional Multi-Layer Flow Consider now the more general case of unidirectional flow of an arbitrary number of N superimposed layers, as illustrated in Fig. 2.2.1. The bottom fluid is labeled 1, and the top fluid is labeled N: The N 2 1 interfaces separating the layers are located at y ¼ yi ; where i ¼ 1; 2; …; N 2 1: The velocity and pressure fields in the ith layer are governed by the following generalized version of Eq. (2.1.3), d2 uðiÞ x þ ri gx x ¼2 ; 2 mi dy

0¼2

›pðiÞ þ ri gy ; ›y

ð2:2:1Þ

where i ¼ 1; 2; …; N; and x is the negative of the streamwise pressure gradient. Integrating the first equation in Eq. (2.2.1) twice with respect to y; we derive the parabolic profile uðiÞ x ðyÞ ¼ 2

x þ ri gx 2 y þ Bi y þ Ai ; 2m i

ð2:2:2Þ

Fig. 2.2.1. Schematic illustration of unidirectional multi-layer flow in an inclined channel.

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185

where Ai and Bi are unknown constants to be determined by requiring (a) the noslip boundary condition at the lower and upper walls, (b) continuity of velocity across the interfaces expressed by ðiþ1Þ uðiÞ ðy ¼ yi Þ; x ðy ¼ yi Þ ¼ ux

ð2:2:3Þ

for i ¼ 1; 2; …; N 2 1; and (c) continuity of shear stress across the interfaces expressed by ! ! ›uðiÞ ›uxðiþ1Þ x mi ¼ miþ1 ; ð2:2:4Þ ›y y¼y ›y y¼y i

i

for i ¼ 1; 2; …; N 2 1: Substituting the profile (2.2.2) in Eq. (2.2.4) and solving for Bi ; we derive the recursion relation   x þ ri gx miþ1 x þ riþ1 gx Bi ¼ yi þ Biþ1 2 yi ; ð2:2:5Þ mi mi miþ1 for i ¼ 1; 2; …; N 2 1: For reasons that will soon become evident, we introduce the shear rate at the upper wall, ! ›uxðNÞ a; : ð2:2:6Þ ›y y¼h Differentiating the profile (2.2.2) for i ¼ N with respect to y; and evaluating the derivative at y ¼ h; we find BN ¼ a þ

x þ rN g x h: mN

ð2:2:7Þ

If we knew the value of a; we would be able to compute the coefficient BN from Eq. (2.2.7), and then evaluate the rest of the coefficients Bi ; for i ¼ N 2 1; …; 2; 1; using the recursion relation (2.2.5). Once this has been accomplished, we could compute the coefficient A1 to satisfy the no-slip boundary condition at the bottom wall using the equation uð1Þ x ðy ¼ 2hÞ ¼ 2

x þ r1 gx 2 h 2 B1 h þ A 1 ¼ U 1 ; 2m 1

ð2:2:8Þ

and then evaluate the rest of the coefficients Ai by requiring continuity of velocity across each interface expressed by Eq. (2.2.3). In the end, the no-slip boundary condition at the upper wall would also be satisfied. Unfortunately, the value of a is a priori unknown.

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An expedient method of producing the value of a and simultaneously computing the unknown coefficients of the velocity profiles may be devised on the basis of the no-slip boundary condition at the upper wall. We begin by expressing this condition in the form f ðaÞ ; uxðNÞ ðy ¼ hÞ 2 U2 ¼ 2

x þ rN gx 2 h þ BN h þ AN 2 U2 ¼ 0; 2mN

ð2:2:9Þ

and note that that a is a root of the function f ðaÞ: A key observation is that f ðaÞ is in fact a linear function of a; and may thus be expressed in the form f ðaÞ ¼ C a þ D;

ð2:2:10Þ

where C ¼ f ð1Þ 2 f ð0Þ and D ¼ f ð0Þ: The linear dependence shown in Eq. (2.2.10) becomes evident by observing that, if we assign a certain value to a; we can use the procedure described in the preceding paragraph to evaluate the coefficients of the velocity profiles across each layer, and then compute the left-hand side of Eq. (2.2.8) by linear algebraic manipulations. Accordingly, the requisite value of a satisfying f ðaÞ ¼ 0 is given by

a¼2

D f ð0Þ ¼ : C f ð1Þ 2 f ð0Þ

ð2:2:11Þ

The solution procedure involves evaluating f ð0Þ and f ð1Þ; and then using Eq. (2.2.11) to obtain a: The algorithm is implemented in the program chan_2d_ml located in the directory 04_various of the fluid dynamics software library FDLIB (Pozrikidis, 2001).

C. Stability Analysis Linear stability analysis of two- and multi-layer arrangements in infinite shear flow, semi-infinite shear flow bounded by a plane wall, and channel flow, has been conducted on several occasions following the pioneering work of Yih (1967). Early and recent literature surveys can be found in the articles of Renardy (1985, 1987a, 1989), Yiantsios and Higgins (1988), Hooper (1989), Anturkar et al. (1990), Su and Khomami (1992), Tilley et al. (1994a), Chen (1995), Severtson and Aidun (1996), South and Hooper (1999), Albert and Charru (2000), and Charru and Hinch (2000). Han (1981, Chapter 8) discusses the stability of non-Newtonian multi-film flow as it relates to polymer processing. Hesla et al. (1986) developed a generalized Squire transformation which appears to obviate the need to study three-dimensional perturbations and allows us to restrict our attention to two-dimensional waves. In particular, they

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187

argued that, in the case of two-layer flow in a horizontal channel with constant surface tension and stable density stratification (the heavy fluid is at the bottom), only two-dimensional perturbations need to be considered in order to determine the critical conditions for instability. Yiantsios and Higgins (1988) pointed out that, in some instances, the existence of the generalized Squire transformation does not necessarily render the study of three-dimensional perturbations redundant. The relation between three-dimensional and two-dimensional disturbances was further discussed and clarified by Joseph and Renardy (1992, Vol. 1) and Tilley et al. (1994a).

1. Two-Layer Flow Linear stability analysis of two-layer channel flow with fluids of equal density reveals that, in the absence of surfactants, the flow is unstable only at non-zero Reynolds numbers. Two general types of instability have been identified: one is a shear-flow instability associated with the global structure of the channel flow, and the second is an interfacial instability associated with the discontinuity in the viscosity across the interface. In the case of Stokes flow, a small-amplitude sinusoidal perturbation decays for any non-zero value of the interfacial tension. The interfacial wave is described by the equation y ¼ yI ðx; tÞ ¼ y0 þ AðtÞ cos½kðx 2 cR tÞ;

ð2:3:1Þ

where y0 is the unperturbed position, k ¼ 2p=L is the wave number, L is the wavelength, cR is the phase velocity, AðtÞ ¼ A0 expðsI tÞ

ð2:3:2Þ

is the instantaneous amplitude of the perturbation, A0 is the initial amplitude, and sI is the growth rate. Here as elsewhere, the subscripts ‘R’ and ‘I’ denote the real and imaginary part, respectively. Relations between cR ; sI and the various geometrical and physical variables of the flow may be derived in closed form, and are implemented in subroutine chan2l0 of directory 08_stab of the fluid dynamics software library FDLIB (Pozrikidis, 2001). The results show that the growth rate is independent of the structure of the base flow, and is identical to that of waves at the interface between two quiescent superposed layers placed in a channel between two parallel walls. The solid lines in Fig. 2.3.1 show graphs of the reduced growth rate, s ; sI m1 h=g; in a horizontal channel, u0 ¼ 0; for a neutrally buoyant arrangement, r1 ¼ r2 ; viscosity ratio (a) l ; m2 =m1 ¼ 1:0; and (b) 0.2, and

C. Pozrikidis

188

0

0

–2 s

s

–2 –4

–4 –6 –6 0 (a)

5

10 kh

15

20

–8

0 (b)

5

10 kh

15

20

Fig. 2.3.1. Reduced growth rate s ; sI m1 h=g of waves in a horizontal channel for r1 ¼ r2 ; viscosity ratio (a) l ¼ 1:0; and (b) 0.20, and interface position y0 =h ¼ 0; 20:10; … 2 0:90 (highest curves). The dashed lines correspond to semi-infinite layers, and the dotted lines correspond to a layer resting on the lower wall underneath a semi-infinite upper fluid.

interface position y0 =h ¼ 0; 2 0.10, 20:2; …; 20:90 (highest curves). Because the growth rate is negative, the configuration is stable for any layer thickness ratio and for all wave numbers. The dashed straight lines passing through the origin correspond to waves at the interface between two semi-infinite fluids, whose reduced growth rate is given by   1 ðr1 2 r2 Þgh s¼2 þ kh : ð2:3:3Þ 2ð1 þ lÞ gk Note that, in this case, the channel semi-width h is relevant only insofar as to provide us with a length scale for non-dimensionalizing the wave number and growth rate. When r2 . r1 ; the first term in the large parentheses on the righthand side of Eq. (2.3.3) introduces the Rayleigh – Taylor instability of unstably stratified fluids. In agreement with physical intuition, all solid lines in Fig. 2.3.1 tend to the dashed lines in the limit of large reduced wave numbers kh; corresponding to wavelengths that are much smaller than the channel width. The dotted lines in Fig. 2.3.1 describe the growth rate for a film whose thickness is equal to that of the lower layer, h1 ; resting on a plane horizontal wall underneath a semi-infinite upper fluid, corresponding to the equivalent limit h1 =h2 ! 0: The reduced growth rate is given by   1 ðr1 2 r2 Þgh1 þ kh1 Fðkh1 ; lÞ; ð2:3:4Þ s¼2 2ð1 þ lÞ gk where

Fðw; lÞ ¼ ð1 þ lÞ

ð1=2Þsinhð2wÞ 2 w þ lðsinh2 w 2 w2 Þ ð1 2 l2 Þw2 þ ðcosh w þ l sinh wÞ2

ð2:3:5Þ

Instability of Multi-Layer Channel and Film Flows

189

(Newhouse and Pozrikidis, 1990). In the limit kh1 ! 1; corresponding to short waves, the function Fðkh1 ; lÞ tends to unity, yielding the growth rate of perturbations on the interface between two semi-infinite fluids stated in Eq. (2.3.3). The three dotted curves in each frame of Fig. 2.3.1 correspond to h1 =h ¼ 1:0; 0:5; and 0.1 (highest curves). The agreement with the results of the stability analysis for channel flow is good for equal layer thicknesses, h1 =h ¼ 1:0; and exceptional for the thinnest layer considered, h1 =h ¼ 0:1:

2. Two-Layer Flow in the Presence of Surfactants Frenkel and Halpern (2002) and Halpern and Frenkel (2003) pointed out that a surfactant may have a destabilizing influence on the two-layer, neutrallystratified, Couette – Poiseuille channel flow. Conversely, a shear flow may destabilize an otherwise stable interface populated by surfactants. The stability of the basic flow is determined by the capillary number Ca ¼ m1 j1 h=g0 ; where j1 is the shear rate on the lower side of the interface given in Eq. (2.1.7). Linear stability analysis reveals the existence of two normal modes. When Ca ¼ 0; both modes are stable independent of the overall velocity profile. On the other hand, when Ca is non-zero, however small, one of the modes associated with Marangoni tractions becomes unstable for a certain range of layer thicknesses and in a certain range of wavenumbers. Blyth and Pozrikidis (2004) reconsidered the linear and non-linear stability problem under the auspices of the lubrication approximation and also in the context of Stokes flow, as will be discussed in Sections II.D and II.E. Their analysis confirmed that the dimensionless growth rate of the normal modes depends on the Capillary number, Ca; Marangoni number, Ma; defined in Eq. (A.13) and expressing the sensitivity of the surface tension to the surfactant concentration, viscosity and density ratios l and d defined in Eq. (2.1.1), and relative layer thicknesses. The irrelevance of the whole of the velocity profile of the base flow is consistent with physical intuition, suggesting that small enough deflections of the interface only feel the effect of the shear term in the local undisturbed velocity profile, as it appears in the surfactant transport equation. Consider shear- or pressure-driven flow in a horizontal channel with fluids of equal density. Figure 2.3.2 shows a graph of the reduced growth rate, s ¼ sI m1 h=g0 ; plotted against the reduced wave number kh for a series of Capillary numbers at a fixed Marangoni number, and zero surfactant diffusivity; g0 is the unperturbed surface tension of the planar interface. These results were obtained by Blyth and Pozrikidis (2004) using the lubrication-flow model

C. Pozrikidis

190 0

0.06 – 0.1 s

y

0.04 –0.2

0.02

–0.3

0 0

(a)

0.2

0.4

0.6 kh

0.8

1

0 (b)

0.2

0.4

0.6

0.8

1

kh

Fig. 2.3.2. Reduced growth rate versus wavenumber for (a) the stable, and (b) the unstable normal mode, both for Ma ¼ 1; l ¼ 0:5; d ¼ 1; h1 =h ¼ 0:25; and u0 ¼ 0: The Capillary numbers are spaced evenly in the range 1023 # Ca # 1; the dashed lines corresponding to Ca ¼ 1023 :

discussed in Section II.D. The graph on the right of Fig. 2.3.2 shows that introducing a surfactant produces instability over a range of wave numbers. Although, strictly speaking, the lubrication analysis is only valid for sufficiently small reduced wave numbers kh; comparison with results obtained from stability analysis for Stokes flow reveals that the predictions are surprisingly accurate even at moderate wave numbers (Blyth and Pozrikidis, 2004). More recently, Blyth and Pozrikidis (2005) considered the effect of inertia on the Yih– Marangoni instability of the two-layer flow by carrying out a normalmode linear stability analysis. In their study, the Orr –Sommerfeld equation describing the growth of small perturbations was solved numerically subject to interfacial conditions that allow for the Marangoni traction. For general Reynolds numbers and arbitrary wavenumbers, the surfactant was found to either provoke instability or significantly lower the rate of decay of infinitesimal perturbations, while inertial effects act to widen the range of unstable wavenumbers. The nonlinear evolution of growing interfacial waves consisting of a special pair of normal modes yielding an initially flat interface was analyzed numerically by a finite-difference method, and the results were found to be consistent with the predictions of the linear theory, while also revealing that the interfacial waves steepen and eventually overturn under the influence of the shear flow.

3. Multi-Layer Flow Several authors have studied the stability of the three- and multi-layer channel flow. Li (1969) carried out the linear stability analysis of the three-layer plane Couette channel flow and reported instability even in the limit of Stokes flow when the interfacial tensions are sufficiently small and the viscosity ratios

Instability of Multi-Layer Channel and Film Flows

191

lie within certain ranges. Anturkar et al. (1990) and Knoester and van der Zanden (1992) formulated the linear stability problem for the general case of N-layer flow, and presented results for two- and three-layer flow. Other authors performed stability analysis for long waves, as will be discussed in Section II.D in the context of the lubrication approximation.

D. Lubrication-Flow Model for Long Waves The non-linear evolution of long waves can be described efficiently working under the auspices of the lubrication approximation, assuming piecewise parabolic velocity profiles. The simplified description allowed Ooms et al. (1985) to derive an evolution equation for interfacial waves whose period is large compared to the unperturbed layer thickness, but the amplitude is not necessarily small compared to the channel width. Numerical solutions suggested the possibility of finite-amplitude interfacial waves with permanent form in twolayer flow. Subsequently, Power and Carmona (1991) replaced the lubrication with the boundary-layer approximation and discussed the effect of fluid inertia. A number of authors have derived non-linear evolution equations for the layer thicknesses in the limit as both the wave number and the amplitude of a perturbation are small compared to the unperturbed layer thicknesses. Examples can be found in the papers of Hooper and Grimshaw (1985), Hooper (1985), Shlang et al. (1985), Renardy and Renardy (1993), and Charru and Fabre (1994). Tilley et al. (1994b) performed a weakly non-linear analysis that produces a Kuramoto –Sivashinsky equation for the wave amplitude. Numerical solutions revealed behavior that is richer and more diverse than that exhibited by the widely studied single-film flow down an inclined plane. However, the physical relevance of these motions was criticized by Barthelet et al. (1995). In Section II.D.1, we present a generalized lubrication-flow model that combines the modular cases of shear-, pressure-, and gravity-driven flow, and also accounts for the presence of an insoluble surfactant inducing variations in surface tension (Pozrikidis, 1998a; Blyth and Pozrikidis, 2004). The integrated formulation provides us with a convenient point of departure for detecting hydrodynamic instability associated with Marangoni tractions. Extension of the basic formulation to three- to multi-layer flow is straightforward, though the analysis involves unwieldy algebraic expressions that discourage analytical treatment.

