MODERN ASPECTS OF ELECTROCHEMISTRY No. 35
Modern Aspects of Electrochemistry Topics from Modern Aspects of Electrochemistry, No. 35 include: Impedance spectroscopy with specific applications to electrode processes involving hydrogen. Fundamentals and contemporary applications of electroless metal deposition. The development of computational electrochemistry and its application to electrochemical kinetics. Transition of properties of molten salts to those of aqueous solutions. Limitations of the Born Theory in applications to solvent polarization by ions and its extensions to treatment of kinetics of ionic reactions. Modern Aspects of Electrochemistry, No. 34: Additivity principles in behavior of redox couples and corrosion processes. Foundation of voltaic measurements at liquid interfaces. Direct methanol fuel cells: current progress and emerging technology. Dynamics of processes in molten salts. Electrochemical techniques and microbially influenced corrosion (MIC). Modern Aspects of Electrochemistry, No. 33: A review of the literature on the potential-of-zero charge. Review and discussion on nonequilibrium fluctuations in corrosion processes. Conducting polymers, their electrochemistry and biomimicking processes. Microwave photoelectrochemistry; from its origins to current research problems. New fluorine cell design from model development through pilot plant tests. The rapidly developing field of electrochemistry of electronically conducting polymers and their applications.
MODERN ASPECTS OF ELECTROCHEMISTRY No. 35 Edited by BRIAN E. CONWAY University of Ottawa Ottawa, Ontario, Canada
and
RALPH E. WHITE University of South Carolina Columbia, South Carolina
KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW
eBook ISBN: Print ISBN:
0-306-47604-5 0-306-46776-3
©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2002 Kluwer Academic/Plenum Publishers New York All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:
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LIST OF CONTRIBUTORS M. ABRAHAM Chemistry Department University of Montreal Montreal, Quebec, Canada
L. BIENIASZ Institute of Phsyical Chemistry PAS Department of Molten Salts 30-318 Krakow, Poland
M.-C. ABRAHAM Chemistry Department University of Montreal Montreal, Quebec, Canada
S. DJOKIC Westaim Biomedical Corporation Fort Saskatchewan Alberta, Canada, T8L 3W4
B. E. CONWAY Chemistry Department University of Ottawa Ottawa, Ontario, Canada, K1N 6N5
A. LASIA Department of Chemistry University of Sherbrooke Sherbrooke, Quebec, Canada, J1K 2R1
A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher.
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An Appreciation Professor J. O’M. Bockris, First and now Retiring Editor of the Modern Aspects of Electrochemistry Series
The publication of the Modern Aspects of Electrochemistry series, now in its
volume, was initiated with its first volume in
1954, edited by Professor John Bockris, then of the Chemistry Department at the Imperial College of Science and Technology, London. At that time, Modern Aspects of Electrochemistry was the first review monograph series aiming to cover, complementarily, the
vii
founding aspects of the subject, including the area of Ionics.
This
tradition is continued in the present volume. The first three volumes were published by the then publishing house, Butterworths Scientific Publications, London, as a sub-series of the more general Modern Aspects of Chemistry, under the then overall editorship of Dr. F.C. Tompkins, FRS. The first volume was, in many ways, seminal in setting the style of the series, encompassing the three main sub-disciplines of electrochemistry:
electrolyte solutions and molten electrolytes,
including ion/solvent interaction; electrode kinetics, welded together (almost literally) through the topic of the double-layer at electrified interfaces. It is of interest to cite the titles and authorship of the five chapters of the first volume which Bockris edited: Physical Chemistry of Synthetic Polyelectrolytes (R.M. Fuoss); (B.E. Conway and J.O’M. Bockris); Electrified Interphases
(R. Parsons);
Ionic Solvation
Equilibrium Properties of Electrode Kinetics (by
J.O’M. Bockris) and Electrochemical Properties of Nerve and Muscle (W.F. Floyd), an early and forward-looking contribution in the field of bio-electrochemistry. These topics, at the time, were seminal in the field and remain so, to a large extent today, being the backbone of modern electrochemistry. Polyelectrolytes are model substances for DNA and proteins, as well as for ion-exchange membranes and conducting polymers. Solvation of ions remains of primary interest in behavior of electrolytes in electrode processes (involving electron-transfer and solvation-shell reorganization), ion discharge and in the study and modeling of the double-layer as well as in hydration of ionic
viii
membranes, e.g. as used in PEM fuel-cells. properties
of
the
double-layer
The structure and
constituting
electrode/solution
interphases (noting the Guggenheim significance of that term) provides the locale of electron-transfer processes in electrode kinetics and related follow-up processes, as in the mechanisms of the cathodic evolution reaction and electrocatalytic oxidation of
pioneered by
Bockris himself. Finally, Bockris’s far-sighted horizons recognized the electrochemical basis of a variety of bio-physical processes as exemplified in phenomena treated in the chapter by Floyd. The wide ranging knowledge and research activities of Bockris in the field of physical chemistry, electrochemistry and interfacial science enabled him to encourage many well known researchers to contribute chapters of high quality and broad literature coverage to the Series. In the third and succeeding volumes of the Series, Bockris was joined by Conway (then of the University of Ottawa) as co-editor while, soon after Bockris had joined the Faculty at Texas A and M University (College Station, TX), he invited Professor Ralph E. White (now Dean of Engineering at University of South Carolina) to become a third co-editor, especially to cover contributions in the applied electrochemistry and electrochemical engineering areas. Following the publication of volume three, production of the Series was taken over by Plenum Publishing Company, New York and has remained in their hands until recently, when Plenum was joined by Kluwer Academic Publishers to form Kluwer Academic Plenum Publishing Company, still to be situated in New York.
ix
The remaining two editors will miss the stimulating discussions we all had with regard to choice of cutting-edge topics for new chapters in various volumes and selection of suitable academic and industrial researchers to provide such contributions, supported by broad coverage of contemporary and recent relevant literature. The series will continue under the joint editorship of Professors Conway and White with the addition of another co-editor to replace Professor Bockris.
B.E. Conway University of Ottawa
R.E. White University of South Carolina
x
Preface The Modern Aspects of Electrochemistry series continues with its thirty-fifth volume but with the regrettable retirement of its founding editor, Professor J.O’M. Bockris (see Appreciation on p. v). As in most previous volumes, a series of five chapters is presented covering both the electrolyte solution and the interfacial electrode process sides of the field of electrochemistry, in fact comprehensively reflecting the actual interphasial situation at the electrode/solution boundary. Chapter 1, by Lasia of the University of Sherbrooke, is complementary to his chapter 2 in the Modern Aspects volume 32 where he wrote on basic aspects of electrochemical impedance spectroscopy with some general applications. In the present volume, he extends his previous contribution to cover specific applications to electrode processes involving hydrogen, especially the behavior of underpotential-deposited H, sorption of H into metals, and the kinetics and mechanisms of cathodic evolution of Chapter 2, by Djokic, provides a comprehensive review of the topic of electroless metal deposition, an area where the fundamentals of metal crystallization, electrocrystallization and autocatalysis meet the advances and requirements of modern technology of non-Faradaic electroplating, especially at non-conductive substrates. This is now a
xi
major area in the technology of metal finishing and Djokic’s chapter provides an account of the fundamentals as well as contemporary applications of this important area of electrochemical and interfacial technology. In chapter 3, Bienasz examines, in extensive detail, the development of computational electrochemistry and its applications to various problems in electrochemical kinetics.
Following a general
conceptual introduction to the methodologies of computational approaches in physical chemistry, he continues with examination of the bases and applications of simulation procedures for treating electrochemical processes, an area pioneered in earlier years, for example by Feldberg, and Nicholson and Shain. From the inception of this Series, the Editors have recognized the close complementarity of research on electrolytic solutions to that on electrode interfacial structure and processes, especially through involvement of the presence and state of ions in the double-layer, the seat of electron-transfer processes in electrode-kinetics. In chapter 4, (M.) Abraham and his wife (M.-C.) Abraham provide a thorough account and analysis of the properties of electrolyte solutions at rather or very high concentrations. Such conditions provide the opportunity for studying the transition from “molten hydrates” to more dilute solutions, through progressive changes of the states of hydration of the ions in the solution. This is a relatively little studied but practically important area of ionic solution electrochemistry, most earlier classical work having been directed to quite dilute solutions and studies on deviations from ideality and interaction effects determining
xii
concentration dependence of activity coefficients and equivalent conductances of salts and acids in solution. Related to the role of solvation in electrolytic solutions, Conway, in the final chapter 5 of this volume, critically reexamines applications of the Born theory to solvent polarization by ions and to its extensions to treatment of kinetics of ionic reactions. The author examines a number of limitations of its applicability beyond those conventionally recognized arising from the “continuum-dielectric” model on which the treatment was historically based. The Editors wish to express their regrets to authors for delays in publication of their contributions to this volume which arose unavoidably from the conjunction of Plenum Publishing Company with Kluwer Academic of Dordrecht, The Netherlands, and from a resulting change over to “camera-ready” publishing technology during the production of the present volume. B.E. Conway University of Ottawa R.E. White University of South Carolina
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CONTENTS
Chapter 1
APPLICATIONS OF ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY TO HYDROGEN ADSORPTION, EVOLUTION AND ABSORPTION INTO METALS A. Lasia I. II. III. IV.
Introduction Determination of Impedances Hydrogen UPD The Hydrogen Evolution Reaction 1. The HER in the absence of mass-transfer effects. 2. In the presence of hydrogen mass-transfer V. Hydrogen absorption into metal electrodes 1. Hydrogen adsorption, absorption and evolution: linear diffusion 2. Absorption of the UPD hydrogen 3. Spherical diffusion 4. Transfer functions approach References
xv
1 2 6 13 13 19 22
22 31 33 37 47
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Contents
Chapter 2 ELECTROLESS DEPOSITION OF METALS D. Djokic I. Introduction II. Electrolessly depositable metals III. Basic definitions, similarities and differences among electroless processes 1 . Displacement deposition 2. Contact depos ition 3. Autocatalytic deposition IV. Solutions for electroless deposition 1. The metal ion sources 2. Complexing agents 3. Reducing agents 4. Stabilizers V. Deposition kinetics and empirical rate laws VI. Mechanistic aspects of electroless deposition 1. The atomic hydrogen mechanism 2. The hydride ion mechanism 3. The electrochemical mechanism 4. The metal hydroxide mechanism 5. The universal mechanism VII. Recent developments 1. Metallization of non-conductive surfaces 2. Electroless deposition of composite coatings 3. Electroless deposition of gold 4. Electroless deposition of other metals 5. New developments and basic research VIII. Conclusions References
51 52 54 54 55 56 56 57 58 59 76 80 87 87 90 91 100 105 107 107 113 114 117 121 124 124
Contents
xvii
Chapter 3 TOWARDS COMPUTATIONAL ELECTROCHEMISTRY – A KINETICIST’S PERSPECTIVE L.K. Bieniasz I. Introduction 135 II. The role of computers in natural sciences 137 1. Computational chemistry 140 2. Chemical informatics 142 3. Chemometrics 144 4. Chemical laboratory automation 144 5. Computational science and engineering 145 6. Modeling of scientific discovery 148 7. The future 149 III. The role of computers in electrochemical kinetics 151 1. Modeling of electrochemical kinetic experiments 151 2. Control of electrochemical instrumentation 155 3. The emerging computational electrochemistry 156 IV. The present approach 159 1. Automatic translation of electrochemical reaction mechanisms into corresponding sets of governing equations 160 2. General formulation of the algorithms of transient kinetic simulations 162 3. Comparative studies of simulation algorithms for kinetic models in one-dimensional spatial geometry 164 4. Formal analysis of numeral stability of the simulation algorithms 166 5. New simulation methods and modeling strategies 172 6. Realization of a PSE for electrochemical kinetic modeling 181 V. Conclusions 184 VI. Acknowledgments 186 References 187
xviii
Contents
Chapter 4 THERMODYNAMIC AND TRANSPORT PROPERTIES OF BRIDGING ELECTROLYTE – WATER SYSTEMS M.A. Abraham and M.-C. Abraham I. Introduction II. Thermodynamic properties 1. Adsorption theory of electrolytes 2. Approaches related to regular solution theories 3. Surface properties 4. Other approaches and observations III. Transport properties 1. Activation energy for viscous flow 2. Transition state theory of viscosity 3. Free volume for viscous flow 4. Equation for fluidity with apparent parameter 5. Activation energy for electrical conductance 6. Transition state theory of electrical conductance 7. Free volume for electrical conductance 8. Equation for equivalent electrical conductance with apparent parameter 9. Relation between viscosity and electrical conductance 10. Other approaches and observations References
Chapter 5 FACTORS LIMITING APPLICATIONS OF THE HISTORICALLY SIGNIFICANT BORN EQUATION: A CRITICAL REVIEW B.E. Conway
197 199 203 231 235 240 243 244 247 258 263 268 272 276 279 282 285 289
Contents
xix
I. General introduction II . Historical introduction III. Basis of derivation of the Born equation for evaluation of energies of ion solvation 1. Principles of the calculation 2. Comparison with experimental data IV. Change of radius of an ion upon entry into solution V. Change of radius of a particle upon charging VI. Comparison of the gas-phase and solution-phase charging energies in the Born equation VII. Structure and volume factors in the solvent co-sphere around an anion VIII. Cases of hydration of the proton and the electron IX. Relation to molecular modeling through ion-dipole interactions X. Born equation as a basis for plotting procedures for evaluation of ionic solvation energies XI. Relation to ionization processes in solution XII. Dielectric polarization effects in kinetics of reactions involving charged transition states
References Index
295 296 298 298 301 303 305 306 307 311 314 315 316 319 321 325
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Applications of Electrochemical Impedance Spectroscopy to Hydrogen Adsorption, Evolution and Absorption into Metals Andrzej Lasia Département de chimie, Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1 Canada
I. INTRODUCTION In the first part of this review1 we have presented fundamental aspects of electrochemical impedance spectroscopy (EIS) and showed how they can be used in data validation and modeling of processes limited by diffusion, electrode kinetics and adsorption for cases of different types of electrode geometries. In this chapter we shall present a general matrix method for impedance determination and apply it to solve electrochemical problems connected with H adsorption and absorption, and evolution of These processes present many problems which are similar to those found in metal underpotential deposition, intercalation, corrosion, etc. Of course, the literature of this subject is very rich and only some selected applications, which illustrate development of various typical impedances, will be presented. Obviously, it is assumed that the impedance was correctly measured and validated. This presentation should help researchers to develop equations for impedances and transfer functions, and correctly model
Modern Aspects of Electrochemistry, Number 35, Edited by B.E. Conway and Ralph E. White, Kluwer Academic/Plenum Publishers, 2002 1
2
Applications of Electrochemical Impedance
the electrode processes to which they are applied. This is the most difficult, but quickly developing, part of the whole field. With a good knowledge of the literature, it is possible to avoid many pitfalls of electrochemical impedance spectroscopy (EIS). It should be stressed again that EIS is a very sensitive technique but it is usually not sufficient to solve all the emergent problems. Good transfer of knowledge from other electro-chemical and nonelectrochemical techniques is a complementary requirement.
II. DETERMINATION OF IMPEDANCES Impedances may be written for any mechanism. The use of matrix notation simplifies complex calculations. Below, a general method using matrices is presented and applied to a complex mechanism. This method will be used throughout this chapter. Let us suppose that species A and C are soluble (diffusing species) and species B is adsorbed on the electrode surface:
Rates of these processes may be described by the following equations:
where subscripts 1 and 2 correspond to reactions (1) and (2), is the fractional surface coverage by B, that is the ratio of the surface concentration of B, to the maximal surface concentration, and are the potentialdependent rates of these reactions, is the symmetry coefficient, the overpotential, and and are surface concentrations of the
A. Lasia
3
soluble species A and C. From the equilibrium condition
arises the following well-known relation between rate constants and concentrations:
where and are the bulk concentrations of species A and C. In order to solve the problem described by Eqs. (l)-(2) it is necessary to write equations for: (i) current as a function of the rate constants, Eq. (7) below, (ii) current as function of fluxes of diffusing species, Eq. (6), and (iii) the mass balance relations for adsorbed species, Eq. (9). The total current flowing in a steady-state is given as:
Where Next, these equations must be written for phasors. Assuming linear, semi-infinite, diffusion, the oscillating current may be written (see Part I, Section III.2, ref. 1) as
and
Moreover, the mass balance for adsorbed species must be added, i.e.
4
where for phasors, is:
Applications of Electrochemical Impedance
is the rate of adsorption of B. Equation (8), written
Equations (6), (7) and (8) may be rearranged into:
or, in a matrix form Y = AX, after division by
The Faradaic admittance is and may be calculated using Cramer’s rule: where B is the determinant of A, and
A. Lasia
5
determinant T is obtained by substitution of the first column in B by Y = det(Y):
and
These determinants may be expanded into:
6
Applications of Electrochemical Impedance
They are polynomials of the second order in impedance may then be calculated and simplified into:
The faradaic
The first term is the inverse of the charge transfer resistance Further rearrangements of this equation are possible and the faradaic impedance may also be easily determined. A general method of treating the electrochemical impedance of multistep mechanisms was presented by Harrington2 and Harrington and Driessche.3
III. HYDROGEN UPD On several noble metals (Pt, Rh, Ru, Ir, Pd) hydrogen ion reduction takes place at the potentials positive to the equilibrium potential for hydrogen evolution. This is so-called hydrogen
A. Lasia
7
underpotential deposition (H UPD) and indicates strong adsorptive interaction between atomic hydrogen and the metal. Similar UPD processes are observed for deposition of metals on metals4. Such a reaction may be written as:
in acid and alkaline solutions, respectively. Assuming Langmuir adsorption isotherm for H, the rate of this reaction is given as:
where Q is the charge corresponding to the adsorption of H, is the charge necessary for attainment of monolayer coverage by adsorbed H, is the standard rate constant, is the surface concentration of free sites (in mol is the surface concentration of adsorbed H and is the standard potential of reaction (15). The fractional surface coverage by adsorbed H is where is the total concentration of adsorption sites. The equilibrium potential for the H UPD reaction can be expressed as: Taking as a reference state the potential at which (corresponding to the peak potential on cyclic voltammograms, In and introducing it into Eq. (16),
8
is obtained, where which also
Applications of Electrochemical Impedance
and include
are potential dependent rate constants concentration terms: and Let us also introduce overpotential, defined here as The impedance of this process was developed by Harrington and Conway5 and discussed by Lasia.6,7 Equation (17) may be linearized as:
or, introducing phasors (see Part I, Section 4.1, ref. 1),
results. These equations may be written in a matrix form:
and the faradaic admittance
is then
A. Lasia
where
9
and
The
faradaic impedance follows as:
This equation corresponds to a connection of the charge transfer resistance, and the capacitance, in series. It is an analog of Eq. (143), Part I, the corresponding complex plane plot represents a semicircle followed by a capacitive line was shown in Figure 23 (Part I). The total impedance consists of the solution resistance, in series with the parallel connection of the double-layer capacitance, Cdl, giving the faradaic impedance:
On solid electrodes, very often must be substituted by a constantphase element (see Part 1, Section V.2, ref.1)
Assuming transfer coefficients as equal to 0.5, the equivalent circuit parameters may be described as:
10
Applications of Electrochemical Impedance
It should be noticed that the pseudocapacitance is independent of the rate constants; hence the kinetic information may be obtained only from the charge-transfer resistance. Assuming that the following values of the equivalent electrical circuit elements are obtained at
A. Lasia
11
The dependences of and log on overpotential are shown in Figure 1 and 2, respectively. In Figure 1 a maximum of and in Figure 2 a minimum of arises at that is for In the case of symmetry coefficients different from 0.5, the minimum of is slightly shifted (see Part I, Section III.2, ref. 1). However, more complex behavior is observed experimentally. On polycrystalline platinum two voltammetric peaks are observed; this indicates a distribution of adsorption energies and/or a more complex adsorption isotherm. Plots of the experimental impedance on polycrystalline Pt show a capacitive behavior6,7 but the analysis allowed only determination of the pseudocapacitance, which was identical with that determined using cyclic voltammetry. The maximal capacitance was which is lower than the maximal value for Langmurian behavior indicating that the experimental isotherm is more complex.
12
Applications of Electrochemical Impedance
Morin et al.9 studied the UPD of H on Pt single-crystal electrodes. The obtained complex plane plots resembling those predicted theoretically; an example obtained on Pt(l00) in is shown in Figure 3 (note that deformation is connected with different scales being used for the two axes). Analysis according to Eq. (22) for an equivalent circuit containing solution resistance, and double-layer capacitance, allowed all of the parameters to be determined. An example of the dependence of and on electrode potential for Pt(100) is shown in Figure 4. It is surprising that is practically potential independent over a wide potential range and and seem to be correlated. It was also found that at more positive potentials on Pt(311) there is an influence of anionic adsorption. In the equivalent circuit, an additional element containing connection in series should be added in parallel with the faradaic impedance, Eq. (22). Further studies of the H UPD adsorption isotherm should explain this complex adsorption
A. Lasia
13
behavior. It should be added that these measurements are difficult because of the high H UPD rate. Harrington10 has applied ac voltammetry to the study of hydrogen UPD and has determined the fastest sweep rate which can be used without affecting the slow ac response. It should be added that similar mathematical treatment might be carried out for the UPD of metals.
IV. THE HYDROGEN EVOLUTION REACTION 1. The HER in the absence of mass-transfer effects
The hydrogen evolution reaction is one of the most often-studied electrocatalytic reactions. It is well accepted that the reaction mechanism usually proceeds through three steps5,7,11,12: (i) electrochemical adsorption or the Volmer reaction, (15), and two possible desorption steps: (ii) electrochemical desorption, the Heyrovsky reaction, (28), and (iii) chemical desorption, the Tafel reaction, (29). They may be written for the reactions in acidic or alkaline solutions:
14
Applications of Electrochemical Impedance
A. Lasia
15
These reaction steps involve H adsorbed on the electrode surface (case of one adsorbed species, see Section IV-1, Part I, ref. 1). Below, kinetic equations will be presented for hydrogen evolution in alkaline solutions but similar equations may be easily written for the reactions in acidic media. Assuming a Langmuir adsorption isotherm for H on the electrode surface, the rates, of these reactions are:
where are the standard reaction rate constants, the standard electrode potentials, and the surface concentrations of adsorbed H and that of free sites (in and symmetry coefficients. Introducing the surface coverages, and and the overpotential the following equations are obtained:
16
Applications of Electrochemical Impedance
where: indicates parameters measured at subsequently be rearranged into:
and index "0" These equations may
where is the reaction rate in each direction at the equilibrium potential. When concentration polarization may be neglected, these equations simplify to:
A. Lasia
17
From the condition
it also follows that:
In the steady-state given by
and the dc surface coverage is
In general, to describe the hydrogen evolution kinetics, it is necessary to determine four (out of six) rate constants and two transfer coefficients. Such a procedure is quite difficult and the results of dc and ac experiments must be used to determine these parameters. In order to evaluate the reaction impedance, a linearization method is used, as described in Section IV-1, Part 1, ref.l:
where and results in
and
Introducing phasors for
i
18
Applications of Electrochemical Impedance
These equations may be represented in matrix form as
giving the faradaic impedance as:
where
The faradaic admittance, Eq. (49), may be written as a corresponding impedance:
A. Lasia
19
see also Part I, Chapter IV.2, ref.1, where:
Equation (49) is identical with Eq. (135), Part I, the only difference is the definition of the parameter As was shown in Part I, this process may produce two semi-circles on the complex plane plots. Such plots have been observed for the HER on Ft,13,14 Ni-Fe,15 Ru,16,17 Rh,18 electrocodeposited Raney Ni.19,20 However, one semicircle is usually observed in the complex plane plots of Ni21 and Ni-based rough and porous electrodes such as Ni-Zn alloy,22 Ni-B,23,24 Ni-P,25 Ni-Zn-P,26 etc. For some electrode materials (Ni-Zn27,28 and Ni-Al29,28 pressed powders), surface porosity causes appearance of two semi-circles in the complex plane plots, with the first one being connected with surface porosity, Section V-4 (iii), Part I. In other cases, de Levie's porous or fractal models have been used.25, 30-33 2. In the presence of hydrogen mass-transfer
Recently, the influence of the mass transfer of evolved hydrogen 34, 35 was evidenced for the HER on single crystal Pt surfaces in Activity of Pt(hkl) depends on the crystallographic orientations36 and effects of hydrogen diffusion from the electrode are observed on the rotating disk electrode. In order to deal with this problem, Eqs. (40) and (41) should be rearranged, using Eqs. (37) and (38). Neglecting the mass-transfer limitations of protons towards the electrode one obtains (concentrated acid solution):
20
Applications of Electrochemical Impedance
and
where and C* are the surface and bulk hydrogen concentrations, respectively. Current, surface coverage changes and the diffusional flux of hydrogen may be written as:
where is the flux of dissolved hydrogen and its diffusion coefficient. For the rotating disk electrode, the flux is: where the diffusion layer thickness is given by: with v being the kinematic viscosity and the rotation frequency. In order to determine the faradaic impedance Eqs. (55)-(57) must be expressed as phasors. The diffusional flux of hydrogen, for the rotating-disk electrode (finite length diffusion, transmissive conditions) is expressed as:
A. Lasia
21
They may be written in a matrix form:
The faradaic admittance is then:
where A, B, and C were defined in Eq. (50) and the parameters D and E are:
Equation (63) differs from Eq. (49) by the presence of the additional term in the denominator. It may be further simplified by assuming the Volmer-Tafel mechanism. It can be rearranged into a faradaic impedance:
22
Applications of Electrochemical Impedance
where
Equation (65) differs from Eq. (49) by the presence of a new parallel branch in the equivalent circuit, containing connection of a desorption resistance and mass transfer impedance in series (note typing errors in the original publications, cited as refs. 34 and 35). Determination of the kinetic parameters of the hydrogen evolution reaction is usually carried out by simultaneous approximation of the dc current and the parameters obtained from the impedance technique (A, B, C) by adjusting the kinetic parameters (rate constants); however, several authors used approximation of the EIS data only. 11, 12, 16, 25, 27-32, 34-40
V. HYDROGEN ABSORPTION INTO METAL ELECTRODES 1. Hydrogen adsorption, absorption and evolution: Linear diffusion We now consider the hydrogen evolution reaction at negative overpotentials, Eqs. (15), (28) and (29), accompanied by the process of H absorption into the cathode material:
A. Lasia
23
followed by diffusion of hydrogen into the bulk metal.41 This process is observed on hydrogen-absorbing metals (Pd) and alloys (e.g. Let it be supposed that a metallic layer of the thickness l is deposited on a non-absorbing support, from which hydrogen cannot escape. The same reasoning may be applied to H deposition on a metallic foil immersed in the solution; in this case the layer thickness l is half the thickness of the foil. The rate of such a reaction is given by:
where is the dimensionless H concentration inside the metal, i.e. the ratio of the H concentration to the maximal possible H concentration, and index 0 indicates the concentration at the electrode surface, x = 0, inside the absorbing metal. The dimensionless H concentration changes between 0 and 1. Under steady-state conditions, defined in this way, the rate of reaction (67) is null, which leads to:
In order to solve this problem the diffusion of H into the metal must be taken into account through the Fick’s second equation:42-48
while the H flux at the surface is given by
24
Applications of Electrochemical Impedance
where is the charge corresponding to the saturation of metal with hydrogen. Eq. (71) may be solved for the oscillating concentration: where is the concentration phasor, see Part 1 , Eq. (41). The equation obtained is analogous to Eq. (47), Part 1; thus
with the boundary conditions:
where J is the flux of H. The solution of Eq. (72) is:
where obtains:
Taking into account the boundary conditions, one
and the surface concentration is
The oscillating flux is represented by:
A. Lasia
25
or
Now, knowing the diffusional flux, it is possible to calculate the total impedance. As in Section IV, the current is given as:
And
by
Similar reasoning should be applied for
i.e.,
where Equations (77), (80) and (81) can be transformed into matrix form as:
26
Applications of Electrochemical Impedance
The system can be simplified because of using Crammer’s rule for is given as
and the admittance is:
The solution, where:
A. Lasia
27
or
where the parameters A, B and C were defined earlier and D and E are defined as: and This equation should be compared with Eq. (49) for the HER; the difference is the additional term in the denominator, related to the hydrogen adsorption/absorption process. In order to relate to an electric equivalent circuit. In order to take this into account, the impedance of a parallel branch involving diffusion of H, from previously electrodeposited H, has to be considered. For relation to an equivalent circuit, it is useful, in the normal way, for Eq. (86) to be rearranged into a further relation in terms of faradaic admittance, through its relation to impedance by means of the defining equation.
28
Applications of Electrochemical Impedance
where and were defined in Eq. (52) and other terms are defined below (see also Part I, Section IV.2, ref. 1):
corresponds to the mass-transfer impedance for finite-length diffusion and a reflecting interface, see Part 1, Eqs. (98)-(99). The units of are and the other elements are expressed in their usual units For large /, and becomes the impedance for semiinfinite diffusion; see Part 1, Eqs. (61) and (63). Equation (87) corresponds to the circuit shown in Figure 5. The only difference between the HER case and the hydrogen evolutionabsorption mechanism is the presence of the additional parallel branch
A. Lasia
29
A case of finite diffusion length and transmissive boundary conditions has also been considered in the literature.42,43 It represents the case of a metallic membrane where, at one side is reduced and H enters the metal and on the other side H is oxidized. The only difference is that in the mass-transfer impedance function coth is replaced by tanh, see also Part 1, Section III.6. When the parameter decreases, that is when frequency is very low or the layer thickness is small, and Eq. (88), then simplifies to:
with represents a simple in the equivalent circuit.
and the Warburg impedance connection in series (see Eq. (100), Part I)
30
Applications of Electrochemical Impedance
Figure 6 presents an example of the complex plane plots obtained in the absence and in the presence of the hydrogen evolution reaction. In the case of hydrogen evolution only (without absorption), two semicircles (continuous line), related to two time constants, and are observed. In the presence of H absorption (dashed line), three semicircles, corresponding to the charge-transfer resistance, absorption resistance, and adsorption resistance, together with H diffusion effects (part of a straight line at 45°) are observable. When the absorption reaction is very fast the semi-circle corresponding to H absorption disappears (dot-dashed line). Finally, when the absorption
A. Lasia
31
reaction is much faster than desorption (dot-dot-dashed line), a depressed semicircle is observed. In the presence of hydrogen evolution, the faradaic impedance changes from (at ) to at This means that the total impedance varies from at to at This is because, at low frequencies, the mass-transfer impedance becomes infinite and the equivalent circuit reduces to that applicable for the HER. 2. Absorption of the UPD hydrogen
For the case when the Volmer reaction is followed by the hydrogen absorption (e.g. in the case of the hydrogen UPD followed by H absorption or reaction at positive overpotentials), the circuit becomes simplified because B = -AC, and is infinite. In this case, the faradaic impedance is described by:
where
and reduces to: Comparison of the complex plane plots for hydrogen UPD, and hydrogen UPD followed by H absorption, is illustrated in Figure 7. The first semi-circle corresponds to the chargetransfer resistance, An additional semi-circle, related to is observed in the case of H absorption. It is followed by the feature corresponding to finite-length diffusion, i.e. a line at 45° and a
32
Applications of Electrochemical Impedance
capacitive line (vertical line). In the absence of the absorption reaction the semi-circle connected with is followed by one arising from a pseudocapacitance. Comparison of the complex plane plots in the case of the H adsorption/absorption processes, for the case of reflecting and transmissive conditions is shown in Figure 8. For the case of transmissive conditions, it was assumed that the concentration of adsorbed H at x = l is equal to zero, i.e. the applied potential is so positive that all the H diffusing across the membrane is immediately oxidized. Further simplification is achieved when the resistance of H absorption is fast. In this case, reaction (68) is in equilibrium and reduces to:
A. Lasia
33
where which reduces to for large values of the equilibrium constant, An example of the complex plane plot obtained for such a case is shown in Figure 9.
3. Spherical diffusion
Very often H absorption is studied on or type alloy electrode materials. They form powders for which a finite-length spherical diffusion tratment must be used.48-50 In such cases, the H diffusion equation (70) must be modified into:
34
Applications of Electrochemical Impedance
This equation may be solved using the substitution: u = X r. For the oscillating concentration,
is obtained for the following boundary conditions:
A. Lasia
35
The solution of Eq. (94) is:
The second boundary condition gives A = -B. Application of the first boundary condition to the solution for gives:
Then the flux at the electrode surface is given by:
and the Warburg impedance in Eq. (87) is:
36
Applications of Electrochemical Impedance
Spherical diffusion changes the shape of the diffusional part of the impedance behavior. Comparison of the results for linear and spherical diffusion is illustrated in Figure 11. It is interesting to compare relative contributions of (equal to the parallel connection of and ) to the total impedance, Such a comparison is presented in Figure 11. When the parameter is small, that is when frequency is sufficiently low or the particle radius is small, x coth(x) simplifies to and Eq. (98), then becomes simplified to:
This represents a purely capacitive behavior of the Warburg impedance, with Such behavior is different from that observed for thin, planar reflective electrodes, Eq. (89), where reduces to at low frequencies. Absorption of H has been studied at Pd both in transmissive42,43 and reflective46,47,51,52 conditions, and at various hydrogen-absorbing 53-55 alloys such as: mishmetals,49, 56-58 and at bilayers.44,59 Not all authors have used the correct equation developed for the H adsorption-absorption process. An example of the impedance curves obtained for a Pd electrode in the case of reflective conditions is shown in Figure 12. It displays features similar to those simulated in Figure 9 or 7. For a very thin Pd layer, no diffusional feature (straight line at 45° in the complex plane plots) was observed. In this case, the Warburg impedance was reduced to a connection, Eq. (89).
A. Lasia
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An experimental impedance complex plane plot for the case of transmissive conditions is shown in Figure 13. In such a case, the impedance at low frequencies becomes a real value connected with the transfer of H across the membrane under such conditions. Hydrogen absorption and phase transitions are accompanied by volume changes leading to a self-induced mechanical stress.60 These 61 effects were taken into account by in description of the impedance behavior for such conditions. 4. Transfer functions approach
Hydrogen absorption in metals may be studied for the conditions of diffusion across a metallic membrane (e.g. Pd, Fe). This process is shown schematically in Figure 14. In this case, using a Devanathan-
38
Applications of Electrochemical Impedance
Stachurski cell,62 it is possible to study other complex functions, different from impedance, using the so-called transfer function approach63-65. In general, response of the electrical system, R(t), depends on the perturbation, P(t), applied to the system. An equation:
can be written, where L is an operator characterizing the system. If the system consists of linear elements, the operator L is linear. Electrochemical systems, are, however, fundamentally nonlinear but they may be linearized for conditions of small perturbation, P(t). For an arbitrary applied signal, the output can be related to input by taking
A. Lasia
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Laplace transforms of the perturbation and the signal. A transfer function, called the impedance, Z(s), is defined as (see Eq. (6), Part I):
where is the Laplace transform operator. The transfer function characterizes response of the system to the applied perturbation. Its knowledge permits prediction of the system response.
40
Applications of Electrochemical Impedance
A. Lasia
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In ac techniques, the perturbation is a periodic function (sin, cos). In order to resolve the problem, one can use the Fourier transform analog of Eq. (100), viz
where and are the Fourier transforms of the perturbation P(t) and the response R(t), and is the transfer function relating 65-68 In the particular case when response to the perturbation. and transfer function called the admittance. Of course, the inverse of this function is the impedance. In general, various transfer functions may be defined, e.g. mass response of conductingpolymer, to the applied ac perturbation, the so-called 69, 70 electrogravimetric transfer function: . Other transfer functions include the electro-optical transfer function i.e. the transfer function for the relation between between current and reflectance, the electrocoulometric transfer function, i.e. transfer function between ring and disk currents, and electrohydrodynamical impedance where the perturbation is a modulation of the angular velocity of the rotating disk electrode, and also magnetohydrodynamical impedance,68 etc. In the case of a metallic membrane, “ordinary” transfer function relating current to the applied potential is an admittance where index 0 denotes parameters at x = 0, i.e. at the entry side of a membrane. However, one can also measure H transfer across the membrane, i.e. currents measured at both sides of the membrane where index l indicates the exit side. It should be noticed that is dimensionless. Another possible transfer function is: which has the units of impedance and characterizes the current measured on the exit side in a response to a sinusoidal potential perturbation on the entry side. Also there is which is a potentiometric transfer function describing variations of the potential at the exit side equals to zero) under perturbation at the entry side. It
42
Applications of Electrochemical Impedance
should be stressed that is always zero or infinity depending on the constant-potential or constant-current conditions. The above defined transfer functions may be measured using a frequency response analyzer. We can now determine transfer functions for the case of H permeation. Let it be supposed that on the entry side a sinusoidal perturbation is superimposed on a dc potential and, at the exit side, the applied potential is sufficiently positive that H arriving from across the membrane is immediately oxidized, Figure 14. In this case we have to solve the diffusion equation, Eq. (72), with the following boundary conditions:
The solution is:
The flux of H in the membrane is given by:
The fluxes at x = 0 and x = l are:
A. Lasia
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Hydrogen (H) concentrations and fluxes at x = 0 and x = l are related by the following equation:64
The transfer function for fluxes, calculated from Eq. (106), is:
This function may be separated into a real and imaginary part and a new expression is obtained:
where The transfer function is dimensionless and normalized. A complex plane plot illustrating the flux transfer function is shown in Figure 15 and the dependence of the real and imaginary parts of this transfer function as a function of the logarithm of the frequency is displayed in Figure 16. However, the H flux is not measurable; only the currents on both sides of the membrane may be determined; the total current on the entry side is and that at the exit side is (if the only reaction at x = 1 is that of H oxidation). The total current flowing to the membrane on the entry side consists of that for double-layer charging, and the faradaic, currents: Then, from purely electrical analysis of the equivalent circuit:
44
Applications of Electrochemical Impedance
In the same way one can find the ratio of the current flowing through the branch, Figure 5, i.e. the current entering the membrane, to the faradaic current,
A. Lasia
45
where Taking into account Eqs. (108), (110) and (111) the transfer function for currents can be obtained as:
46
Applications of Electrochemical Impedance
An example of the current transfer function for H transfer across a Pd membrane is shown in Figure 17. Because the frequencies studied were below 0.2 Hz, only the first term in Eq. (112), containing cosh(sl) was important, that is Eq. (108) was sufficient to approximate the results. In a similar way it is possible to determine the transfer function of the exit current to the applied potential, It has the dimensions of admittance and can be obtained from Eq. (112) as:
A. Lasia
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Because the membrane thickness is typically only diffusional effects are observed at very low frequencies. Hydrogen transfer functions have been studied for iron65, 71 and palladium.72 It should be added that intercalation of metal ions into solid matrices, e.g. in Li-ion electrode systems, may be formally described by equations similar to those for electrochemical absorption of H. REFERENCES 1. A. Lasia, Electrochemical Impedance Spectroscopy and its Applications in Modern Aspects of Electrochemistry, vol. 32, R. E. White, B. E. Conway, and J. O’M. Bockris, Eds., Plenum Press, New York, 1999, p. 143. 2. D.A. Harrington, J. Electroanal. Chem., 403 (1996) 11; 449 (1998) 9, 29. 3. D. A. Harrington and P. van den Driessche, Electrochim. Acta, 44 (1999) 4321. 4. E. Leiva, Electrochim. Acta, 41 (1996) 2185. 5. D. A. Harrington and B. E. Conway, Electrochim. Acta, 32 (1987) 1703. 6. A. Lasia, Polish J. Chem., 69 (1995) 639. 7. A. Lasia in Proceedings of the Symposium on Electrochemistry and Materials Science of Cathodic Hydrogen Absorption and Adsorption, B.E. Conway and G. Jerkiewicz, Eds., The Electrochemical Society, 94-21, 1994, p. 261. 8. D. A. Harrington and B.E. Conway, J. Electroanal. Chem., 221 (1987) 1. 9. S. Morin, H. Dumont, and B.E. Conway, J. Electroanal. Chem., 412 (1996) 39. 10. D. A. Harrington, J. Electroanal. Chem., 355 (1993) 21. 11. A. Lasia, Current Topics Electrochem., 2 (1993) 239. 12. E. R. Gonzalez, G. Tremiliosi-Filho, and M. J. de Giz, Current Topics Electrochem., 2 (1993) 167. 13. L. Bai, D. A. Harrington, and B. E. Conway, Electrochem. Acta, 32 (1987) 1713. 14. P. Gu, L. Bai, L. Gao, R. Brousseau, and B. E. Conway, Electrochim. Acta, 37 (1992) 2145. 15. R. Simpraga, G. Tremiliosi-Filho, S. Y. Qian, and B. E. Conway, J. Electroanal. Chem., 424 (1997) 141. 16. J. Fournier, P. K. Wrona, A. Lasia, R. Lacasse, J.-M. Lalancette, and H. Ménard, J. Electrochem. Soc., 139 (1992) 2372. 17. H. Dumont, P. Los, A. Lasia, and H. Ménard, J. Appl. Electrochem., 23 (1993) 684.
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Applications of Electrochemical Impedance
18. H. Dumont, P. Los, L. Brossard, A. Lasia, and H. Ménard, J. Electrochem. Soc., 139 (1992) 2143. 19. Y. Choquette, L. Brossard, A. Lasia, and H. Ménard, J. Electrochem. Soc., 137 (1990) 1723. 20. Y. Choquette, L. Brossard, A. Lasia, and H. Ménard, Electrochim. Acta, 35 (1990) 1251. 21. A. Lasia and A. Rami, J. Electroanal. Chem., 294 (1990) 123. 22. L. Chen and A. Lasia, J. Electrochem. Soc., 138 (1991) 3321. 23. P. Los and A. Lasia, J. Electroanal. Chem., 333 (1992) 115. 24. J. J. Borodzinski and A. Lasia, J. Appl. Electrochem., 24 (1994) 1267. 25. R. Karimi Shervedani and A. Lasia, J. Electrochem. Soc., 144 (1997) 511. 26. R. Karimi Shervedani and A. Lasia, J. Electrochem. Soc., 144 (1997) 2652. 27. L. Chen and A. Lasia, J. Electrochem. Soc., 139 (1992) 3214. 28. C. Hitz and A. Lasia, J. Electroanal. Chem., 500 (2001) 213. 29. L. Chen and A. Lasia, J. Electrochem. Soc., 140 (1993) 2464. 30. L. Chen and A. Lasia, J. Electrochem. Soc., 139 (1992) 3458. 31. A. Rami and A. Lasia, J. Appl. Electrochem., 22 (1992) 376. 32. P. Los, A. Rami, and A. Lasia, J. Appl. Electrochem., 23 (1993) 135. 33. P. Los, A. Lasia, L. Brossard, and H. Ménard, J. Electroanal. Chem., 360 (1993) 101. 34. J. Barber, S. Morin, and B. E. Conway, Proceedings of the Symposium on the Electrochemical Surface Science of Hydrogen Adsorption and Absorption, G. Jerkiewicz and P. Marcus, Edts., The Electrochemical Society, vol. 97-16, 1997, p. 101. 35. J. H. Barber, S. Morin, and B. E. Conway, J. Electroanal. Chem., 446 (1998) 125. 36
B. N. Grgur, and P. N. Ross, J. Phys. Chem. B, 101 (1997) 5405. 37. M. J. de Giz, G. Tremiliosi-Filho, and E. R. Gonzalez, Electrochim. Acta, 39 (1994) 1775. 38. M. J. de Giz, G. Tremiliosi-Filho, E. R. Gonzalez, S. Srinivasan, and A. J. Appleby, Int. J. Hydrogen Energy, 20 (1995) 423. 39. E. B. Castro, M. J. de Giz, E. R. Gonzalez, and J. R. Vilche, Electrochim. Acta, 42 (1997) 951. 40. N. A. Assunção, M. J. de Giz, G. Tremiliosi-Filho, and E. R. Gonzalez, J. Electrochem. Soc., 144 (1997) 2794. 41. A. Lasia and D. Grégoire, J. Electrochem. Soc., 142 (1995) 3393. 42. C. Lim and S.-I. Pyun, Electrochim. Acta, 38 (1993) 2645. 43. C. Lim and S.-I. Pyun, Electrochim. Acta, 39 (1994) 363. 44. J.S. Chen, R. Durand, and C. Montella, J. Chim. Phys., 91 (1994) 383. 45. T.-H. Yang and S.-I. Pyun, J. Power Sourc., 62 (1996) 175. 46. T.-H. Yang and S.-I. Pyun, J. Electroanal. Chem., 414 (1996) 127. 47. T.-H. Yang and S.-I. Pyun, Electrochim. Acta., 41 (1996) 843. 48. C. Wang, J. Electrochem. Soc., 145 (1998) 1801. 49. L. O. Valøen, S. Sunde, and R. Tunold, J. Alloys Comp., 253-254 (1997) 656. 50. B.S. Haran, B.N. Popov and R.E. White, J. Power Sources, 75 (1998) 56.
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51. G. Jorge, R. Durand, R. Faure, and R. Yvari, J. Less-Common Met., 145 (1988) 383. 52. S.-I. Pyun, T.-H. Yang, and C.-S. Kim, J. Appl. Electrochem., 26 (1996) 953. 53. P. Agarwal, M.E. Orazem, and A. Hiser in Proceedings of Symposium on Hydrogen Storage Materials, Batteries and Electrochemistry, D.A. Corrigan and S. Srinivasan, Eds., Electrochemical Society, Pennington, N.J., 1991, p. 120. 54. T.-H. Yang and S.-I. Pyun, J. Power Sourc., 62 (1996) 175. 55. P. Millet and P. Dantzer, J. Alloys Comp., 253-254 (1997) 542. 56. N. Kuriyama, T. Saki, H. Miyamura, I. Uehara, and T. Iwasaki, J. Alloys Comp., 202 (1993) 183. 57. W. Zhang, M. P. S. Kumar, S. Srinivasan, and H.J. Ploehn, J. Electrochem. Soc., 142 (1995) 2935; see also Y. Leng, ibid., 144 (1997) 2941 and W. Zhang and S. Srinivasan, ibid., 144 (1997) 2942. 58. X. Gao, J. Liu, S. Ye, D. Song, and Y. Zhang, J. Alloys Comp., 253-254 (1997) 515. 59. Y.-G. Yoon and S.I. Pyun, Electrochim. Acta, 40 (1995) 999. 60. B. Baranowski in Flow, Diffusion, and Rate Processes, (Advances of Thermodynamics, vol. 6), S. Sieniutycz and P. Salamon, Edts., Taylor and Francis, New York, 1992, p. 168. 61. Electrochim. Acta, 44 (1999) 4415; J. Electroanal. Chem., 501 (2001) 89. 62. M. A. V. Devanathan and Z. Stachurski, Proc. Roy. Soc., A270 (1962) 90; J. Electrochem. Soc., 111 (1964) 619. 63. N. Boes and H. Züchner, J. Less Common Metals, 49 (1976) 223. 64. C. Montella, J. Electroanal. Chem., 462 (1999) 73; 480 (2000) 150, 166. 65. P. Bruzzoni, R. M. Carranza, J. R. Collet, and E. A. Crespo, Electrochim. Acta, 44 (1999) 2693. 66. P.-P. Grand, PhD Thesis, Université de Sherbrooke, 2001. 67. C. Gabrielli, M. Keddam, and H. Takenouti, Electrochim. Acta, 35 (1990) 1553. 68. C. Gabrielli, B. Tribollet, J. Electrochem. Soc., 141 (1994) 1147. 69. C. Gabrielli, M. Keddam, N. Nadi, and H. Perrot, Electrochim. Acta, 44 (1999) 2095. 70. C. Gabrielli, M. Keddam, H. Perrot, M.C. Pham, and R. Torresi, Electrochim. Acta, 44 (1999) 4217. 71. P. Bruzzoni, R.M. Carranza, J.R. Collet Lacoste, and E.A. Crespo, Int. J. Hydrogen Energy, 24 (1999) 1093. 72. P.P. Buckley, J.A. Fagan, and O.C. Searson, J. Electrochem. Soc., 147 (2000) 3456.
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2
Electroless Deposition of Metals and Alloys
Stojan S. The Westaim Corporation, Fort Saskatchewan, Alberta, T8L 3W4, Canada
I. INTRODUCTION Electroless deposition of metals and alloys has a very significant practical importance in modern technology especially in the production of new materials for applications in electronics, wear and corrosion resistant materials, medical devices, battery technologies, etc. This process supplements and in some cases replaces electrodeposition for several practical reasons. The solutions for electroless deposition have excellent throwing power and allow plating on articles of very complex shapes and plating through holes. Deposits obtained by electroless deposition are more dense (more pores-free) and exhibit better properties for corrosion and electronics applications. Other important advantages of electroless deposition over electrodeposition include its applicability for metallization of nonconductive surfaces (glass, ceramics, polymers, etc.) and the ability to selectively deposit thin metal films only on catalyzed areas of the substrate. Finally, for electroless deposition processes, an external current source is not needed. In spite of relatively intensive studies in recent decades, electroless deposition is still not clearly understood. Attempts have been made to Modern Aspects of Electrochemistry, Number 35, Edited by B.E. Conway and Ralph E. White, Kluwer Academic/Plenum Publishers, 2002 51
52
Electroless Deposition of Metals and Alloys
explain the mechanisms and kinetics of electroless deposition; however, a fully accepted model is not yet established. It is obvious that much more research is needed to advance scientific understanding in order to ensure successful operations of the process. The aims of this chapter are to review achievements in the field of electroless deposition of metals and alloys, and to examine in some detail the mechanisms that have been proposed for the processes involved.
II. ELECTROLESSLY DEPOSITABLE METALS AND ALLOYS
The first scientific description of electroless deposition processes by von Liebig in 1835 was related to the reduction of Ag(I) salts to Ag metal using aldehydes.1 Later in 1844, Wurtz observed that ions 2 can be reduced by hypophosphite ions. However, he obtained only a black powder. This discovery can be considered as the beginning of the development of the electroless deposition process. In spite of its early start, process development in this field remained slow until 1946, when Brenner and Riddell described the proper conditions for electroless deposition of smooth Ni films from an aqueous solution containing nickel and hypophosphite ions. 3,4 Over the past few decades, significant progress in this field has been made because of rapid developments in modern technology related to electronics, corrosion protection, batteries, decorative purposes, biomedical applications, etc. It seems that all metals electrochemically depositable from aqueous solutions can also be deposited electrolessly under proper conditions (bath composition, pH, temperature, and corresponding catalytic surface), using suitable reducing agents. Table 1 presents a survey of metals and alloys that have been deposited electrolessly hitherto. In the first group are listed commonly deposited single metals such as Ni, Co, Cu, Ag, Au and Pd. Other metals from this group do not have significant applications at the
S. Djokic
53
present, but it should be noted that there are reports on their deposition in the published literature. For example, it has been reported that Cd can be deposited using as a reducing agent5 and Pb can be 6 deposited using thiourea or titanium chloride7 as reducing agents. Bi8 and Sb9 have been deposited by In10 by and Pt11, Ru12 and Rh13 by hydrazine. Although there are no other reports, it is worth noting the claim that Cr can be electrolessly deposited from an aqueous solution of and sodium hypophosphite as the reducing agent.14 The second group lists elements, which cannot be deposited alone. However, they can easily be codeposited with nickel or cobalt. Typical examples are Mo and W. The phenomenon is somehow analogous to induced electrodeposition as Brenner defined it.15 The third group represents alloys based on the first and/or second group of elements. These alloys have been deposited for various applications, mainly in the electronics industries. There is a high probability that other alloys (which are commonly electrodeposited) can also be electrolessly deposited, but there are no published data so far.
54
Electroless Deposition of Metals and Alloys
III. BASIC DEFINITIONS, SIMILARITIES AND DIFFERENCES AMONG ELECTROLESS PROCESSES Brenner and Riddell3,4 were the first authors to introduce the term electroless metal deposition when describing an autocatalytic process of depositing a metal in the absence of an external source of electrical current. Since there are other metal depositions from aqueous solutions that are carried out without an external current, this process can be divided into three main groups: 1. displacement deposition 2. contact deposition 3. autocatalytic deposition 1. Displacement Deposition
Displacement deposition is a heterogenous galvanic process in which the noble metal ions are reduced and deposited at the surface of an active metal, as a consequence of dissolution of that metal. The process is sometimes called immersion plating, although this term it is not a specific description, and therefore should be avoided, or cementation. The overall displacement reaction is quite simple16 :
and involves the displacement half-reaction of a more active metal
by a more noble metal,
Typical cementation systems in practice are Ag/Zn, Ag/Cu, Cu/Zn, Cu/Fe, Cu/Al, Sn/Cu etc. The displacement reaction stops immediately
S. Djokic
55
after the reduced metal (more positive metal) covers the surface of the immersed metal (more negative metal). Accordingly, the thickness of the deposited metal is always limited. The time of immersion is particularly critical for achieving a uniform coating layer. Very often, the adhesion of the deposited films is not as good as that of films prepared by electrodeposition or by autocatalytic deposition. The displacement deposition differs from all other plating processes from aqueous solutions without an external source of electrical current, because it does not require a reducing agent. Similarly to autocatalytic processes, displacement deposition has the advantage of nearly unlimited throwing power. Because of lower quality and thinner coatings, displacement deposition has found applications mainly in the refining metals. To a certain extent, however, there are other applications such as coatings for porcelain enamelling17, zincate coatings18, decorative finishing, soldering19, etc. 2. Contact Deposition
Contact deposition is equivalent to electrochemical deposition with the exception that the current is derived from the chemical reaction and not from an outside source. The metal on which deposition takes place, and the auxiliary metal with which it is in contact, form a galvanic element. In this galvanic element, the auxiliary metal acts as an anode and dissolves; the other metal is a cathode. Consequently, the dissolved metal is deposited on the cathode (metal on which deposition takes place) at a mixed potential. The importance of contact deposition for industrial applications is relatively small. Sometimes the process is used with autocatalytic Ni to initiate Ni deposition on copper and its alloys.18 This is achieved by coupling the Cu or Cu-alloys with Al, Fe or Ni. The contact deposition is applicable only to a limited extent, and uniform thicker deposits cannot be obtained. On the other hand, the constant increase of dissolved metal concentration in the solution may cause instability of the solution. It should be noted, though, that contact deposition is very suitable for the initiation of autocatalytic deposition on some surfaces,
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Electroless Deposition of Metals and Alloys
(e.g. Ni on Cu). The metals that can be plated by the contact deposition process include Ni, Co, Cu, brass, Ag, Pt, Sn.18 3. Autocatalytic Deposition
Autocatalytic deposition is the most commonly used chemical method for the deposition of metallic films from aqueous solution without an external source of electrical current. The metal films are formed only on catalytically active surfaces without an external source of electrical current and by the chemical reduction of metallic ions in an aqueous solution containing a reducing agent. Autocatalytic deposition is defined as a process for deposition of metallic films by a controlled chemical reaction that is catalyzed by the metal or alloy being deposited.20,21 If the metal ion, is reduced by the reducing agent ion, the process can be simply described by the following reaction:
Although the term electroless deposition broadly describes all processes of metal and alloy deposition without an external source of electrical current, it should be noted that this term is commonly used the for autocatalytic deposition process. Consequently, in this chapter, the term electroless deposition is used only for the autocatalytic deposition processes. The development of electroless deposition is mainly connected with Ni or Cu deposition. However, other electrolessly depositable metals and/or alloys such as Ag, Au, Co, Sn, AuSn, NiWP, etc. have also been studied because of their important applications.