C. Pozrikidis

192

1. Two-Layer Flow Referring to Fig. 2.4.1, we describe the interface by the equation y ¼ yI ðx; tÞ and approximate the flow within each layer with a locally unidirectional flow whose streamwise velocity profile is given by uðxjÞ ¼ 2

xj þ rj gx ðy 2 yI Þ2 þ jj ðy 2 yI Þ þ uI ; 2mj

ð2:4:1Þ

where 2h , y , yI ðx; tÞ for j ¼ 1; yI ðx; tÞ , y , h for j ¼ 2; and j1 ðx; tÞ; j2 ðx; tÞ are the shear rates of the lower or upper fluid evaluated at the interface. The position-dependent and time-varying coefficients xj ðx; tÞ ¼ 2›pð jÞ =›x express the negative of the local and instantaneous streamwise pressure gradient within each layer. Integrating the y-component of the equation of motion, we obtain the pressure distributions pð jÞ ¼ Pj 2

ðx 0

xj ðx 0 Þdx0 þ rj gy y;

ð2:4:2Þ

for j ¼ 1; 2; where Pj are unspecified constants. Requiring the no-slip boundary condition at the walls, we set uð1Þ x ðy ¼ 2hÞ ¼ ð2Þ U1 and ux ðy ¼ hÞ ¼ U2 ; and find "

# hj uI 2 Uj ; jj ¼ ^ 2ðxj þ rj gx Þ þ 2m j hj

ð2:4:3Þ

Fig. 2.4.1. Schematic illustration of Couette–Poiseuille-gravity-driven two-layer flow in an inclined channel with a slowly undulating interface, used to develop the lubrication-flow model.

Instability of Multi-Layer Channel and Film Flows

193

where the plus (minus) sign applies for the lower (upper) layer, j ¼ 1 (2), and h1 ðx; tÞ ¼ yI ðx; tÞ þ h;

h2 ðx; tÞ ¼ h 2 yI ðx; tÞ;

ð2:4:4Þ

are the local and instantaneous lower and upper layer thicknesses. To compute the interfacial velocity, we substitute the right-hand sides of Eq. (2.4.3) into the interfacial shear stress balance expressed by ›g ; ð2:4:5Þ m1 j1 ðx; tÞ ¼ m2 j2 ðx; tÞ þ ›x where g is the interfacial tension; the last term on the right-hand side of Eq. (2.4.5) represents the Marangoni traction. Rearranging, we obtain

 rU1 þ lU2 2h2 r uI ¼ x1 þ x2 r þ r1 gx ð1 þ drÞ þ 2 m1 ð1 þ rÞ ðl þ rÞ lþr þ

4h2 r ›g ; 2 m1 ð1 þ rÞ ðl þ rÞ ›x

ð2:4:6Þ

where rðx; tÞ ¼

h2 ðx; tÞ h1 ðx; tÞ

ð2:4:7Þ

is the local and instantaneous layer thickness ratio, l ¼ m2 =m1 ; and d ¼ r2 =r1 : It is reassuring to confirm that, when x1 ¼ x2 and ›g=›x ¼ 0; expression (2.4.6) agrees with its flat-interface counterpart for unidirectional flow given in (2.1.8). The normal stress undergoes a jump across the interface due to the surface tension. Neglecting the viscous contribution and approximating the normal stress with the negative of the pressure given in Eq. (2.4.2) evaluated at the interface, we obtain ðx ðp1 2 p2 Þy¼yI ¼ P1 2 P2 2 ðx1 2 x2 Þdx0 þ r1 ð1 2 dÞgy yI 0

2

¼ gk . 2g

› yI : ›x 2

ð2:4:8Þ

The negative of the second derivative in the last expression of Eq. (2.4.8) is an approximation to the curvature. Rearranging, we obtain     ðx ðx ›2 y 0 0 P2 2 x2 dx ¼ P1 2 x1 dx þ r1 ð1 2 dÞgy yI þ g 2I : ð2:4:9Þ ›x 0 0 The expressions enclosed by the square brackets on the left- and right-hand side represent modified pressures distinguished by the exclusion of hydrostatic variations normal to the channel walls. Differentiating Eq. (2.4.9) with respect to x; we find

x2 ¼ x1 2 r1 ð1 2 dÞgy y0I 2 g 0 y00I 2 gy000I ;

ð2:4:10Þ

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where a prime denotes a partial derivative with respect to x at constant time. Using Eq. (2.4.10) to eliminate x2 in favor of x1 from Eq. (2.4.6), we derive the following expression for the interfacial velocity,

uI ¼

h 2h2 r ð1 þ rÞx1 2 r1 ð1 2 dÞgy ry0I 2 r g 0 y00I m1 ð1 þ rÞ2 ðl þ rÞ i rU þ lU h2 1 2 þ 2 r gy000 g 0: I þ r1 gx ð1 þ drÞ þ lþr m1 ðl þ rÞ

ð2:4:11Þ

To derive an evolution equation for the interface, we first integrate both sides of Eq. (2.4.1) with respect to y over its domain of definition and substitute expressions (2.4.3) for the interfacial shear stress to find the following expressions for the flow rates in terms of the interfacial velocity,

Qj ¼

x j þ rj gx 3 1 hj þ ðuI þ Uj Þhj : 2 12mj

ð2:4:12Þ

An overall mass balance for the individual layers over the channel cross-section requires

›h1 ›Q ¼2 1; ›t ›x

›h2 ›Q ¼2 2; ›t ›x

ð2:4:13Þ

and since h1 þ h2 ¼ 2h is a constant, it must be

›Q1 ›Q 2 þ ¼ 0: ›x ›x

ð2:4:14Þ

Upon integration, we find

Q1 þ Q2 ¼

x1 þ r1 gx 3 1 x þ r2 gx 3 h1 þ ðuI þ U1 Þh1 þ 2 h2 2 12m1 12m2 1 þ ðuI þ U2 Þh2 ¼ wðtÞ; 2

ð2:4:15Þ

where the function wðtÞ is to be determined as part of the solution. Using once again Eq. (2.4.10) to eliminate x2 in favor of x1 ; substituting the right-hand side

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of Eq. (2.4.6) for the interfacial velocity and rearranging, we obtain    r1 ð1 2 dÞgy ›pð1Þ m1 l ; 2 x1 ¼ 2 h h22 y0I h2 þ 6 lþr ›x D m1 2

    g l r1 gx l h h22 y000 hh1 h2 h2 þ 6 lh31 þ dh32 þ 6ð1 þ drÞ I þ m1 lþr lþr m1

þ12lh

þ



U1 þ rU2 rU1 þ lU2 þ 1þr lþr



   12 l 1 l hh2 g 0 2 þ h2 h22 g 0 y00I þ f ðtÞ ; 6h lþr m1 l þ r m1

ð2:4:16Þ

where D ¼ lh31 þ h32 þ 12

l 2 h h2 ; lþr

ð2:4:17Þ

and f ðtÞ is a rescaled version of wðtÞ: If the flow is periodic with period L; the function f ðtÞ is evaluated by specifying the pressure drop over one period, Dpð1Þ ðtÞ ¼ Dpð2Þ ðtÞ ¼ pðiÞ ðx þ L;y;tÞ 2 pðiÞ ðx;y;tÞ;

ð2:4:18Þ

for i ¼ 1 or 2. In the case of pure shear- or gravity-driven flow, we require Dpð1Þ ðtÞ ¼ Dpð2Þ ðtÞ ¼ 0: Substituting the expression for x1 shown in Eq. (2.4.17) into Eqs. (2.4.6) and (2.4.12) for j ¼ 1; and then putting the emerging expression for the flow rate into the first equation in Eq. (2.4.13), we derive a non-linear evolution equation for the interface position which can be expressed in the symbolic form

›y I 0000 ¼ FðyI ; y0I ; y00I ; y000 I ; yI Þ: ›t

ð2:4:19Þ

The right-hand side is a strongly non-linear function of the arguments of the rate-of-change function F: When the amplitude of the perturbation is small compared to the wavelength and the surface tension is sufficiently high, Eq. (2.4.19) reduces to the Kuramoto –Sivashinsky equation as discussed, for example, by Charru and Fabre (1994, Section IV). It is instructive to note that the y-component of the interfacial velocity, denoted by vI ; does not participate explicitly in the mathematical formulation. If desired, it can be computed a posteriori from the kinematic condition

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C. Pozrikidis

D½y 2 yI ðx; tÞ=Dt ¼ 0; where D=Dt is the material derivative, yielding vI ¼

›y I ›y þ uI I : ›t ›x

ð2:4:20Þ

Under the auspices of the lubrication approximation, the evolution equation for the surfactant concentration, stated in its general form in Eq. (A.1), simplifies to the one-dimensional convection – diffusion equation

›G ›ðuI G Þ ›2 G þ ¼ Ds 2 : ›t ›x ›x

ð2:4:21Þ

Solving the system of equations (2.4.19) and (2.4.21) allows us to compute the evolution of a periodic interface from a specified initial condition. Pozrikidis (1997b, 1998a) implemented finite-difference methods for solving the non-linear differential equation (2.4.19) in the absence of surfactants. The spatial derivatives required for the computation of the flow rate Q1 are evaluated by centered differences, while ›Q1 =›x is computed by central or backward upwind differences. As an example, Fig. 2.4.2 shows the evolution of a sinusoidal wave in a channel that is inclined at an angle u0 ¼ p=4 and is open at both ends, corresponding to gravity-driven flow. Numerical experimentation shows that the upwind scheme is capable of describing the development and propagation of steep profiles by suppressing

Fig. 2.4.2. Evolution of the interface between two layers in an inclined channel that is open at both ends for viscosity ratio l ¼ 0:2; density ratio d ¼ 0:5; vanishing surface tension, unperturbed layer thickness ratio h2 =h1 ¼ 3; and wavelength L ¼ ph: The simulations are based on the lubrication-flow model.

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the growth of spurious oscillations, though significant numerical diffusivity may compromise the physical relevance of the computed interfacial shapes. Blyth and Pozrikidis (2004) recently performed simulations in the presence of surfactants, confirmed the occurrence and described the non-linear stages of the instability induced by Marangoni tractions.

2. Three-Layer Flow Than et al. (1987), Renardy (1987b), and Kliakhandler and Sivashinsky (1995, 1996) applied the lubrication approximation to study the evolution of long waves in the three-layer Poiseuille flow under constant interfacial tension. The results of Kliakhandler and Sivashinsky (1995, 1996) appear to suggest that surface tension may have a destabilizing influence on the evolution of long waves. Soliton-like interfacial structures were found to develop under certain conditions.

E. Numerical Simulations for Stokes Flow The boundary-integral formulation provides us with an efficient method of computing the evolution of interfacial waves under the auspices of Stokes flow, beyond the constraints on the amplitude of the perturbation imposed by linear theory. In the limit of vanishing Reynolds number, the flow in the two fluids is governed by the linear equations of Stokes flow, 0 ¼ 27pð jÞ þ mj 72 uð jÞ þ rj g;

7·uð jÞ ¼ 0;

ð2:5:1Þ

where the lower and upper layer correspond to j ¼ 1; 2: The velocity is required to be continuous across the interface, but the traction undergoes a discontinuity due to the surface tension, given by Df ; ðsð1Þ 2 sð2Þ Þ·n ¼ gkn 2 t

›g ; ›l

ð2:5:2Þ

where s is the Newtonian stress tensor, k is the curvature of the interface in the xy-plane reckoned to be positive when the interface is upward parabolic, as shown in Fig. 2.5.1, n is the unit vector normal to the interface pointing into the lower fluid labeled 1, and t is the unit vector that is tangential to the interface and points in the direction of increasing arc length l:

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Fig. 2.5.1. Schematic illustration of periodic two-layer flow in a channel flow showing the control area used to develop the boundary-integral formulation.

1. Integral Formulation To develop the boundary-integral formulation for a flow that is periodic with period L; as depicted in Fig. 2.5.1, we decompose the velocity and pressure fields in the two fluids into a reference component denoted by the superscript R; and a complementary disturbance component denoted by the superscript D: The reference flow satisfies the equations of Stokes flow with the gravitational force included, whereas the disturbance flow satisfies the unforced equations of Stokes flow with the gravitational force absent. The reference velocity respects the noslip boundary condition at both channel walls; accordingly, the disturbance velocity is required to vanish over the two walls. To simplify the formulation, we stipulate that the disturbance flow does not induce a pressure drop across each period. A convenient choice for the reference velocity is ð jÞ x þ rj gx 2 ð jÞ uRx ¼ jðy 2 yR Þ þ ðh 2 y2 Þ; uRy ¼ 0; ð2:5:3Þ 2mj for j ¼ 1; 2; where

j¼

U2 2 U1 ; 2h

yR ¼ 2h

U2 þ U1 ; U2 2 U1

ð2:5:4Þ

and x is the negative of a specified pressure gradient. The associated reference pressure field is given by ð jÞ

pR ¼ 2xx þ rj gy y þ Pj ; where Pj are constants.

ð2:5:5Þ

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199

The reference velocity undergoes a discontinuity across the interface given by ð1Þ

ð2Þ

ð1Þ uRy

ð2Þ uRy

DuRx ; uRx 2 uRx ¼ DuRy

;

2

     1 1 d x 12 þ r1 gx 1 2 ðh2 2 y2 Þ; 2m1 l l

ð2:5:6Þ

¼ 0;

evaluated at the interface, where l ¼ m2 =m1 is the viscosity ratio and d ¼ r2 =r1 is the density ratio. The interfacial traction of the reference flow also undergoes a discontinuity given by ð1Þ

ð2Þ

Df R ; ðsR 2 sR Þ · n " # 2Drgy y m1 ð1 2 lÞj 2 Drgx y ¼ · n 2 ðP1 2 P2 Þn; 2 D rgy y m1 ð1 2 lÞj 2 Drgx y

ð2:5:7Þ

evaluated at the interface, where Dr ¼ r1 2 r2 ¼ r1 ð1 2 dÞ: To facilitate the forthcoming algebraic manipulations, we introduce the single- and double-layer Stokes flow potentials IjSLP ðx0 ; f; CÞ ;

1 ð G ðx; x0 Þfi ðxÞdlðxÞ; 4pm1 C ij

ð2:5:8Þ

1 ð u ðxÞTijk ðx; x0 Þnk ðxÞdlðxÞ; 4p C i

ð2:5:9Þ

and IjDLP ðx0 ; u; CÞ ;

where Gij is the periodic velocity Green’s function of two-dimensional Stokes flow representing the flow induced by a periodic array of point forces in a channel confined between two parallel walls, and Tijk is the associated stress tensor (Pozrikidis, 1992; Zhou and Pozrikidis, 1993). A Fortran program that produces the Green’s function and illustrates its properties is available in the subdirectory sgf_2d of the directory 06_stokes of the fluid dynamics software library FDLIB (Pozrikidis, 2001, 2002). It is important to point out that a degree of freedom is afforded in the definition of the Green’s function, as an arbitrary Hagen – Poiseuille flow may be added to the flow produced by the point forces to alter the axial flow rate and consequently modify the pressure drop across each period. This flexibility will be exploited to simplify the integral representation. Applying now the boundary-integral formulation for the disturbance flow in the lower fluid (e.g., Pozrikidis, 1992), we obtain the following integral

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200

representation for the velocity at a point x0 located at the interface 1 2

ð1Þ

ð1Þ

ð1Þ

SLP uD ðx0 ; f D ; IÞ 2 IjSLP ðx0 ; f D ; C1ð1Þ Þ j ðx0 Þ ¼ 2Ij ð1Þ

ð1Þ

2 IjSLP ðx0 ; f D ; C2ð1Þ Þ þ IjDLP – PV ðx0 ; uD ; IÞ ð1Þ

ð1Þ

þ IjDLP ðx0 ; uD ; C1ð1Þ Þ þ IjDLP ðx0 ; uD ; C2ð1Þ Þ;

ð2:5:10Þ

where I denotes one period of the interface. C1 and C2 are the periodic segments depicted in Fig. 2.5.1, f ¼ n·s is the traction, and PV denotes the principal value of the double-layer potential. The unit normal vector n over the various boundaries of the control area is oriented as shown in Fig. 2.5.1. Repeating the derivation for the upper fluid keeping in mind that the singlelayer potential has been defined with respect to the viscosity of the lower fluid, we find ð2Þ ð2Þ 1 Dð2Þ 1 h SLP uj ðx0 Þ ¼ Ij ðx0 ; f D ; IÞ 2 IjSLP ðx0 ; f D ; C1ð2Þ Þ 2 l i ð2Þ ð2Þ 2 IjSLP ðx0 ; f D ; C2ð2Þ Þ 2 IjDLP – PV ðx0 ; uD ; IÞ ð2Þ