IV. SOLUTIONS FOR ELECTROLESS DEPOSITION
All solutions for electroless metal deposition have many similarities, but depending on the metal or alloy to be deposited, there are also some
S.Djokic
57
differences. Typically, the constituents of a solution for electroless metal deposition are:
1. 2. 3. 4.
Source(s) of metal ions Complexing agent(s) Reducing agent(s) Stabilizer(s) and inhibitor(s) 1. The Metal Ion Sources
Table 2 presents sources of metal ions in electroless deposition of common metals. Generally speaking, the metal ion sources can be any water-soluble salts such as sulfates, chlorides, acetates, cyanides, etc. The nature of the metal ion source is usually determined by the stability of the solution, properties of the deposited films and also by environmental issues. Chloride ions are used for limited applications, since they act deleteriously when these salts are used to coat Al or ferrous based alloys.20 In the case of Al, the chloride ion can promote its dissolution, and the coating will exhibit poor adhesion. Similarly, in the case of ferrous metals, adsorbed chloride ions at the substrate surface can provoke corrosion. A recent report22 on the effect of nickel salts used in electroless deposition showed the best overall results (in terms of rate of deposition and desirable properties of the deposit) had been accomplished using and then or The best nitric acid resistance and the highest hardness of electroless deposited nickel coatings was obtained using nickel fluoride as the source of nickel ions. In electroless copper deposition, is used mainly as the source of copper ions. In electroless gold or silver deposition, cyanide salts are still employed. However, environmental problems related to the use of cyanide solutions, create permanent pressure towards using other non-toxic solutions.
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Electroless Deposition of Metals and Alloys
2. Complexing Agents The majority of complexing agents used in electroless metal deposition are organic acids or their salts, with a few exceptions of inorganic ions such as or Cyanides have never been used in electroless nickel deposition. Pyrophosphates are exclusively used in alkaline electroless nickel solutions. Ammonia and ions, in the case of nickel solutions, are mainly used for pH control. The choice of the complexing agents is dependent, first of all, on the nature of the metal ion used for deposition. The principal functions of complexing agents are: buffering action, prevention of precipitation of hydroxides and salts, and reduction of the concentration of free (aquo) metal ions. In addition, complexing agents affect the rate of reduction and the properties of metal deposits. In some cases, complexing agents apparently form strong complexes with metallic contaminants, thereby making them less susceptible to react with reducing agents. Complexing agents used for the electroless deposition of common metals are listed in Table 3. Commercial solutions for nickel electroless deposition operate in the pH range 4.5 to 6. The complexing agents are most effective in this pH range. However, in the
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electroless deposition of Cu, Au, Ag, Pd and in some cases Ni, solutions with pH > 8 are used. In these cases, different complexing agents are needed (Table 3). For an extensive review of the stability of solutions and the effect of complexing agents on hydrolysis of cations, the reader is referred to field-related monographs (e.g. see References 23- 25).
3. Reducing Agents
Choice of reducing agent depends on conditions of electroless deposition and, of course, on the metal or alloy being deposited, including their physico-chemical properties. Use of reducing agents containing phosphorus or boron leads unavoidably to the incorporation of these elements, which can dramatically affect the properties of the metal deposit. On the other hand, electroless deposition of pure metals is also possible, using reducing agents such as hydrazine or
Electroless Deposition of Metals and Alloys
60
formaldehyde. Individual reducing agents, used for electroless deposition, are discussed in the following sections. (i)
Hypophosphite
Hypophosphite is mainly used for the electroless deposition of Ni, Co, Pd and their alloys. The deposits are not purely metallic as they usually contain phosphorus. Utilization of hypophosphite in electroless metal deposition is considerably less than 100 %. The reduction reaction takes place only at certain surfaces - such as metals belonging to group VIII of the periodic table (Fe, Co, Ni, Rh, Pd, and Pt). It also takes place on Au. Initial reactions in electroless plating depend on the nature of the metal being deposited and on the nature of the substrate on which deposition takes place. For example, when electroless Ni plating takes place on metals such as Fe or Al, the initial reaction in the deposition process is the displacement reaction:
or
At the formed Ni layer, further reduction of with hypophosphite proceeds independently. A similar initial process takes place when Cu or its alloys are plated with gold. The first reaction is displacement of Cu with
On the other hand, when Cu or its alloys are plated with Ni, pretreatment with catalyst or contact deposition20 is required. The most studied reaction among electroless processes is definitely deposition of Ni with hypophosphite. The overall reaction for nickel deposition with hypophosphite can be represented as:
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Similar reactions are applicable for Co or Pd deposition with hypophosphite. Generally, if the concentration of hypophosphite is increased, the phosphorous content in NiP alloy is increased.26-30 The concentration in most acid-type solutions (pH 4 to 6) is 0.09 to and molar concentration for sodium hypophosphite is 0.18 to With a constant concentration of the other constituents and constant pH, the Ni deposition rate is proportional to the concentration of hypophosphite.26,31 Similar behaviour has been reported for alkaline solutions: the rate of deposition increased with an increase in hypophosphite concentration.20,32 Lloyd and Mallory investigated kinetics of Ni electroless deposition with hypophosphite from an acidic solution (pH=4.8) in the presence of hydroxyacetate, lactate, malate and citrate.33 They found for all investigated complexes that increasing hypophosphite concentration up to leads to an increase in the deposition rate. Typical dependence of the deposition rate on the hypophosphite concentration for the case of used as a complexing agent is presented in Figure 1. The reaction order in respect to the hypophosphite concentration depended on the nature of the complexing agent. An increase in the hypophosphite ion concentration in the solution leads to an increase in phosphorus content in electroless deposited CoP films. 34 Feldstein and Lanscek reported linear relationships between the deposition rates of Co or Ni and the steady-state potential in the electroless deposition of NiP or CoP alloys with hypophosphite as the reducing agent. 35 Similarly, the rate of phosphorus deposition increased linearly as the steady-state potential increased. An electrochemical investigation of electroless NiP deposition from ammoniacal solutions shows the existence of strong interactions between the reduction of ions and oxidation of hypophosphite 36 ions. The rate of reduction of ions increases the rate of hypophosphite oxidation. This behavior is attributed to deprotonation
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Electroless Deposition of Metals and Alloys
of the hypophosphite anion, which occurs on the catalytic surface of the growing layer. Electroless deposition of Cu with hypophosphite is still doubtful. In the presence of nickel, however, NiCuP alloy films have been deposited successfully. 37-42 In Ref. [38], it was claimed that in spite of small amounts of Ni was not deposited (i.e. "pure" CuP films were obtained). The reaction of Cu deposition with hypophosphite takes place in the alkaline region (pH 8 to 10) and was described as:
Chassaing et al. reported that an increase in hypophosphite concentration from 0.1 to during autocatalytic deposition of NiCuP alloys at pH=9 leads to an increase in both the phosphorus 42 content (2 to 9 %) and the rate of deposition (0 to It seems, according to these results, that further increase in the hypophosphite concentration leads to a decrease in the rate of alloy deposition, which is explicable by the increase in bath instability. Electroless Au deposition solutions, with hypophosphite as reducing agent, are described in literature. These solutions are first of all convenient for Au deposition onto Ni-based substrates.43-46 As shown by Brenner46 and later by Kurnoskin et al.44, these processes are not autocatalytic. Kurnoskin et al. showed that in the initial stages, the process primarily proceeds by the galvanic displacement mechanism. In the later stages, exposed areas of nickel serve as the catalyst surface for the hypophosphite oxidation, causing gold deposition onto the remaining areas of the surface. Solutions for electroless deposition of silver with hypophosphite have been reported in the literature, although, it seems that they provide deposition rather by the galvanic displacement mechanism than by a truly autocatalytic process.46
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There are several solution formulations for electroless Pd deposition using hypophosphite as the reducing agent.47-51 All these solutions are alkaline with pH higher than 8. The results show that only 31 % of hypophosphite used in the process is consumed for the reduction of the remainder decomposes with evolution of gaseous hydrogen from the solution.48 The rate of deposition increases with an increase in hypophosphite concentration up to Above this concentration the solution is unstable. (ii) Boron-containing reducing agents Boron-containing reducing agents used in electroless Ni deposition are mainly borohydrides and amine boranes. Deposits usually contain 90 to 99 % metallic phase, depending on the composition of the
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Electroless Deposition of Metals and Alloys
solution and operating conditions. The rest is usually boron and other occluded reacting agents. The-boron containing reducing agents are used for electroless deposition of common metals, such as Ni, Co, Pd, Pt, Au, Ag and their alloys. Borohydride ion The electroless deposition of Ni with borohydride takes place in alkaline solutions. Theoretically, each borohydride ion can reduce four nickel ions:
However, experimental results show that one mole of borohydride reduces approximately one mole of nickel ion.20 The reduction to boron, according to literature data, proceeds according to the following reactions:20,52
or
or
or
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Gorbunova et al. investigated the conditions for electroless nickelboron deposition using sodium borohydride as a reducing agent.52,53 They found that an increase in the concentration in solution without a stabilizer or with stabilizers such as lead chloride, 2mercaptobenzothiazole or thallium nitrate, leads to an increase in the rate of Ni-B deposition (Figure 2). Using a solution containing as a stabilizer gives a faster rate of Ni-B deposition. In the electroless deposition of gold, Okinaka assumed that the reducing agent of is not itself, but the species formed as a result of hydrolysis according to the reactions:54
and
Since the utilization efficiency of borohydride in electroless Au deposition is very low (less than 2 %), Efimof et al55,56 proposed that another hydrolysis product, participates as a reducing agent This result is attributed to the decomposition of borohydride. The overall reactions for Au deposition were described as follows:55,56
and
The maximum rate of Au deposition with borohydride is obtained with 57 concentration of about This unusual dependence is attributed to the competitive adsorption of and
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Electroless Deposition of Metals and Alloys
on the gold surface. An increase in concentration acts as a catalytic poison, thus preventing the adsorption of the reducing agent.
If is formed because of hydrolysis, according to a polarographic investigation carried out by Gardiner and Collat58-60, the overall reaction describing gold deposition from a borohydride solution can be presented as
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Using borohydride as the reducing agent, gold-based alloys such as Au-Ag61 and Au-In62 have been produced. Other metals deposited using borohydride as a reducing agent are Pt63 and In64. Electroless cobalt deposition with borohydride has also been reported. 20,65,66
Amine Boranes Whereas borohydrides such as are completely ionic, the amine boranes are covalent compounds. The electrons in the amine boranes are displaced toward the boron atom, giving the boron atom negative character, while the nitrogen atom displaces positive charge as is illustrated by the following formula: In practice, the application of aminoboranes is limited to dimethylamine borane, Dimethylamine borane (DMAB) is used for the electroless deposition of Ni67-71, Cu72,72, Co74-76 and Ag77. In alkaline and neutral solutions, the preceding chemical reaction of dimethylamine borane with ions can be represented as:57,78
The acid-catalyzed hydrolysis of dimethylamine borane occurs according to the following equation:79
Most authors believe that the major species supplying electrons for 67,68,71,72 metal-ion reduction is The investigation of hydrolysis 71 of DMAB shows that hydrolysis is pronounced at pH’s below 5. In
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Electroless Deposition of Metals and Alloys
this way, a significant amount of DMAB is wasted by the hydrolysis, and consequently the electroless deposition in this region should be avoided. In the pH region above 5, the consumption of DMAB by hydrolysis approaches a minimum. The rate of Ni deposition increases with an increase in DMAB for all investigated pH's within the range 6 to 11.71 However, it should be noted that an increase in pH within this range leads to a decrease in the rate of Ni deposition. This can be attributed to the increase in the solution stability (probably because of the tendency of to hydrolyze at very high pH). Under these conditions, the reduction reaction may start in the bulk solution and the rate of deposition (at the corresponding substrate) decreases. Consequently, the deposition efficiency decreases. Mallory suggested that the preferred operating pH range for Ni deposition with DMAB is 6 to 7 (near neutral). 20 DMAB has three active hydrogen atoms bonded to the boron, and theoretically should reduce three metal ions for each ion of
The boron reduction can be represented by the following reaction:71
Based on the experimental results, in the electroless nickel deposition the molar ratio of nickel ions reduced to DMAB molecules consumed during the process is approximately 1:1.71 Parlstein and Weightman investigated electroless deposition of Co with DMAB from acid solutions containing cobalt sulfate, sodium succinate, sodium sulfate and DMAB.76 Cobalt-deposits obtained at 70 °C and pH 5.0 contained 96 % Co, 1.7 % B, 0.97 % C and 0.05 % N. Dependence of deposition rate on DMAB concentration is presented in Figure 3. As illustrated in this figure, the rate of Co deposition increases almost linearly up to
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DMAB concentration of about A further increase in DMAB concentration results in a rapid decomposition of the solution.
(iii) Formaldehyde Formaldehyde is mainly used for electroless copper and silver deposition; however, there are reports that this reducing agent can also be used for electroless deposition of AuCu alloy80 or Co81. An overall reaction for electroless copper deposition with formaldehyde was proposed by Lukes82
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Electroless Deposition of Metals and Alloys
Dumesic et al. studied electroless copper deposition from an EDTA alkaline solution, using formaldehyde as a reducing agent.83 They reported that an increase in the formaldehyde concentration from 0.03 to leads to a linear increase in the initial deposition rate (Figure 4).
On the other hand, the final rate of copper deposition, according to the results of these authors, is almost independent of formaldehyde concentration. Paunovic investigated the effect of complexing agents such as and on electroless copper deposition.84 The rate of copper deposition as a function of ligand, increased in the order: tartrate, EDTA, quadrol,
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CDTA. The addition of a second complexing agent to the solution that already contains alters the rate of deposition.85 In this case, the second complexing agent is treated as an additive. The dependence of deposition rate on formaldehyde concentration passes through a maximum, when only CDTA (cyclohexane-1,2-diamine-N,N,N',N'tetraacetic acid) is used as a complexing agent of ions.86 However, when 50 %of CDTA and 50 % of EDTA are present, a linear increase in the deposition rate with an increase in the formaldehyde concentration is observed (see Figure 5).
Electroless Ag deposition with formaldehyde is fast, but either a cloudy film of silver metal is obtained or peeling occurs. Silver
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Electroless Deposition of Metals and Alloys
deposition with formaldehyde can be described by the following reactions:
or
It seems that the last reaction is more probable, since the hydrogen evolution reaction has not been observed during the electroless deposition. From other metals, as mentioned earlier, the electroless Co deposition is carried out using formaldehyde as a reducing agent.81 The rate of cobalt deposition increased in a linear way, with an increase in concentration within the range to On the other hand, the dependence of the rate of Co deposition on the formaldehyde concentration is found to pass through a maximum for the formaldehyde concentration at about In the alkaline solution (NaOH concentration 7 to is formed.81 Reduction to Co is accompanied by hydrogen gas evolution. The overall reaction for Co deposition is described by the following equation:
(iv) Hydrazine Hydrazine has long been recognized as a very powerful reductant of metallic ions.87-89 Hydrazine and its salts are excellent reducing agents, as indicated by their standard redox potentials:90
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(E° =1.16 V, alkaline conditions) and
(E° =0.23 V, acidic conditions). As shown by these reactions, the hydrazine is a better reducing agent under alkaline than under acidic conditions. Hydrazine has been used for electroless deposition of metals and alloys onto metallic and non-metallic surfaces. Examples include electroless Cu90,91, Ni92-94 , Au95, Ag87,89,96,97, Pt-group of metals and their alloys98-102, NiSnW89, NiFe89 and alloys resembling stainless steel.103 There are also reports on electroless deposition of Co with hydrazine.104,105 Levy proposed the following reaction for the reduction of with hydrazine.93
However, this reaction does not account for the hydrogen evolution reaction, which is unavoidable during electroless Ni deposition with hydrazine. In order to explain the hydrogen production, Mallory20 proposed the following reaction:
The above reaction does not explain a decrease in pH21 (accumulation of during electroless deposition. The overall reaction accounting for the formation and consequent decrease in pH, can therefore be proposed as:
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Electroless Deposition of Metals and Alloys
The rate of electroless deposition of Co with hydrazine, increases with an increase in concentration, which is presented in Figure 6. 105 The net reaction for electroless deposition of Co with hydrazine is described as:
Similarly to other reducing agents, increasing hydrazine concentration leads to an increase in the rate of electroless deposition. Moskvichev et al. studied electroless Au deposition with hydrazine at Ni substrates.106-108 They showed that the first reaction is galvanic displacement:
The continuous, substrate-catalyzed deposition was described by the following reactions:
and
Finally, the overall autocatalytic reaction for gold deposition is described as:
In the initial stages, when the displacement and substrate-catalyzed reactions are in progress, the rate of deposition is faster. The rate of deposition slows down when the surface is covered with Au and the deposition becomes entirely autocatalytic.
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Tetramine Pd(II) complex solutions were used for Pd electroless deposition with hydrazine by Rhoda.98 The overall reaction is:
The rate of deposition linearly increased with temperature and the deposit contains 99.4 % Pd. Hydrazine is often used in the spray method for mirror production as the deposition rate is fast.109 The deposition of Ag from an complex solution can be described with the following reaction:110
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Electroless Deposition of Metals and Alloys
4. Stabilizers Stabilizers are chemical compounds used in electroless deposition of metals in order to avoid the decomposition of the solution. Addition of these compounds to the plating solution assures, under proper conditions, operations over an extended period of time. Bath decomposition occurs as a precipitation of metallic particles in the bulk solution. These particles act as a highly efficient catalyst for further metal reduction because of their large surface area. The choice of a stabilizer depends on the metal being deposited and its compatibility with the process. Stabilizers, used in the electroless deposition of Ni, have been divided into the following classes: 20
I) II) III) IV)
compounds containing elements such as S, Se, Te; compounds containing oxygen heavy metals cations and unsaturated organic acids (maleic, itaconic acid, etc.)
The concentration of stabilizers is very important, since it determines the rate of deposition. An increase in the concentration of stabilizers of classes I or II above 2 ppm may completely inhibit the deposition reaction. The concentration of class III stabilizers is in the range to and the concentration of class IV stabilizers is in the range to Feldstein and Lancsek35 investigated the effect of three distinct classes of accelerators on electroless deposition of Ni and Co from hypophosphite-based solutions. Thiourea and thyocianate were investigated as accelerator-inhibitor substances (class AI). The dependence of the rate of deposition on the additive concentration reached a maximum at about 300 ppm. A further increase in the additive concentration led to a decrease in the rate of deposition, which was explained by inhibition due to an adsorption-poisoning
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mechanism. In the class API (accelerator with partial inhibition) glycine was investigated. An initial acceleration in deposition rate with an increase in the concentration of glycine was observed. With further increase in glycine concentration, incomplete inhibition in the rate of deposition was observed. A plateau was reached at higher accelerator concentrations. In the third class were investigated compounds designated as AO (accelerator only). Contrary to the behavior of classes AI and API, the rate of deposition approached a maximum value asymptotically with increasing additive concentration. This type of behavior was noted for formate and fluoride ions. A presence of oxyanions such as etc. may inhibit electroless deposition of Ni.111 With an increase in concentration from 0 to 400 ppm, the steady-state potential sharply decreases, resulting in a decrease in the rate of deposition. This behavior is explained by a surface adsorption mechanism. Substances reported as stabilizers in electroless deposition of Cu are mercaptobenzothiazole, thiourea, cyanide or ferrocyanide salts, mercury compounds, methyl butynol, propionitrile, etc.112 These compounds are usually employed at low concentrations, typically 1 to 100 ppm. Solutions for electroless deposition of Ag are usually very unstable. There are many different stabilizers used in the electroless deposition 113 of Ag. They include metal ions ,sodium-2,3114 mercapto-propane sulfonate , cystine, cysteine and NaSCN115, tetrabutylammonium nitrate and dodecylammonium acetate. 116 Small amounts of 3-iodotyrosine or 3,5-diiodo tyrosine significantly increased the stability of solutions for electroless deposition of Ag (pH 10 to 10.5).117 In electroless deposition of Pd, compounds such as and N,N'-p-phenylsulfonic acid-cmercaptoformazan are reported to act as stabilizers. 101 During electroless deposition of Au with borohydride, the loss of borohydride due to hydrolysis is expected. The accumulation of the hydrolysis products will cause the instability of the gold-borohydride
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Electroless Deposition of Metals and Alloys
solution. 118 Compounds recommended as bath stabilizers in electroless deposition of Au are nitrilotriacetic acid and mercaptosuccinic acid. 119 Kato et al.120 investigated the effect of dithizone, 1,10phenanthroline hydrochloride, sodium N,N-diethyldithiocarbaminate and mercapto compounds on the rate of deposition from a non-cyanide solution. They found that solution for electroless deposition of Au in the presence of these compounds is stable for over 20 hours.
However, all other additives, except mercapto compounds, rapidly decreased the rate of deposition, suggesting that they cannot be used as stabilizers. Mercapto compounds such as 2-mercaptobenzothiazole (MBT), 2-mercaptobenzoimidazole (MBI) and 6-ethoxy-2mercaptobenzothiazole (EMBT) significantly stabilized solution for
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electroless Au deposition. The effect of MBT concentration on the rate of Au deposition and bath stability is presented in Figure 7. The optimum concentration of MBT, recommended for electroless Au deposition is about 2 ppm. Metallic ions such as and are known to catalyze the reduction of Au through their specific adsorption on the Au surface and subsequent underpotential deposition. 120,121 The results on the effect of metal ions on the rate of deposition show that only is effective as an accelerator (Figure 8).120 On the other hand, electrochemical measurements show that all three ions, and accelerate the cathodic production of Au. The rate of Au deposition increased by a factor of 2, with a concentration of 1 ppm.
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Electroless Deposition of Metals and Alloys
V. DEPOSITION KINETICS AND EMPRICAL RATE LAWS The rate of electroless deposition can be determined by several techniques, such as gravimetric, electrical resistance and optical transmission measurements. The later two procedures have been adopted for process control in order to continuously determine rates of deposition. There are also methods such as electrochemical techniques (polarization resistance122, a.c. impedance123 and coulostatic methods124,125), which authors claim may be useful tools for automatic control of various electroless processes. The empirical rate law for the electroless deposition, can be expressed as:
where K is the frequency factor from the Arrhenius law, the activation energy, T the absolute temperature and R is the universal gas constant; etc. are concentrations of the species A, B, C acting as reagents or products in the deposition reaction; and etc. are the reaction orders of the respective species. The individual time derivatives, etc., are referred to as the rate of the reaction with respect to the species A, B, C, etc., and is the overall reaction order. Based on literature data, the exact empirical rate laws for the electroless deposition of a given metal differ. These differences are attributed to the use of different experimental conditions such as bath composition, pH, types of reducing agents, etc. On the other hand, there are also some similarities. The rate of deposition is independent of nickel ion concentration in the range where most hypophosphite-based solutions are employed.20 Gutzeit126 reported that the rate of electroless Ni deposition is the first order in respect to the hypophosphite ion concentration. Mallory and Lloyd127 showed that for several solutions, reaction order, in
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81
respect to the concentration, was -0.4. The rate of electroless Ni deposition, r, for this case can be expressed as:
Mimani and Mayanna studied electroless nickel deposition from an acidic tartrate solution.128 On the basis of the observed data, they proposed the following kinetic equation for the electroless nickel deposition:
For the activation energy, the authors reported a value of 68.6 kJ/mol. Lloyd and Mallory reported that the reaction order and the activation energy depend on the nature of the complexing agent.129 Their results are summarized in Table 4. The activation energy for all complexing agents had values between 50 and 100 kJ/mol. However, some differences in the reaction order, in respect to the corresponding species, appeared. Differences in the activation energy and reaction orders are obviously due to use of different solutions (different complexing agents) in the experiments.
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Electroless Deposition of Metals and Alloys
Mallory and Lloyd investigated electroless Ni deposition with dimethylamine borane (DMAB).130 The overall deposition reaction for this system can be represented by the following equation:
The deposition rate was considered to be a function of the concentrations of complexing agent, and temperature. Since and form a complex131, the number of parameters was reduced from seven to six, and deposition rate, v, was studied in terms of the following equation:
where DMAB and L are dimethylamine borane and complexing agent respectively, and AB is the dimethylamine/boric acid complex. For their studies Mallory and Lloyd used three different complexing agents: glycolate, malate and citrate.130 They found that the glycolate has no effect on the deposition rate. In cases of malate or citrate, the values for reaction order, are near zero or negative. The effect of complexing agent concentration on the deposition rate is shown in Figure 9. Three empirical rate laws were developed for three electroless nickel-boron deposition systems, depending on the nature of the complexing agent. These are represented below:
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Glycolate:
Malate:
83
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Electroless Deposition of Metals and Alloys
Citrate:
As can be seen from these equations, the deposition rate is independent of which was confirmed experimentally, but only for pH ranging from 6.0 to 7.0. This is also illustrated in Figure 10, which shows that the rate of deposition is independent of pH within the range 6.0 to 7.0. On the other hand, increasing in pH from 4.0 to 6.0 leads to
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85
an increase in the rate of deposition. Accumulation of the byproducts of DMAB oxidation (AB concentration) has little or no effect on the deposition rate. These laws are valid only for a limited range of DMAB and concentrations, pH, and only for the respective complexing agents. However, the method suggested by Mallory and Lloyd130 may be applied to any other electroless deposition reaction to optimize the behavior of the system of interest. For the overall Cu electroless deposition reaction:
according to data obtained by Shippey and Donahue132, for the case of alkaline tartrate solutions, the following empirical rate law equation can be written:
where [L] is the tartrate concentration, d is the empirical reaction order in respect to tartrate at 40 °C), k is the rate reaction constant; and is the activation energy (E = 49 kJ/mol).132 Molenaar et al. 133 investigated electroless deposition of Cu from a solution containing EDTA. They found that at high and high ECHO concentrations, the deposition rate is controlled by the concentration in the range from to For this case, Molenaar et al.133, proposed the following kinetic equation:
where L is the EDTA concentration and E is the activation energy At lower and HCHO concentrations, the deposition rate is controlled by the concentration of these species, and is independent of
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Electroless Deposition of Metals and Alloys
concentration. The empirical rate under these conditions was expressed as:
Values for k, d and E for this case were not reported. For the partial anodic and cathodic processes during the electroless Cu deposition, Donahue and Shippey134 proposed a method for deriving the rate laws. Based on measurements of mixed potential, and corresponding current, for various concentrations of the reactants, they reported the following rate laws for the respective anodic and cathodic partial reactions:
and
where and are rates and constants of anodic and cathodic reactions respectively. These equations demonstrate that is not involved in the anodic partial reaction but that and HCHO are involved in both partial reactions (anodic as well as cathodic). For different solutions used in electroless deposition of Cu, based on mixed potential theory, Donahue et al. obtained the following empirical rate law equation:135
Their results showed that predicted and measured values of deposition rate agreed to within 20 %.
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According to these results, it is obvious that different conclusions have been reached by different authors. This is probably a consequence of using different reaction conditions, as well as different bath compositions, including the presence of complexing agents and additives. VI. MECHANISTIC ASPECTS OF ELECTROLESS DEPOSITION
In spite of relatively intensive study of the electroless deposition of metals and alloys, there is still some disagreement in the treatment of mechanistic aspects of these processes. In order to explain electroless deposition of metals and alloys, five different mechanisms have been proposed, as follows:
1. "atomic hydrogen" mechanism 2."hydride ion" mechanism 3. "pure electrochemical" mechanism 4. "metal hydroxide" mechanism 5. "uniform" mechanism These mechanisms involve various attempts to explain electroless deposition. However, according to some experimentally observed characteristics, it is difficult to use any one of these mechanisms for a general explanation of an electroless deposition process. The following are discussions of the proposed mechanisms.
1. The Atomic Hydrogen Mechanism
This mechanism was originally postulated by Brenner and and later supported by others.52,126,136-140 The atomic hydrogen mechanism was developed for electroless Ni deposition with hypophosphite. Brenner and Riddell postulated that the atomic hydrogen reduces and acts by heterogenous catalysis at the
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Electroless Deposition of Metals and Alloys
catalytic Ni surface. Atomic hydrogen is generated by the reaction of hypophosphite with water, and is then desorbed at the catalytic surface according to the equation below:
At the catalytic surface, the adsorbed hydrogen reduces
ions:
The hydrogen evolution reaction, accompanying the catalytic nickel reduction is described as the recombination of two H atoms, according to the equation:
Gutzeit136 supported Brenner's assumption that ions are reduced by atomic hydrogen. He attributed the formation of H atoms to the dehydrogenation of hypophosphite during the formation of metaphosphite ion:
According to the atomic hydrogen mechanism, all of the evolved hydrogen gas must originate from H bonded directly to phosphorus in hypophosphite. Formation of elemental phosphorus (its deposition) is explained by a secondary reaction between hypophosphite and atomic hydrogen:
In general terms, the atomic hydrogen mechanism, according to literature data137-139, can be described by the following equations:
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89
Anodic:
where RH is formaldehyde, hypophosphite or borohydride Cathodic:
Recombination:
The reduction of ions by H is theoretically possible in a certain pH range for which the calculations are published. 141 However, the atomic hydrogen mechanism fails to explain many of the features of electroless deposition such as the simultaneous hydrogen evolution reaction. In this mechanism, deposition of phosphorus and involvement of hydrogen evolution reactions are explained as side reactions. Furthermore, this scheme does not explain why the stoichiometric utilization of hypophosphite is always less than 50 %. In an attempt to explain why the utilization of hypophosphite on reduction cannot exceed 50 %, Ivanovskaya and Gorbunova142 proposed the following reaction for electroless Ni deposition from alkaline solutions:
According to this equation, the pH of the reacting solution will decrease, and the reduction of one Ni atom leads to production of one molecule. The atomic hydrogen formed in this reaction does not reduce but is evolved as gas. These authors claimed that, in agreement with their experimental results, the above equation predicts
Electroless Deposition of Metals and Alloys
90
that the hypophosphite utilization for reduction cannot exceed 50%. Ivanovskaya and Gorbunova142 did not examine reduction paths of phosphorus codeposition, or discuss electroless Ni deposition in acidic solutions. 2. The Hydride Ion Mechanism The hydride ion mechanism was first suggested by Hersch. 143 According to Hersch’s hypothesis, the hypophosphite acts as the donor of hydride ions. The hydride ion is the reducing agent of both and ions. This mechanism, originally proposed by Hersch143, was later modified by Lukes 144 who applied it both acidic and alkaline solutions. In acidic solutions, formation of the hydride ion was described by the reaction:
In alkaline solutions, the formation of following reaction:
Lukes described by the
Two hydride ions from the above reactions can then react with or one ion with either a hydrogen ion or water:
Acid solutions:
Alkaline solutions:
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91
In broad terms, the hydride ion mechanism can then be described by the following general equations:
where RH is formaldehyde, hydrazine or hypophosphite as previously. The proportion of hydrogen gas evolved that can be attributed to hydrogen originally bonded to phosphorus in hypophosphite will vary from 50 to 100 %. Lukes did not discuss the reactions and mechanism by which phosphorus is included in the electroless Ni deposition by the hypophosphite reduction. He suggested that phosphorus codeposition should not materially affect the hydride generation and reduction reactions. From Lukes’ theory arises a question of the reality of a hydride ion formation having a standard reduction potential of - 2.08 V in a hypophosphite solution with standard potential of -1.57 V. 142 Both potentials are reported for pH=0. The change-over from standard conditions to those in which metals are reduced by hypophosphite does not alter the difference between these potentials. On the other hand, the existence of hydride ions in an alkaline medium, even in an intermediate state, appears very unlikely.128 3. The Electrochemical Mechanism The so-called electrochemical mechanism was first proposed by Brenner and Riddell3,4 and later modified by other researchers.145-150 In
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Electroless Deposition of Metals and Alloys
this mechanism, electroless deposition is considered to result from mixed anodic and cathodic reactions. In the case of electroless Ni deposition, the oxidation of hypophospliite with water generates electrons, and is considered as the partial anodic process:
with The electrons formed in the above reaction are utilized in the coupled cathodic processes for deposition of Ni and P:
with
and
with According to the electrochemical mechanism, the evolution of hydrogen gas is a result of the secondary reaction which follows:
with The electrochemical mechanism does not explain reduction of metal ions in the bulk solution (i.e. without the presence of a metallic substrate). It also does not explain the reduction of metal hydroxides (formed as precipitates) to a metallic state. As experimental results show, the presence of any metallic surface is not a sufficient condition
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to start electroless deposition. It is very well known that electroless deposition takes place only at catalytically active surfaces. Examples include non-conductive Pd activated surfaces. In terms of mixed potential theory, electroless deposition was first described by Paunovic.145 According to this theory, electroless metal deposition can be considered as the superposition of anodic and cathodic curves crossing at the mixed potential, Electroless deposition of metals takes place at the mixed potential. The mixed potential, and the deposition current, are obtained by the intersection of the partial anodic and cathodic polarization curves, as it is schematically shown in Figure 11. This theory predicts that the rates of anodic reactions do not depend on the cathodic processes occurring simultaneously at the cathodic surface. The rates of separate reactions (anodic and cathodic) depend only on the mixed potential at which they have the same values. For the case of electroless deposition of copper from an alkaline solution of copper sulfate, containing EDTA as a complexing agent and using formaldehyde as a reducing agent, Paunovic145 determined the rate of Cu deposition in three different ways. Firstly, the rate of Cu deposition was determined from the intersection of the polarization curve for the oxidation of formaldehyde and polarization curve for the reduction of at single electrodes. This value of was which corresponds to the value of a mixed potential of 0.65 V vs SCE. On the basis of Faraday's law and the mixed potential theory, the rate equation for electroless copper deposition was written as:
Using the value for obtained for the single electrodes (1.9 mA and the above equation, the rate of electroless copper deposition was derived as In the second experiment, the rate of electroless copper deposition was determined from the extrapolated current density at the mixed potential obtained from the cathodic polarization curve for the mixed
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Electroless Deposition of Metals and Alloys
electrode. The value for was at the mixed potential of -0.66 V vs SCE, and the consequent rate of Cu deposition was
Finally, the rate of electroless copper deposition was determined gravimetrically, and had a value of The comparison of these three values 2.2, 2.1, and showed that there was very close agreement among the methods used in this work. The application of the mixed potential theory to the electroless deposition of Ni by hypophosphite can be described by the following reactions. The oxidation of hypophosphite:
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95
is an anodic process, with the hypophosphite oxidation current, designated here as The cathodic processes are related to the deposition of Ni and P, and hydrogen evolution, as described by the following reactions:
with the Ni deposition current
with the P deposition current
and
with the H evolution current At the mixed potential, the rate of reduction rate of oxidation, i.e.
Since the rate of reduction,
is equal to the
is:
Using Faraday's law the rate of Ni deposition is:
The equation (83) is derived for the case of deposition of Ni (the constant 1.09 is calculated from M(Ni)/zF for Ni only, and does not take into account the deposition of P), although,
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Electroless Deposition of Metals and Alloys
Furthermore, the same equation was used to describe electroless deposition of Ni with DMAB but did not consider the deposition of boron. 68 In order to propose mechanism of electroless deposition, van den Meerakker151 investigated anodic oxidation of formaldehyde in alkaline solutions. Based on the assumption that the anodic oxidation of formaldehyde and the cathodic reduction of metal ions are independent processes in electroless plating solutions, he investigated the effect of electrode material, pH and potential. The experimental results were explained by a mechanism in which methylene glycol anions are dehydrogenated at the electrode surface, thereby producing adsorbed hydrogen atoms. The adsorbed hydrogen can either be oxidized to water or be desorbed as a gas. He reported that electroless Cu, Pt and Pd solutions behave according to the same mechanism. By the applying the mixed potential theory it was suggested that the mechanism can be predicted from the polarization curves for the partial processes. However, the extrapolation of partial polarization curves and application of the mixed potential theory is not often realized, since the two partial processes are independent of each other.152 An application of the mixed potential theory153 led to the development of a technique by which electroless process may be classified according to their overall mechanisms.154 The overall mechanism of electroless Cu deposition was determined by a technique based on the application of the mixed potential theory to a rotating disc electrode. In this theory, the mixed potential is considered to be a function of agitation, concentration of the reducing agent and, concentration of metal ions. 154 Based on the assumption that each partial reaction is either under electrochemical or under mass-transfer control, Bindra and Roldan154 developed four cases. Case1. Cathodic partial reaction is diffusion-controlled and the anodic partial reaction is electrochemically controlled. The equation which describes this case is:
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where is the mixed potential, is the standard potential for HCHO, is the Tafel slope for the anodic reaction, is the exchange current density for HCHO oxidation, is diffusion parameter for complex, is the rotation rate of electrode and is the bulk concentration of ions. This equation shows that mixed potential is a linear function of and ln The Tafel slope , may be obtained by plotting vs or vs ln Case 2. Cathodic partial reaction is electrochemically controlled and the anodic partial reaction is diffusion controlled. The equation describing this case is:
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Electroless Deposition of Metals and Alloys
where is the standard electrode potential for the Cu deposition reaction, the Tafel slope for cathodic partial reaction, the exchange current density for metal deposition, is the diffusion parameter for HCHO and is the bulk concentration of HCHO. Case 3. Both partial reactions are electrochemically controlled Case 4. Both partial reactions are diffusion controlled Cases 3 and 4 were not discussed by the authors; they explain that in both cases mixed potential is independent of the rotation rate, and that such cases are rarely encountered in electroless processes. Measurements were performed in the complete electroless bath, as well as in the catholyte and in the anolyte separately. The theory developed by Bindra and Roldan154 was experimentally verified by a comparison of mixed potentials for different rotation rates and concentrations. As shown in Figures 12 and 13, there are linear relationships between and and between and which is in agreement with equations (84) and (85). For the oxidation of formaldehyde in complete solution (with all constituents), the polarization curve is illustrated in Figure 14. The Tafel slope for formaldehyde oxidation obtained from Figure 14 has the value +210 mV per decade at 70 °C. The large value of the Tafel slope is attributed to the electrochemical oxidation of formaldehyde under Temkin adsorption conditions. For electroless Cu deposition, they showed that the formaldehyde partial reaction (oxidation) is electrochemically-controlled, while the metal deposition partial reaction is diffusion-controlled which is in an agreement with their theoretically proposed model. Mital et al.70 investigated the validity of mixed potential theory for electroless Ni deposition. The results for Ni deposition with DMAB show the predominance of an electrochemical mechanism. These authors stated that the mixed potential theory is applicable only for
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systems where the reduction of metal ions and oxidation of the reductant is electrochemically feasible at the mixed potential, perhaps an obvious requirement. According to Mital et al. 70, in the case with DMAB, reduction of metal ions and oxidation of the reductant is electrochemically feasible at the mixed potential. However, in the case of they reported that a chemical mechanism predominates. Although the results obtained for the electroless Cu deposition showed strong agreement with the theoretical approach based on the mixed potential theory68,131, there is no strong evidence that this theory works for the case of electroless Ni deposition with hypophosphite (see above discussion) . It is the opinion of this author that the mixed potential theory works perhaps for the case when cathodic processes are related only to the deposition of single metals (e.g. electroless Cu
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Electroless Deposition of Metals and Alloys
deposition or electroless deposition of metals by hydrazine). The mixed potential theory does not explain reduction of metal ions in bulk solution. Also, when the cathodic processes represent a combination of several reactions such as electroless Ni deposition with hypophosphite, or electroless deposition of alloys, a simple combination of mixed potential theory and Faraday's law does not work. To prove this, however, further intensive experimental studies are required.
4. Metal Hydroxide Mechanism
The metal hydroxide mechanism was originally proposed by Salvago and Cavallotti. 27,32 For the example of electroless Ni
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deposition with hypophosphite, they proposed a mechanism based on experimental observation. Electroless deposition of Ni is possible in both acidic and alkaline solutions. In the acidic solutions, the reaction is experimentally significant only at pH>3, and its rate increases sharply with pH. The process occurs in a pH range where hydrolysis phenomena are involved. Hydrolyzed species of Ni ions can also react directly with hypophosphite. The metal hydroxide mechanism proposed by Salvago and Cavallotti27,32 can be described briefly by the following scheme. At the catalytic Ni surface, the ionization of water takes place according to the reaction:
Hydrolysis of as follows:
and formation of hydroxo-complexes takes place
and
The hypophosphite ions interact directly with hydrolyzed species, as is indicated below:
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Electroless Deposition of Metals and Alloys
Deposition of phosphorus was explained in terms of the reaction:
This reaction also explains non-homogenous distribution of P and lamellar morphology of deposits. The hydrolyzed Ni(I) species interact directly with water:
Evolution of hydrogen can be explained as:
or by the reaction of hypophosphite ions with water:
According to the reactions (91) and (92) Savago and Cavallotti27,32 explained lamellar morphology of electroless NiP deposits. It is obvious that any periodicity between the reactions (91) and (92) will produce deposits having layers richer with P, and then layers richer with Ni (lamellar morphology). Cavallotti and Savago reported that when nickel hydroxide is precipitated, inhibition phenomena are evident. The results of calorimetric studies of electroless deposition of Ni, obtained by Randin and Hintermann155, supported the mechanism proposed by Cavallotti and Savago.27,32 The molar ratio ] in the overall reaction for electroless nickel deposition is 0.25, according to the following equation:
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supported the metal hydroxide mechanism for electroless deposition of Co, using hydrazine as the reducing agent. 105 In this work, electroless deposition of Co powders is explained by the metal hydroxide mechanism, according to the following reactions:
or
Finally, the overall reaction for electroless Co deposition with hydrazine is represented by the following equation:
The observations in this work105 support the metal hydroxide mechanism as a means of explaining the electroless deposition of Co by hydrazine. Hydrolyzed species of can react directly with hydrazine producing metal powder. The reaction occurs in bulk electrolyte, when a precipitate of cobalt hydroxide is formed and production of Co powder takes place. The SEM micrographs show that the Co powder produced under these conditions was dendritic in terms of surface morphology (Figure 15).
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Electroless Deposition of Metals and Alloys
XRD patterns in Figure 16 show that the Co powder contained the 70 % Co having the hcp structure and 30 % Co the fcc. Co powder of similar morphology and structure can also be obtained by an electroless reduction of alkaline Co(III) solutions.143 Although the metal hydroxide mechanism explains most of the characteristics of electroless Co deposition by hydarzine, particularly the reduction of precipitated cobalt hydroxides and deposition of dendritic Co powder, there are still some doubts about this mechanism. For example, it does not explain the oxidation of at a Pd-activated surface in Co(II)-free solutions. In order to explain deposition of shiny 105 suggested that and smooth coatings at flat surafces, contributions from both the electrochemical and the metal hydroxide mechanisms should be considered.
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5. The Universal Mechanism
Based on similarities among electroless processes, van den Meeraker proposed a mechanism that accounts for both the electrochemical and the catalytic nature of the process.157 This mechanism was developed according to the following features which are common to different electroless systems: (a) The electroless deposition process proceeds only on certain catalytic metals that are known as effective hydrogenationdehydrogenation catalysts; (b) Electroless deposition is always accompanied by evolution of hydrogen gas;
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Electroless Deposition of Metals and Alloys
(c) Poisons for hydrogenation-dehydrogenation reactions, such as thiourea and mercaptobenzothiazole, act as stabilizers in practically all electroless processes; and (d) The deposition rate increases with an increase in pH. The reactions taking place during electroless deposition were described as follows: Anodic:
Dehydrogenation: RH = R + H Oxidation: Recombination: Oxidation: Cathodic:
Metal Deposition: Hydrogen Evolution: In this scheme, RH represents the reducing agent. It dissociates to a radical R and atomic hydrogen. The electrons for reduction of metal ions are supplied by the oxidation of R and/or reaction of H with The universal mechanism is not adequate for the explanation of all electroless processes. It fails to explain electroless deposition of metals on non-conductive surfaces, and also deposition of metal particles in solution. In spite of intensive studies on mechanistic aspects, it is obvious that there is still not enough experimental data to confirm proposed theoretical approaches for electroless deposition of metals and/or alloys. The proposed mechanisms explain most of the characteristics of electroless deposition. However, as discussed above, there are some characteristics which cannot be explained using these mechanisms. It seems that major problems arise when attempting to generalize the proposed models for electroless deposition. A more realistic approach would be to look for specific reactions, for particular conditions and
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substrates. It is very unlikely, in spite of the similarities of electroless processes, that a general mechanism will be developed explaining features for all electroless deposition of metals. Nevertheless, this literature review suggests that intensive experimental studies are required in order to develop and/or prove existing models for electroless deposition of metals.
VII. RECENT DEVELOPMENTS Development of electroless deposition of metals and alloys in the past years has been remarkable and still continues. This process was investigated for various applications such as magnetic disks158-159, printed circuits160-165, selective plating and semiconductors166-174, batteries175-177, medical devices178, etc. Most of these applications are related to electroless deposition of copper or nickel. It is beyond the scope of this section to review all published literature related to the topic of electroless deposition. However, a few important points related to new developments and basic research will be presented here. Considering the fact that many of these applications use similar approaches and in a way overlap, the further discussion in this section is presented as follows: 1. metallization of non-conductive surfaces 2. electroless deposition of composite coatings 3. electroless deposition of gold and other metals 4. new developments and basic research 1. Metallization of Non-Conductive Surfaces
Metallization of non-conductive surfaces (polymers, ceramics and glass) requires specific treatments prior to electroless plating. Usually, these surfaces are first etched, then sensitized by a simple immersion in an solution. During the sensitization process, the adsorption of ions takes place. Senzitized surfaces are then exposed to a
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Electroless Deposition of Metals and Alloys
solution containing and HCl. This process is called activation. The activation process can be described by the following equation:
The Pd sites formed during the activation step allow chemical deposition of Ni or Cu. In some cases, sensitization and activation steps are combined in one step. In other words, solutions for the sensitization and activation are combined and represent mixtures of and HCl. Other processes recommended for the activation of non-conductive surfaces for metal deposition from aqueous solutions (electroless deposition or electrodeposition) are carbon/graphite systems, conductive polymers, and non-formaldehyde based electroless processes. For details on these processes, the reader is referred to Ref. [179] and references therein. The metallization of alumina surfaces with electroless NiP is used in the electronics industry for printed circuits and sensors applications; however, on metallized alumina substrates, failure is often observed at the metal-ceramic interface. In order to increase adhesion of the metallic layer to the ceramic substrate, several approaches are found in the literature. In one example180, alumina substrates were cleaned with a surfactant (amino perfluoroalkylsulfonic acid), etched with HF, sensitized with solution and activated first with then with solutions. Metallization of pre-treated alumina substrates was carried out using an acidic, chloride-based solution. The thin NiP films on alumina substrates exhibited a columnar structure. This layer plays a crucial role in adhesion for all cases where interfacial failure is observed. The same authors181 recommended a very different approach to solve adhesion-related problems. On clean alumina substrates, a thin layer of Ti (about 20 run) was deposited by vacuum deposition. A thin Pd film of approximately the same thickness (20 nm) was then deposited on the Ti layer, also by vacuum deposition. On these surfaces, NiP was deposited by electroless deposition; a thick layer of Ni was then electrodeposited from a sulfamate solution.
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The resulting very strong adhesion was attributed to chemical binding of the first Ti monolayer with the substrate oxygen atoms. For the metallization of piezoelectric ceramic surfaces, based on lead titanium zirconate, electroless deposition of NiP182 or Cu183 was recommended. Surfaces were senzitized with and activated with in the standard way. Electroless deposition of palladium on a ceramic substrate consisting of 70 % 29 % and 1% was carried out from a solution containing EDTA, and hydrazine as the reducing agent. 184 Prior to the electroless deposition of Pd, the ceramic substrate was sensitized and activated with and acidic solutions. For applications where metallization of advanced devices on titanium silicide substrates takes place, Cu is deposited by a low pressure CVD from copper (II) hexafluoroacetylacetonate. 185 The substrates with Cu films were then ready for further metallization. Graphite samples were sensitized and activated with and and plated with Cu or Ni using formaldehyde or hypophosphite as reducing agents, respectively. l86 A complete coverage of graphite was not observed and this result was attributed to the porosity of the graphite material. However, when electroless plating of Cu followed by electroless plating of Ni was applied, the surface became completely covered. An interesting approach for the pre-treatment of non-conductive surfaces prior to electroless plating was recommended by Calvert et al. 187-191 This process is based on the application of tin-free, aqueous Pd(II) catalyst in conjunction with chemisorbed ligating organosilane films. Typical ligating systems include organosilane precursors containing pyridine, alkylamine and arylphosphine ligands. Most substrates of technological importance, such as silicon, metals, metal oxides, ceramics, plastics, polymers and diamonds, are suitable for silane attachment (adsorption). 190,191 A self-assembled monolayer is used as a ligating surface to bind a Pd(II) catalyst. The Pd(II) catalyst can covalently bind to ligands containing nitrogen, sulphur and phosphorous atoms. The Sn-free, Pd catalyst is made by dissolving 11.4 mg of sodium tetrachlorpalladate in 1 mL of 1M NaCl. To this
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Electroless Deposition of Metals and Alloys
mixture MES (2-morpholinethane sulfonic acid) buffer (0.1M, 10 mL) at pH 5 is added, followed by dilution with water up to 100 mL. The solution is left at 22-23 °C for about 20 hours. Under these conditions, the polymerization takes place, and Pd-oligomers are formed. Silicon substrates are first immersed in 1:1 solution for 30 minutes, followed by a rinse in water and immersion in for 30 min. After rinsing, the substrates are immersed in boiling water for 10 minutes, dried with and then transferred into 1 % organosilane solution. The organosilane compounds used in this work were 2-(trimethoxysilyl)ethyl-2-pyridine, N-(2-aminoethyl)-3-aminopropyltrimethoxy silane, and (aminoethylaminomethyl)phenethyltrimethoxy silane. Silicon samples are immersed in organosilane solutions for 1 hour at room temperature. Substrates are then removed from solutions and baked on a hot plate for 3 to 4 minutes at 120 °C. A 20 % catalyst coverage is attained when substrates with organosilane films are immersed for 10 min in the Pd catalyst solution. 192 For treatment times over 2 hours, 100 % coverage is observed. On activated substrates, the NiB alloy is electrolessly deposited from a commercial solution. Metallized surfaces exhibited broad particle size distribution with the maximum Ni particle size exceeding 120 nm. A similar approach for the activation of surfaces with Pd(II) catalyst dispersion solution with HC1, hydrolyzed at pH 7) for electroless Ni deposition in nanolithography applications is recommended by Brandow et al. 193 The metallization of catalyzed surfaces with Ni led to the deposition of Ni particles with an average size of 21 ± 5 nm. The first stages of platinum electroless deposition on Si substrates were investigated using atomic force microscopy. 194 Silicon substrates are etched in 40 % HF solution. To this solution, is added to yield a 1.0 mM concentration of Pt(IV) in solution. During the immersion of silicon in an aqueous HF solution containing Pt(IV), platinum is nucleated at the surface, while the silicon is etched. Deposited nuclei are polycrystalline and highly pure. The process is represented by the following reaction:
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The etching of silicon with HF solution containing galvanic displacement process composed of two reactions:
111
ions is a
cathodic
with and anodic
with A similar process for electroless deposition of Pd onto Si substrates has also been published.195 This process, based on a solution, is represented by the following reaction:
Metallization of aluminum nitride ceramics is necessary for applications in the electronics industries. Prior to electroless deposition of copper, A1N substrates are etched with a 4 % NaOH solution. 196 The etching process is described by:
A1N substrates are sensitized and activated with and respectively, then metallized with electroless copper from an alkaline solution using formaldehyde as a reducing agent. Adhesion of Cu to a substrate increases as the etching time is increased.
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Electroless Deposition of Metals and Alloys
et al. 197 reported a procedure for metallization of ceramic substrates, including AIN. After proper substrate preparation, they recommended direct electroless NiP deposition onto A1N substrates. Cu was electrodeposited up to a thickness of onto these substrates. Adhesion of the Cu layer plated onto A1N substrates was very good. The activation of diamond surfaces for electroless metal plating (Cu and Ni) is achieved by laser-induced modification of the film surface during ablative etching. 198 Encapsulation of the electroless copper lines into diamond films is obtained. Metallization of polymers and plastics has also attracted significant attention from researchers because of its various industrial applications. Different pre-treatment steps before metallization of polymeric substrates were disclosed. Polymers of interest included polyimides199203 , polystyrene204,205, olycarbonates205-207, polyetherimides208, ABS plastics206,209, aramid fibres210,211, A-PPE (allylated poly-2,6-dimethyl1,4-phenileneether) resin212, fluoropolymers213 etc. Two different ways of activation of polyimide resins for the electroless plating will now be discussed. According to patents granted to IBM201 and Monsanto Company202, a precursor polyamic acid film is deposited at the surface. The film of polyamic acid is exposed to a solution containing Pd(II) ions then heat treated to imidize the polymer. Samples with Pd seeds are used for electroless deposition. This procedure is further recommended as a pre-treatment step for electroless deposition on substrates such as metals, ceramics, glass, silicon, high heat polymers, and thermoset polymers. Direct metallization of polyimide resins is proposed according to the following procedure.199 Polyimide resin substrates are immersed in fuming sulfuric acid at 25 °C for 15 to 30 seconds. The sulfonated test samples are then dipped in solution for 3 minutes to adsorb ions. After a careful water rinse, the samples are exposed to solution to reduce ions and form a thin copper film. The thickness of the thin Cu film formed after the reduction process is recommended for the metallization ofA-PPE resin. 212
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Prior to metallization, polycarbonate and polystyrene polymers are treated with and plasma. 205 After treatment with plasma the amount of oxygen significantly increases, while treatments with or plasma increases the amount of nitrogen. When the surface of polymers contains nitrogenated and oxygenated groups, it can chemisorb palladium directly. However, when the surface of polymers contains only oxygenated groups it chemisorbs palladium only through previously chemisorbed stannous ions. This sequence is used for direct or treatments, in order to eliminate the use of This process is recommended not only for polycarbonates and polystyrene, but for any other polymers, including fluorinated ones. 2. Electroless Deposition of Composite Coatings
Codeposition of solid, inert particulates within a metal matrix during electroless deposition of that metal matrix (single metal or alloy) leads to production of composite coatings. In typical composite coatings, the fine particulates' diameter size ranges from 0.1 to 20 Their content in the coating can exceed up to 40 vol.%. The metal matrix in this class of composite coatings is usually electroless NiP or NiB. These materials are used for the improvement of wear and corrosion resistances, friction coefficient and hardness. Inert particulates, depending on applications, may include chromium carbide, alumina, titanium carbide, silicon carbide, boron carbide, diamonds, PTFE etc. There is renewed interest in coatings with exceptional hardness, wear and friction properties for automotive and other mechanical applications.214 High performance coatings include alloys of cobalt and tungsten, composites with fluoropolymers etc. These coatings have potential for replacement of hard chromium. Composite NiPSiC coatings are obtained by electroless deposition of NiP alloy from a solution containing dispersed SiC particles (1 to 215 Such particles can be homogenously incorporated into a deposit through a proper agitation of the plating solution. The amount of SiC incorporated into the deposit is estimated to be about 30 vol. %.