ð2Þ

þ IjDLP ðx0 ; uD ; C1ð2Þ Þ þ IjDLP ðx0 ; uD ; C2ð2Þ Þ:

ð2:5:11Þ

Next, we multiply Eq. (2.5.11) through by the viscosity ratio l; and add the result to Eq. (2.5.10) to find 1 2

ð1Þ

ð2Þ

D ½uD j ðx0 Þ þ luj ðx0 Þ

¼ 2IjSLP ðx0 ; Df D ; IÞ 2 IjSLP ðx0 ; f D ; C1 Þ 2 IjSLP ðx0 ; f D ; C2 Þ ð1Þ

ð2Þ

ð1Þ

þ IjDLP – PV ðx0 ; uD 2 luD ; IÞ þ IjDLP ðx0 ; uD ; C1ð1Þ Þ ð1Þ

ð2Þ

þ IjDLP ðx0 ; uD ; C2ð1Þ Þ þ l½IjDLP ðx0 ; uD ; C1ð2Þ Þ ð2Þ

þ IjDLP ðx0 ; uD ; C2ð2Þ Þ:

ð2:5:12Þ

Significant simplifications occur by stipulating that the flow due to the periodic array of point forces underlying the Green’s function does not induce a pressure drop across each period. Exploiting the periodicity of the disturbance velocity, we find that the last two pairs of terms on the right-hand side of Eq. (2.5.12) vanish due to cancellation of integrals over the periodic segments C1 and C2 : Recalling our earlier stipulation that the disturbance flow does not induce a pressure drop across each period, we find that the second and third terms on the right-hand side of Eq. (2.5.12) also vanish due to cancellations. The simplified

Instability of Multi-Layer Channel and Film Flows

201

integral equation involves integrals over one period of the interface I alone, i ð2Þ 1 h Dð1Þ uj ðx0 Þ þ luD j ðx0 Þ 2 ð1Þ

ð2Þ

¼ 2IjSLP ðx0 ; Df D ; IÞ þ IjDLP – PV ðx0 ; uD 2 luD ; IÞ:

ð2:5:13Þ

Continuity of velocity at the interface requires ð2Þ

ð1Þ

D R uD j ðx0 Þ ¼ uj ðx0 Þ þ Duj ðx0 Þ:

ð2:5:14Þ

ð2Þ

Using this relation to eliminate uD j ðx0 Þ from Eq. (2.5.13) and rearranging, we obtain a Fredholm integral equation of the second kind for the interfacial velocity, ð1Þ

uD j ðx0 Þ ¼ 2

l 2 SLP 12l DuRj ðx0 Þ 2 Ij ðx0 ; Df D ; IÞ þ 2 1þl 1þl 1þl ð1Þ

 IjDLP – PV ðx0 ; uD ; IÞ 2 2

l DLP – PV I ðx0 ; DuR ; IÞ: 1þl j

ð2:5:15Þ

The strength density of the single-layer potential is given by Df D ¼ Df 2 Df R ¼ gkn 2 t

›g 2 Df R : ›l

ð2:5:16Þ

It is instructive to recognize three special cases: Shear-driven flow. In this case, x ¼ 0; gx ¼ 0; the reference velocity is continuous at the interface, DuR ¼ 0; and the disturbance velocities are equal, ð1Þ ð2Þ uD ¼ uD ; uD : The master integral equation (2.5.15) becomes uD j ðx0 Þ ¼ 2

2 SLP 1 2 l DLP – PV Ij ðx0 ; Df D ; IÞ þ 2 I ðx0 ; uD ; IÞ: 1þl 1þl j

ð2:5:17Þ

When l ¼ 1; the disturbance velocity is represented by a single-layer potential. Pressure-driven flow. In this case, U1 ¼ 0; U2 ¼ 0; j ¼ 0; and gx ¼ 0: ð1Þ Equation (2.5.6) shows that DuR ¼ ð1 2 1=lÞuR which can be substituted into the master integral equation (2.5.15) to yield an integral equation for the total interfacial velocity, uj ðx0 Þ ¼

ð1Þ 2 2 SLP 12l uRj 2 Ij ðx0 ; Df D ; IÞ þ 2 1þl 1þl 1þl

 IjDLP – PV ðx0 ; u; IÞ:

ð2:5:18Þ

As in the case of shear-driven flow, when l ¼ 1; the disturbance velocity is represented by a single-layer potential.

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Gravity-driven flow. In this case, U1 ¼ 0; U2 ¼ 0; j ¼ 0; and x ¼ 0: Equation ð1Þ (2.5.6) shows that DuR ¼ ð1 2 d=lÞuR which can be substituted into the master integral equation (2.5.15) to yield an integral equation for the total interfacial velocity, uj ðx0 Þ ¼

1 þ d Rð1Þ 2 SLP 12l u 2 I ðx0 ; Df D ; IÞ þ 2 1þl j 1þl j 1þl  IjDLP – PV ðx0 ; u; IÞ 2 2

ð1Þ 1 2 d DLP – PV I ðx0 ; uR ; IÞ: 1þl j

ð2:5:19Þ

When l ¼ 1 and d ¼ 1; the disturbance velocity is represented by a single-layer potential.

2. Boundary-Element Methods The integral equation (2.5.15) can be solved accurately and efficiently by boundary-element methods. In the simplest implementation, one period of the interface I is divided into N boundary elements, and all functions defined over the interface are assumed to have constant values over the individual elements, yielding the approximate equation ð1Þ

uD j ðx0 Þ ¼ 2

N l 2 X DuRj ðx0 Þ 2 A ðx Þ½DfiD m 1 þ l m¼1 ijm 0 1þl

þ2

N N ð1Þ 12l X l X Bijm ðx0 Þ½uD B ðx Þ½DuRi m ; i m 2 2 1 þ l m¼1 1 þ l m¼1 ijm 0

ð2:5:20Þ where ½·m denotes an mth-element quantity, and summation over i ¼ 1; 2 is implied on the right-hand side. We have introduced the influence coefficients 1 ð G ðx; x0 ÞdlðxÞ; 4pm1 Em ij

ð2:5:21Þ

1 ðPV T ðx; x0 Þnk ðxÞdlðxÞ; 4p Em ijk

ð2:5:22Þ

Aijm ðx0 Þ ; and Bijm ðx0 Þ ;

where Em stands for the mth element.

Instability of Multi-Layer Channel and Film Flows

203

Mass conservation requires that the single-layer influence matrix satisfies the identity N X Aijm ðx0 Þ½ni m ¼ 0; ð2:5:23Þ m¼1

where ½ni m is the unit vector normal to the mth element evaluated at the element midpoint, and summation over i ¼ 1; 2 is implied on the left-hand side. Equilibrium requires that the x-component of the force exerted on the interface due to an array of point forces pointing in the y-direction vanish, and vice versa, N X m¼1

Bxym ðx0 Þ ¼ 0;

N X

Byxm ðx0 Þ ¼ 0:

ð2:5:24Þ

m¼1

Placing the evaluation point x0 at the middle of the qth boundary element, denoted by xM q ; and rearranging, we recast Eq. (2.5.20) into the form  N  N X ð1Þ 12l 2 X D dqm dij 2 2 Bijm ðxM Þ ½uD Aijm ðxM i m ¼ 2 q q Þ½Dfi m 1 þ 1 þ l l m¼1 m¼1 N h i X l R ð2:5:25Þ 2 d d þ 2Bijm ðxM q Þ ½Dui m ; 1 þ l m¼1 qm ij where dqm and dij are Kronecker’s deltas. Applying Eq. (2.5.25) at all element mid-points, we obtain a system of linear equations for the element disturbance velocities. Once the solution has been found, the position of nodes defining the boundary elements can be advanced in time using, for example, a Runge –Kutta method.

3. Finite-Volume Method for Surfactant Transport The convection– diffusion equation for the surfactant concentration must be integrated in time simultaneously with the equations describing the evolution of the interface. In the finite-volume method, the surfactant transport equation (A.1) is integrated over the mth boundary element defined by the m and m þ 1 end-nodes, yielding ð Em

ð ð dG dl ¼ w dG 2 ðut G Þmþ1 þ ðut G Þm 2 Gkun dl dt Em Em     ›G ›G þ Ds 2Ds ; ›l mþ1 ›l m

ð2:5:26Þ

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C. Pozrikidis

where Ds is the surfactant diffusivity. Next, the surfactant concentration, tangential and normal components of the velocity, and curvature are approximated with constant functions over each element, denoted by an overbar, yielding dGm Dlm ¼ w m ðGmþ1 2 Gm Þ 2 ðut G Þmþ1 þ ðut G Þm 2 GEm km u nm Dlm dt     ›G ›G þ Ds 2Ds : ›l mþ1 ›l m

ð2:5:27Þ

Moreover, the end-node values and their derivatives are expressed in terms of the element values by numerical interpolation and differentiation writing, for example,   ›G Gm ¼ w0;m Gm21 þ w1;m Gm ; ¼ c0;m Gm21 þ c1;m Gm ; ð2:5:28Þ ›l m where w0;m ; w1;m are interpolation coefficients, and c0;m ; c1;m are differentiation coefficients. Equation (2.5.27) then becomes dGm Dlm ¼ cm Gm21 þ am Gm þ bm Gmþ1 ; dt

ð2:5:29Þ

where the coefficients am ; bm ; and cm are defined in terms of the node velocity and surfactant diffusivity. The tridiagonal system of ordinary differential equations arising by applying Eq. (2.5.29) at all elements can be integrated in time using a standard method. For example, the first-order semi-implicit Euler method yields i Dt h n  nþ1 nþ1 cm Gm21 þ anm Gmnþ1 þ bnm Gmþ1 ¼ Gmn ; Gmnþ1 2 ð2:5:30Þ Dlm where the superscripts n and n þ 1 designate two successive time levels separated by the interval Dt: The system (Eq. (2.5.30)) can be solved efficiently at each time step using Thomas’s algorithm (Pozrikidis, 1998c).

4. Stokes Flow Simulations Pozrikidis (1997b) performed boundary-integral simulations of the two-layer Couette and Poiseuille flow in the absence of surfactants. Linear stability analysis predicts that small-amplitude interfacial waves are stable or neutrally stable in the presence or absence of surface tension. The simulations addressed situations where the magnitude of a disturbance is so large that non-linear effects may no

Instability of Multi-Layer Channel and Film Flows

205

Fig. 2.5.2. Evolution of the interface between two layers in horizontal Couette flow for viscosity ratio l ¼ 0:2; density ratio d ¼ 1; zero surface tension, unperturbed layer thickness ratio h2 =h1 ¼ 2; and wavelength L ¼ 4h: The simulations are based on the boundary-element method for Stokes flow.

longer be neglected, and the interface does not maintain a sinusoidal shape during the evolution. The results revealed that the finite-amplitude motion is unstable when the perturbation is sufficiently strong, and the morphology of the interfacial patterns emerging for the instability depend strongly on the ratio of the viscosities of the two fluids. Examples of evolutions in horizontal Couette and Poiseuille flow computed using the boundary-element method discussed in Section II.E.2 are shown in Figs. 2.5.2 and 2.5.3. Subsequently, Pozrikidis (1998a) studied the instability two-layer flow in an inclined channel that is open at both ends, in the absence of a mean pressure gradient. The gravity-driven flow is an interesting hybrid of the shear- and pressure-driven flow encompassing a rich family of unidirectional base flows upon which perturbations may grow or decay. The profile of the base flow is

Fig. 2.5.3. Same as Fig. 2.5.2, but for pressure-driven flow.

206

C. Pozrikidis

parametrized by the relative viscosities and densities of the two fluids. Linear stability analysis predicts that, in the limit of Stokes flow and when the fluids are stably stratified, the flow is either stable or neutrally stable under any conditions. The numerical simulations revealed permanent interfacial deformation for sufficiently strong initial perturbations. More recently, Blyth and Pozrikidis (2004) performed boundary-integral simulations in the presence of surfactants and confirmed the occurrence of an instability due to Marangoni tractions.

F. Numerical Simulations for Navier –Stokes Flow The significance of fluid inertia on the finite-amplitude motion has been investigated by several authors. Coward et al. (1997) and Li et al. (1998) presented numerical simulations of two-layer Couette flow at moderate and low Reynolds numbers using the VOF method. Their results confirmed the destabilizing influence of inertia, and revealed the occurrence of wave steepening and subsequent saturation due to finite-amplitude interactions. Similar numerical simulations for selected case-studies were presented by Yiu and Chen (1996), Zaleski et al. (1996), Tryggvason and Unverdi (1999), and Zhang et al. (2002). Pozrikidis (2005) recently developed a numerical method for simulating the motion in the presence of an insoluble surfactant. The algorithm combines Peskin’s immersed-interface method with the diffuse-interface approximation, wherein the step discontinuity in the fluid properties is replaced by a transition zone defined in terms of a mollifying function. A finite-difference method was employed for integrating the generalized Navier– Stokes equation incorporating the jump in the interfacial traction, and a finite-volume method was implemented for solving the surfactant transport equation over the evolving interface. Results for selected case studies suggest that the surfactant-induced Marangoni instability persists at non-zero Reynolds numbers, though inertial effects have a mild effect on the growth rates. Other authors simulated the dynamics of axisymmetric interfacial waves in core-annular flow. The core-annular flow is similar in many respects to the two-layer channel flow, though there are important differences associated with the destabilizing effect of surface tension in axisymmetric and threedimensional configurations. For example, Bai et al. (1996) computed steadily propagating waves under the assumption that the inner fluid viscosity is so large that the core translates as a rigid body. An updated review of the relevant literature can be found in the recent work of Kouris and Tsamopoulos (2002).

Instability of Multi-Layer Channel and Film Flows

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G. Experimental Experimental studies of the two-layer channel flow are limited. Charles and Lilleleht (1965) investigated stratified oil– water flow in a rectangular channel, and observed that the onset of interfacial waves coincides with transition to turbulence. Kao and Park (1972) established critical conditions for the onset of instability in turbulent pressure-driven flow. Nearly 15 years later, Yiantsios and Higgins (1988) compared the laboratory observations with theoretical predictions based on linear stability analysis. Han (1981, Chapter 8) discussed three-dimensional instabilities of multi-layer non-Newtonian flow and emphasized the effect of interfacial instability on the layer thickness non-uniformity and manufactured product quality. The most systematic experimental study of the two-layer Couette flow instability is due to Barthelet et al. (1995) who discovered that, above a critical shear rate, the interface develops a fundamental slowly growing long wave whose wave length is equal to the channel perimeter, and its harmonics.

H. Oscillatory, Non-Isothermal, and Non-Newtonian Flow The basic problem reviewed in this section has been generalized to include the effect of flow oscillations. Coward and Papageorgiou (1994) considered the behavior of long waves in two-layer Couette flow subject to small-amplitude oscillations generated by the in-plane vibrations of one of the walls, and Coward et al. (1997) studied the combined case of Couette –Poiseuille flow for arbitrary perturbations. A Floquet stability analysis revealed that the oscillations may have either a stabilizing or a destabilizing influence on the base flow. Other authors studied the stability of two-layer flow in the presence of heating or cooling causing viscosity variations perpendicular to the direction of the mean flow, and the significance of non-Newtonian fluid properties. The relevant literature is reviewed by Pinarbasi and Liakopoulos (1996) and Pinarbasi (2002).

III. Film Flow In the second part of this article, we consider the single- and multi-layer, gravity-driven film flow down an inclined or wavy wall. We begin in Section III.A by describing the structure of the single- and two-layer unidirectional flow, and outline an algorithm for computing the velocity profile across an arbitrary

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208

number of layers. In Section III.B, we discuss the linear stability in the limit of Stokes flow, and present a method of deducing the properties of the normal modes. To describe the non-linear motion for long interfacial waves, in Section III.C we formulate the problem in the context of the lubrication approximation and discuss stability analysis and numerical solutions. Finally, in Section III.D we outline a boundary-integral formulation that allows us to simulate the evolution of arbitrary waves.