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Electroless Deposition of Metals and Alloys
During the heat treatment of electroless deposited NiPSiC coatings at 650 °C for 1 hour, a new phase of is identified by structural analysis. Electroless deposition of NiB matrix composite coating, containing Al/SiC particles, is also reported.216 These composites have very good adhesion after treatment at temperatures above 400 °C. The inclusion of fluoropolymeric materials in nickel-based coatings requires the addition of surfactants which block the aggregation of the particles and provide conditions for their incorporation in the coating. 217 The influence of the nature of different surfactants in electroless deposition of NiP-PTFE composite coatings is reported by Nishira and Osamu.218 They found that anion-active and cation-active surfactants have no effect on the formation of composite coatings. If a solution for electroless NiP deposition contains only non-ionic surfactants, incorporation of PTEF into deposit is not observed. Uniform coatings are obtained when combined cation-active and non-ionic surfactant are added to the solution. The relationship between the Zeta potential and codeposition of PTFE particles suspended in electroless Ni solution was investigated by Hu et al. 219. Their results showed that PTFE particles, with more positive potential, codeposit more easily with NiP. Consequently, their content in coatings is greater. These composite coatings contain about 25 vol. % of PTFE. Commercial acceptance of composite coatings has increased in a number of applications, especially since productivity, quality and environmental concerns continue to expand at increasing rates. 220 3. Electroless Deposition of Gold
Electroless deposition of Au is used for applications in the electronics industries (deposition on semiconductors and circuit patterns), as well as for decorative purposes. Solutions for electroless deposition and properties of deposited Au films were reviewed recently. 221 The classical electroless Au solutions utilize as a source of Au and or DMAB as reducing agents. These solutions are
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autocatalytic, and it is possible to deposit sufficiently thick layers of gold. In order to replace cyanide-based solutions, two different systems for electroless deposition of Au were developed in recent years. Both use gold thiosulfate as their source of gold. The main difference is the reducing agent. The first system uses thiourea, while the second uses ascorbic acid. Typical baths for electroless Au deposition contains 120
The electroless deposition is carried out at 60 °C at pH 6, using ascorbic acid as the reducing agent. In the presence of thiosulfate and sulfite, Au(III) from is reduced to Au(I), according to the reactions below:
and
According to the results from polarization curves, Au(I) is reduced to the metallic state from the thiosulfate complex. If the plating solution contains sulfite alone Au can still be deposited; however, the deposition rate is very slow. In a search for a reducing agent Koto et al. 120 found that the reducing power of boron-containing compounds DMAB and tributhylamine borane) is so strong that the solution decomposed immediately upon the addition of these compounds at near-neutral pH. The reducing power of other agents decreases in the following order: ascorbate>thiourea>hydrazine. Hydroxylamine and formaldehyde are not effective as reducing agents in the system investigated. Deposition of Au using ascorbate as the reducing agent, is explained as the combination of an anodic reaction (oxidation of ascorbic acid):
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Electroless Deposition of Metals and Alloys
coupled with the cathodic reaction (reduction of Au(I) to Au):
The effect of ascorbate concentration on the rate of Au deposition is shown in Figure 17. In deposition of Au with thiourea as the reducing agent, the anodic reaction is:
while the cathodic reaction is described by equation (113). Heterocyclic mercapto compounds, especially 2-mercapto-benzothiazole (MBT), stabilize the solution and do not significantly influence the rate of deposition. Sullivan and Kohl222 investigated the autocatalytic deposition of Au from thiosulfate solution using sodium L-ascorbate as the reducing agent (at pH 6.4 and 30 °C). They found that ascorbic acid reduces gold thiosulfate at a deposition rate of approximately The rate of deposition decreases with time which was attributed to the accumulation of thiosulfate in the reacting solution. The activity of the bath and the deposition rate are partially restored by adding to the reacting solution. decomposes thiosulfate to trithionate and sulfate. In this way, the accumulation of thiosulfate is slowed, and the rate of deposition is restored. Other developments include improvements of non-cyanide solutions (gold(I) thiosulfate with ascorbic acid as a reducing agent) for the electroless deposition of gold, which prevents the formation of any precipitates during the storage of the bath223, and solutions with chelating agents, such as diethylenetetraaminepentaacetic acid224, dimethylamine225, etc.
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4. Electroless Deposition of Other Metals
Among other metals, studied from the aspect of electroless deposition, should be mentioned Ag226-230, Sn231-234, Sb-Pb235-237, Bi 8 , Sn-Bi solder238,239, Pd240, Ni-Sn-P alloys241, etc. Molenaar reported that the autocatalytic Sn deposition is possible by a disproportionation reaction of tin(II) ions in alkaline solutions.231 An increase in the mass of an autocatalytic Sn layer on a Cu substrate, as a function of plating time, is observed when the deposition is carried out from a solution containing 3.85 M NaOH, and t =75 °C (Figure 18).
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Electroless Deposition of Metals and Alloys
The deposition reactions were described as follows: (a) displacement reaction:
(b) reduction of tin(II) by hypophosphite:
(c) disproportionation of tin (II) ions:
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(i)
(ii)
(iii)
Concentrations of tin (II) and hypophosphite determine the plating rate. In the later work232, Molenaar reported that the autocatalytic deposition of Sn can be carried out through a disproportionation reaction in alkaline solutions without hypophosphite. The thickness of deposited tin layer as a function of time, when the deposition is carried out from the solution containing 2.5 M NaOH, at 75 °C is presented in Figure 19. They suggested that other metal derivatives such as Pb(II), Pt(II), Pd(II), Hg(I), Cu(I), In(I), T1(I) and Au(I) will give this type of disproportionation reaction. It is first reported by Rutkevich et al., that Bi can be reduced by Ti(III) complexes in an autocatalytic mode.8 The main characteristics of the process are explained in terms of pure electrochemical mechanism. The reduction of Bi(III) to Bi is represented as:
The authors also claimed a possibility of using Ti(III) complexes to reduce Ni(II) and Co(II), as well as application of V(III) for autocatalytic Cu(II) reduction. NiSnP and NiSnB alloys are deposited from alkaline solutions containing as a complexing agent, using sodium
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Electroless Deposition of Metals and Alloys
hypophosphite and DMAB as reducing agents, respectively. 241 For electroless deposition of NiSnP a source of Sn is and for NiSnB, a source of Sn is Sn(IV) gluconate.
The maximum contents of Sn in the deposit are estimated at 30 at.% for NiSnP and 42 at.% for NiSnB alloys. The crystallinity of alloys increases as the Sn content increases.
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5. New Developments and Basic Research
Several publications related to new developments and basic research in the field of electroless deposition will be discussed here. Hatsukawa et al. 242 reported that during electroless deposition of NiP films, carbon can be codeposited only if the solution contains complexing agents having or =NH groups (e.g. diethylenetriamine or -alanine). Carbon cannot be incorporated, if the solution contains only -COOH or other carbon-containing groups. The deposition of carbon is explained by the adsorption of or =NH groups, followed by incorporation of carbon from the main chain of the compound via codeposition with nickel. Deposited NiPC films contain about 3 atom. % of C when electroless deposition is carried out from solutions containing diethylenetriamine or at pH 6 (90 °C).243 These films exhibit very high resistivity. Heat treatment of NiPC films leads to crystallization of as-deposited amorphous alloys. Crystallization occurs at temperatures higher than the crystallization of pure NiP alloys. For electroless deposition of Cu, the glyoxilic acid is used as an alternative reducing agent of ions. 244 Electroless deposition is carried out from a solution containing 0.24 M EDTA, 0.2 M lactic acid, at 60 °C and pH 12.5 (adjusted with NaOH). The solution also contains 10 ppm of 2, The reaction for deposition of Cu is described as:
with a standard redox potential of 1.01 V for the reaction:
The deposition rate and solution stability are superior to that of commonly used formaldehyde solutions. The rate of deposition
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Electroless Deposition of Metals and Alloys
increased with an increase in temperature, pH, concentrations of and reducing agent (Figure 20). The use of KOH for pH adjustment, instead of NaOH, significantly suppressed the reaction. A low-temperature electroless deposition of Ni from an alkaline solution is recommended by Chen. 245 A decrease in plating temperature was attributed to the addition of sodium tungstate to the plating solution. The composition of deposited films was not reported. It seems, according to this work, that the authors deposited NiWP alloy, rather than NiP. A process for electroless deposition of metals and alloys on large parts is disclosed. 246 This process is based on sequences of simultaneous spraying of two aqueous solutions; one containing the reducing agent and the second containing metallic ions. The authors claimed the process can be carried out at room temperature, because the electron exchange reactions can occur in a very thin aqueous layer at the surface. Metals that can be deposited by this method are classified in four groups: a) pure metals (Ni,Co, Cu), b) binary alloys (NiB, CoB), c) ternary alloys (CuCoB, CuNiB and NiCoB) and d) quaternary alloys such as CuNiCoB. There are several other publications related to mechanistic aspects of electroless deposition. 247-250 Mishra and Paramguru247 investigated mechanisms of electroless deposition of copper from an alkaline solution containing tartrate and using formaldehyde as a reducing agent. In this study, they used potentiostatic and galvanostatic curves, as well as steady-state plots. They found that the mechanism of electroless Cu deposition changes from anodic to cathodic control, as well as from diffusion to activation, depending on the concentration of and HCHO in this solutions. Based on the morphological evolution via AFM for the electroless deposition of Co with hypophosphite as the reducing agent on highly activated pyrolytic graphite, Hwuang and Lin248 concluded that this process follows the electrochemical mechanism. Electroless deposition of NiP alloys is described by a mathematical model which is based on steady-state convective diffusion equations with non-linear boundary conditions and overpotential equations. 249 The results are explained in terms of the mixed potential theory. On
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the other hand, Abrantes and Correia250 found that the mixed potential theory is unable to describe the electroless deposition of NiP alloys.
They investigated the initial stages of electroless deposition of NiP alloys from a chloride solution by means of potentiodynamic and opencircuit potential measurements. The authors assumed that the process was initiated by adsorption of hypophosphite at the surface and the homolysis of its hydrogen bond with formation of radicals. These radicals promote and hypophosphite reduction in order to form the NiP deposits. It is obvious, according to these papers that there is still disagreement among researchers on the mechanistic aspects of
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Electroless Deposition of Metals and Alloys
electroless deposition. To fully understand the process of electroless deposition, further investigations are required. Several patents in the field of electroless deposition have appeared recently. They are related to electroless deposition of Pd251, activating solution based on a mixture of zinc lactate, copper lactate and palladium chloride in alkaline medium252, Cu or Ag plating on textile for electronics and biomedical applications253, and a method for depositing NiP alloys254.
VIII. CONCLUSIONS In spite of development of other competitive technologies, it is obvious that applications of electroless deposition of metals and alloys will continue to grow in the future. Applications of electroless deposition of metals and alloys are related to developments in the electronics industries, battery technologies, medical devices, and protective and decorative coatings. According to literature review, more work is required to understand fundamental issues related to the reaction mechanisms of electroless deposition, the influence of processing parameters on properties of deposited coatings, etc. This knowledge is needed to ensure the successful operation of the process. The new electroless deposition-based technologies should improve selectivity, satisfy quality requirements of deposited coatings, and assure the consistency of the process. The environmental concerns related to the solutions used for electroless deposition must also be investigated, in order to develop environmentally-friendly technologies, and to allow successful competition with other available processes.
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Y. Okinaka and T. Osaka, "Electroless Depeosition Processes: Fundamentals and Applications" in Advances in Electrochemical Science and Engineering, VCH Publishers Inc., New York, H. Gerischer and C. Tobias, Editors, New York, 1994.,p.55. 222 A. M. Sullivan, and P.A Kohl, J. Electrochem. Soc., 142 (1995) 2250. 223 M. Kato, Y. Yazawa, S. Hoshino, USA Patent 5,470,381 (1995). 224 H. Wachi, Y. Otani, USA Patent 5,660,619 (1997). 225 H. Wachi, Y. Otani, USA Patent 5,560,764 (1996). 226 N. Koura, A. Kubota, J. Metal Finish. Jpn., 36 (19850 182. 227 S. USA Patent 5,945,158 (1999). 228 S. R. H. Lepard, M.L. Dubois, J.C. Currie, in Proceedings of the Symposium on Reliability of Metals in Electronics, Vol. 95-3, The Electrochemical Society, Inc., Pennington, N.J., 1995., p.178. 229 S. in Proceedings of AESF SUR/FIN '98, Annual Technical Conference, June 22-25, 1998, Minneapolis, MN, AESFS, Inc., Orlando FL (1998). 230 S. Masaki, H. Inoue, H. Honma, Met. Finish., 96 (1) (1998) 16. 231 A. Molenaar and J.J. Coumans, Surface Technol., 16 (1982) 265. 232 A. Molenaar and J.W.G. de Baker, J. Electrochem. Soc., 136 (1989) 378. 233 J.H. Dailey, J.R. Morgan, K.I. Saad, USA Patent 5,534,048 (1996). 234 J. H. Dailey, J. R. Morgan, K.I. Saad, USA Patent 5,562,950 (1996). 235 Y. Nishikara, A. Oharada, USA Patent 5,147,454 (1992). 236 H. Uchida, M. Kubo, M. Kiso, T. Hotta, T. Komitamari, USA Patent 5,248,527(1993). 237 Y. Takano, A. Senda, USA Patent 5,360,471 (1994). 238 C.M. Melton, A. Growney, H. Ferhaupter, USA Patent 5,391,402 (1995). 239 C.M. Melton, A. Growney, H. Ferhaupter, USA Patent 5,435,838 (1995). 240 R.E. Buxbaum, P.C. Hsu, USA Patent 5,149,420 (1992). 241 H. Shimauchi, S. Ozawa, K. Tamura, T. Osaka, J. Electrochem. Soc., 141 (1994) 1471. 242 T. Hatsukawa, T. Higashikawa, T. Osaka and H. Nakao, J. Surf. Finish. Jpn., 47 (1996) 779. 243 T. Osaka, T. Higashikawa, A. Iizuka, M. Takai, M. Kim, J. Electrochem. Soc., 145 (1998) 2419. 244 H. Honma and T. Kobayashi, J. Electrochem. Soc., 141 (1994) 245 K. Chen, Y. Chen, Plat. and Surf, Finish., 84 (9) (1997) 80. 246 A.F. Karam and G. Stremsdoerfer, Plat. and Surf. Finish., 85 (1) (1988) 88. 247 K.G. Mishra and R.K. Paramguru, J. Electrochem. Soc., 143 (1996) 510. 248 B.J. Hwang and S.H. Lin, J. Electrochem. Soc., 142 (1995) 3749. 249 Y. -S. Kim, H.J. -Sohn, J. Electrochem. Soc., 143 (1996) 505. 250 L.M. Abrantes and J.P. Correia, J. Electrochem. Soc., 141 (1994) 2356. 251 L. Stein, H. Mahlkow, W. Strache, USA Patent 5,882,736 (1999).
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Towards Computational Electrochemistry - a Kineticist’s Perspective L. K. Bieniasz Institute of Physical Chemistry of the Polish Academy of Sciences, Molten Salts Laboratory, ul. Zagrody 13, 30-318 Cracow, Poland.
I. INTRODUCTION The recent decade has brought an unprecedented development of computer technology and its applications. One of the areas of the human activity, in which the above development is particularly noticeable and intensive, is science. Computers are already employed at many stages of scientific investigations but we still have an impression of being in the initial phase of the computer revolution in science. Computers and computing are also increasingly utilized in electrochemistry, as can be seen from several representative reviews, focused on simulation techniques,1-12 computer-aided data analysis,13-16 computer-driven instrumentation,17-20 or various such subjects jointly.21-24 Figure 1 demonstrates a systematic increase of the number of publications related to various computer applications in electrochemistry. The main objective of the present article is to argue that some of the above trends occurring in natural sciences, and in electrochemistry in particular, indicate an emergence and justify a creation of a new branch of electrochemistry that might best be called “computational electrochemistry.” The term “computational electrochemistry” has already been sporadically used in the literature (see, for example Ref. Modern Aspects of Electrochemistry, Number 35, Edited by B.E. Conway and Ralph E. White, Kluwer Academic/Plenum Publishers, 2002 135
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25), and on the Internet. “compu-
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Research groups devoted entirely to
tational electrochemistry” are beginning to appear, such as the Computational Electrochemistry Group of Prof. Deconinck at the Vrije Universiteit Brussel, Belgium.26 (The group mission is to ”stay beyond the state-of-the-art in numerical modeling of electrochemical systems”.) “Computational electrochemistry” is also declared as a part of the activities of other research groups, like that of Prof. Compton in Oxford, UK,27 or Prof. Voth in Utah, USA.28 However, the present author is not aware of any publications that would actually define the meaning and scope of this emerging field, in contrast to well established disciplines of computational chemistry, computational physics, computational biology, etc., the meaning and scope of which have been discussed in numerous texts (see the following section II). For this reason, in sections II-III of the present paper, an attempt will be undertaken to formulate a definition and suggest a research program for computational electrochemistry, as relevant to the particular area of electrochemical kinetics. The latter choice results from the present author’s scientific interests, as he has devoted the past ten years to the development of computer-aided approaches to electrochemical kinetics.
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The discussion in sections II-III has partially the character of a review, but in view of the above objective it was necessary to achieve a compromise between the breadth and depth of the review. On the one hand, an effort has been made to provide a solid justification for the views expressed. This has resulted in a considerable number of references that span a very wide range of topics related to the application of computers to natural sciences, with special emphasis laid on chemistry and electrochemical kinetics. This aspect of the present article may interest readers beyond just the circle of computationally oriented electrochemists. On the other hand, owing to such a broad, unifying view of the subject discussed, the review is of necessity selective. A more complete review would require a substantially larger volume, too extensive for this monograph series. Readers interested in detailed accounts of the past and present advances in computer applications in electrochemistry are referred to the review publications1-24 cited above. A second objective of this article is to summarize the present author’s work in a systematic way, and to compare it with similar recent developments of other authors. This work has been motivated by the vision of computational electrochemistry, outlined in sections II-III, which explains why the summary is included in this paper. However, this work has been so far limited to the computer-aided modeling of electrochemical transient methods for kinetic problems in no more than one spatial dimension, so that the summary, contained in section IV, is also subject to this limitation.
II. THE ROLE OF COMPUTERS IN NATURAL SCIENCES When trying to compile an overview of the role of computers in natural sciences, one is struck with three observations: (a) everything in this area changes very fast; (b) there exist differences (often large) between opinions of experts; (c) The terminology used is often inconsistent and there are terms that mean different things to different people, while different terms sometimes denote similar or identical things. This state of affairs is associated with an enormously rapid progress in computer hardware and software technology, a rapidly expanding scope and diversity of computer applications and a certain degree of
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competition between various lobbies in the scientific community, interested in magnifying their contribution to the progress in this area, or in taking over control over developments. In this situation, being objective is not easy. One essential aspect of the role played by computers and computational approaches in science, has been convincingly formulated by Jackson29 in a recent study of the fundamental changes that are occurring in the foundations, methods and objectives of science. According to Jackson, during the period of approximately 1570-1790, the first metamorphosis of science occurred, and transformed the operational foundations of science (that were largely the heritage from the time of Aristotle) into its modern form. The first metamorphosis was due in large part to Brahe, Kepler, Galileo, Newton, Leibnitz, Euler, and Lagrange, and resulted, among other things, in a widely held belief about our ability to deduce information about nature only through the use of (a) physical experiments and (b) mathematical models, as basic operational methods of science. Over the past century, however, the character and structure of science has been going through a second metamorphosis, associated with the discovery of our inherent limitations to make analytic mathematical deductions, deterministic physical predictions and structurally stable models of closed systems. In addition, mathematical discoveries (such as the Gödel theorem) were made, which shook the basic idea that mathematical systems are consistent and can establish any result which is true. In these circumstances, the development of the electronic digital computer around 1950 enabled the emergence of a new operational basis for obtaining knowledge about nature, which luckily proved to be particularly useful for investigating phenomena that are the most difficult to study by physical experiments and mathematical models alone. The new operational basis can be called by the general name of (c) computer experiments. Jackson points out that computer experiments have a vast potential to uncover the wonders of nature, which is only vaguely appreciated at present. Many scientists still believe that computers are only good for obtaining approximate solutions of the mathematical models, being unaware of the many discoveries already being made in different ways. However, Jackson indicates a number of new opportunities generated by rational computer experiments, and argues that computer experiments need to be recognized as a coequal operational method, complementary to physical experiments and mathematical models, in
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the process of understanding processes in nature. Furthermore, operations carried out in each of the above three areas (physical experiments, mathematical models, and computer experiments) can yield independent knowledge, and can interface directly with either of the other two areas. Opinions similar to those of Jackson29 are presented increasingly by many scientists, aware of computer capabilities. Refs. 30-33 may serve as typical examples of such statements, made by the representatives of various scientific disciplines, including physics, chemistry, biology, and computer science. In particular, Bunker31 had described computer experiments relevant to chemistry already in 1964. He concluded that ”computer ... has provided insight into chemical problems that cannot be achieved by laboratory experiments,” and ”the computer is employed not merely as a high-speed adding machine but as an experimental tool in itself.” One may possibly argue with various particular statements made by Jackson.29 For example, it is possible to be critical about the emphasis that Jackson lays on the mathematical models. Although models are frequently used in scientific practice, they are usually regarded as simplified theoretical descriptions of reality, and cannot be a substitute for (unsimplified) theories. Therefore, it may seem more appropriate to consider physical theories and models jointly (and not only mathematical models alone) as the second operational method of science. Also, one may ask why Jackson limits the role of computers in science to computer experiments. There exists a variety of computer applications, such as the use of text editors, graphical programs, information retrieval or data acquisition systems, automated laboratory instrumentation, electronic mail, etc., which rather cannot be classified as examples of computer experiments, but nevertheless have a considerable impact on science through the growth of productivity of scientists, and through the improvement of the reliability of the scientific research that they conduct. Apart from the above, it has to be realized that the role of computers and computing in contemporary science is more than just instrumental. The development of the conceptual foundations of computing is closely related to our understanding of complex, emergent phenomena, such as life or the human mind, and hence determines our perception of science as a complex product of intelligent human activities. (See, for example, the recent excellent
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book by Coveney and Highfield,34 which also provides many interesting examples of computer experiments.) The development of relevant computer/computational tools for natural sciences is no longer limited to the sphere of computer science, numerical and applied mathematics. Conversely, over the past decades a number of new scientific disciplines, focused on various aspects of computer uses in natural sciences, has emerged from natural sciences, mathematics, computer science, and philosophy of science. These new disciplines are: computational physics, computational chemistry, computational biology, etc., chemometrics, biometrics, etc., chemical informatics, bioinformatics, etc., computational science and engineering, and modeling of scientific discovery. The new disciplines, although still evolving, are now firmly established, have their own journals, conferences, associations, divisions in scientific societies, etc. In sections II.1-II.6 the new, partially interrelated disciplines associated with the use of computers in chemistry, are briefly characterized. Predictions of further changes to be caused by the use of computers in science are contained in section II.7. The purpose of sections II.1-II.7 is to enable understanding of the current status of computer uses in electrochemical kinetics, which are discussed in section III. In view of all the above facts and arguments, the conviction about the increasing importance of computers in science is well justified, and is central to the present article. However, to make this report complete it should not be forgotten that in spite of evident progress resulting from the use of computers in science, there are still individual scientists or entire scientific communities having negative attitudes towards the use of computers in science. Some of the critical attitudes have rational, albeit arguable, elements (see, for example, Ref. 35). Such views should be carefully analyzed and taken into account in the process of further application of computers in science. However, most of the negative attitudes are irrational and should be eliminated by education.36 1. Computational Chemistry
The utilization of computing and computers in chemical research, and the possibilities opened by them, have been long awaited and anticipated by many prominent chemists and physicists. Already in 1808, Gay-Lussac37 predicted that ”We are perhaps not far removed from the
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time when we shall be able to submit the bulk of chemical phenomena to calculation.” Dirac38 complained: ”The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.” Mulliken, the winner of the 1966 Nobel Prize in chemistry, said in his frequently cited speech (see, for example, Ref. 39): ”I would like to emphasize strongly my belief that the era of computing chemists, when hundreds if not thousands of chemists will go to the computing machine instead of the laboratory for increasingly many facets of information, is already at hand.” Boyd40 mentions a number of other prominent chemists, including Nobel Prize winners, who have practiced the computational approach to chemistry in their research and laid foundations for the discipline of computational chemistry. Historically, computational chemistry ”began as a few computer applications by a small number of quantum chemists. Then, over the course of several decades, as more and more applications and users appeared, it coalesced into a full-fledged discipline.” 39 Computational chemistry is still evolving and although many authors use this term in the literature, they often assume different definitions. Lipkowitz and Boyd, editors of the prestigious Reviews in Computational Chemistry, collected a number of definitions in the preface to volume 1 of the reviews.41 Some of these definitions tend to restrict the scope of computational chemistry to various kinds of quantum-mechanical calculations in chemistry, or to computational molecular modeling. In fact, these subjects are most frequently discussed at conferences and in journals devoted to computational chemistry (such as the Journal of Computational Chemistry, Computers and Chemistry and a number of other journals listed in Ref. 40). However, it is hard to understand why other branches of chemistry, which do not directly use quantum mechanics, or which are based on the continuum approximation to the structure of matter, but nonetheless rely heavily on computational investigations (in particular, chemical and electrochemical kinetics, clearly belong to this category, as Readers may see from representative reviews such as Refs. 42-55 and 1-24, respectively), should not have a proper place in computational chemistry. It is worth pointing out that first uses of computational techniques in chemical/biochemical kinetics date back to 1943 (see, for example,
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Ref. 56), and in electrochemical kinetics back to about 1948 (see, for example, Refs. 57 and 58), i.e. they belong to the pre-computer era, similar to first uses of computational techniques in quantum chemistry.39 Some other definitions, in turn, actually put an equality sign between computational and theoretical chemistry, which is another exaggeration in view of the distinction between mathematical models and computer experiments,29 previously discussed. Lipkowitz and Boyd41 choose to proffer a wide definition: ”computational chemistry consists of those aspects of chemical research that are expedited or rendered practical by computers,” and consistently use this definition for the selection of the material published in their review series. A recent chapter by Larter and Showalter55 in their review series leaves no doubts about their seeing chemical kinetic computations as a part of computational chemistry. For the present author, the definition from Ref. 41 appears the most adequate. Weber et al.59 proposed an even wider definition. (”Computational chemistry is synonymous with computer-assisted chemistry: it consists of all the areas of chemistry which benefit from the use of computers”.) However, since a situation can easily be imagined when the whole of chemistry benefits from the use of computers, this seems to be a too general specification. Definitions similar to that in Ref. 41 have been formulated for computational physics (which is physics done by means of computational methods60), and computational biology (which is a computation-based research devoted to understanding basic biological questions61). With regard to the actual type of work performed by computational chemists, it should be recognized that at present there is a core of ”ca. 2000 to 6000 scientists who consider themselves to be computational chemists and who practice computational chemistry on a fulltime basis, working on methods, theory, software, and/or applications,” 62 and also that ”all the people who develop software packages ... are computational chemists.” 39 Computer experiments play a significant role in the research activity of computational chemists.39 Although opinions are sometimes encountered that computational chemistry is a multidisciplinary area of research (see, for example, Refs. 40 and 41), from an organizational point of view it has become a branch of chemistry. In particular, divisions of computational chemistry exist in a number of chemical societies. 2. Chemical Informatics
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The terms ”chemical informatics” and more recent mutations such as ”cheminformatics” or ”chemoinformatics,” have been used for some time, yet there does not seem to be any general agreement with regard to the meaning of these terms. According to one definition available from Internet,63 ”chemical informatics encompasses everything from traditional library science methods of organizing chemical information to modern computer-based techniques for the storage and retrieval of chemical information.” In view of this definition, and other examples of the use of this term, chemical informatics might be regarded as a part of computational chemistry, as defined in section II.1, or even an independent area, concentrated on handling chemical information. Such a meaning of the term is analogous to the traditional meaning of bioinformatics (the research and development work required to build the information infrastructure needed in modern biology).61 The Division of Chemical Information of the American Chemical Society represents such understood chemical informatics, as opposed to the Division of Computers in Chemistry, which represents computational chemistry. However, the above Internet resource contains further specifications that complicate this interpretation: ”The application of computer-based tools to the problems of chemistry is being carried out most intensively by those in the emerging field of chemical informatics (also known as ‘chemical information science and technology’ or ‘computer chemistry’ ... The practitioners of chemical informatics must now deal with heavily technology-based areas such as molecular modeling and computational chemistry. As a reflection of the broadened scope of chemical informatics, the American Chemical Society recently enhanced the title of the Journal of Chemical Information and Computer Sciences with the phrase ‘Includes Chemical Computation and Molecular Modeling.’ ” These additions seem to equate chemical informatics with computational chemistry, or even tend to view computational chemistry as a part of chemical informatics. Such an extended meaning of chemical informatics is similar to the recently suggested extended meaning of bioinformatics (a scientific discipline that encompasses all aspects of biological information acquisition, processing, storage, distribution, analysis and interpretation, that combines the tools and techniques of mathematics, computer science, and biology with the aim of understanding the biological significance of a va-