A. Unidirectional Base Flow Consider the unidirectional flow of a single film down a plane that is inclined at the angle u0 with respect to the horizontal. Elementary derivations yield the Nusselt velocity profile ux ðyÞ ¼

rgx yð2h 2 yÞ: 2m

ð3:1:1Þ

This semi-parabolic profile is half the complete parabolic profile of pressure- or gravity-driven flow in a channel of width 2h; where the free surface is located at the centerline. The shear stress varies linearly from a certain value at the wall to the required value of zero at the free surface. The flow rate arises by integrating the velocity across the film, and is given by Q;

ðh 0

ux ðyÞdy ¼

grh3 2 ¼ hUs ; 3 3m

ð3:1:2Þ

where Us ; ux ðhÞ is the maximum velocity occurring at the free surface. In the case of two-layer flow, elementary derivations yield the composite Nusselt velocity profile

r1 gx h21 y^ ð2 þ 2dR 2 y^ Þ; 2m1     r2 gx h21 1 2 ¼ l 2R þ 2 2R 2 1 þ 2ð1 þ RÞ^y 2 y^ ; 2m2 d

uð1Þ x ¼ uð2Þ x

ð3:1:3Þ

where the bottom layer labeled 1 is located next to the plane, the top layer labeled 2 is exposed to the constant ambient pressure, h1 and h2 are the film thicknesses, y^ ¼ y=h1 ; R ¼ h2 =h1 ; l ¼ m2 =m1 ; and d ¼ r2 =r1 : The interfacial velocity arises from either one of the preceding equations by setting y^ ¼ 1: When d ¼ 1 and l ¼ 1; or when one of the film thicknesses h1 ; h2 vanishes, we recover the earlier results for single film flow.

Instability of Multi-Layer Channel and Film Flows

209

Consider now the more general case of N superposed liquid layers, as illustrated in Fig. 3.1.1. The bottom layer labeled 1 is located next to the inclined plane, and the top layer labeled N is exposed to the constant ambient pressure. The N 2 1 interfaces separating the films are located at y ¼ yi ; where i ¼ 1; 2; …; N 2 1; and the free surface is located at y ¼ yN : The density and viscosity of the ith fluid are denoted by ri and mi ; with the understanding that mNþ1 ¼ 0 and rNþ1 ¼ 0; the surface tension of the ith interface is denoted by gi : The no-slip boundary condition requires that the velocity vanish at the plane located at y ¼ 0; while the free-surface condition requires that the shear stress vanish at the free-surface located at y ¼ h: The velocity and pressure fields in the ith layer are governed by the following simplified components of the equation of motion, d2 uðiÞ rg x ¼2 i x; mi dy2

›pðiÞ ¼ ri gy : ›y

ð3:1:4Þ

Integrating the first equation in Eq. (3.1.4) twice with respect to y; we derive the parabolic profile uðiÞ x ðyÞ ¼ 2

ri gx 2 y þ Bi y þ Ai ; 2m i

ð3:1:5Þ

where Ai and Bi are unknown constants to be determined by requiring (a) the no-slip boundary condition at the inclined plane, (b) continuity of velocity at the interfaces expressed by Eq. (2.2.3), (c) continuity of shear stress at the interfaces expressed by Eq. (2.2.4), and (d) the condition of zero shear stress at the free surface; with reference to Eq. (2.2.6), a ¼ 0: Knowledge of the value of the shear stress at the free surface allows us to evaluate the coefficients Bi and Ai ; working as described in Section II.B for channel flow. The numerical method has been

Fig. 3.1.1. Multi-layer flow of a film down an inclined plane. The interface labels are shown on the right.

210

C. Pozrikidis

implemented in subroutine films located in directory 04_various of the software library FDLIB (Pozrikidis, 2001, 2002). One noteworthy property of the multi-layer film flow is that the wall shear stress and velocity profile across the first layer, adjacent to the wall, are independent of the viscosities of the rest of the films. To see this, we write the velocity profile (3.1.5) for i ¼ 1; and require the no-slip boundary condition to find A1 ¼ 0: To compute B1 ; we perform a force balance over the whole crosssection of the composite film confined between two parallel planes located at x ¼ x1 and x2 : The balance requires that the force exerted by the shear stress at the wall and at the free surface counterbalance the streamwise component of the weight of the fluid residing within the control volume. We note that the shear stress at the free surface is equal to zero, and obtain

sð1Þ xy ðy ¼ 0Þ ¼

N X

ri gx ðyi 2 yi21 Þ;

ð3:1:6Þ

i¼1

with the understanding that y0 ¼ 0 and yN ¼ h: Using the profile (3.1.5) for i ¼ 1; we find ! ›uð1Þ x ð1Þ sxy ðy ¼ 0Þ ¼ m1 ¼ m 1 B1 : ð3:1:7Þ ›y y¼0 Setting the right-hand side of Eq. (3.1.6) equal to the right-hand side of Eq. (3.1.7), solving for B1 ; and substituting the result into the profile (3.1.5) for i ¼ 1 proves the stated independence of the first velocity profile and wall shear stress on the viscosity of the overlying fluids.

B. Stability The stability of the single-film flow down an inclined plane has been the subject of numerous investigations following the seminal analysis of Yih (1963), as reviewed by Chang (1994) and Oron et al. (1997). The stability of the multifilm flow has also been studied on many occasions following the pioneering analysis of Yih (1967) for two-layer channel flow. Motivation has been provided by applications in multi-layer coating of photographic emulsions where a sharp contrast in the viscosities of the layers may lead to two- or three-dimensional wavy patterns that are detrimental to the quality of the manufactured product (Kistler and Schweizer, 1997). In photographic film manufacturing, as many as 13 films may flow down an inclined plane to be deposited onto a rapidly moving support.

Instability of Multi-Layer Channel and Film Flows

211

1. Single Film Flow In the limit of Stokes flow and in the absence of surfactants, a small-amplitude sinusoidal perturbation on the surface of a single film decays for any non-zero value of the interfacial tension. The interfacial wave is described by the equation y ¼ yI ðx; tÞ ¼ h þ a0 cos½kðx 2 cR tÞ expðsI tÞ;

ð3:2:1Þ

where h is the unperturbed film thickness, k ¼ 2p=L is the wave number, L is the wavelength, cR is the phase velocity, sI is the growth rate, and a0 is the initial amplitude of the perturbation. Yih (1963) derived the following expressions for the dimensionless phase velocity v ; cR =Us and dimensionless growth rate s ; sI h=Us ; v¼1þ

1 ; cosh k^ þ k^ 2 2

s¼2

^ 2 2k^ t sinhð2kÞ ; 2 2k^ cosh k^ þ k^ 2

ð3:2:2Þ

where k^ ¼ kh is the reduced wave number, Us ¼ rgx h2 =ð2mÞ is the unperturbed surface velocity, and

t ¼ cot u0 þ

2p2 Ca

is a composite dimensionless parameter. The Capillary number   mUs L 2 rgx L2 Ca ¼ ¼ ; h g 2g

ð3:2:3Þ

ð3:2:4Þ

also regarded as a Bond number, expresses the ratio between the magnitude of the viscous stresses, mUs =h; and the capillary pressure, hg0 =L2 ; where the surface curvature has been scaled with h=L2 : The first expression in Eq. (3.2.2) shows that long waves, corresponding to the limit k^ ! 0; travel with twice the fluid surface velocity. The solid lines in Fig. 3.2.1 display graphs of the reduced growth rate s ; sI h=Us ; plotted against the reduced wave number for inclination angle u0 ¼ p=4; and reduced surface tension g=ðrgh2 Þ ¼ 0 (thin line), 0.5. and 1.0 (thickest line). In the absence of fluid inertia, the growth rate is negative and the flow is stable under any conditions, even for zero surface tension. The dashed lines represent the analytical results for small wave numbers obtained using the lubrication approximation to be discussed in Section III.C. a. Effect of Surfactants The effect of an insoluble surfactant was discussed recently by Pozrikidis (2003). Linearizing the equations governing fluid flow and surfactant transport,

C. Pozrikidis

212

0

s

–1

–2

–3 0

0.5

1

1.5 kh

2

2.5

3

Fig. 3.2.1. Stability of single-film flow at vanishing Reynolds number: reduced growth rate s ; sI h=Us for inclination angle u0 ¼ p=4; and g=ðrgh2 Þ ¼ 0 (thin line), 0.5, and 1.0 (thickest line). The dashed lines represent the predictions of lubrication theory for long waves.

we find that the complex growth rate c ¼ cR þ icI arises by setting the determinant of the following matrix equal to zero, 2

2k^

1

1

0

3

7 6 6 2p2 i 2k^ 7 2 2 7 6 ð1 2 qÞðzk^ 2 þ 1Þ ^ ^ ^ ^ z þ z z 2 z z e k k þ 1 qð k k þ 1Þ 2 7 6 ^ 7 6 Ca k 7; N¼6 7 6 ^2 2 2 ^ 2 it ^ þ itÞ 7 6 zk ð1 þ qÞ 2 itð1 2 qÞ z 2qð z 0 k k 7 6 7 6   5 4 2pi ^ ^ þ qÞ ^ ^ Ma kð1 Mað1 þ kÞ Ma qð1 2 kÞ 2 z2 e2k Pe ð3:2:5Þ ^ Ma is the where z ¼ 1 2 c=Us ; ‘i’ is the imaginary unit, q ¼ expð22kÞ; Marangoni number defined in Eq. (A.13), Pe ¼ Us L=Ds is the surfactant Pe´clet number, and Ds is the surfactant diffusivity. Specifically, setting the determinant of N equal to zero provides us with a cubic algebraic equation for the computation of the shifted and reduced complex phase velocity z: One trivial root is given by z ¼ 0 corresponding to c ¼ Us ; and the other two roots may be computed in terms of the coefficients of the factorized binomial using the quadratic formula. In practice, the coefficients of the binomial are extracted by solving a system of linear equations for three trial values of z: Thus, in the presence of surfactants, the flow admits two normal modes.

Instability of Multi-Layer Channel and Film Flows

213

To illustrate the effect of the surfactants, we introduce the alternative Capillary number and property group  2 mUs h Ca ¼ ¼ Ca ; L g0 0

a0 ¼

g0 h ; mDs

ð3:2:6Þ

both defined using as length scale the film thickness alone. Figure 3.2.2(a) shows graphs of the reduced growth rate s for u0 ¼ p=4; Ca0 ¼ 2; a0 ¼ 100; and a range of values of the surfactant sensitivity parameter b defined in Eq. (A.12) and satisfying Ma ¼ b=ð1 2 bÞ: The dashed lines correspond to the first normal mode, and the solid lines correspond to the second normal mode. In the limit as k^ tends to infinity while Ca0 is held constant, the growth rate of the Yih mode behaves like

s.2

t k^ ; .2 2Ca0 k^

ð3:2:7Þ

while the growth rate of the complementary Marangoni mode behaves like k^ s.2 0 Ca

! Ma k^ : þ 2 a0

ð3:2:8Þ

The latter is described by the dotted lines in Fig. 3.2.2. The asymptotic functional ^ the forms shown in Eqs. (3.2.7) and (3.2.8) reveal that, to leading order in k; growth rate of the Yih mode is independent of the properties of the surfactant, that is, the presence of the surfactant has a negligible effect on the behavior of the corresponding normal mode. On the other hand, the growth rate of the Marangoni mode is determined by the surfactant diffusivity and is independent of the Marangoni number. The quadratic dependence of the Marangoni growth rate on k^ shown in Eq. (3.2.8) suggests that, unless Ds ¼ 0 and Ma , 1; this mode will decay faster than the Yih mode at sufficiently high wave numbers. Accordingly, the dashed lines in Fig. 3.2.2 (a) must switch from the Yih branch corresponding to Eq. (3.2.7) over to the Marangoni branch corresponding to Eq. (3.2.8) at a certain value of b; the converse behavior is expected for the solid lines. This transition is confirmed in the amplified scale of Fig. 3.2.2 (b) corresponding to b ¼ 0:5: The heavy broken line represents the growth rate of the Yih mode in the absence of surfactants, and the dotted line represents the asymptotic prediction (Eq. (3.2.8)).

C. Pozrikidis

214 0

s

–1

–2

–3

–4 0

1

2

3

4

5

6

kh

(a) –1.2

s

–1.4

–1.6

–1.8

–2 5 (b)

6

7

kh

Fig. 3.2.2. (a) Reduced growth rate s of surface waves for u0 ¼ p=4; Ca0 ¼ 2; a0 ¼ 100; and b ¼ 0:001 (heavy solid and broken lines), 0:1; 0:2; …; 0:9: The dotted lines represent the asymptotic form of the Marangoni mode. (b) Magnified view for b ¼ 0:5; showing the switching of branches of the two normal modes.

2. Two-Layer Flow Kao (1965a,b, 1968) first addressed the stability of the multi-layer configuration. Much later, Kobayashi and Scriven (1981, 1982 Spring AIChE Meeting) Kobayashi, Nohjo, Chino, and Yoshimura (1986 Spring AIChE Meeting), and Lin (1983) formulated and solved the linear stability problem for

Instability of Multi-Layer Channel and Film Flows

215

time and space growing perturbations, and compared their results with laboratory observations of film flow on a slide coating apparatus (see also Kobayashi, 1992, 1995). Their results suggested that the flow can be unstable even at vanishing Reynolds number. Loewenherz and Lawrence (1989) tackled the linear stability problem in the limit of Stokes flow for fluids of equal density and vanishing interfacial and freesurface tension. Their results showed explicitly that the two-film flow can be unstable when the less viscous fluid is next to the wall. Their work was subsequently generalized by Weinstein (1990) and Chen (1992) to nonNewtonian fluids and non-zero Reynolds numbers, and by Chen (1993) to a broader range of relative film thickness, non-zero surface tension, and non-zero Reynolds numbers. Chen (1993) emphasized that free-surface deformability is necessary for the growth of small perturbations and attributed the unstable behavior to a resonance between the interface and free-surface waves. When the free surface is maintained flat by infinite surface tension, the instability does not appear. More recently, Jiang et al. (2004) discussed the physical reason for the Stokes-flow instability on the basis of an energy budget, and compared their predictions with numerical solutions based on a spectralelement method. 3. Multi-Layer Flow Akhtaruzzaman et al. (1978), Wang et al. (1978), and Lin (1983) formulated the linear stability problem for an arbitrary number of layers and presented results for three-layer film flow, but did not identify instability. The three-layer flow was discussed in detail by Weinstein and Kurz (1991) and Weinstein and Chen (1999) who pointed out that instability is predicted even in the context of the lubrication approximation when the viscosity of the middle layer is lower than the viscosity of the layers at the top and next to the wall. In contrast, the lubrication formulation for two-film flow erroneously predicts that the motion is stable under any conditions. More recently, Kobayashi (1995) compared theoretical predictions with laboratory observations.

4. Identification of Normal Modes Consider interfacial waves on a composite film consisting of N superposed layers, as depicted in Fig. 3.1.1. Carrying out a normal-mode stability analysis,

216

C. Pozrikidis

we find that exponentially growing or decaying waves along the jth interface or free surface are described by the equations y ¼ yj ðx; tÞ ¼ Yj þ ANM;l ðtÞ cos½kðx 2 cNM;l tÞ 2 DfNM;l ; j j R

ð3:2:9Þ

for j ¼ 1; …; N; where Yj are the unperturbed positions, k ¼ 2p=L is the wave number, the superscript ‘NM’ stands for Normal Mode, cNM;l is the phase R velocity, and DfNM;l is the phase lag of the wave on the jth interface with respect j to the wave on the first interface, all corresponding to the lth normal mode. By definition then, DfNM;l ¼ 0: The amplitudes ANM;l ðtÞ are exponential functions of j 1 time given by ANM;l ðtÞ ¼ ANM;l ðt ¼ 0Þ expðsNM;l tÞ; j j I

ð3:2:10Þ

where sNM;l is the growth rate of the lth normal mode shared by all interfaces. I Negative, zero, and positive growth rate sNM;l implies, respectively, stable, I neutrally stable, and unstable behavior. In the case of Stokes flow, and only then, for each N-layer configuration there are N normal modes in the absence of surfactants, and 2N normal modes in the presence of surfactants. A complete description of the lth normal modes in the absence of surfactants requires specification of the following 2N quantities: † Ratio of the amplitudes and phase lags of the interfacial waves with respect to the first wave rjNM;l ;

ANM;l ðtÞ j ðtÞ ANM;l 1

;

; DfNM;l j

ð3:2:11Þ

for j ¼ 2; …; N: † Phase velocity cNM;l and growth rate sNM;l : R I For example, in the case of single-film flow, there is only one normal mode whose phase velocity and growth rate are given in Eq. (3.2.2). When surfactants are present, a set of unknowns corresponding to Eq. (3.2.11) must be introduced regarding the surfactant concentration along each interface. In computational studies of non-linear stability, it is both expedient and physically meaningful to specify an initial condition that corresponds to the most unstable normal mode. But since the complete set of properties of this mode are available in analytical form only in the case of the single-film flow, a method of extracting them from the results of a numerical simulation with

Instability of Multi-Layer Channel and Film Flows

217

arbitrary initial conditions in the limit of small interfacial deformations is required. We begin by considering the motion when the interfaces and free surface are perturbed with arbitrary sinusoidal small-amplitude waves whose relative amplitude and phase shift does not necessarily correspond to a normal mode. Without loss of generality, we can assume that, at the origin of time, the interfaces and free surface are described by the equations y ¼ yj ðx; t ¼ 0Þ ¼ Yj þ Aj cosðkx 2 fj Þ;