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riety of data61), which considers computational biology as a part of bioinformatics. 3. Chemometrics
According to Massart et al.,64 the term ”chemometrics” denotes ”a chemical discipline that uses mathematical, statistical, and other methods employing formal logic (a) to design or select optimal measurement procedures and experiments, and (b) to provide maximum relevant chemical information by analysing chemical data.” Chemometrics has found particularly widespread application in analytical chemistry. Practical chemometrics is a matter of carrying out computations. Hence, a computer is involved.64 The scope of, and subjects of interest to chemometrics, have been discussed in numerous reviews (see, for example, Refs. 65 and 66, and references cited therein), and they include: statistics, optimization, signal processing, resolution, calibration, parameter estimation, structure/activity relationships, pattern recognition, library searching, and artificial intelligence.65 Recently, the concept of an integrated intelligent laboratory,64 in which all steps of the analytical process are computer-controlled, becomes increasingly important for developments of chemometrics (see also next section II.4). In accordance with their definitions of computational chemistry, provided in section II.1 of this chapter, Weber et al.59 and Boyd67 view chemometrics as a part of computational chemistry. Chemometrics has also counterparts in other sciences, for example, biometrics (or biometry) in biology and related disciplines.68,69 Biometry has a long history but it is mainly limited to the application of statistical methods. 4. Chemical Laboratory Automation
Development of analytical chemistry and chemometrics remains in close relationship with efforts towards chemical laboratory automation70 which, in recent years, has become a research field in itself, with its own journals such as Laboratory Automation and Information Management, or Automatic Chemistry. According to McDowall,71 who reviews, compares, and criticizes a number of former definitions of laboratory automation, laboratory automation should be understood as ”apparatus, instrumentation, communications or computer applica-
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tions designed to mechanize or automate the whole or specific portions of the analytical process in order for a laboratory to provide timely and quality information to an organization.” In most cases ”organization” means here an industrial organization, but academic and governmental research laboratories are also in the sphere of interests of laboratory automation. McDowall distinguishes four main areas of interest to laboratory automation: (a) instrument automation (for example autosamplers, autoanalyzers, robots, batch analysers, control devices), (b) communications (for example serial and parallel communication, networks, multimedia, client-server architectures), (c) data-toinformation conversion (for example, chromatographic data acquisition and analysis software, chemometric techniques, digital filtering), (d) information management (for example, database management systems, laboratory information management systems, expert systems, and neural network systems). An important activity within the laboratory automation area is the development of strategies according to which laboratory automation systems should be designed and introduced in particular laboratories/organizations.72 Refs. 73-75 discuss various issues important for the future of laboratory automation. Chemical laboratory automation and computational chemistry now evolve rather separately, although they were once closer: the American Chemical Society Division of Computers in Chemistry was originally directed by chemists interested in laboratory automation, and only later became dominated by computational chemists who came over from theoretical chemistry and physical-organic chemistry.67 At present, opinions are encountered that laboratory automation should become more an engineering than a scientific discipline.76 5. Computational Science and Engineering
According to Sameh and Riganati77 the term ”computational science” (which should not be confused with computer science) was first used by Kenneth Wilson (who received a Nobel prize for his work in physics in 1982) to refer to those activities in science and engineering that exploit computing as their main tool. In the past decade, computational science (now termed ”Computational Science and Engineering,” or CSE) has emerged as a rapidly developing field of study that in some sense unifies and generalizes activities represented jointly by computational physics, computational chemistry, computational biol-
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ogy, and other similar disciplines associated with the use of computations in science. Although the exact definition and scope of the CSE are still opened to debate, the number of declared computational scientists grows systematically, conferences and meetings on CSE are organized, and in 1994 the prestigious American IEEE Computer Society launched a new magazine called IEEE Computational Science and Engineering, devoted specifically to CSE.77 According to Sameh78,79 CSE should generally be viewed as a multifaceted field that includes not only the study of computational aspects of science and engineering disciplines, but concerns the whole computational process. It seeks to advance science and engineering disciplines through better understanding of advanced computers and computational methods, as well as advancing the state of the art in computer architecture system software and algorithm design through better understanding of science and engineering applications. Thus, CSE covers both computation in science and the science of computation. In brief, CSE is applicationsalgorithms-architectures, or (see, for example, Ref. 80). Computational experiments are recognized as a fundamental component of CSE.81 It is difficult to avoid an impression that there is much overlap between the goals and activities of computational physics, computational chemistry, etc., and CSE. Individual professional careers, affiliations and research interests of the scientists seem to play the main role in deciding with which discipline they identify themselves. Those who moved into the field from physics, chemistry or biology, tend to call themselves computational physicists, chemists or biologists, while those who came from computer science or applied or numerical mathematics, prefer to say they are computational scientists. There are also scientists who do not actually make a distinction between, e.g., computational physics and CSE (for example, Landau and Páez Mejiá82 often use the terms ”computational physics” and ”computational science” interchangeably), or who recognize that much can be gained by a close relationship between these disciplines. (For example Truhlar83 indicates relationships between computational chemistry and CSE.) Some researchers view CSE as an interdisciplinary field. For example, Stevenson33 stresses that constituent disciplines (physical sciences, engineering, mathematics, computer science) maintain their autonomy. According to this view, within CSE a computer scientist
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retains expertise in computer science but emphasizes applications in science or engineering. Surprisingly though, Stevenson concludes that computational science should be made a part of computer science and vice versa. In a similar spirit, O’Leary84 emphasizes the role of interdisciplinary teams for successful solving of difficult problems by CSE. Some other researchers seem to be less conservative in this respect. For example, Sameh85 argues that CSE flourishes when walls separating classical disciplines are removed or made permeable. He points out a recommendation made in a report by the (US) National Academy of Sciences, that calls for the redesigning of graduate education in the USA. He also complains that ”entrenched vested interests make it difficult for some academic institutions to replace arcane programs in science and/or engineering colleges, that ceased to be effective decades ago, by genuine CSE programs.” Independently of which of these somewhat contradictory opinions currently prevails, facts seem to indicate that the second may finally dominate. A number of American universities has already started educational programs in CSE as a separated discipline (with computational chemistry being also represented).86,87 This trend is likely to develop (see section II.7 later). According to a number of CSE experts (see, for example, Ref. 88), the main goal of CSE should be the conception, design and implementation of problem-solving environments (PSEs) for science and engineering. The term PSE denotes a computer program that provides all the computational facilities necessary to solve a target class of problems, conveniently and efficiently.89-95 These features include90,91,94: advanced solution methods, automatic or semiautomatic selection of solution methods, and ways to easily incorporate novel solution methods. Moreover, PSEs use the language of the target class of problems, so that users can run them without specialized knowledge of the underlying computer hardware or software. The PSEs should enable more people to solve more problems more rapidly, and they should enable many people to do things that they could not otherwise do.90 Clearly, computer experimentation is a category of problem solving. Probably most of the research efforts expended hitherto has concentrated on the development of PSEs for the solution of partial differential equations (PDEs). Such PSEs are of importance for all natural sciences. Prominent examples of such systems are described in Refs. 90 and 93. Contemporary research involves among other developments: extension of these PSEs to parallel computer architectures90
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and distributed (network) systems,96 and incorporation of the elements of artificial intelligence.97 Further developments are expected to concentrate on elaborating specialized PSEs for selected, small application areas, and on elaborating multidisciplinary PSEs.98,99 The latter PSEs should allow detailed simulation of complex physical systems with a large number of components that have different shapes, obey different physical laws and/or manufacturing constraints, and interact with each other through geometric and physical interfaces (for example engines, cars, airplanes, perhaps even entire living organisms). Collaborating PDE solvers100,101 belong to this category of PSEs. It is also expected that the World Wide Web will gradually evolve into a global PSE.94 There are predictions that by the beginning of the next century computer technologies will enable anyone having access to a computer to find an answer to any question that has a known or effectively computable answer.90,99 6. Modeling of Scientific Discovery
An important point raised about PSEs is that the nature of science and engineering problems for which PSEs can be developed must be well understood and standardized, since one cannot expect a powerful and reliable PSE in an area where no one yet knows how to solve the principal underlying problems.93 Thus, PSEs can be viewed as advanced tools with a clearly pre-defined scope of application and capabilities. This is a certain limitation. However, de Jong and Rip102 have recently put forward an idea of creating ”computer-supported discovery environments” (CSDEs) which seem to have the potential of surpassing the above limitation. Firstly, de Jong and Rip envisage CSDEs as extensions of the so-called ”discovery programs”103 that can discover scientific laws or concepts not known before. A number of prototype programs of this kind has already been created in the course of the research on the modeling of scientific discovery, which aims at developing a ”normative theory of human discovery” by combining advances of artificial intelligence with the philosophy of science,104 and thereby enable ”machine discovery”105 to be achieved. Among them are programs useful for chemistry, such as, for example, MECHEM106 designed for making discoveries in chemical kinetics. Secondly, CSDEs are defined as ”large scale sociotechnical systems embedded in a research practice,” 102 which means that they involve
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not only relevant computer tools (shared through a global network), but also a system of institutions/organizations and research procedures, together with the scientists coordinating the functioning of the system. Although the concept of CSDEs may look like science fiction at present, de Jong and Rip102 provide a realistic scenario of how CSDEs might be used in future research practice, taking molecular biology as an example. The scenario presents the scientific activity as a ceaseless interaction of scientists with computers. Almost every action (communication between scientists, information/knowledge storage and retrieval, physical experimentation, hypothesis generation, process simulation, result dissemination, etc.) is computerized or automated, and controlled by an integrated CSDE. The role of the scientists is practically limited to deciding which problems merit attention, which tools are to be used, what the scientific value of the results is, and which results are to be disseminated and/or incorporated into the CSDE. The scientists are also responsible for a continuous development of the CSDE. Many components needed for creating CSDEs are already available, or should be at hand soon, although further research in the area of discovery programs is still necessary. De Jong and Rip indicate a number of guidelines for the development of CSDEs, and suggest, among others, that tools in a discovery environment should be developed cooperatively, possibly involving the scientific community as a whole. Undoubtedly, a combination of the research on PSEs with the research on CSDEs, which so far seem to have proceeded independently, might open entirely new perspectives for the role of computers in natural sciences. 7. The Future
Any attempts to predict the future of the role of computers in science must be equally risky and uncertain, as do other kinds of prophesying. The present author is not going to speculate about the development of computer and computing technologies, nor about future discoveries that could be made by means of computers. (Interested Readers may be guided by Refs. 107-112 to find examples of such predictions.) However, he finds it tempting to discuss changes that the continued computer revolution is likely to have on the disciplinary structure of science. There is a growing conviction that a stiff separation of
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science into disciplines, both from the point of view of the contents, organization, and education, does not, in fact, promote progress. This is particularly noticeable at the junction between natural sciences and computer science. At present, this conviction finds expression mostly in calls for changes in educational programs that are heard from many sides. The calls have already resulted in new curricula, as was noted in section II.5. However, changes in the traditional areas of interest of scientific disciplines are also envisaged. For example, Browne,113 physicist and computer scientist, argues that traditional department structure of universities has become obsolete: ”Many research problems and product-development problems require multiple disciplines or knowledge that spans multiple traditional university disciplines and computer-science knowledge. ... the entire discipline-oriented structure of education in today’s technical universities should be seriously reconsidered with a view towards regarding all training as establishing foundations on which future competence can be developed, rather than establishing someone as a discipline-knowledge specialist.” In turn, Gershenfeld,114 physicist, argues that ”physicists have been... guilty... in maintaining a culture that can inhibit people from working in areas that are new, interesting, and also relevant to current problems.” Ugi et al.,115 computational chemists, criticize the present status quo of (German) computational chemistry, and the quality of research, in their opinion unsatisfactory. They argue that knowledge of mathematics, computer science and programming among computational chemists is insufficient. They call for appropriate changes in educational programs, more interdisciplinary style of research, as well as for improvements in scientific infrastructure for computational chemistry. Hood,116 biologist, expects that interdisciplinary scientists will play a major leadership role in biology and medicine of the twenty-first century. He calls for establishing educational programs comprising biology and computer science. The programs would develop interdisciplinary (PhD) scientists, those with expertise in both disciplines, and the ability to forge interdisciplinary collaboration. Similar voices are heard from the side of computer scientists. For example, Tsichritzis,117 professor of computer science and informatics, suggests: ”If we (computer scientists) want to achieve new, spectacular results, we have to move to new boundaries between computer science and other totally new areas within biology, medicine, chemistry, etc.” Probably the most
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dramatic changes have been expected by Newell, an expert in artificial intelligence. He has argued118: ”We should, by the way, be prepared for some radical, and perhaps surprising, transformations of the disciplinary structure of science (technology included) as information processing pervades it. In particular, as we become more aware of the detailed information processes that go on in doing science, the sciences will find themselves increasingly taking a metaposition, in which doing science (observing, experimenting, theorizing, testing, archiving, ...) will involve understanding these information processes, and building systems that do the object-level science.” It seems important to be aware of these predictions, to take them into account in planning scientific research, and to adapt research practices, organizational structures of the scientific community, and educational standards, to the ongoing and expected changes.
III. THE ROLE OF COMPUTERS IN ELECTROCHEMICAL KINETICS He and Faulkner22 discern six different roles for digital computers in electrochemistry: (a) sample preparation, (b) experimental control, (c) preparation of data for interpretation, (d) model building, (e) interpretation, and (f) tactical and strategic decision making. A slightly different classification was suggested by Ridgway and Mark.17 For the purpose of the present discussion, focused on electrochemical kinetics, points (a)-(f) will be combined into two basic categories of computer applications: modeling of kinetic experiments and control of electrochemical instrumentation. These two categories are discussed in sections III.l and III.2, leading to the conclusion presented in section III.3. 1. Modeling of Electrochemical Kinetic Experiments
Electrochemical kinetic phenomena can be investigated experimentally by means of a large variety of techniques that have been described at length in a number of textbooks (see, for example Refs. 2 and 119-122). By the modeling of electrochemical kinetic experiments, performed by such techniques, we shall understand here all
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activities relevant to the construction and analysis of mathematical models and to the design and performing of computer experiments, together with those activities relevant to physical experiments that refer to experimental data analysis (physical experiments, mathematical models, and computer experiments mean here the basic operational methods of science, as discussed by Jackson29). A similar definition of modeling is often accepted in (non-electrochemical) chemical kinetics (see, for example, Refs. 52, 123, and 124). The modeling of electrochemical kinetic experiments has been for a long time dominated by a practice that in the opinion of the present author might most conveniently be called ”the Nicholson and Shain paradigm,” after the names of the authors of fundamental works on interpretive aspects of transient methods in electrochemistry, including the most frequently cited paper of 1964.125 This paradigm can be expressed as follows: Check up in the literature if the theory for the experimental method you use, and reaction mechanism you believe to occur in your system, has been already described. If yes, then interpret your experimental data by means of the theoretical diagnostic criteria, working curves or other theoretical results contained in the literature. If no, then formulate and solve relevant mathematical model equations, generate diagnostic criteria, working curves, etc., publish them, and apply the results to your data. In this paradigm the computer plays a modest role of a ”number cruncher.” Computer experimentation is necessary, because model equations usually do not have analytical solutions. However, the main goal of the theoretical and computational investigations undertaken is to produce a summary of the theoretical properties of an electrochemical system, together with a set of hopefully simple interpretive rules, easily applicable to experimental data. Once such a summary is obtained, the role of the computer experiment is finished. As a result, the vast potential of the computer experiments is not accessible to those who are the most interested: electrochemists seeking to understand experimental systems they study. Furthermore, details of the computational methodology are of secondary importance in this paradigm. The Nicholson and Shain paradigm has played an important role in the development of electroanalytical chemistry (see for example, Refs. 2, 120-122, and references cited therein), but a question arises how long can we proceed using this approach. Hundreds of theoretical papers analyzing new kinetic models are published every year and
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nobody is able to effectively acquire and make use of this sophisticated knowledge any longer. Also, even if a desired theoretical report is available in the literature, an experimentalist often finds that the results contained therein are not particularly useful for his/her purpose, because e.g. they correspond to different parameter values, different intervals of independent variables, or are normalized in a different way, etc. Consequently, a need arises for repeating computations already published. Apart from this, one may ask a question about the rationale for publishing longish theoretical/computational reports for increasingly complex kinetic models. Taking into account the contemporary advancement of CSE and chemical informatics, it becomes more and more realistic to think about designing PSEs that might even generate reports similar to Ref. 125 automatically, or with a little human intervention. Publishing such machine-generated reports in scientific journals would become equally controversial as publishing detailed accounts of arithmetical operations in the era of commonly available pocket calculators. It would be more relevant and interesting to see publications describing foundations and design details of the PSEs that can do the job. In view of the above it is encouraging to see signs showing that a new modeling paradigm is beginning to emerge. Among the studies that particularly clearly lay foundations for this change should be mentioned the entire area of digital simulation in electrochemistry,1-12 and advances in the area of computer-aided experimental data analysis (see reviews 13-16, and Refs. 126-135 that contain representative examples of various alternative approaches to the computer-aided data analysis in electrochemical kinetics). In a narrow sense the term “digital simulation” is usually used to denote modeling activities concentrated around direct numerical solution on a computer, of mathematical equations that describe electrochemical kinetic experiments. Feldberg1 and Britz6,11 speak here about the solution of relevant PDEs only, but other possibilities exist. In particular, direct numerical solution is often necessary also in the case of electrochemical kinetic models independent of spatial coordinates, in which transport PDEs are not considered at all (for example, models of processes involving chemisorption of intermediates, such as, e.g., the and evolution, electrocatalytic oxidation of small molecules, etc.) but the governing equations take the form of, e.g., ordinary differential equations (ODEs). Coupled systems of PDEs, and mathe-
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matical equations of other kinds, are another possibility (see later). Digital simulation based on PDE solving has become popular in large part owing to the works of Feldberg,1 beginning with the introductory paper by Feldberg and Auerbach,136 although other electrochemists used such direct numerical solution techniques earlier, notably Randles57 and Hale.137 Numerical solution of electrochemical governing equations independent of spatial coordinates, was initiated at about the same time (see, for example, Refs. 138-140). In a still wider sense, “digital simulation” may also denote combined analytical-numerical solution methods used in electrochemical kinetics, such as integral evaluations by numerical summation (see, for example Refs. 58 and 141-143), numerical solution of integral equations (see, for example Refs. 144-146), approximate summation of infinite series (see, for example Refs. 147-150), semi-integration and semi-differentiation algorithms (see, for example Refs. 151 and 152), etc. In view of the discussion in section II, every instance of digital simulation should be regarded as a category of computer experiments. From a mathematical point of view digital simulation is limited to solving the “direct” or “forward” problems, as opposed to solving ”inverse” problems (see, for example, Ref. 153), that play an equally important role in chemical kinetics.154 Solution of inverse problems is usually regarded as a (modeling) activity belonging to experimental data analysis. It should be stressed, however, that the above meanings of modeling and simulation are by no means generally accepted in science, and other definitions are also in use (cf., for example, the compilation of definitions prepared by Pritsker155). Digital simulation represents a beginning of a new modeling paradigm at least for two reasons. Firstly, digital simulation is now perceived as a part of electrochemistry and not (as it might seem 20-30 years ago) as a part of numerical mathematics and/or computer science. Over the past decades many electrochemists contributed to this field, trying to work out numerical approaches most suitable for dealing with intricate peculiarities of electrochemical kinetic models, and to apply them to real-life problems (Refs. 1-12, and references cited therein). Most of the literature on these subjects has been published in electrochemical journals. The present author’s attempts to publish his papers in computationally oriented journals were several times commented by reviewers with questions ”why is this paper not submitted to an electrochemical journal?” Secondly, more and more attention has
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recently been paid to the development of general software tools that make use of the experience gathered by electrochemical kinetic simulationists, and that bring the power of computer experiments to every electrochemist. Such general simulation programs or libraries have been, or are currently developed, by Klinger et al.,156 Cook and Landau,157 Mager,158 the EG & G Company (Condesim Simulation Software),159 Speiser (EASIEST),160,161 Sánchez et al. (SIMULA),162 Gosser and Zhang (CVSIM and CVFIT),163-165 the present author (ELSIM),166-169 Penar et al.,170 Rudolph and Feldberg (DigiSim),171-173 Verbrugge and Gu,174 Huang and Hibbert (POLAR),175,176 He and Chen,177 Villa and Chapman,178 Kolár (TRANSIENT),179 Nervi (ESP),180 Biader Cerpidor et al.,181 the Computational Electrochemistry Group (ELEMD),26,182 and Alden and Compton.27,183,184 Possibly there are more programs of this kind but they have not been published nor made generally available. A comparison of some of the above programs has been recently presented by Speiser.10 The development of general simulation programs shows a clear relation to the goals and activities of CSE (see section II.5), although it seems that only the present author has explicitly expressed this connection in his papers.167-169 Similar to digital simulation, electrochemists have been at the forefront of the development of computer-aided electrochemical data analysis methods, which shows relation to chemometrics and laboratory automation (see sections II.3 and II.4). In particular, the methodology of parameter estimation from electrochemical responses has been the subject of a number of studies,13-16,126-135 and possibilities for parameter estimation have been included into several of the above general simulation program158,160,161,163-169,171-173,178, Recently developed expert systems for the automatic elucidation of mechanisms of electrode processes,185-189 and for the voltammetric determination of trace metals,190-195 or the WWW-based data-analysis program of Alden and Compton,196-198 are other examples of general software tools, this time, products of increased efforts towards automated electrochemical data analysis (see also Ref. 19 for some earlier examples). The WWWDA program196-198 offers a foretaste of how globally accessible PSEs or CSDEs for electrochemical kinetics may serve the electrochemical community in the future. 2. Control of Electrochemical Instrumentation
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Computer applications to electrochemical instrumentation also show relation to research directions of chemometrics and laboratory automation, being strongly characterized by trends towards automation.17-22 According to the review by Gunasingham,19 the history of the development of electrochemical instrumentation should be divided into three periods: the early history, the electronic age, and the contemporary computer age. Gunasingham provides a number of references confirming an opinion that a major area of the present activity in computer-aided electrochemical instruments has been the development of software that enables a computer-aided interpretation of electrochemical data, to provide analytical, mechanistic and thermodynamic information on the measured electrode process. For example, Faulkner and co-workers state in their paper devoted to the elaboration of a ”cybernetic potentiostat”199 that the essential features of their potentiostat include ”an ability to interpret results, at least by partially reducing data,” and ”an ability to make decisions based on experimental results.” They predict that ”we are not far from seeing instruments that will carry out scientific investigations, not just measurements.” Increasingly many laboratory-made or commercial electrochemical instruments are equipped with software systems that combine computer control of the measurements with elements of the simulation and data analysis. 3. The Emerging Computational Electrochemistry
In view of all the above arguments and circumstances, it is the thesis of this review that the time is ripe for realising that we should replace the Nicholson and Shain paradigm, or at least complement it with a new ”computational paradigm” of electrochemical kinetic modeling. The computational paradigm should be consistent with analogous paradigms already accepted in other branches of natural sciences, that rely on the concept of computer experiments, and it should continue and creatively expand achievements discussed in sections II. 1II.6, III.l and III.2. It appears desirable that the electrochemical community consciously initialize and stimulate efforts towards the development of a new discipline of ”computational electrochemistry” which would be responsible for realizing the computational paradigm in practice. Taking into account trends described in sections II.1-II.7, III.l, and III.2, it can be suggested that the activities of computational
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electrochemistry within electrochemical kinetics should include in particular the following procedures: (a) Collecting, systematizing, and improving electrochemical knowledge, from the point of view and in correlation with the development of computer algorithms and systems that can efficiently utilize this knowledge for the purpose of computational modeling of electrochemical kinetic phenomena, including possible artificial intelligence approaches. (b) Development of computational strategies (including numerical, symbolic, and other methods) most suitable for solving a wide spectrum of problems occurring in electrochemical kinetics. Attention should preferably be paid to strategies potentially suitable for solving (when necessary) any conceivable kinetic problem, in contrast to previous efforts to actually solve and publish every particular problem. (c) Design and development of PSEs, and in further perspective CSDEs, serving for a widely understood modeling and computer-aided research in electrochemical kinetics. These systems should ultimately allow electrochemists to conveniently investigate the behavior of any kinetic models that may occur in electrochemical studies, without a necessity to look for and rely on published materials, and without the necessity to publish detailed reports of the results of the computations. It appears important that the development of software systems serving for computer experimentation become a legitimate research area of computational electrochemistry, in the same way as the development of experimental methods, and design and construction of experimental set-ups, is a part of experimental electrochemistry. (d) Performing computer experiments and other computer-aided operations relevant to electrochemical kinetic studies.
The particular task of designing PSEs for electrochemical kinetic modeling must take into account the complex character of this research process, which usually comprises several stages analogous to those of the research process in (non-electrochemical) chemical kinetics (listed, for example, in Refs. 52, 123, and 124), which can also be regarded as example steps of a general problem-solving procedure of a
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computational scientist, outlined in Refs. 90 and 93. Thus, one can distinguish at least the following stages: 1. Suggesting a set of chemical/electrochemical reactions (often called a reaction scheme or a reaction mechanism) supposedly occurring in the given system studied. 2. Formulating the mathematical governing equations that describe spatial and/or temporal evolution of the system studied, under conditions of a selected experimental method and for the reaction mechanism suggested in stage 1. 3. Solving the governing equations with the aim of obtaining simulated responses (to externally imposed perturbations such, for example, potential or current changes) of the system studied, and understanding their properties. 4. Comparing simulated and experimental responses to external perturbations, with the aim of verifying the reaction mechanism suggested, and determining model parameters. In the case of an unsatisfactory agreement between the properties of the model and experimental results one usually returns to stage 1, modifies the reaction mechanism or other assumptions, and repeats the entire process.
PSEs for electrochemical kinetic modeling should support investigators in performing all the above stages. Representation of stage 1 in the form of a computer algorithm can be very difficult, owing to the multiplicity of types of information and argumentation used for suggesting reaction mechanisms, and also a partially intuitive or subjective character of the reasoning. Future attempts to provide computer support to stage 1 might follow guidelines of the modeling of scientific discovery (see section II.6). Examples of works and opinions partially consistent with the above research program have already been noted above.26, 27, 156-198 Computational electrochemistry could unify all such efforts. General simulation programs, ”simulators,” expert systems, etc., that are products of these efforts, could benefit from the experience already available in the well established line of research on PSEs within CSE, and from conforming to the quality standards expected from the PSEs. These programs should also be started to be called PSEs, as their purpose is consistent with that of PSEs.