ð3:2:12Þ

for j ¼ 1; …; N; where fj are the phase shifts with respect to the designated origin of the x-axis. One way to describe the evolution of these waves is to express them as a linear combination of the N normal modes, writing yj ðx; t ¼ 0Þ ¼ Yj þ

N X

ANM;l ðt ¼ 0Þ cosðkx 2 fNM;l Þ: j j

ð3:2:13Þ

l¼1

The arguments of the trigonometric functions in Eq. (3.2.13) evolve according to Eq. (3.2.9), whereas the amplitude functions ANM;l ðtÞ evolve according to j Eq. (3.2.10). Combining these expressions, we derive the following evolution equation for arbitrary linear waves in terms of the properties of the normal modes, yj ðx;tÞ ¼ Yj þ

N X

ANM;l ðt ¼ 0Þcos½kðx2cNM;l tÞ2 fNM;l expðsNM;l tÞ j j R I

l¼1

¼ Yj þ

N X

ANM;l ðt ¼ 0Þcos½kx2ðsNM;l t þ fNM;l ÞexpðsNM;l tÞ j j R I

l¼1

"

¼ Yj þ cosðkxÞ

N X

ANM;l ðt ¼ 0ÞcosðsNM;l t þ fNM;l Þ j j R

l¼1

þsinðkxÞ

N X

ANM;l ðt ¼ 0ÞsinðsNM;l t þ fNM;l Þ j j R

# expðsNM;l tÞ; I

ð3:2:14Þ

l¼1

¼ kcNM;l : To compute the requisite 2£N 2 constants ANM;l ðt ¼ 0Þ where sNM;l j R R NM;l and fj for l ¼ 1;…;N and j ¼ 1;…;N; we set the right-hand side of Eq. (3.2.13) equal to the right-hand side of Eq. (3.2.14), and require that sums of coefficients of like trigonometric functions balance to zero, thereby obtaining a system of 2N algebraic – trigonometric equations. If the normalmode wave amplitude ratios and phase shifts DfNM;l defined in Eq. (3.2.11) j were available for l ¼ 1;…;N and j ¼ 2;…;N 21; we could have invoked their

C. Pozrikidis

218

definition to obtain an additional set of 2NðN 21Þ equations, thereby matching the number of the unknowns the total number of equations. Confronted with the unavailability of the complete properties of the normal modes, we develop a method of extracting them from a numerical solution subject to arbitrary small-amplitude sinusoidal perturbations. As a first step, we express Eq. (3.2.14) in the form yj ðx; tÞ ¼ Yj þ Fj;c ðtÞ cos kx þ Fj;s ðtÞ sin kx;

ð3:2:15Þ

where Fj;c are cosine Fourier coefficients, given by Fj;c ¼

N X

ANM;l ðt ¼ 0Þ cosðsNM;l t þ fNM;l Þ expðsNM;l tÞ; j j R I

ð3:2:16Þ

l¼1

and Fj;s are sine Fourier coefficients given by Fj;s ¼

N X

ANM;l ðt ¼ 0Þ sinðsNM;l t þ fNM;l Þ expðsNM;l tÞ: j j R I

ð3:2:17Þ

l¼1

The key idea is that these coefficients may be expressed as sums of 2N complex exponentials, and the 2N 2 real unknowns, ANM;l ðt ¼ 0Þ; fNM;l ; may be recovered j j by performing N-mode complex exponential fitting using a method developed by Prony (e.g., Hildebrand, 1974, pp. 457– 463; Kay and Marple, 1981; Marple, 1987, pp. 303– 349). To implement the method, we write

Fj;c ¼

N p 1X NM;lp ½cðlÞ expð2isNM;l tÞ þ cðlÞ tÞ; j;c expð2is 2 l¼1 j;c

N p 1X NM;lp Fj;s ¼ ½cðlÞ expð2isNM;l tÞ þ cðlÞ tÞ; j;s expð2is 2 l¼1 j;s

ð3:2:18Þ

where cj;c ; cj;s are complex coefficients, sNM;l are complex growth rates, and an asterisk denotes the complex conjugate. Comparing Eqs. (3.2.16) and (3.2.17) with Eq. (3.2.18), we find NM;l ðt ¼ 0Þ expð2ifðlÞ cðlÞ j;c ¼ Aj j Þ;    p NM;l ðlÞ cðlÞ ¼ A ðt ¼ 0Þ exp 2i f 2 : j j;s j 2

ð3:2:19Þ

Instability of Multi-Layer Channel and Film Flows

219

Assume now that we have available a time series for Fj;c ðtÞ with constant sampling time Dt; and denote, for brevity, zq ; Fj;c ðqDtÞ;

ð3:2:20Þ

where q ¼ 0; 1; 2; … Prony’s method proceeds in five stages: (1) At the first stage, we solve the following generally overdetermined linear system of M £ 2N equations 2 6 6 6 6 6 6 6 4

z1

z2

···

z2

z3

···

.. .

.. .

zM

zMþ1

z2N

32

a2N

3

2

z2Nþ1

3

76 7 7 6 7 7 6 6z z2Nþ1 7 76 a2N21 7 6 2Nþ2 7 76 7 7 6 76 . 7 ¼ 26 . 7; .. .. 76 . 7 6 . 7 76 . 7 6 . 7 . . 54 5 5 4 a · · · z2NþMþ1 z2NþM 1

ð3:2:21Þ

for the 2N real unknowns a1 ; a2 ; …; a2N ; where M $ 2N: (2) At the second stage, we compute the roots of the 2N-degree characteristic polynomial P2N ðzÞ ¼ z2N þ a1 z2N21 þ a2 z2N22 þ · · · þ a2N21 z þ a2N ;

ð3:2:22Þ

call them z1 ; z2 ; z3 ; …; z2N : Since the coefficients of P2N ðzÞ are real, complex roots appear in pairs of complex conjugate values, ðz1 ; z2 ¼ zp1 Þ;

ðz3 ; z4 ¼ zp3 Þ;

…;

ðz2N21 ; z2 N ¼ zp2N21 Þ:

ð3:2:23Þ

(3) At the third stage, we extract the complex growth rates from the relations p

z2 ¼ expðisNM;1 DtÞ;

z3 ¼ expð2isNM;2 DtÞ;

z4 ¼ expðisNM;2 DtÞ;

z2N21 ¼ expð2isNM;N DtÞ;

p

ð3:2:24Þ

…

…

z1 ¼ expð2isNM;1 DtÞ;

p

z2N ¼ expðisNM;N DtÞ

(4) Next, we recover the requisite coefficients cðlÞ j;c by solving a generally overdetermined linear system that arises by applying the first equation in Eq. (3.2.18) at the data points, using the complex growth rates computed in step 3.

220

C. Pozrikidis

(5) Finally, we recover the initial normal mode amplitudes ANM;l ðt ¼ 0Þ and j ðlÞ phase shifts fj from the Euler decomposition (3.2.19).

There is an intrinsic ambiguity in the definition of the complex growth rates s ; stemming from our freedom to interchange the complex conjugate roots on the left-hand sides of Eq. (3.2.24), which amounts to replacing s NM;l with its complex conjugate on the right-hand side. To eliminate this ambiguity, we also perform the Prony fitting of the sine coefficient Fj;s ðtÞ by introducing a time series for Fj;s ðtÞ; setting zq ; Fj;s ðqDtÞ; and working in a similar fashion with the second ðt ¼ 0Þ; instead of the first equations in Eq. (3.2.18) to recover an 2N-tuple ANM;l j ðlÞ NM;l fj ; for l ¼ 1; …; N; for each interface. The proper values of s are the ones ðlÞ ðt ¼ 0Þ; f ; for l ¼ 1; …; N; computed from that give the same values of ANM;l j j the two Prony fittings. Combining the results for the interfaces and for the free surface, we deduce the wave amplitude ratio and phase lag of the normal modes. The procedure was applied by Pozrikidis (1998a,b) for two-layer flow using numerical data obtained by the boundary-integral method described in Section III.D, and for three-layer flow using a lubrication flow model described on Section III.C. A similar application for two-layer channel flow in the presence of surfactants was reported by Blyth and Pozrikidis (2004). The success of the results encourage the broader application of the method to other areas of hydrodynamic stability for the purpose of identifying the properties of the normal modes from measurements or using the results of numerical computation. NM;l

C. Lubrication-Flow Model for Long Waves Consider the flow of N superposed layers down an inclined plane wall with small-amplitude periodic undulations of wavelength L; as illustrated in Fig. 3.3.1. The position of the wall is described by the equation y ¼ y0 ðxÞ; and the position of the ith interface is described by the equation y ¼ yi ðx; tÞ; where i ¼ 1; …; N: We shall assume that the wall, interface, and free-surface slopes dyi =dx are uniformly small, and the effect of fluid inertia is insignificant inside all layers. When these conditions are met, we may work under the auspices of lubrication-flow theory applicable for long waves, and approximate the generally two-dimensional flow within each film with a locally unidirectional flow having a parabolic velocity profile uðiÞ x for i ¼ 1; …; N (e.g., Pozrikidis, 1997a).

Instability of Multi-Layer Channel and Film Flows

221

Fig. 3.3.1. Schematic illustration of multi-layer film flow down an inclined plane wall with small-amplitude undulations. The interfaces are labeled with bold numbers shown on the left.

The governing equations of Stokes flow simplify to

›pðiÞ d2 uðiÞ x ¼ mi þ ri gx ; ›x dy2

›pðiÞ ¼ ri gy ; ›y

ð3:3:1Þ

for i ¼ 1; …; N; where pðiÞ is the ith layer pressure field, and gx ¼ g sin u0 ; gy ¼ 2g cos u0 are the components of the acceleration of gravity. The following kinematic and dynamic boundary conditions are required: † Continuity of the streamwise velocity across the interfaces, ðiþ1Þ uðiÞ ðy ¼ yi Þ; x ðy ¼ yi Þ ¼ ux

ð3:3:2Þ

uð1Þ x ðy ¼ y0 Þ ¼ 0:

ð3:3:3Þ

for i ¼ 1; …; N 2 1: † No-slip at the wall,

† Continuity of shear stress at the interfaces, ! ! ›uðiÞ ›uxðiþ1Þ x ¼ li ; ›y y¼y ›y y¼y i

ð3:3:4Þ

i

for i ¼ 1; …; N; where li ; miþ1 =mi are the viscosity ratios, with the understanding that lN ¼ 0: † Normal stress balance at the interfaces, pðiÞ ðy ¼ yi Þ ¼ pðiþ1Þ ðy ¼ yi Þ þ gi ki ;

ð3:3:5Þ

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222

for i ¼ 1; …; N; where gi is the tension of the ith interface, ki ¼ 2y00i =ð1 þ y02i Þ3=2 is the curvature of the ith interface, and a prime denotes a derivative with respect to x: It is reckoned that pðNþ1Þ is equal to the constant ambient pressure prevailing above the free surface. The statement of the problem is now complete, and we may proceed to formulate the solution. For simplicity, time t will be omitted in the list of dependent variables in the following discussion. First, we integrate the second equation in Eq. (3.3.1) with respect to y and use the normal stress condition (3.3.5) to derive an expression for the pressure distribution within the ith layer, pðiÞ ðx; yÞ ¼ pðiþ1Þ ðx; y ¼ yi Þ þ gi ki þ ri gy ½y 2 yi ðxÞ;

ð3:3:6Þ

for i ¼ 1; …; N: Differentiating Eq. (3.3.6) with respect to x; we obtain the recursion relation

›pðiÞ ›pðiþ1Þ ›pðiþ1Þ ›yi ›k ›y ¼ þ þ gi i 2 ri gy i ; ›x ›x ›y ›x ›x ›x

ð3:3:7Þ

with the understanding that ›pðNþ1Þ =›y ¼ 0: Using the second equation in Eq. (3.3.1) to evaluate the y derivative on the right-hand side, and rearranging, we find

›pðiÞ ›pðiþ1Þ ›k ›y 2 ¼ gi i 2 ðri 2 riþ1 Þgy i ; ›x ›x ›x ›x

ð3:3:8Þ

for i ¼ 1; …; N; with the understanding that rNþ1 ¼ 0: Algebraic manipulation produces the preferred form 2

N N X ›kj X ›y j ›pðiÞ ¼ 2 gj þ : ðrj 2 rjþ1 Þgy ›x ›x ›x j¼i j¼i

ð3:3:9Þ

Now, the velocity profile within the ith layer arises by integrating the first equation in Eq. (3.3.1), finding 2 uðiÞ x ¼ Ai ðxÞ þ Bi ðxÞy 2 Gi ðxÞy ;

ð3:3:10Þ

where Ai ðxÞ and Bi ðxÞ are to be determined by imposing boundary conditions, and ! 1 ›pðiÞ Gi ðxÞ ; þ ri gx : 2 2m i ›x

ð3:3:11Þ

Instability of Multi-Layer Channel and Film Flows

223

The right-hand side of Eq. (3.3.11) may be evaluated from knowledge of the instantaneous interfacial profiles using Eq. (3.3.9). In particular, when the interfaces are flat, the functions Gi ðxÞ are all constant. Integrating the profiles (3.3.10) over their domain of definition, we find that the streamwise flow rate within the ith layer is given by Qi ;

ð yi yi21

uðiÞ x dy ¼ Ai ðxÞðyi 2 yi21 Þ þ 2

1 B ðxÞðy2i 2 y2i21 Þ 2 i

1 G ðxÞðy3i 2 y3i21 Þ: 3 i

ð3:3:12Þ

To evaluate the functions Ai ðxÞ and Bi ðxÞ; we use the interfacial and wall conditions Eqs. (3.3.2) – (3.3.4), obtaining Ai ðxÞ þ Bi ðxÞyi 2 Gi ðxÞy2i ¼ Aiþ1 ðxÞ þ Biþ1 ðxÞyi 2 Giþ1 ðxÞy2i ;

ð3:3:13Þ

for i ¼ 1; …; N 2 1; A1 ðxÞ þ B1 ðxÞy0 2 G1 ðxÞy20 ¼ 0;

ð3:3:14Þ

Bi ðxÞ 2 2Gi ðxÞyi ¼ li ½Biþ1 ðxÞ 2 2Giþ1 ðxÞyi ;

ð3:3:15Þ

and

for i ¼ 1; …; N: A straightforward rearrangement can be made to replace the recursion relation (3.3.15) with the explicit formula Bi ðxÞ ¼ 2Gi ðxÞyi þ 2

N X mk G ðxÞðyk 2 yk21 Þ; mi k k¼iþ1

ð3:3:16Þ

for i ¼ 1; …; N: Once the functions Bi have been evaluated, the function A1 ðxÞ follows from Eq. (3.3.14), and the rest of the functions Ai ðxÞ follow from Eq. (3.3.13). A mass balance for each film requires

›Qi ›ðyi 2 yi21 Þ þ ¼ 0; ›x ›t

ð3:3:17Þ

for i ¼ 1; …; N: Combining these expressions, we find i X ›Q j ›y i ¼2 : ›t ›x j¼1

ð3:3:18Þ

Substituting Eq. (3.3.9) into Eq. (3.3.11) and the result into Eq. (3.3.10) and (3.3.12), and then inserting the resulting expressions into Eq. (3.3.18), we obtain

C. Pozrikidis

224

a system of fourth-order non-linear partial differential equations governing the evolution of the interfaces and free surface.

1. Single-Film Flow In the case of a homogeneous film consisting of a single layer, N ¼ 1, and a flat support located at y0 ¼ 0; we readily compute B1 ðxÞ ¼ 2G1 ðxÞy1 and A1 ðxÞ ¼ 0; and derive the non-linear evolution equation " #! ›y 1 rgx › 3 ›y 1 L2 ›3 y1 ¼2 y 1 2 cot u0 þ ; ›t 3 m ›x 1 ›x 2Ca ›x3

ð3:3:19Þ

where L is a specified length, Ca is the Capillary number Ca ¼

  mUs L 2 rgx L2 ¼ ; h g 2g

ð3:3:20Þ

and h is the unperturbed film thickness. For convenience, we have partially suppressed the subscript 1 denoting the first layer. To study the linear evolution of long waves, we set y1 ðx; tÞ ¼ hð1 þ e  exp½ikðx 2 ctÞÞ; where e is a dimensionless coefficient whose magnitude is much less than unity, k ¼ 2p=L is the wave number, L is the period, and c is the complex phase velocity. Substituting this expression into Eq. (3.3.19) and linearizing with respect to e ; we find ^ c ¼ 2Us ð1 2 i 13 tkÞ;

ð3:3:21Þ

where k^ ¼ kh; and the dimensionless group t was defined previously in Eq. (3.3.23) as t ¼ cot u0 þ 2p2 =Ca: The phase velocity and growth rate of long waves are thus given by cR ¼ 2Us ;

sI ; kcI ¼ 2

2 Us ^ 2 tk ; 3 h

ð3:3:22Þ

which are the asymptotic forms of Yih’s general expressions (3.2.2) arising in the limit as the reduced wave number kˆ tends to zero. The difference between the asymptotic prediction for the growth rate represented by the dashed line in Fig. 3.2.1, and Yih’s exact expression is less than 1% when the reduced wave number k^ is approximately less than 0.05.

Instability of Multi-Layer Channel and Film Flows

225

2. Two-Layer Flow In the case of two films, N ¼ 2; and a flat support located at y0 ¼ 0; we compute B2 ðxÞ ¼ 2G2 ðxÞy2 ; A1 ðxÞ ¼ 0;

B1 ðxÞ ¼ 2G1 ðxÞy1 ðxÞ þ 2lG2 ðxÞðy2 2 y1 Þ;

ð3:3:23Þ

A2 ðxÞ ¼ G1 ðxÞy21 þ G2 ðxÞy1 ½2lðy2 2 y1 Þ 2 2y2 þ y1 ;

where l ¼ m2 =m1 is the viscosity ratio. The flow rates are given by   Q1 ¼ y31 23 G1 y1 þ lG2 ðy2 2 y1 Þ ;  Q2 ¼ ðy2 2 y1 Þ G1 y21 þ 2G2 ðy2 2 y1 Þ½ly1 þ

1 3

 ðy2 2 y1 Þ ;

ð3:3:24Þ

where

rg G1 ¼ 2 x 2m 1

G2 ¼

r2 gx 2m 2

"

! g1 ›3 y1 ›3 y2 þ g2 ›x3 ›x 3      ›y 2 1 ›y 1 1 2 21 ; 2cot u0 þ ›x d ›x d ! g2 ›3 y2 ›y2 þ1 ; 2 cot u0 r2 gx ›x 3 ›x

g2 r2 gx

ð3:3:25Þ

and d ¼ r2 =r1 is the density ratio. The evolution of the interface and the free surface is described by the equations

›y 1 ›Q ¼2 1; ›t ›x

›y 2 ›Q ›Q2 ¼2 1 2 : ›t ›x ›x

ð3:3:26Þ

3. Numerical and Asymptotic Solutions The system of partial differential equations developed in this section may be solved for an arbitrary number of layers using standard finite-difference methods. Pozrikidis (1998a,b) presented numerical solutions for two- and three-layer flow down an inclined plane, and analyzed the results of the simulations for small amplitude waves using Prony’s method of exponential fitting to deduce the properties of the normal modes. As an example, Fig. 3.3.2 shows the initial and an advanced stage in the evolution of a four-layer film down an inclined plane, illustrating the possible amplification of perturbations. Asymptotic forms of evolution equations for small-amplitude long waves were derived and discussed by Kliahandler and Sivashinsky (1997) and Kliakhandler (1999).

C. Pozrikidis

226

Fig. 3.3.2. Evolution of a four-layer film down an inclined plane computed using the lubrication approximation applicable for long waves; initial (left) and an advanced (right) stage of the motion.

D. Boundary-Integral Formulation for Stokes Flow The non-linear evolution of interfacial waves down an inclined plane can be computed efficiently using the boundary-integral formulation (Pozrikidis, 1998a). To develop the integral formulation, we decompose the velocity and pressure within the ith layer into a reference and a disturbance component, denoted, respectively, by the superscripts ‘R’ an ‘D’, writing uðiÞ ¼ uRðiÞ þ uDðiÞ ;

pðiÞ ¼ pRðiÞ þ pDðiÞ ;

ð3:4:1Þ

where i ¼ 1; …; N: The reference pressure is chosen to be constant and equal to the ambient pressure, whereas the reference velocity is defined as

uxRðiÞ ¼

ri g x yð2h 2 yÞ; 2m i

uyRðiÞ ¼ 0;

ð3:4:2Þ

where h is the total film thickness in unidirectional flow. Note that the reference velocities are related by uRðiÞ ¼ ðmk =mi Þðri =rk ÞuRðkÞ : Following a standard procedure, we find that the velocity at a point x0 located at the mth interface or free surface, m ¼ 1; …; N; satisfies the

Instability of Multi-Layer Channel and Film Flows

227

integral equation 1 þ dm RðmÞ u ðx0 Þ 1 þ lm j N ð 1 2 X 2 ðDf ðlÞ 2 DfiRðlÞ ÞGij ðx;x0 ÞdlðxÞ 4pmm 1 þ lm l¼1 Il i

uðmÞ j ðx0 Þ ¼

þ

N ðPV

 1 1 X ml ð1 2 ll Þui ðxÞ 2 ð1 2 dl Þui ðxÞRðlÞ ðxÞ 2pmm 1 þ lm l¼1 Il

 Tijk ðx; x0 Þnk ðxÞdlðxÞ;

(3:4:3Þ

where Il denotes one period of the lth interface or free surface, subject to the following definitions: † li ¼ miþ1 =mi and di ¼ riþ1 =ri are the viscosity and density ratios, with the understanding that lN ¼ 0 and dN ¼ 0: † Gij ðx;x0 Þ is the periodic Green’s function of two-dimensional Stokes flow in a semi-infinite domain bounded by a plane wall, and Tijk ðx;x0 Þ is the associated stress tensor. The four scalar components of Gij ðx;x0 Þ represent the velocity at the point x induced by a periodic array of point forces that are deployed along the x-axis above the plane wall, and are separated by the distance L which is equal to the period of the film flow; one of the point forces is located at the point x0 : As the observation point x moves away from the wall, all components of Gij and Tijk tend to vanish with the exception of Gxx that tends to a constant value dependent on the distance of the point forces from the wall. As the point forces approach the wall, this constant tends to vanish. A FORTRAN program that produces the Green’s function is available in directory 06_stokes/sgf_2d of the fluid dynamics software library FDLIB (Pozrikidis, 2001, 2002). † The unit vector n is normal to the lth interface and points into the underlying lth layer. † The strength density of the single-layer potential involves (a) the jump in traction across the lth interface and free surface corresponding to the reference flow, given by " # gy gx RðlÞ Df ¼ rl ð1 2 dl Þðy 2 hÞ ·n; ð3:4:4Þ gx gy and (b) the jump in traction due to the surface tension given by Df ðlÞ ¼ gl kðlÞ n;

ð3:4:5Þ

228

C. Pozrikidis

where k is the curvature of the interface or free surface in the xy-plane, reckoned to be positive when the normal vector n points away from the center of curvature. † The qualifier PV; signifying the principal value of the double-layer potential, applies only when l ¼ m: Writing Eq. (3.4.3) for m ¼ 1; …; N; we obtain a system of linear integral equations of the second kind for the interfacial velocities. To describe the flow of a single film, we set N ¼ 1; l1 ¼ 0; and d1 ¼ 0; and obtain an integral equation for the disturbance velocity defined over one period of the free surface I; 1 ð uD ðDfiðlÞ 2 DfiRðlÞ ÞGij ðx; x0 ÞdlðxÞ j ðx0 Þ ¼ 2 4pm I 1 ðPV D þ u ðxÞTijk ðx; x0 Þnk ðxÞdlðxÞ: ð3:4:6Þ 2p I i Pozrikidis (1998a) discusses the spectrum of eigenvalues of the double-layer operator on the right-hand side of Eq. (3.4.6). Pozrikidis (1998a) studied the two-layer gravity-driven channel flow and the two-layer film flow down an inclined plane. The reason for considering these two flows alongside is that they both reduce to the single-film flow down an inclined plane in special limits. In the case of channel flow, the single-film flow emerges when the density and viscosity of one of the layers are negligible compared to those of the other layer. In the case of film flow, singe-film flow emerges when the thickness of the upper film vanishes, or the physical properties of the two fluids are the same and the interfaces have identical tension. In spite of this similarity, the channel flow is stable in the limit of Stokes flow, whereas the film flow is unstable when the viscosity of the upper fluid is higher than that of the fluid next to the wall. The simulations confirmed that free-surface deformability is necessary for the growth of unstable waves.

IV. Discussion Multi-layer channel and film flows are susceptible to three general types of instability: a shear-flow inertial instability similar to that arising in a homogeneous fluid; an interfacial instability associated with viscosity stratification; and an instability due to Marangoni tractions arising in the presence of surfactants. The second and third types of instability arise even under conditions of Stokes flow provided that the ratios of the layer viscosities lie within certain

Instability of Multi-Layer Channel and Film Flows

229

ranges, the interfacial tensions are sufficiently small but not necessarily equal to zero, and the Marangoni number expressing the sensitivity of the surface tension to the surfactant concentration falls within a certain range. The presence of multiple interfaces is necessary for the onset of the inertialess instability in the absence of surfactants: in Stokes flow, the two-layer channel flow is stable, whereas the two-layer film flow down an inclined plane is unstable when the viscosity of the fluid next to the plane is lower than that of the overlying fluid. In this article, a tutorial on several aspects of multi-layer channel and film flow has been given, and recent advances on several topics have been reviewed. A satisfactory understanding of the multi-layer instability requires further progress on several fronts: linear stability analysis including the effect of soluble surfactants for Stokes and Navier – Stokes flow, effect of non-Newtonian fluid properties, investigation of the instability for thin layers with viscoelastic properties resembling membranes, and significance of non-planar wall geometry.

Acknowledgments This work has been supported by a grant provided by the National Science Foundation. I am grateful to Mark G. Blyth for insightful comments on the manuscript. Appendix A: Surfactant Transport When an interface is populated with a surfactant, the surface tension, g; is a function of the local surface surfactant concentration, G; defined as the number of moles per unit surface area of the interface. In reality, a thin surfactant-free sublayer of thickness h ¼ G=cs lines the interface on either side, where cs is the bulk surfactant concentration evaluated at the edge of the sublayer. Inside and across this sublayer, surfactant is depleted by adsorption and is released to the bulk of the fluids by desorption at rates that are determined by appropriate adsorption– desorption kinetics. Far from the sublayer, the bulk surfactant concentration evolves according to the usual convection –diffusion equation for the concentration of a passive species, as will be discussed in Appendix A.2. Accounting for the effect of a surfactant thus requires the integrated description of interfacial and bulk surfactant transport, as well as the availability of a constitutive equation relating the surface tension to the surface surfactant concentration.

230

C. Pozrikidis A.1. Interface Transport

Consider surfactant transport over a two-dimensional interface in the xy-plane, as illustrated in Fig. A.1. Mass conservation requires that the evolution of G is governed by the inhomogeneous convection– diffusion equation dG ›ðut G Þ ›G ›2 G þ 2w ¼ 2Gkun þ Ds 2 þ qn ; dt ›l ›l ›l

ðA:1Þ

(e.g., Li and Pozrikidis, 1997; Pozrikidis, 1998a –c, 2001; Yon and Pozrikidis, 1998) subject to the following definitions: † ut ¼ u·t is the tangential velocity along the interface, t is the tangential unit vector, and l is the arc length measured in the direction of t: † The derivative d=dt on the left-hand side of Eq. (A.1) expresses the rate of change of a variable following interfacial marker points moving with the component of the fluid velocity normal to the interface, and with an arbitrary position-dependent tangential component given by wðlÞt: When the marker points move normal to the interface, w ¼ 0; whereas when the marker points are Lagrangian point particles moving with the fluid velocity, w ¼ ut : The second and third terms on the left-hand side of Eq. (A.1) express convective transport. † un ¼ u·n is the normal velocity, n is the normal unit vector, and k is the interface curvature reckoned to be positive when the normal vector n points away from the center of curvature, as shown in Fig. A.1. The first term on the right-hand side of Eq. (A.1) expresses changes in the surfactant concentration due to interface compression or dilation. † Ds is the surface surfactant diffusivity, that is, the diffusivity over the interface. The second term on the right-hand side of Eq. (A.1) expresses diffusion over the generally curved interface.

Fig. A.1. Illustration of surfactant transport along a two-dimensional interface.

Instability of Multi-Layer Channel and Film Flows

231

† qn is the net flux of the fluid from the bulk of the fluids toward the interface on either side. At steady state and when w ¼ 0; we obtain the simplified equation

›ðut G Þ ›2 G ¼ Ds 2 þ qn : ›l ›l

ðA:2Þ

expressing a balance between interfacial convection, in-plane diffusion, and flux toward the stationary interface. An equation similar to Eq. (A.1) involving the mean curvature can be written for transport over a three-dimensional interface (e.g., Li and Pozrikidis, 1997; Yon and Pozrikidis, 1998).

A.2. Bulk Transport The concentration of the surfactant in the bulk of the fluids, c; satisfies the convection– diffusion equation

›c þ u · 7c ¼ Db 72 c; ›t

ðA:3Þ

where Db is the bulk surfactant diffusivity. A mixed interfacial condition for c arises by balancing (a) the diffusive fluxes normal to the interface at the edges of the sublayer on either side, given by qD^ ¼ ^Db n · 7c;

ðA:4Þ

where the plus sign applies for the side on the interface in the direction of the normal vector, and the minus sign for the other side, and (b) the flux due to adsorption and desorption occurring across the sublayer, denoted by qa=d : The normal flux shown on the right-hand side of Eq. (A.1) is given by qn ¼ qDþ þ qD2 ; and mass conservation requires qn ¼ qa=d :

ðA:5Þ

Assume, for illustration, that transport occurs only on the side of the interface indicated by the direction of the normal vector. If the adsorption – desorption process is described by first-order kinetics corresponding to Henry’s isotherm, to be discussed in Section A.3.1, qa=d ¼ Ka cs 2 Kd G;

ðA:6Þ

where Ka and Kd are, respectively, adsorption and desorption rate constants, and the subscript ‘s’ denotes evaluation at the edge of the sublayer. Combining

232

C. Pozrikidis

Eqs. (A.4) and (A.5) with Eq. (A.6), we obtain the interfacial condition n · 7c ¼

Kd ðKcs 2 G Þ; Db

ðA:7Þ

where the normal derivative on the left-hand side is evaluated at the edge of the sublayer. The constant K ¼ Ka =Kd is the partition coefficient at equilibrium. When the interfacial transport is absorption –desorption controlled, the bulk concentration is nearly uniform, and the concentration cs on the right-hand side of Eq. (A.7) is nearly constant. On the other hand, when the interfacial transport is diffusion-controlled, the surface concentration, G; is at equilibrium with the bulk concentration at the edge of the sublayer, cs ; and the surfactant is partitioned according to an equilibrium isotherm, as will be discussed in Section A.3. Insoluble surfactants arise under two conditions: the time scale of the adsorption– desorption flux, qa=d ; is long compared to the time scale of the hydrodynamics; or the diffusive flux qD is negligible compared to the flux due to surface convection (Pawar and Stebe, 1996). In both cases, the normal flux qn may be neglected on the right-hand side of the surface transport equation (A.1).

A.3. Gibbs Equation and Constitutive Laws Gibbs’ adsorption equation provides us with a rational point of departure for deriving a relation between the surface tension and the surfactant concentration at equilibrium (e.g., Adamson, 1990; Chang and Franses, 1995). In the case of a non-ionic or uni-univalent ionic surfactant, thermodynamics requires that quasistationary (equilibrium) changes in the bulk and surface concentration of the surfactant at constant temperature occur such that dg ¼ 2RT G

dcs ; cs

ðA:8Þ

where R is the ideal gas constant and T is the absolute temperature. Adopting an adsorption model that relates the bulk to the surface surfactant concentration along an isotherm, allows us to integrate Eq. (A.8) and thereby derive the desired surface equation of state. Strictly speaking, Gibbs’ equation and its derivatives are valid under quasistationary conditions. A common approximation is to extend the range of applicability to local and time-dependent conditions. For simplicity, in the remainder of this section we shall continue to assume that the surfactant is soluble in only one of the phases indicated by the direction of the normal vector.