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The above discussion of the emerging field of computational electrochemistry has been limited to electrochemical kinetics. There are also other areas of electrochemistry where computational approaches prove indispensable. Quantum-mechanical calculations of metallic surfaces (see, for example, Refs. 200 and 201), solvent free energies for heterogeneous charge transfers,202 or molecular structures at electrodes (see, for example, Refs. 203 and 204), Monte Carlo simulations of electrochemical interfaces (see, for example, Refs. 205 and 206), molecular dynamic studies of the interfaces (see, for example, Refs. 207-209) and of ion hydration, lattice-gas models of adsorption,210 computer models of porous electrodes211 or computer-aided electrochemical process design,212 are just a few arbitrarily chosen examples. These areas should also find their place in computational electrochemistry, presumably with similar research programs. Finally, from an organizational point of view, the natural place of computational electrochemistry appears to be within electrochemistry, including relevant representations in electrochemical societies, as well as (possibly) specialized periodicals and educational programs. However, in the interest of the quality of their work, computational electrochemists should be encouraged to follow trends discussed in sections II.1-II.7, and to cross boundaries of traditional scientific disciplines and make use of the experience of contemporary computational chemistry, chemical informatics, chemometrics, laboratory automation, computational physics, CSE, computer science, and other relevant domains. Therefore, computational electrochemistry would also benefit from becoming a part of computational chemistry and a part of CSE.
IV. THE PRESENT APPROACH This section contains a summary of the investigations performed by the present author in partial cooperation with others, that correspond to the research program formulated in section III.3. These investigations have been concentrated so far on attempts to computerize stages 2 and 3 (and in part stage 4) of the modeling process (section III.3). They have been further limited to the modeling of controlled-potential and controlled-current transient methods, mostly those in which electric current or electrode potential are the only observables, and to electrochemical models involving a single working electrode, and described by at most one spatial coordinate (i.e.
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and described by at most one spatial coordinate (i.e. models independent of spatial coordinates or in one-dimensional space geometry). Under all these restrictions, efforts undertaken by the present author can be divided into the following groups: (a) Elaboration of an algorithm for an automatic translation of the electrochemical reaction mechanisms into corresponding sets of governing equations. (b) Elaboration of a new approach to formulating simulation algorithms, that would enable an effective implementation in a PSE, of various numerical solution methods for electrochemical governing equations, in a manner independent of specific kinetic problems, ensuring maximum generality. (c) Comparative studies of various simulation methods previously suggested in the electrochemical literature, aimed at determining the degree of their efficiency and reliability, as well as their usefulness for a PSE. (d) Theoretical analysis of the numerical stability of the practically used simulation methods, in the context of typical kinetic problems. (e) Search for new, more effective, and more general modeling and simulation methods and strategies. (f) Practical realization of a PSE for the modeling of electrochemical transient methods. (g) Applications of the elaborated programs and methods to the solution of various representative kinetic problems.
Results obtained for tasks (a)–(g) are summarized in sections IV. 1-IV.6 below. 1. Automatic Translation of Electrochemical Reaction Mechanisms onto Corresponding Sets of Governing Equations
Systems investigated by transient methods can be conveniently characterized by specifying and discussing reaction mechanisms. On the other hand, the main purpose of digital simulation is to solve the governing equations in order to obtain insight into the spatio-temporal dependencies of the solutions and, in consequence, to understand the transient responses of the electrochemical systems and to describe
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them quantitatively. Hence, it is also natural and inevitable to use the language of mathematical equations and symbols while specifying electrochemical kinetic models. The issue of the relationships between these two formal descriptions of electrochemical kinetic systems is therefore of theoretical and practical importance, and needs to be elucidated within point (a) of the research program for computational electrochemistry (section III.3). From the theoretical point of view, it is important to understand how the structure and other features of the reaction mechanisms find expression in the structure and features of the mathematical formalism. From the practical point of view, the design of a PSE requires that both alternatives for specifying kinetic problems be available interchangeably for the users, since a PSE should use the language of the class of problems to be solved (see section II.5). Although it is usually not difficult to formulate the governing equations in the case of very simple reaction mechanisms involving only a few reactants/products, this task becomes gradually more and more complicated and prone to errors, when the number of reactants and reactions in a reaction mechanism increases. In Refs. 213 and 214 a theoretical analysis has been performed, of the connections between electrochemical reaction mechanisms and corresponding sets of governing equations describing controlledpotential and controlled-current transient methods. It has been assumed that reaction mechanisms may include bulk species (species distributed in the electrolyte and subject to diffusional and convectional transport, as well as to homogeneous reactions), and also interfacial species (species localized at the electrode surface, e.g. adsorbed). Concentrations of certain selected species may have remained practically constant, while for the remaining species governing equations have been formulated. Various types of reactions, not necessarily elementary, have been allowed to occur in the mechanisms: e.g. electrochemical, heterogeneous non-electrochemical, and homogeneous types. In addition, the reactions may have been equilibrium reactions, non-equilibrium reversible or irreversible. Reaction rates have been assumed to obey a particular form of the power rate law, often used in electrochemical kinetics. Under the above assumptions, the governing equations for bulk species take the form of the reaction-convection-diffusion PDEs, accompanied by initial and boundary conditions. It has been shown214
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that in the presence of interfacial species these PDEs are additionally coupled with equations that govern the kinetics of the concentration changes of interfacial substances, that, in the general case, take the form of differential-algebraic equations (DAEs), and in particular cases the form of ODEs, or purely algebraic equations (AEs). The connections of the electrochemical kinetics with the formalism of the DAEs have not so far been recognized in electrochemistry. However, the awareness of these connections is essential for the proper understanding of kinetic models and for the direction of the further development of simulation methods. An automatic algorithm for translating reaction mechanisms into the above governing equations has been elaborated, together with a number of testing procedures, serving for verifying the correctness of the reaction mechanisms. These tests include: a check for non-conservative (stoichiometrically inconsistent) reaction mechanisms, a check for inconsistent assumptions regarding the presence/absence of various species in the initial equilibrium state of the reaction mechanisms, and a check for the consistency in specifying irreversible non-electrochemical reactions with the assumptions regarding formal potentials of electrochemical reactions in closed loops of dependent reactions. The algorithms and tests take into account the procedure of detecting dependent reactions, reported by Luo et al.,215 which has been considerably extended in Ref. 213. The extension includes, among other matters, discussion of the dependencies between initial concentrations, ignored in Ref. 215, as well as adaptation to other reaction types. The discussion of Refs. 213 and 214 has also been extended onto kinetic models independent of spatial coordinates, in which all dynamic species are interfacial, showing that in such a case the governing equations take the form of the AE, ODE, or DAE systems. 2. General Formulation of the Algorithms of Transient Kinetic Simulations
A traditional approach to digital simulations included writing a respective simulation program in a selected programming language, removing possible errors, compiling and linking with required library procedures, and finally making the actual calculations. The process was tedious and inefficient, easily prone to errors, especially when a complex numerical method was chosen for the simulation. Programs
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very similar to one another were elaborated independently by many electrochemists, which was undoubtedly a waste of time. Despite claims encountered in the literature, stating that adapting such and such program to simulating a new kinetic problem requires just a few little changes, application of every program was practically limited to one or very few kinetic problems. One can say that traditional simulation algorithms were problem-oriented. An alternative, method-oriented approach to the formulation of simulation algorithms, suitable for designing a PSE, and consistent with point (b) of the research program for computational electrochemistry (section III.3), has been proposed in Refs. 216-218. The methodoriented approach consists in treating mathematical expressions that describe kinetic problems as text input data for a simulation program. A special formula translator converts the text of the expressions to the form of an intermediate internal code that is later ”executed” during simulation with the aim of calculating expression values. The intermediate code is also analyzed by the program, in order to reveal dependencies of the expressions on various variables and parameters. According to the results of this analysis an appropriate sequence of elementary operations that realize a solution method chosen by the program user is constructed internally by the program which selects these operations from a large set included into the simulation system. In this way computational cost is dynamically adapted to the complexity of problems to be solved. A user of the program designed in this way can simulate a practically unlimited number of kinetic models belonging to a pre-selected class, without the necessity of making any changes in the program. It is sufficient to change the text input data only. A certain disadvantage of the present design of the formula translator is a relatively slow execution of the intermediate code, although a significant improvement has already been achieved compared with the initial version of the translator.218 However, this disadvantage can surely be eliminated in the course of further research. For example, machine code could be generated in the way employed in the compilers of higher level programming languages. Similar design assumptions have already been accepted in various computer programs serving for scientific and technical computations (see, among others, the recent Ref. 219 for a chemical example). Refs. 216-218 have introduced them into electrochemical modeling.
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3. Comparative Studies of Simulation Algorithms for Kinetic Models in One-Dimensional Spatial Geometry
Experience gathered in the course of numerous simulation studies performed over the past decades1-12 shows that while solving the governing equations pertaining to transient methods and kinetic models in one-dimensional spatial geometry, characteristic difficulties are encountered, caused, among other factors, by: (a) Discontinuous boundary conditions associated with the modeling of transient methods such as the potential-step method (chronoamperometry) or the current-step method (chronopotentiometry); (b) Fast homogeneous reactions that are responsible for extremely short relaxation times and/or for the presence of extremely thin reaction layers (at the electrode surface), or thin moving reaction zones in the electrolyte (at finite distances from the electrode surface). While comparing various simulation methods, one has to take into account their ability to overcome the above difficulties, and also features such as accuracy and efficiency (that is the ability of a method to produce accurate results in the shortest possible computational time).6 Until recently it has also been regarded as important that simulation methods be simple (see, for example, Ref. 6, p. 135). At present, taking into account the fact that the method-oriented approach described in section IV.2 does not require the users of a simulation system to write their own programs, this argument loses its importance. On the contrary, a single effort associated with implementing a complicated method may give rewards, many times over, in solving various different problems, if a given simulation method (owing to its complexity) possesses advantageous numerical properties. Simulation methods used for solving one-dimensional PDEs of electrochemical kinetics have been compared a number of times in the electrochemical literature (see Ref. 6 and references cited therein, and Ref. 220 for a more recent example of this kind of electrochemical studies). The purpose of the comparisons to be described below was: (a) To complete the above results6, 220 with properties of the Saul’yev finite-difference method.221 The Saul’yev method has been suggested by several electrochemists222-225 as an advantageous alternative to simulation methods used earlier; (b) To explain inconsistent conclusions obtained by different electrochemists226, 227 with regard to the relative advantages and disadvantages of the finite-difference methods
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and the orthogonal collocation method, which is a variant of the socalled weighted-residual methods.5, 6 In Refs. 217 and 228 various finite-difference simulation methods (classic explicit,6, 230 Runge-Kutta second-order explicit method of lines,6, 231 Crank-Nicolson,6, 232 and Saul’yev221) and orthogonal collocation,5, 6 have been used to simulate potential-step chronoamperometric (under limiting-current conditions) and linear potential scan voltammetric transients2, 125 for the standard catalytic reaction mechanism (pertinent to homogeneous redox catalysis of electrochemical reactions233):
in which reaction (2) is a pseudo-first-order homogeneous regeneration reaction, and linear potential scan voltammetric transients for the Pedersen and Svensmark234 mechanism:
involving second-order homogeneous reactions (4) and (5). Planar electrodes and semi-infinite diffusion conditions have been assumed. The above examples represent typical reaction-diffusion models that allow the simulation methods to be tested under conditions of thin reaction layers at the electrodes, and/or of temporal discontinuities in boundary conditions at the electrodes. With respect to the Saul’yev method, it has been found217 that at small (less than unity) values of parameter (here and h are intervals of the uniform discrete grids along the axes of the dimensionless time and space coordinates, respectively) this method can be competitive from the point of view of accuracy and efficiency, with such finite-difference methods as the classic explicit method,6, 230 the Runge-Kutta second-order explicit method of lines,6, 231 and even the implicit Crank-Nicolson method.6, 232 However, when the parameter
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takes larger values, a significant error increase is observed, rendering the Saul’yev method less useful for simulating problems involving thin reaction layers. There are also limitations associated with the numerical stability of the Saul’yev method (see next section IV.4). In general, it has been found that, contrary to the opinions previously presented in the electrochemical literature, the Saul’yev method is not particularly reliable and its use should be limited to problems without fast homogeneous reactions and discontinuous boundary conditions. In the orthogonal collocation method, PDEs for the concentrations of bulk species are converted to a set of ODEs for the concentration values at selected collocation points in space.5 Comparisons of finitedifference (classic explicit,6, 230 Runge-Kutta second-order,6, 231 CrankNicolson6, 232) and orthogonal collocation simulation5, 6 methods, presented in Ref. 228, revealed that, depending on the choice of the integrator applied to solve these ODEs, and on the selection of the integrator parameters, the collocation method can be much more accurate and efficient than are the finite-difference methods. However, the simplest possible variant of the collocation method (which uses Legendre collocation polynomials over the entire spatial domain of interest) yields less reliable simulation results than finite-difference methods, in the case of problems with discontinuous boundary conditions and fast homogeneous reactions, in particular if these are not first-order reactions. In order to improve the reliability of orthogonal collocation, it is necessary to employ more sophisticated collocation variants, such as e.g.: the spline collocation, in which the spatial interval of interest is decomposed into sub-intervals, and the collocation conditions are formulated separately for every sub-interval, together with appropriate continuity conditions for the solution at the sub-interval boundaries; the collocation using the expanding simulation space, in which the locations of the collocation points vary in time according to a pre-defined function; the collocation using various transformations of the spatial coordinate; etc.229, 235 4. Formal Analysis of the Numerical Stability of the Simulation Algorithms
One of the elementary requirements imposed on any numerical algorithm is that discretization errors made at successive steps of the algorithm be not magnified or accumulated during the calculations,
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increasingly deteriorating the accuracy of the solution obtained. The errors should be damped or, in the worst case, they should remain at a constant, low level. This property, called numerical stability, is necessary while solving differential equations,236-239 hence it is important for electrochemical simulations. This fact has been pointed out in several simulation studies. However, statements of the stability of simulation methods used were not supported by formal proofs of stability, although it is known that the numerical stability of finite-difference methods depends not only on the type of a method but also on the problems solved, as well as on a variety of additional assumptions or approximations made in the calculations. For example, there exist various definitions of numerical stability. All this has to be taken into account while formulating statements about stability. Furthermore, the stability of numerical methods should not be mistaken for accuracy. Stability cannot be heuristically deduced from the observed good accuracy of the calculations. Although both properties are associated with numerical errors, they are not equivalent. It is possible to obtain imprecise results using stable methods and also, in certain cases, accurate results using unstable methods. There exists a relatively simple way of verifying the accuracy of the solutions (assuming that the stability has been ascertained earlier) that consists in gradually diminishing discrete temporal and spatial intervals, and in concluding an asymptotic convergence of the solutions in this limit. However, it is difficult to indicate similar ways for practical verification of stability. In view of the above, formal proofs of stability are necessary. Such proofs are not easy, which explains in part why they have largely been absent in the literature on electrochemical kinetic simulations. The formal analysis of stability becomes particularly important for the development of PSEs. Users of such programs should be supplied with adequate knowledge or criteria regarding numerical methods implemented, that would allow a rational evaluation of the reliability of their simulation results. Future PSEs should include automated stability analysis. This is a difficult, and so far unresolved problem.240,241 Therefore, one should remain cautious about statements expressed by the authors of some of the general simulation programs (see, for example, Refs. 164, 171, and 180), telling that their programs enable simulation of almost every kinetic problem, whereas the program user is not given much control over methods or algorithms that are running ”behind the scene.” The example of numerical stability shows that a
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specific minimum of expertise in numerical methods is still indispensable for making and understanding simulations, even if they are highly automated. The present author has initiated a discussion of the numerical stability of the finite-difference simulation methods used traditionally in electrochemical kinetics. In Ref. 242 the Von Neumann method of stability analysis243,244 has been used to investigate the effect of the homogeneous kinetic term in the one-dimensional electrochemical reaction-diffusion equation
on the so-called practical (or “stepwise”) stability (Ref. 236, p. 202; Ref. 245) of several finite-difference methods: classic explicit,6,230 sequential explicit (explicit with sequential treatment of diffusion and homogeneous kinetics),6, 136, 246 Runge-Kutta second-order explicit method of lines,6, 231 Du Fort-Frankel,247 fully implicit,6, 248 CrankNicolson,6, 232 and Saul’yev.221 Equation (6) is a simple example of reaction-diffusion equations that arise in electrochemical kinetics, and it directly describes the Reinert-Berg system249 under limiting-current conditions. Here c(x, t) denotes concentration of a bulk species, as dependent on spatial coordinate x and time t, D is a diffusion coefficient, and k is the rate constant of a first-order homogeneous reaction. It has been shown that, in contrast to previous thoughts, the practical stability depends on k. In particular, the classic explicit method is stable for (where is the dimensional time step), and not, as was previously believed (see, for example, Refs. 1 and 6, and references cited therein), for This implies that the simulation of problems with fast homogeneous reactions by means of this method is difficult, not only as a result of troubles with ensuring a proper resolution of the discrete spatial grid inside thin reaction layers or zones, but also as a result of the stability limitation. A similar result has been obtained for a coupled system of two reaction-diffusion equations:
and
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which corresponds to the catalytic reaction mechanism (1) and (2) in section IV.3. Equal diffusion coefficients of A and B are assumed in Eqs. (7) and (8), and and denote forward and backward rate constants of reaction (2), respectively. In this case the classic explicit method is stable for Similar stability criteria have been obtained for other simulation methods discussed. The formal predictions obtained in Ref. 242 were later confirmed in calculations by Britz and Østerby.250 Refs. 251-254 concentrated mostly on investigating the effect of the discretization of the mixed boundary conditions with timedependent coefficients on the practical stability of several finitedifference simulation methods. Mixed boundary conditions (i.e. conditions that depend both on the concentrations of bulk species and on their first spatial derivatives) at electrodes are typical for electrochemical kinetic problems. Simulation of cyclic voltammetry for an irreversible or quasi-reversible charge-transfer reaction has been discussed as a representative example of such boundary conditions. The so-called matrix method of stability analysis237-239 has been used, because this method takes into account boundary conditions, in contrast to the Von Neumann method applied in Ref. 242. The simplest, two-point forward-difference approximation (cf. Ref. 6, p. 45) to the concentration gradient at the electrode was assumed in Ref. 251. This approximation has been most frequently used by electrochemical simulationists. It has been shown, among other matters, that the common opinion about the unconditional stability of the Crank-Nicolson method is imprecise under the conditions discussed. In particular cases that depend on the discretization of the mixed boundary conditions, this method may become unstable. For example, the instability may occur during the reverse potential sweep in the simulation of cyclic voltammetry for an irreversible charge-transfer process, if it happens that where k(t) is the dimensionless rate constant of the electrochemical reaction, and h and are discrete, uniform grid steps along the dimensionless space and time coordinates. In Ref. 252 this discussion has been extended onto multipoint forward-difference
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multipoint forward-difference concentration gradient approximations.255,256 In Ref. 253 conclusions on conditional stability have been obtained for the Saul’yev method, also previously considered unconditionally stable,221-225 assuming the two-point forward-difference gradient approximation. The instability of Saul’yev arises for the left-right variant of the method221 (assuming that the mixed boundary conditions occur at the electrode that is located at the left boundary of the spatial domain), when parameter exceeds a certain limiting value, that depends in a complex way on the parameter and the grid spacings. Fortunately, it is quite difficult to enter these instability regions, as this requires a selection of grid spacings that is unreasonable from the point of view of accuracy requirements. Finally, Ref. 254 concentrated on the search for alternative discretizations of the boundary conditions, that would ensure unconditional stability of the Crank-Nicolson method. Two particular stable discretizations based on central-difference and backward-difference approximations to the concentration gradient at the electrode have been found (these approximations use a fictitious spatial grid point, and corresponding solution values ”inside” the electrode). However, the new stable discretizations render the Crank-Nicolson method more oscillatory than the one previously used (based on the forward-difference gradient approximation). The instability predicted in Refs. 251, 252, and 254 should not be confused with the well known problem of oscillations of the CrankNicolson method (see, for example, Ref. 257 for a discussion of this problem). In fact, almost no oscillations were observed in Refs. 251 and 254 under unstable regimes. If the oscillations occur under a stable regime, they are damped and do not normally cause instability. Only in the case of certain kinetic models (for example models involving second-order homogeneous reactions) stable oscillations are not harmless, as they may lead to the occurrence of negative concentrations, resulting in a non-physical evolution of the numerical solution (see, for example, Ref. 258). Another problem so far neglected in electrochemical simulations, of numerical stability, has been pointed out in Ref. 218. This problem refers to the discretization of the coupled systems of one-dimensional PDEs and ODEs, that arise in the modeling of electrochemical systems
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involving simultaneously bulk and interfacial species (see section IV. 1). Using the example of a reversible anodic dissolution
of a fractional monolayer of metal M by means of linear potential scan voltammetry,259 it has been shown that in the simulation of this problem by explicit finite-difference methods, the temporal discrete grid spacing must be selected in such a way that the discretizations of both the PDEs for bulk species and ODEs for interfacial species are stable. Here the discretization of the ODEs is more demanding, since it is possible to select a for which the discretization of PDEs is stable, but instability occurs for ODEs, yielding incorrect results for the coupled system. It appears that the results obtained in Refs. 242 and 251-254 are of interest not only for the purpose of electrochemical simulations, but also for the theory of numerical methods. However, in view of the limitation to simple examples of kinetic problems, and the simplifications accepted (e.g. the assumption of uniform spatial and temporal grids), these considerations should be regarded as an introduction to a desired, much wider, discussion of these questions. It is encouraging to note that other authors are continuing this important line of investigations.260-262 The problem with the Crank-Nicolson oscillations, mentioned above, has motivated Mocak and Feldberg,263 and Feldberg and Goldstein,264 to suggest that the fully implicit finite-difference algorithm with Richtmyer modification265 should be a method of choice for electrochemical kinetic simulations. This algorithm has also formed a basis for the DigiSim program.171-173 Indeed, this multilevel method, based on the temporal backward-differentiation formulae (BDF), does not produce oscillations nor negative concentrations, has an accuracy comparable to or better than Crank-Nicolson (starting from the threetime-level BDF approximation), and it has been shown to be unconditionally stable (in the Von Neumann sense) up to the seven-time-level BDF approximation.261-262 Unfortunately, problems occur with starting protocols.266 At present, the extrapolation method of Lawson and Morris,267 and Gourlay and Morris,268 suggested for electrochemical simulations by Strutwolf and Schoeller269 appears to avoid the latter diffi-
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culty, while retaining all advantages of the BDF over the CrankNicolson scheme. However, in the opinion of the present author, the search for methods most suitable for electrochemical kinetic simulations is not yet completed. An ideal method (if it can be devised at all), apart from being stable and accurate, should also possess a number of other features. In particular: possibilities for inexpensive and reliable temporal/spatial grid adaptation, solution of coupled PDE/AE, PDE/ODE and PDE/DAE systems, and efficient calculation of sensitivity information. Some of the latter issues are discussed in the next section. 5. New Simulation Methods and Modeling Strategies
The program of creating PSEs for electrochemical modeling, outlined in section III.3, point (c), imposes increased requirements on simulation methods, and modeling strategies, in general. Features such as high efficiency, reliability and maximal generality of the methods are desired. Currently used methods are not entirely satisfactory in this respect. They do not guarantee reaching the ideal such as could be a completely automatic simulation and data analysis, synchronously with the collection of experimental data. To contribute to progress in this area, two new simulation methods have been proposed by the present author: a fast solution method for integral equations occurring in the modeling of linear and cyclic voltammetry, and an advanced finite-difference method based on the idea of adaptive grid strategies. Also, an extension of the traditional finite-difference algorithms for electrochemical governing PDEs, has been suggested onto coupled systems of PDEs and AEs, ODEs or DAEs that also arise in electrochemical kinetics (see section IV. 1). Finally, a new modeling approach to the age-long problem of examining the mutual relationships between simulated transients and model parameters has been introduced, based on the concept of sensitivity analysis. These four proposals are summarized below in sections IV.5.i-IV.5.iv, respectively. (i) An Efficient Method of Solving Electrochemical Integral Equations
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Integral equations play an important role in the theory of linear potential scan and cyclic voltammetry. In particular, for a number of simple electrochemical reaction mechanisms, the dimensionless linear voltammetric current function obeys the Abel integral equation
where is a dimensionless time variable (often denoted by at or bt in the literature2,120,125), is a pre-defined function, and is a pre-defined weighting function, different from For example, according to Nicholson and Shain125, for a Nernstian charge-transfer reaction and where parameter determines the position of the starting potential with respect to the formal potential of the charge-transfer reaction, and is a sawtooth function: for and for with denoting a dimensionless time at which the potential is switched. A more complicated integral equation is also frequently encountered:
where
is a pre-defined, generally non-linear function of its vari-
ables,
and
dots express a possible occurrence of several (more than one) integrals that differ by functions Finally, coupled systems of two integral equations for two current functions and are also encountered:
where and are pre-defined functions of their variables, analogous to function in Eq. (11), and integrals associated with a second current function are indicated by a prime.