Instability of Multi-Layer Channel and Film Flows

233

A.3.1. Henry’s Isotherm and Small Perturbations When the surfactant concentration is well below the saturation level, we may adopt Henry’s law isotherm

G ¼ Kcs ;

ðA:9Þ

which arises by setting the right-hand side of Eq. (A.6) to zero. Substituting Henry’s law into Eq. (A.8) and integrating the resulting equation, we derive the linear relation

gc 2 g ¼ EG;

ðA:10Þ

where gc is the surface tension of a clean interface that is devoid of surfactants, and E ; 2›g=›G ¼ RT is the surface elasticity. Rearranging Eq. (A.10), we obtain the linear law     G g0 G g ¼ gc 1 2 b 12b ¼ ; ðA:11Þ G0 G0 12b where G0 is a reference surfactant concentration corresponding to the reference surface tension g0 ¼ gc ð1 2 bÞ; and

b¼

E G0 RT G0 ¼ gc gc

ðA:12Þ

is a dimensionless coefficient defined in terms of the reference surface concentration. The effect of the surfactant can be expressed either by the coefficient b or by the Marangoni number Ma ;

E G0 b ¼ : g0 12b

ðA:13Þ

The linear law may also be used to describe the behavior for small deviations of the surfactant concentration from the uniform equilibrium value G0 ; and is appropriate for carrying out linear stability analysis for small perturbations.

A.3.2. Langmuir’s Isotherm Langmuir’s isotherm applies to ideal systems involving non-interacting molecules, and is described by Kcs ¼

G G1 2 G

or

G Kcs ; ¼ G1 Kcs þ 1

ðA:14Þ

where G1 is the maximum packing surface concentration of the surfactant.

C. Pozrikidis

234

The physical origin of Eq. (A.14) becomes evident by considering the precursor equation qa=d ¼ Ka cs ðG1 2 G Þ 2 Kd G;

ðA:15Þ

describing the flux across the sublayer due to adsorption and desorption. The first term on the right-hand side of Eq. (A.15) is the rate of adsorption expressed by first-order kinetics in the surfactant concentration at the edge of the sublayer and in the number of available molecular surface sites, G1 2 G; whereas the second term is the rate of desorption expressed by first-order kinetics in the surface concentration G: Langmuir’s isotherm arises by setting qa=d ¼ 0 and identifying K ¼ Ka =Kd with the partition coefficient at equilibrium. Using the first equation in Eq. (A.14) to eliminate cs from Eq. (A.8), and integrating the resulting equation, we obtain Langmuir’s equation of state, also called the von Szyckowski equation,      G b G g ¼ gc þ RT G1 ln 1 2 ¼ gc 1 þ ln 1 2 c G1 c G0    g0 b G ¼ ; ðA:16Þ 1 þ ln 1 2 c b c G0 1 þ lnð1 2 cÞ c where the constant b is defined in Eq. (A.12), and c ; G0 =G1 : Linearizing with respect to G; we recover the linear law (A.10).

A.3.3. Frumkin’s Isotherm A more advanced model that takes into consideration the energetics between the surfactant macromolecules results in Frumkin’s adsorption isotherm described by Kcs ¼

G expð2AG=G1 Þ G1 2 G

or ðA:17Þ

G Kcs ; ¼ G1 Kcs þ expð2AG=G1 Þ where the dimensionless constant A is a measure of the surfactant non-ideal behavior. When A is positive or negative, the surfactant molecules exhibit, respectively, cohesive or repulsive interactions. Langmuir’s isotherm describing ideal behavior corresponds to A ¼ 0:

Instability of Multi-Layer Channel and Film Flows

235

Using Eq. (A.17) to eliminate cs from Eq. (A.8), and integrating the resulting equation, we derive Frumkin’s equation of state "     # G A G 2 g ¼ gc þ RT G1 ln 1 2 þ G1 2 G1 "    # b G A 2 G 2 ¼ gc 1 þ ln 1 2 c ; þ c c G0 2 G0

ðA:18Þ

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Author Index

Numbers in italics refer to pages on which the complete references are cited

Bower, A. F., 97, 119, 136, 176, 177 Burke, J. E., 84, 172

A Abeyaratne, R., 7, 172 Adalsteinsson, D., 136, 175 Adamson, A. W., 232, 235 Aidun, C. K., 186, 238 Akhtaruzzaman, A. F. M., 215, 235 Albert, F., 186, 235 Alexander, J. I. D., 7, 172, 174 Andreussi, F., 108, 172 Angenent, S., 64, 70, 74, 82, 84, 85, 87, 91, 172 Angenent, S. B., 84, 86, 172 Anturkar, N. R., 186, 191, 235 ˚ gren, J., 12, 19, 147, 172 A Asaro, R. J., 5, 119, 136, 172 Asta, M., 122, 176 Atkinson, C., 52, 172 Aziz, M. J., 122, 136, 172, 175

C Cahn, J. W., 7, 12, 17, 18, 19, 20, 26, 37, 38, 41, 84, 86, 91, 97, 104, 120, 141, 147, 151, 172, 175, 176 Cammarata, R. C., 108, 172 Carmona, C., 191, 237 Caroli, B., 27, 173 Caroli, C., 27, 173 Cermelli, P., 71, 147, 173 Chang, C.-H., 232, 235 Chang, H.-C., 210, 235 Charles, M. E., 207, 235 Charru, F., 186, 191, 195, 207, 235 Chen, K., 180, 186, 215, 235, 236 Chen, K. P., 206, 215, 238 Chen, Y.-G., 84, 86, 173 Cheung, S. Y., 191, 237 Chuang, T. J., 5, 119, 175 Coleman, B., 12, 26, 173 Coward, A. V., 206, 207, 236

B Babchin, A. J., 191, 238 Bai, R., 206, 235 Bankoff, S. G., 186, 187, 191, 206, 210, 237, 238, 239 Barles, G., 84, 86, 172 Barthelet, P., 191, 207, 235 Barvosa-Carter, W., 122, 136, 172, 175 Beck, P. A., 6, 172 Bhattacharya, K., 88, 176 Bimberg, D., 108, 117, 119, 123, 132, 136, 175 Blyth, M.G., 189, 190, 191, 197, 206, 220, 235

D Danescu, A., 119, 136, 173 Davı´, F., 9, 15, 101, 102, 106, 173 Davis, S. H., 91, 97, 119, 139, 173, 174, 176, 186, 187, 191, 210, 237, 238 DeHoff, R. T., 17, 173

241

Author Index

242 Desai, R. C., 119, 136, 175 DiCarlo, A., 91, 97, 137, 139, 173 E

E, W., 119, 136, 177 Eckart, C., 15, 173 Eshelby, J. D., 7, 8, 52, 63, 151, 172, 173 F Fabre, J., 191, 195, 207, 235 Frank, F. C., 86, 173 Franses, E. I., 232, 235 Frenkel, A. L., 189, 191, 236, 238 Freund, L. B., 5, 108, 118, 119, 136, 173, 175 Fried, E., 1, 9, 10, 26, 108, 141, 161, 165, 173 G Gao, H., 108, 119, 136, 173 Gao, H.-J., 119, 136, 173 Gibbs, J. W., 34, 173 Giga, Y., 84, 86, 173 Gjostein, N. A., 86, 91, 173 Goldenfeld, N., 91, 139, 176 Golovin, A. A., 91, 139, 173, 174 Goto, S., 84, 86, 173 Gray, L. J., 136, 175 Grenet, G., 108, 175 Grilhe, J., 119, 136, 174 Grimshaw, R., 191, 236 Grinfeld, M., 7, 136, 151, 174 Gurtin, M. E., 1, 7, 8, 9, 10, 15, 26, 32, 52, 53, 55, 57, 59, 64, 70, 71, 74, 78, 82, 84, 85, 87, 88, 91, 97, 101, 102, 106, 108, 117, 137, 139, 141, 147, 148, 151, 152, 161, 165, 167, 172, 173, 174 Guyer, J. E., 97, 174 H Halpern, D., 189, 236 Han, C. D., 186, 207, 236 Handwerker, C. A., 84, 176 Heidug, W., 7, 174 Helenbrook, B., 215, 236 Herring, C., 5, 8, 52, 81, 82, 84, 88, 91, 103, 105, 118, 174 Hesla, T. I., 186, 236

Higgins, B. C., 180, 186, 187, 207, 238 Hildebrand, F. B., 218, 236 Hilliard, J. E., 97, 172 Hinch, J. E., 186, 235 Hoffman, D. W., 86, 91, 172 Hooper, A. P., 186, 191, 236, 238 I Ibach, H., 108, 174 J Jabbour, M. E., 139, 174 James, R. D., 7, 151, 174 Jana, S., 181, 236 Jaumann, G., 15, 174 Jiang, W. Y., 215, 236 Johnson, W. C., 7, 141, 172, 174, 175 Jonsdottir, F., 119, 136, 173 Joseph, D. D., 187, 197, 206, 235, 236, 238 K Kao, M. E., 207, 236 Kao, T. W., 214, 236 Kaplan, T., 136, 175 Kay, S. M., 218, 236 Kelkar, K., 206, 235 Khomami, B., 186, 238 Kirkwood, J. G., 35, 175 Kistler, S. F., 210, 236 Kliakhandler, I. L., 197, 225, 236 Knoester, H., 191, 236 Knowles, J. K., 7, 172 Kobayashi, C., 215, 236, 237 Koehler, J. S., 8, 52, 175 Kouris, C., 206, 237 Kurz, M. R., 180, 215, 238 L Larche´, F. C., 7, 12, 17, 18, 19, 20, 26, 37, 38, 41, 104, 120, 147, 151, 172, 175 Lawrence, C. J., 180, 215, 237 Lehner, F. K., 7, 174 Leo, P., 7, 175 Le´onard, F., 119, 175 Li, C.-H., 180, 190, 237 Li, J., 206, 237, 239

Author Index Li, X., 230, 231, 237 Liakopoulos, A., 207, 237 Lilleleht, L. U., 207, 235 Lin, S. P., 214, 215, 235, 236, 237, 238 Lin, S. Y., 235, 237 Liu, F., 139, 175 Loewenherz, D. S., 180, 215, 237 Lohr, E., 15, 175 Lu, W., 124, 175 M Maldarelli, C., 235, 237 Marple, L., Jr., 218, 236 Marple, S. L., 218, 237 Marty, A., 108, 175 Maugin, G. A., 8, 63, 175 McKeigue, K., 235, 237 Meiron, D. I., 97, 119, 136, 176 Meixner, J., 15, 175 Metcalfe, G., 181, 236 Miksis, M. J., 137, 176, 206, 239 Mu¨ller, I., 15, 175 Mullins, W. W., 6, 12, 17, 81, 84, 88, 103, 106, 175 Murdoch, I., 117, 161, 167, 174 N Nepomnyashchy, A. A., 91, 139, 173, 174 Newhouse, L. A., 189, 237 Nix, W. D., 108, 119, 136, 173 Noll, W., 12, 26, 173, 176 Norris, A. N., 5, 11, 118, 175 O Oliemans, R. V. A., 191, 237 Ooms, G., 191, 237 Oppenheim, I., 35, 175 Oron, A., 210, 237 Ottino, J. M., 181, 236 Otto, F., 97, 176

P Papageorgiou, D. T., 207, 236, 238 Papanastasiou, T. C., 186, 191, 235 Park, C., 207, 236

243

Pawar, Y., 232, 235, 237 Peach, M. O., 8, 52, 175 Phan, A.-V., 136, 175 Pinarbasi, A., 207, 237 Podio-Guidugli, P., 53, 91, 97, 137, 139, 173, 175 Politi, P., 108, 175 Ponchet, J., 108, 175 Power, H., 191, 237 Pozrikidis, C., 179, 181, 186, 187, 189, 190, 191, 196, 197, 199, 204, 205, 206, 210, 211, 220, 225, 226, 227, 228, 230, 231, 235, 237, 238, 239; Pranckh, F. R., 186, 236 Preziosi, L., 186, 236

R Rastelli, A., 136, 137, 176 Reik, H. G., 15, 175 Renardy, M., 191, 206, 207, 236, 237, 238 Renardy, Y., 186, 187, 191, 197, 236, 238 Renardy, Y. Y., 206, 207, 236, 237 Rice, J. R., 5, 52, 119, 175 Richards, J. R., 206, 207, 236 Robin, P.-Y. F., 7, 151, 175 Rosso, F., 197, 238 Roulet, B., 27, 173

S Scardovelli, R., 206, 239 Schweizer, P. M., 210, 236 Seaborg, J. J., 215, 238 Segal, A., 191, 237 Sekerka, R. F., 7, 12, 17, 175 Sethian, J. A., 136, 175 Severtson, Y. C., 186, 238 Shchukin, V. A., 108, 117, 119, 123, 132, 136, 175 Shenoy, V. B., 108, 175 Shewmon, P. G., 17, 175 Shih, C. F., 119, 136, 176 Shlang, T., 191, 238 Siegel, M., 137, 176 Sieradzki, K., 108, 172 Simha, N. K., 88, 176 Sivashinsky, G. I., 191, 197, 225, 236, 238 Smoluchowski, R., 6, 176 Soner, H. M., 84, 86, 172, 176

Author Index

244

Souganides, P. E., 84, 86, 172 South, M. J., 186, 238 Spaepen, F., 108, 176 Spencer, B. J., 45, 97, 119, 124, 127, 134, 136, 137, 154, 176 Srolovitz, D. J., 119, 136, 176, 177 Stebe, K. J., 232, 235, 237 Stewart, J., 91, 139, 176 Stringfellow, G. B., 108, 176 Struthers, A., 8, 52, 174 Su, Y. Y., 186, 238 Suo, Z., 88, 119, 124, 136, 175, 176

T Taylor, J., 84, 176 Tersoff, J., 45, 119, 124, 127, 134, 136, 137, 176 Than, P. T., 197, 238 Tiller, W. A., 5, 119, 136, 172 Tilley, B. S., 186, 187, 191, 238 Toupin, R., 16, 26, 176 Truesdell, C., 16, 26, 174, 176 Truskinovsky, L. M., 7, 176 Tryggvason, G., 206, 238, 239 Tsamopoulos, J., 206, 237 Turnbull, D., 6, 84, 172, 173, 176

U Uhuwa, M., 84, 176 Unverdi, S. O., 206, 238

V Van de Walle, A., 122, 176 van der Zanden, J., 191, 236 Vargas, A. S., 15, 174

Villain, J., 108, 175 von Ka¨nel, A., 136, 137, 176 Voorhees, P. W., 9, 45, 97, 119, 122, 124, 127, 134, 135, 136, 137, 141, 148, 152, 154, 174, 175, 176

W Wang, C. K., 215, 235, 238 Wang, W., 119, 136, 176 Watson, S. J., 97, 176 Weinstein, S. J., 180, 215, 238 Wilkes, J. O., 186, 191, 235 Wu, C. H., 5, 118, 176

X Xia, L., 119, 136, 176 Xiang, Y., 119, 136, 177

Y Yang, W. H., 119, 136, 177 Yiantsios, S., 180, 186, 187, 207, 238 Yih, C. S., 186, 210, 211, 238 Yiu, R. R., 206, 238 Yon, S., 230, 231, 239

Z Zaleski, S., 206, 239 Zanetti, G., 206, 239 Zang, Y.-W., 177 Zhang, J., 97, 206, 239 Zhou, H., 199, 239

Subject Index

An italic f (or t) following a page number indicates that the material referred to is in a figure caption (or table)

Boundary-integral formulation, 226 –228 Bulk transport, 231– 232

A Accretion, 53–55 Adatom densities and chemical potentials, 130–133 and equations for solid-vapor interfaces, 124–129 neglecting for solid-vapor interface equations, 129 –134 for solid-vapor interface, 113 Adatom diffusion and lattice constraints, 119 Anti-lubrication, 180 Arc velocity, 65 Atom-vacancy exchange, 19f Atomic balance, 22 and configurational forces in bulk, 57 equation for mechanically simple material, 51 for solid-vapor interface, 112–113 Atomic flows for coherent phase interfaces, 144–145 for grain-vapor interfaces, 99–100 Axisymmetric interfacial waves, 206

C Capillary instability, 179 Channel flow, 187–189 lubrication-flow model for long waves, 191 –197 multi-layers, 190–191 Navier-Stokes flow, 206 non-isothermal flow, 207 non-Newtonian flow, 207 numerical simulations for Stokes flow, 197 –206 oscillatory flow, 207 stability analysis for multiple layers, 190 –191 stability analysis for two layers, 187 –189 stability analysis for two layers with surfactants, 189– 190 two-layer flow, 198f unidirectional multi-layer, 184–186 unidirectional two-layer, 182 –184 Chemical compatibility equations, for solid-vapor interface, 126– 127 Chemical potential in absence of adatom densities, 130–133 and flow of atoms, 15 –16 and free energy imbalance, 20, 21

B Balance law for atoms, 14 Balance of energy, and configurational forces in bulk, 58 Boundary-element method, 202 –203, 205f

245

Subject Index for mechanically simple substitutional alloys, 47 relative for substitutional alloys, 19–20 at zero stress, 34 Coherent phase interfaces atomic transport, 144–145 configurational forces, 142–143 constitutive equations, 152–153 free energy imbalance, 145–148 general equations, 153– 154 global theorems, 148–150 normal configurational force balance, 150–152 Commutator relation, 67 –68 Compliance tensor, 29 Compositional strain, for cubic material, 33 Compressibility, for cubic material, 32 Configurational bulk tension, 59 Configurational force balances, 52–53 absence of unifying principle, 8– 9 for coherent phase interfaces, 150–152 for grain boundaries, 75 –76 for grain-vapor interfaces, 98 importance of kinetic term, 134 –135 mechanical potential, 118 –119 and small deformation theory, 163–164 for solid-vapor interface, 10–11, 108 Wu-Norris-Freund relation, 11 Configurational forces in bulk configurational force balance, 52 –53 Eshelby relation, 59–60 migrating control volumes, 53–55 power expended on migrating control volume, 55 –56 role of constitutive equations, 62– 64 thermodynamics for migrating control volumes, 56 –62 Configurational surface tension, 9 Configurational torque balance, 92 –94 and interfacial couple stress, 137–138 Constitutive equations for coherent phase interfaces, 152–153 and configurational forces in bulk, 62 –64 for grain boundaries, 81 –84 for grain-vapor interfaces, 102–103 for interfacial couples, 96 for solid-vapor interface, 130–133 without regard to torque balance, 164–165

246

Constitutive theory and cubic symmetry, 32 –34 for elastic material with Fickean diffusion, 25– 27 and free enthalpy, 29–30 linear dissipative response, 88–91 for mechanically simple materials, 30 –32 for mechanically simple substitutional alloys, 45– 49 for multiple atom species without lattice constraint, 25–34 for small deformations, 166–168 for solid-vapor interface, 121–123 for substitional alloy, 36 and thermodynamic restrictions, 27–29 at zero stress, 35 Constitutive theory for substitutional alloys constraints on response functions, 40 cubic symmetry, 49 –50 free enthalpy, 43–45 Larche-Cain derivatives, 36–39 mechanically simple, 45 –49 mobility constraints, 40–41 thermodynamic restrictions, 41–43 Control-volume equivalency theorem, 157 –160 Control volumes, 53 –55 Convective transport, 180 Core-annular flow, 206 Couette channel flow experimental studies, 207 numerical simulations, 206 Stokes flow simulations, 204 –206 Couette-Poiseuille channel flow, 189 using lubrication-flow model, 192f Couette-Poiseuille-gravity-driven flow, 182– 184 Cubic material, 32–34 substitutional alloys, 49–50 Curvature, 64–67 allowance for energy dependence on, 135 –140 of crystal surface, 91 –97 evolving subcurves, 68 –70

D Deformation and atomic transport in bulk balance law for atoms, 14

Subject Index constitutive theory for elastic material, 25–26 constitutive theory for substitutional alloys, 36 –49 and cubic symmetry, 32–34 and free enthalpy, 29 global decay theorem, 24–25 global theorems, 22–25 governing equations, 50–51 mechanically simple materials, 30 –32 substitutional alloys, 17–22 thermodynamic restrictions, 27–28 Dissipation inequalities, 17 for coherent phase interfaces, 146–147 and constitutive equations, 25 for grain boundaries, 81 for interfacial couples, 95 for solid-vapor interface, 116–118 for substitutional alloys, 20 thermodynamic restrictions, 88–91

E Elastic material, constitutive theory (with Fickean diffusion), 25–27 Elasticity tensor, 28 for cubic material, 32 Entropy imbalance, 15–16 and configurational forces in bulk, 58 Epitaxy, 107f Equilibrium chemical potential, 135 Equivalency theorem, for substitutional alloys, 120 –121 Eshelby relation, 59–60 and bulk configurational stress, 118 interfacial analogy, 117 modified for energy dependence on curvature, 138 and small deformation theory, 163 Evaporation-condensation, 140–141 External body force, 63

F Faceted islands, 135–138 Facets (and grain boundary equations), 86–87

247

Fickean diffusion, 6, 12 and constitutive theory for elastic material, 25– 27 Fick’s law, for cubic material, 34 Film flow identification of normal modes, 215 –220 numerical and asymptotic solutions, 225 single-film flow, 211 stability and surfactants, 211–213 unidirectional, 208– 210 Fingering instability, 180 Finite-deformation theory basic equations, 168 isotropic strain energy equations, 169 –170 kinematics, 160 –161 power, 161–162 Finite-volume method, 203–204 Flow. see Channel flow; Film flow Fluid dynamics software library FORTRAN program for Green’s function, 226– 228 Stokes flow integral, 199 two-layer flow, 187 unidirectional multi-layer flow, 186 Fluid inertia and boundary layer approximation, 191 and finite-amplitude motion, 206 Flux constraint, for substitutional alloys, 18 –19 Frank diagram, 86, 87 Free energy curvature-dependent, 91–97 for mechanically simple substitutional alloys, 45– 49 Free energy imbalance, 9 for coherent phase interfaces, 145–148 dissipation inequalities, 17 and entropy imbalance, 15 –16 finite-deformation theory, 162 –163 global theorems, 22 –23 for grain boundaries, 78 –81 for grain-vapor interfaces, 100–102 for interfacial couples, 94–96 isothermal conditions, 16– 17 for migrating control volumes, 62 and migrational balance laws, 61–62 for solid-vapor interface, 115–118 at zero stress, 34 Free enthalpy, 29–30 and constitutive theory for substitutional alloys, 43– 45

Subject Index Free surface flow, 222– 224 Frumkin’s isotherm, 234–235

G Gibbs-Duhem equation, at zero stress, 34–35 Gibbs equation, 232– 235 Gibbs relation, at zero stress, 34–35, 154–157 Grain boundaries backward parabolicity, 86–87 configurational force balance, 75 –76 curvature-flow equation, 84 dissipation inequalities, 81 evolution equation, 84 –86, 96– 97 global free-energy imbalance, 78–81 growth and decay of isolated grain, 79 interfacial couples, 91–97 junctions, 87–88 power, 77–78 surface tension and interfacial free energy equality, 80 tangential component of internal force, 80 theory (neglecting deformation and atomic transport), 74 –91 Grain-vapor interfaces atomic flows, 99–100 basic equations, 103–104 configurational force balance, 98 constitutive equations, 102–103 flat interface at equilibrium, 104–107 free energy imbalance, 100–102 motion by evaporation-condensation, 107 motion by surface diffusion, 106 power, 99 Gravity-driven flow stability in limit of Stokes flow, 228 and Stokes flow integral, 202 Green’s function, and Stokes flow integral, 199

H Hagen-Poiseuille flow, 199 Hele-Shaw cell, 180 Helmholtz free energy, 16–17 Henry’s isotherm, 233

248

Herring’s equation, 5–6 Hydrodynamic instability, 179

I Immersed-boundary methods, 206 Inertia and finite-amplitude motion, 206 lubrication-flow model for long waves, 191 in solid state problems, 13 and Yih-Marangoni instability, 190 Instability channel flow, 186 –191 due to Marangoni tractions, 206 identification of normal modes for film flow, 215– 220 multi-layer film flow, 215 Rayleigh-Taylor instability of unstably stratified fluids, 188 single-film flow, 210 two-layer film flow, 214–215 for two-layer flow, 187 types in film flow, 228–229 types in multi-layer channel flow, 228– 229 Interface conditions and configurational force balances, 8–9 Herring’s equation, 5–6 Kinetic Maxwell equation, 6–7 Leo-Sekerka relation, 7–8 Mullin’s equations, 6 Interface kinematics, 64–74 commutator relation, 67–68 curvature, 64– 67 deformation of the interface, 70–73 interfacial pillboxes, 73– 74 normal time-derivative, 64 –67 normal velocity, 64– 67 transport identities, 67–68 transport theorem for integrals, 70 Interfacial couples, 91–97 configurational torque balance, 92–94 constitutive equations, 96 dissipation inequalities, 95 evolution equation for grain boundary, 96–97 free energy imbalance, 94–96 power, 93–94 for solid-vapor interface, 135–138

Subject Index Interfacial driving traction, 151 Interfacial flow. see Channel flow; Film flow Interfacial instability. see Instability Interfacial pillboxes, 73–74 alpha-atom transport, 113f atomic flows, 99–100 free energy imbalance, 78–81, 94–96 power, 93–94 power for solid-vapor interfaces, 110–112 standard forces for solid-vapor interfaces, 109f Interfacial strain, 72– 73 Interfacial velocity, 195–196 for Stokes flow integral, 201 for two-layer flow, 193 Internal body force, 63 Interstitial alloys, 17–18 Intrinsicality hypothesis, 58–59 Island formation, 135 –138 Isolated body, deformation and atomic transport in bulk, 22 –23 Isolated two-phase body, 149–150 Isothermal conditions, and migrational balance laws, 61 –62 Isotropic grain-boundary interface, 6 Isotropic strain energy equations, 169– 170

J Junctions of grain boundaries, 87–88

K Kelvin-Helmholtz instability, 179 Kinematics. see also Interface kinematics finite-deformation theory, 160 –161 Kinetic Maxwell equation, 6–7 Kinetic modulus, 11 Kuramoto-Sivashinsky equation, 191, 195

L Langmuir’s isotherm, 233 –234 Larche-Cain derivatives, 36–39 Lattice constraint, 12 and adatom diffusion, 119 elimination for substitutional alloys, 22

249

and free energy imbalance, 21 and Larche-Cain derivative, 37 and substitutional alloys, 17–18 Leo-Sekerka relation, 7–8 applied to dynamical problems, 11 Liquid films. see Film flow Liquid layers, 209 Liquid layers, hydrodynamic instability, 179 –180 Long waves, lubrication-flow model, 191 –197, 220 –225 Lubrication-flow four-layer film flow, 226f for long waves, 191 –197, 220 –225 two-layer interface evolution, 196f

M Marangoni instability, 180, 191, 228 surfactant-induced, 206 Marangoni number, 189, 229 and effect of surfactant on film flow stability, 212–213 Marangoni traction, 197 and Stokes flow simulations, 206 Materials with cubic symmetry, 32 –34 Materials, unconstrained, mechanically simple, 30–32 Mechanical potential, 118–119 Mechanically simple materials, 30–32 chemical compatibility equations, 127 constitutive equations, 50–51 Mechanics of deformation and atomic transport, 13 –14 Migrating control volume, 53– 55, 53f control-volume equivalency theorem, 157 –160 free energy imbalance, 62 intrinsicality hypothesis, 58 –59 thermodynamic laws, 56–62 Migrational balance laws, 61 Mobility matrix, 26 Mobility tensor, 25 for mechanically simple substitutional alloys, 51 for substitutional alloys, 40–41 Motion velocity, 13 Mullin’s equations, 6 Multi-layer channel flow, 186–187

Subject Index Multi-layer film flow, 209f stability analysis, 210, 215 Multi-layer flow, 190–191. see Channel flow; Film flow N Navier-Stokes equation, 183 Navier-Stokes flow, 206 Non-isothermal flow, 207 Non-Newtonian flow, 207 Normal time-derivative, 64–67 Normal velocity, 64 –67 Nusselt velocity, for unidirectional film flow, 208 O Orr-Sommerfeld equation, 190 Oscillatory flow, 207 P Parabolicity facets and wrinkling for grain boundaries, 86– 87 and grain boundary equation, 84–86 Peskin’s immersed-interface method, 206 Photographic film manufacturing, 210 Poiseuille flow, 197 Stokes flow simulations, 204 –206 Power for coherent phase interfaces, 143–144 finite-deformation theory, 161 –162 for grain boundaries, 77 –78 grain-vapor interfaces, 99 interfacial couples, 93–94 for solid-vapor interface, 110–112 Pressure-driven flow experimental investigations, 207 in horizontal channel, 189 interface evolution, 205f simulations, 205f and Stokes flow integral, 201 Q Quadratic strain energy, 131– 133

250 R

Rayleigh-Taylor instability of unstably stratified fluids, 188 Relative chemical potentials constitutive equation for substitutional alloys, 39 for substitutional alloys, 19–21 Reynolds numbers and instability between liquid layers, 180 and stability of two-layer flow, 187 and Stokes flow, 197

S Saffman-Taylor instability, 180 Shear-driven flow destabilization by surfactant, 189 and Stokes flow integral, 201 Shear-flow inertial instability, 228 Shear stress for grain boundaries, 75 for unidirectional film flow, 208– 210 Single-film flow, 211 identification of normal modes, 216 lubrication-flow model, 224 stability analysis, 210 Small-deformation theory comparisons, 170 –172 constitutive equations, 166–168 Small-strain estimates, 166 Software. see Fluid dynamics software library Solid-solid interfaces, 71 Solid state problems balance law for atoms, 14 mechanics of deformation and atomic transport, 13 –14 Solid-vapor interfaces, 107 allowance for evaporation-condensation, 140 –141 atomic balance, 112 –113 chemical compatibility, 126– 127 chemical potential and curvature, 5 chemical potential and deformation, 6 configurational forces, 108 constitutive equations, 121–123 constitutive relations neglecting adatom densities, 130–133 dissipation inequalities, 116–118 equations neglecting standard surface stress, 127–129, 133–134

Subject Index equations when adatom densities are neglected, 129–134 faceted islands, 135–138 free-energy imbalance, 115 –118 general relations, 124 –126 governing equations, 124–129 interfacial couples, 135–138 kinematics, 71 kinetic term of configurational force balance, 134 –135 mechanical potential, 118 –119 normal configurational force balance, 10–11 power, 110–112 and quadratic strain energy, 131 –133 standard forces for solid-vapor interfaces, 109–110 Soliton-like interfacial structures, 197 Solute-expansion modulus, 33 Squire transformation, 186– 187 Stokes flow, 187 boundary-integral formulation, 226– 228 convection-diffusion equation for surfactant concentration, 203 –204 destabilization by surfactant, 189 intergral formulation, 198 –202 numerical simulations, 197 –206 simulations, 204 –206 solving using boundary-element methods, 202–203 Strain-displacement relation, 13 Strain, interfacial, 72 –73 Stress-composition tensor, 28 Stretch rate, 73 Subcurves, 68–70 Substitutional alloys chemical compatibility equations, 126– 127 chemical interface-conditions, 115 chemical potential for coherent phase interface, 151 constitutive equations, 51 constitutive relations, 39 elimination of lattice constraint, 22 equations modified for energy dependence on curvature, 139– 140 equations neglecting standard surface stress, 128–129 equivalency theorem, 120– 121 Eshelby relation, 59 flux constraint, 18– 19 free energy imbalance and chemical potential, 21

251

Herring’s equation, 5–6 interfacial chemical potentials and relative chemical potentials, 119–121 interfacial flux constraint, 147–148 Larche-Cain derivatives, 36–39 mechanically simple materials, 45–49 relative chemical potentials, 19 –21 Surface tension configurational, 9 and interfacial free energy for grain boundaries, 80 and solid-vapor interface, 133 –134 Surfactant convection-diffusion equation for concentration, 203– 204 and film flow stability, 211 –213 finite-volume method for transport, 203 –204 Frumkin’s isotherm, 234 –235 and Gibbs equation, 232 Henry’s isotherm, 233 Langmuir’s isotherm, 233–234 and Marangoni instability, 206 and stability of two-layer flow, 189– 190 and surface tension, 229 –231 transport mechanism, 229– 231 T Thermodynamic restrictions, 27–29 Thermodynamics isothermal conditions and free energy imbalance, 16 –17 for migrating control volumes, 56–62 restrictions and constitutive theory for substitutional alloys, 41 –43 second law and entropy imbalance, 15–16 Three-layer flow, 197 Torque balance, 92–94 for coherent phase interfaces, 142–143 and constitutive equations, 164–165 for solid-vapor interface, 109–110 Transport identities, 67 –68 Transport theorem for integrals, 70 Two-layer channel flow, 182–184 and boundary-integral formulation, 198f experimental studies, 207 instability, 205– 206 lubrication-flow model, 192 –197 stability analysis, 187–189 Stokes flow simulations, 204 –206

Subject Index Two-layer film flow lubrication-flow model, 225 stability, 214–215 stability in limit of Stokes flow, 228 Two-phase body, 149– 150

U Unconstrained materials, 12 chemical compatibility equations, 126–127 chemical interface-conditions, 115 constitutive equations, 51 dissipation inequalities, 146– 147 equations modified for energy dependence on curvature, 139 equations neglecting standard surface stress, 128 Eshelby relation, 59 Unidirectional base flow, 208– 210

252 V

Vacancies, and lattice constraint, 18 Vapor. see Grain-vapor interface Velocity normal velocity, 64– 67 for unidirectional film flow, 208– 209 Viscosity stratification, 180, 228 Volume-of-fluid methods, 206 von Szyckowski equation, 234

W Wrinkling (and grain boundary equations), 86 –87

Y Yih-Marangoni instability, 190