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Until recently, for solving integral Eqs. (10)-(13), algorithms based on the so-called product-integration method270 have been commonly used in electrochemistry.144 A weakness of these methods is their large computational cost: thus, the computational time increases with the square of the number, N, of integration steps along the interval of in which the solution is sought. This property results from the non-separable dependencies of the functions contained under integrals on the variables and z. The new method suggested in Ref. 271 consists in replacing expression by an approximation composed of a sum of expressions in which variables and z are separated. The particular formula
has been proposed, where and M are numerical constants selected to minimize approximation error in a given interval of This approach is analogous to the so-called degenerate kernel method that is widely used for solving Fredholm integral equations (see, for example Refs. 272 and 273). Owing to the separation of the variables one achieves a proportionality of the computational time to the number N of integration steps. This speeds up computations by a factor of one to three orders of magnitude compared with the traditional productintegration methods that use the same N. Ref. 271 has concentrated on the solution of Eq. (10) for reversible cyclic voltammograms. Further extensions of this method onto Eqs. (11)-(13) have been discussed in Ref. 274, assuming slowly varying functions which in practice corresponds to slow and medium speed homogeneous reactions. Analogous algorithms might well be used for efficient semi-integration of controlled-potential electrochemical transients, that is used to facilitate experimental data analysis (see, for example Refs. 131, 151, and 152). (ii) An Adaptive Grid Strategy for Solving Electrochemical Governing Equations
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A successful application of the traditional methods of the direct integration of the governing PDEs (such as the finite-difference methods discussed in sections IV.3 and IV.4), is still much more an art than a routine. One needs to have the (sometimes quite far reaching) ability of predicting the solutions of kinetic problems studied, and of selecting the method parameters in such a way that the solutions can be obtained with satisfactory accuracy over the entire relevant domain of independent variables. In particular, the occurrence of discontinuous boundary conditions, short relaxation times, and thin reaction layers or moving reaction zones (see section IV.3) requires such a selection of the time and space integration steps so that an increased number of grid points is present in spatial and temporal regions in which considerable variations of the concentrations are expected. In some cases, by anticipating an approximate form of the solution, appropriate transformations of the independent variables can be introduced, thereby facilitating and speeding up the solution process (see Refs. 275-279 for examples of such transformations, relevant to problems in onedimensional space geometry). The ability of noticing possibilities for simplifying governing equations (for example by using a quasi-steadystate approximation for certain intermediates) is also of significance, allowing solutions to be more easily obtained in some limiting cases. Taking into account the features expected from PSEs (see section II.5), this state of affairs is unsatisfactory. Although it is known that there are no numerical methods that can solve any problem, we would like to have available simulation algorithms that are automatic to a much higher degree, similar to many existing automatic ODE solvers (see, for example, Ref. 280). It would be desirable to eliminate, or at least to considerably reduce, arbitrary decisions associated with the predictions of the solutions and selections of the method parameters. Such decisions should be replaced by the decisions of the algorithm itself, based on rigorous numerical criteria. In recent years, some considerable progress has been achieved with respect to the development of such automatic solution methods for PDEs, although difficulties occurring here are incomparably greater than in the case of ODEs. These methods, known under the name of adaptive grid strategies,281, 282 have not been previously applied to electrochemical kinetic problems, although they can be regarded as a natural generalization of the previously mentioned algorithms that use transformations of the coordinates and that are frequently used in electrochemistry.6, 275-279
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The essence of the generalization is that the transformations are not arbitrarily chosen but they are dynamically and automatically adapted to the solutions obtained, at every point of space and time. In Refs. 283–286 one variant of adaptive grid strategies has been elaborated, and the possibility of its use for electrochemical simulations has been analyzed. The strategy selected utilized the idea of an adaptive moving spatial grid, according to which the positions of the spatial grid nodes are moved in the course of the simulation in such a way that they automatically concentrate in the regions requiring higher resolution. The particular moving grid technique chosen has been previously suggested in the numerical literature by Verwer, Blom, and Sanz-Serna.287, 288 The choice of this variant of grid adaptation was dictated mainly by the ease of implementation, because it was to a large extent possible to make use of existing procedures for traditional fixed grid calculations. However, extensive computations performed revealed the necessity of introducing many changes and improvements into the original strategy in order to maximally adjust it to the needs and peculiarities of electrochemical kinetics. Spatial grid adaptation to reaction layers at the electrodes has been tested using examples of the simulation of linear potential scan voltammetry for the catalytic reaction mechanism (1) and (2), and for the mechanism:
involving a homogeneous dimerization reaction (16). Spatial grid adaptation to moving reaction zones away from the electrodes has been tested using examples of the simulation of double potential step chronoamperometry for the simple mechanism of electrochemically generated chemiluminescence:
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and of the linear potential scan voltammetry for the reaction mechanism:
assuming a large difference between formal potentials of reactions (20) and (21). Temporal grid adaptation has been tested using examples of the simulation of the (cyclic) square-wave controlled-potential transient for a single charge-transfer reaction under pure diffusion conditions (which exhibits discontinuous boundary conditions), the potential-step transient for the catalytic mechanism (1) and (2) under limiting-current condition (which exhibits short relaxation time), and linear potential scan voltammetry for mechanism (15) and (16). It has been shown that the strategy examined is potentially very powerful in dealing with the computational difficulties associated with discontinuous boundary conditions, short relaxation times, and extremely thin reaction layers or moving reaction zones. In particular, the strategy allows one to obtain complete solutions of problems with moving reaction zones, i.e. solutions that are valid simultaneously in the large scale of the entire diffusion layer, and in the small scale of the reaction zone. This feature deserves special attention. It is impractical, or even impossible, to obtain such solutions by means of traditional finite-difference methods that use fixed grids, if the homogeneous reactions are fast, so that reaction zones are very thin. An opinion can be encountered (see, for example Ref. 171) that fixed spatial grids with non-uniform, exponentially expanding spacing are sufficient to provide solutions to problems characterized by moving reaction zones. However, such simulations may easily become incorrect, when it happens that the local grid spacing in the area of the reaction zone exceeds dimensions of the reaction zone, so that the structure of the zone cannot be resolved, and the resulting error must propagate further out on the concentrations, rendering the result unreliable. For example, figure 5d in Ref. 171 appears to reveal such a low resolution in the region of the maximum of the concentration of species B.
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Despite these achievements, further work on the adaptive grid strategy of Refs. 283-286 appears necessary before it can be recommended for common use and inclusion into PSEs. It would be desirable to improve the accuracy of the spatial discretization, to eliminate negative concentrations, sporadically occurring in the solution, and to elucidate the question of numerical stability. Alternative adaptive grid strategies should also be examined. (iii) Extension of the Traditional Finite-Difference Simulation Algorithms
A major part of the literature on the simulation algorithms used in electrochemical kinetics6 has been previously limited to problems in which only bulk species are involved. However, in many important kinetic problems, interfacial species also arise. For this reason it is necessary to extend the simulation algorithms onto sets of PDEs coupled with AEs, ODEs or DAEs which describe such problems (as was discussed in section IV.1). In Ref. 218 relevant extensions have been proposed, and elaborated for a number of traditional finite-difference algorithms (classic explicit,6, 230 sequential explicit,6, 136, 246 the RungeKutta second-order explicit method of lines,6, 231 Saul’yev,221 fully implicit,6, 248 and Crank-Nicolson6, 232), and for orthogonal collocation5, 6 (PDE/ODE systems only). The essence of this proposal is as follows. If the changes of concentrations of interfacial species are described by explicit ODEs, then for solving these equations a finite-difference method can be used that corresponds to the method used for solving PDEs (for example, the forward Euler method corresponds to the classic explicit method, and the trapezium integration rule corresponds to the Crank-Nicolson method). In other cases, it is convenient to consider the equations for interfacial species as an extension of the set of equations that describe boundary conditions at the electrode for bulk species. If the additional equations for interfacial species take the form of AEs, then the discretized extended set of boundary conditions forms a system of (often non-linear) AEs for the discrete concentrations at a new time level. For solving such AE systems one can use iterative methods such as the discrete Newton or secant methods.289, 290 If the additional equations for interfacial species take the form of DAEs, then the temporal derivatives of the concentrations of interfacial species in these equations
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can be replaced by difference quotients, leading again to non-linear AEs for the discrete concentrations at a new time level. The latter discretization amounts to using the backward Euler integration of the equations for interfacial species, independently of the method chosen for solving PDEs for bulk species. Such extended algorithms have been examined and found useful for practical calculations, although the selection of the grid spacings is not automatic in these algorithms. A satisfactory accuracy and efficiency of the algorithms has been obtained for a number of typical examples of the governing equations. Algorithms based on the implicit finite-difference methods prove to have particularly valuable properties, since they combine good accuracy and efficiency with numerical stability. (iv) Sensitivity Analysis of Electrochemical Transients
The task of elucidating the effect of model parameters on transient curves is central to stages 3 and 4 of the modeling process (see section III.3). Electrochemists usually use the simplest, ”brute force,” approach to perform this task: the parameters are varied over a domain of interest, model equations are solved, and simulated transients obtained for different combinations of the parameter values are compared. Such a practice is neither conceptually satisfactory nor efficient. In Refs. 291-293 an alternative, more mathematically rigorous, approach to this task has been proposed, based on the concept of the so-called sensitivity analysis, that has already proven useful in various branches of chemical kinetics,294-301 but it has not been previously applied to the modeling of electrochemical transients. The so-called ”local direct method of sensitivity analysis” 52, 124, 294-301 has been chosen for the study. In this method, by formally differentiating the governing equations with respect to various parameters, one derives a system of equations for the sensitivity coefficients of the concentrations of chemical species (the sensitivity coefficients are derivatives of the concentrations with respect to the parameters). The equations for the sensitivity coefficients are then solved numerically, simultaneously with the governing equations for the species concentrations. The solution of the (always linear) sensitivity equations is often more efficient than the solution of the governing equations for the concentrations (if they are non-linear) in the ”brute force” method. The efficiency further depends on the solution method used. Analysis of the sensitivity coeffi-
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cients offers an increased insight into the behavior of kinetic models.291 For example, one can establish a rank ordering of various parameters with respect to their effect on transient curves or their features. One can also study and quantify interdependencies between different parameters and/or dependent variables (concentrations, or observables such as electric current or electrode potential). These properties of the sensitivity analysis have been demonstrated using examples of cyclic voltammetry for a quasi-reversible charge-transfer reaction with a potential-dependent charge-transfer coefficient, and of chronopotentiometry for a reaction mechanism:
The sensitivity analysis is also useful for performing ”expansion” or ”reduction” of kinetic models292 within stage 4 of the modeling process (section III.3), owing to the identification of the relative importance/unimportance of various candidate or inherent model components. By confronting sensitivity coefficients with the deviations of the simulated transients from experimental curves, decisions can be taken whether a particular component (e.g. a reaction step in a reaction mechanism, or a transport mode) should be added to or removed from the model. Such a procedure may become a part of an automated computer-aided strategy of developing electrochemical kinetic models. The usefulness of sensitivity analysis for model expansion/reduction has been demonstrated by its application to selected cyclic voltammetric data for two example kinetic problems: anodic oxidation of in dichloromethane at a Pt stationary disk electrode (model expansion from to meaning an addition of a follow-up homogeneous reaction step to the reversible charge-transfer reaction step), and cathodic reduction of bis paracyclophane) Ru (II) bis (tetrafluoroborate) in propylene carbonate at a glassy carbon stationary disk electrode (model reduction from to meaning an elimination of the homogeneous disproportionation reaction step, DISP, from the reaction mechanism, resulting in a mechanism involving two reversible chargetransfer steps only).
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Further benefits resulting from the use of the sensitivity information are possible.291 In particular, calculation of sensitivity coefficients by the direct method permits one to easily perform a detailed analysis of the statistical error/uncertainty propagation in simulation, and in parameter estimation by non-linear least-squares fitting of the simulated transients to experimental curves. A relevant formalism has been elaborated in Ref. 293. Uncertainty analysis of the simulated transient curves has not been so far practiced in electrochemical kinetics, although the experience gathered in other branches of natural sciences shows that it is not sufficient to develop seemingly correct models of the physical systems. It is also important to know how reliable they are in view of the uncertainties in the available data and model parameters. Error analysis of the parameters estimated using direct search (e.g. simplex) or other types of least-squares minimization, pointwise analysis of the variances of estimated parameters, least-squares gradient-based minimization, or identification of deterministic correlations between parameters, can all be performed easily with the help of sensitivity information, and/or more efficiently or reliably compared to formerly used techniques. Some of these features have been demonstrated by making a detailed error/uncertainty analysis of synthetic linear potential scan and cyclic voltammetric transients for a quasireversible charge-transfer reaction subject to Butler-Volmer kinetics.293 6. Realization of a PSE for Electrochemical Kinetic Modeling
Parallel to the investigations described in previous sections, work has progressed on consecutive versions of the PSE named ELSIM. Until now, three versions of this program have been created and comprehensively described in Refs. 166–169. The current version is designed for IBM-compatible personal computers operating under the control of the MS DOS operating system. The program allows solution of the following types of problems:
1. Integral Eqs. (10)-(13) for linear potential scan and cyclic voltammetry. 2. Governing equations independent of spatial coordinates, describing arbitrary controlled-potential electrochemical transient methods, for electrochemical systems involving interfa-
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cial species that have variable concentrations, and possibly bulk species that have constant concentrations. 3. Governing equations analogous to those in point 2, but for controlled-current transients. 4. Governing equations in one-dimensional spatial geometry, describing arbitrary controlled-potential electrochemical transient methods, for electrochemical systems involving bulk species that have variable or constant concentrations, and possibly interfacial species that have variable or constant concentrations. 5. Governing equations analogous to those in point 4, but for controlled-current transients. Governing equations corresponding to categories 2 and 3 are allowed to take the form of the AEs, ODEs, or DAEs. Governing equations corresponding to categories 4 and 5 are allowed to take the form of reaction-convection-diffusion PDEs, optionally coupled with AEs, ODEs or DAEs. Boundary conditions at both ends of the spatial solution domain can be non-differential, differential or mixed, with generally non-linear dependence on concentrations or their gradients, and they can depend explicitly on time. The theory discussed in section IV.1 has been utilized to design a “reaction compiler” that reads user-written electrochemical reaction mechanisms together with other kinetic assumptions, and generates corresponding systems of governing equations, corresponding to categories 2-5 above, in textual form. The full text of the generated governing equations is made available to the user, and it can be modified, if desired, before using the equations for simulation. This program, together with the proposed principles of the language serving for formulating reaction mechanisms, has been described in Ref. 168, and it is now included in ELSIM.169 The formula translator and other solutions previously discussed in section IV.2 have also been utilized in the ELSIM program. Hence, kinetic problems can be entered to ELSIM either in terms of user-written reaction mechanisms, or in terms of user-written mathematical equations. None of the other currently available programs156-165, 170-181 seems to be equipped with a more comprehensive and flexible system of formulating kinetic problems and checking their correctness. In particular, CVSIM,163-165 DigiSim,171-173 the programs of He and Chen177 and of Villa and Chap-
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man,178 and ESP,180 allow kinetic models to be entered exclusively by writing electrochemical and chemical reactions (using various conventional notations), and by specifying values of the parameters such as rate and equilibrium constants and charge-transfer coefficients, diffusion coefficients, etc. The class of problems that can be dealt with by these programs is substantially smaller than the one that can be dealt with and solved by ELSIM. Programs such as the Condesim Simulation Software,159 program of Mager,158 EASIEST,160, 161 SIMULA,162 POLAR,175, 176 and the program of Biader Cerpidor et al.181 are equipped with library-type collections of pre-defined problems and only these can be solved. Finally, the programs of Klinger et al.,156 Cook and Landau,157 Penar et al.,170 Verbrugge and Gu,174 and Kolár,179 require that user-written program modules be written, compiled and linked with the programs, which is the least convenient solution. Relevant details regarding the ELEMD system26, 182 and the program of Alden and Compton183, 184 have not been published yet. Problems belonging to categories 1-5 can be solved in ELSIM by various numerical methods, including those previously discussed in sections IV.2–IV.5, with the exception of the adaptive grid technique, not yet regarded as sufficiently mature. The sensitivity analysis is also not yet included in ELSIM. Parameter estimation can be accomplished either by means of the working curve approach, or by least-square fitting of the simulated transients, using the simplex method. Further details regarding the program are presented in a comprehensive (250 pages) user manual.302 Although ELSM has not yet been transferred to the MS Windows platform, so that it suffers from limitations inherent in the DOS system, the program is one of the first ever written, the most comprehensive, and the most sophisticated among the currently available programs designed for the modeling of electrochemical transient methods and kinetic problems in no more than one spatial dimension. ELSIM has been tested using a specially selected collection of about 50 different examples taken from the literature, that included various categories of kinetic models and/or transient methods, and represented various difficulties for simulation (see Refs. 166-169 and 302 for lists of the examples). In the experience of the present author, the application of ELSIM to all these examples fully confirms the usefulness of the program. Traditional ways of solving these problems
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required months of work. Using ELSIM, such problems can be solved and comprehensively analyzed within hours or even minutes.
V. CONCLUSIONS Changes that occur in methods, objectives, and disciplinary structure of natural sciences as a result of the computer revolution, and some recent trends in computer applications in electrochemistry, suggest that a new discipline of computational electrochemistry can be formally established. While traditional disciplines of experimental electrochemistry and theoretical electrochemistry are concerned with physical experiments and mathematical modeling, respectively, computational electrochemistry would concentrate around all aspects of computer experiments and other computer-aided research activities in electrochemistry. In particular, it is postulated that computational electrochemistry can lay foundations for, and develop problem solving and discovery environments for electrochemical kinetic modeling. A number of electrochemists have already reported contributions consistent with such formulated research program. However, it appears that in order to make further advances possible, computational electrochemistry ought to be recognized as a legitimate research area within electrochemistry, with all consequences regarding scientific organizations, funding, career opportunities, etc. It is hoped that this review will help to initiate discussions and changes leading in this direction. Much can be gained from such an active participation of electrochemists in the computer revolution in science. Problem solving and discovery environments that encapsulate electrochemical knowledge can substantially facilitate and enhance many aspects of electrochemical research and education. Development and availability of such programs can eliminate a need for certain kinds of published information, thereby reducing effort currently required for literature searching and result publishing. Also, some kinds or phases of electrochemical research can be automated or replaced by less tedious or less expensive computer experiments. Easy access to, and widespread use of computer experimentation is likely to result in a better understanding of phenomena studied. Developing more powerful computational methods can enable more complex or otherwise difficult problems to be solved.
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New kinds of electrochemical investigations may become possible, that so far could not be done at all. This particular vision of computational electrochemistry has motivated the present author to develop various computational approaches to the modeling of transient methods, leading to the following results relevant for kinetic models limited to no more than one spatial dimension: (a) A theoretical discussion of the connections between electrochemical reaction mechanisms and corresponding sets of mathematical governing equations has been performed and utilized for designing a reaction compiler. It has been proven that the governing equations take the general form of the PDEs coupled with DAEs, which was hitherto not realized. (b) A new approach to the formulation of simulation algorithms has been elaborated, based on the idea of the formula translation and automatic generation of simulation procedures. (c) The usefulness of the Saul’yev method for electrochemical simulations has been investigated, and limitations of this method, so far unknown, have been revealed. (d) The relative advantages and disadvantages of the orthogonal collocation method, as compared to finite-difference methods, have been revealed. (e) The influence of the homogeneous reactions on the numerical stability of the traditional finite-difference simulation methods for electrochemical reaction-diffusion equations, so far unknown, has been demonstrated. (f) The limitations of the numerical stability of the CrankNicolson and Saul’yev methods, associated with the discretization of the mixed boundary conditions, and previously unknown, have been revealed. A new discretization of the boundary conditions, that does not suffer from this drawback, has been proposed for the Crank-Nicolson method. Opinions regarding the stability of some other traditionally used methods have been confirmed by means of a formal analysis. (g) A new, fast method of solving integral equations for linear and cyclic voltammetry has been elaborated, based on the idea of a degenerate kernel approximation.
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(h) A new, advanced simulation method has been elaborated, based on the idea of adaptive grid strategies. The method is particularly useful for solving kinetic problems that involve discontinuous boundary conditions and extremely fast homogeneous reactions, and it operates in a nearly automatic way. (i) An extension of several simulation algorithms used in electrochemical kinetics has been proposed, that permits solution of problems involving simultaneously bulk and interfacial species. (j) A new, mathematically defined approach to the problem of investigating the effect of model parameters on electrochemical transients has been proposed, based on the concept of a local direct method of sensitivity analysis. This approach offers a number of advantages over the traditionally used ”brute force” method that simply relies on examining differences between transients corresponding to various combinations of parameter values. (k) A problem solving environment ELSIM has been elaborated for a comfortable modeling of transient methods for userdefined electrochemical kinetic models in no more than one spatial dimension. This is one of the first, so far the most general and the most extended program of this type, currently available.
VI. ACKNOWLEDGEMENTS The author wishes to thank Dr. D. Britz, Dr. O. Østerby, and Prof. B. Speiser for many instances of valuable help and advice, and for fruitful cooperation. He also thanks Profs: A. M. Bond, D. Boyd, M. J. C. Crabbe, M. Frenklach, J. T. Hwang, R. H. Landau, R. Larter, K. B. Lipkowitz, J. R. Rice, and R. E. Valdés-Perez, and Drs. C. Bureau, H. de Jong, R. Kutner, R. D. McDowall, P. Mendes, M. Rudolph, and I. Ruisánchez, for providing him with reprints of their works, and/or for valuable correspondence and discussions. Useful discussions with the subscribers of the Computational Chemistry List, Chemical Information List and Electrochemical Discussion List (ELETQM) on the Internet, are gratefully acknowledged. The author is
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also grateful to Prof. J. Kitowski, for reading the manuscript and for helpful comments. The WWW sites cited here were operative prior to submitting this article. Readers interested in obtaining a copy of the ELSIM system are kindly requested to contact the author, preferably by e-mail (
[email protected]; http://www.cyf-kr.edu.p1/~nbbiema). The former distributor’s address, indicated in Refs. 166-169, is no longer valid.
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C. P. Andrieux, J. M. Dumas-Bouchiat, and J. M. Savéant, J. Electroanal. Chem. 87 (1978)39. 234 S. U. Pedersen and B. Svensmark, Acta Chem. Scand. A40 (1986) 607. 235 B. Speiser, Acta Chem. Scand. 47 (1993) 1238., and references cited therein. 236 F. B. Hildebrand, Finite-Difference Equations and Simulations, Prentice-Hall, Englewood Cliffs, N. J., 1968, p. 202. 237 A. R. Mitchell and D. F. Griffiths, The Finite Difference Method in Partial Differential Equations, Wiley, Chichester, 1985. 238 L. Lapidus and G. F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering, Wiley, New York, 1982. 239 G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods (3 rd edn.), Clarendon Press, Oxford, 1985. 240 M. Thuné in: PDE Software: Modules, Interfaces and Systems, B. Engquist and T. Smedsaas, eds., p. 89, Elsevier, Amsterdam, 1984. 241 E. V. Vorozhtsov and S. I. Mazurik, Sov. Math. Dokl. 39 (1989) 564. 242 L. K Bieniasz, J. Electroanal. Chem. 345 (1993) 13. 243 J. Von Neumann and R. D. Richtmyer, J. Appl. Phys. 21 (1950) 232. 244 G. G. O’Brien, M. A. Hyman, and S. Kaplan, J. Math, and Phys. 29 (1950) 223. 243 O. Østerby, in: Proceedings of the 2nd International Colloquium on Numerical Analysis, D. Bainov and V. Covachev, eds., p. 165, VSP, Plovdiv, Bulgaria, 1994. 246 and D. Britz, Acta Chem. Scand. 45 (1991) 1087. 247 E. C. Du Fort and S. P. Frankel, Math. Tables Other Aids Comput. 7 (1953) 135. 248 P. Laasonen, ActaMath. 81 (1949) 309. 249 K. E. Reinert and H. Berg, Monatber. Deutsch. Akad. Wiss. Berlin 4 (1962) 26. 250 D. Britz and O. Østerby, J. Electroanal Chem. 368 (1994) 143. 251 L. K. Bieniasz, O. Østerby, and D. Britz, Comput. Chem. 19 (1995) 121. 252 L. K. Bieniasz, O. Østerby, and D. Brilz, Comput. Chem. 19 (1995) 351. 253 L. K. Bieniasz, O. Østerby, and D. Britz, Comput. Chem. 19 (1995) 357. 254 L. K. Bieniasz, O. Østerby, and D. Britz, Comput. Chem. 21 (1997) 391. 253 G. W. Batten, Jr.,Math. Comput. 17 (1963) 405. 256 D. Britz, Anal. Chim. Acta 193 (1987) 277. 257 W. L. Wood and R. W. Lewis, Int. J. Num. Meth. Eng. 9 (1975) 679. 258 M. Störzbach and J. Heinze, J. Electroanal. Chem. 346 (1993) 1. 259 M. M. Nicholson, J. Am. Chem. Soc. 79 (1957) 7. 260 S. W. Feldberg, C. I. Goldstein, and M. Rudolph, J. Electroanal. Chem. 413 (1996) 25. 261 D. Britz, Comput. Chem. 21 (1997) 97. 262 K. Johannsen and D. Britz, Comput. Chem. 23 (1999) 33. 263 J. Mocak and S. W. Feldberg, J. Electroanal. Chem. 378 (1994) 31. 264 S. W. Feldberg and C. I. Goldstein, J. Electroanal. Chem. 397 (1995) 1. 265 R. D. Richtmyer and K. W. Morton, Difference Methods for Initial Value Problems (second edn.), Interscience, New York, 1967. 266 D. Britz, Comput. Chem. 22 (1998) 237. 267 J. D. Lawson and J. Ll. Morris, SIAM J. Numer. Anal. 15 (1978) 1212. 268 A. R. Gourlay and J. Ll. Morris, SIAM J. Numer. Anal. 17 (1980) 641. 269 J. Strutwolf and W. W. Schoeller, Electroanalysis 9 (1997) 1403. 270 C. T. M. Baker, The Numerical Treatment of Integral Equations, Clarendon Press, Oxford, 1978. 271 L. K. Bieniasz, Comput. Chem. 16 (1992) 311. 272 M. Krasnov, A. Kiselev, and G. Makarenko, Problems and Exercises in Integral Equations, Mir Publishers, Moscow, 1971, p. 166.
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4
Thermodynamic and Transport Properties of Bridging Electrolyte-Water Systems Maurice Abraham and Marie-Christine Abraham Département de Chimie, Université de Montréal, Montréal, Canada
I. INTRODUCTION For a long time, most efforts in the physical chemistry of solutions were made to obtain information on the structure of aqueous electrolytic solutions. But, at the same time as our knowledge of these solutions was expanding, especially regarding dilute solutions, another field of research attracted attention, the physical chemistry of molten salts, with increasing interest in new technologies. Due to various reasons of scientific, technological and even historical nature, most investigations in these two fields of research were made almost independently. According to Braunstein classification, 1,2 between dilute aqueous solutions and molten salts lie the hydrates, with complete or incomplete water shells around the ions, covering a very large water mole fraction range, from about 0 to 0.9. Obviously, investigations on hydrates are very important for the knowledge of transition properties between dilute aqueous solutions and molten electrolytes. As a matter of fact, these electrolyte-water systems became gradually the object of intense attention and their study is now seriously expanding as much for technical applications as for theoretical reasons. Hydrates are used, or planned to be, in tech-
Modern Aspects of Electrochemistry, Number 35, Edited by B.E. Conway and Ralph E. White, Kluwer Academic/Plenum Publishers, 2002 197
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Bridging Electrolyte-Water Systems
nologies such as ore leaching and extraction processes, waste-water treatment, chemical and electrochemical manufacturing, absorption-type refrigeration machines. Due to concern about pollution and security problems, areas of technological interest include energy storage and generation. For example, solar energy storage, exploitation of geothermal energy sources, molten salts based fluids in nuclear reactors which could contain more or less water, deliberately introduced or not. The knowledge of thermodynamic and transport properties, for example water vapor pressure, heat of vaporization, viscosity, electrical conductance..., as well as the influence of water concentration, temperature... on these properties, over the ranges of anticipated operating conditions, are essential in the design and operation of technical systems in which they are utilized. With regard to scientific interest, the suggestion was made, now and 1-8 then, that more progress in the understanding of very concentrated aqueous solutions could come from the consideration of solutions obtained by adding water to fused electrolytes rather than concentrating dilute aqueous solutions. Reciprocally, one would expect the structure of solutions where water plays the role of the solute, its mole fraction being less than about 0.5, be akin to that of the anhydrous molten electrolytes so that in the limit of vanishing water mole fraction the properties of the solutions would tend to those of the anhydrous electrolytes. From this point of view, any theoretical bringing-in regarding those solutions could contribute to more progress concerning the anhydrous electrolytes themselves. Recently, electrolyte-water systems were investigated in the liquid phase over practically the whole water concentration range. Since these systems bridge the gap between anhydrous electrolytes and dilute aqueous solutions, they are designated by the expression “bridging electrolytewater systems”, the electrolyte being a single one or a mixture of several components. Bridging systems lie at the heart of the present chapter. Electrolyte-water systems which do not cover the entire concentration range will also be examined or taken into account inasmuch as they cover a sufficiently large water mole fraction region, starting especially near the anhydrous electrolyte, so that they can give information on the
M. Abraham and M.-C. Abraham
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transition of properties between those at the two extremes of the concentration scale. It now appears that experimental and theoretical investigations on bridging electrolyte-water systems have produced results with regard to thermodynamic and transport properties significant enough to justify upto-date review and appraisal.
II. THERMODYNAMIC PROPERTIES One important aim of the thermodynamics of electrolyte-water systems is to obtain information on the water activity The values of in the liquid system are generally obtained from water vapor pressure measurements by the equation
where is the fugacity of pure water and the fugacity of the water vapor in equilibrium with the electrolyte-water system. The water activity is related to the water mole fraction on an unionized basis by
where is the water activity coefficient. In the case of a mixed electrolyte, is defined by
with the number of moles of water and the number of moles of the electrolyte i. The corresponding mole fraction of the electrolyte is given by
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Bridging Electrolyte-Water Systems
The composition of an anhydrous mixed electrolyte is expressed by
where is the mole fraction of the electrolyte i in the anhydrous melt. The first ever determinations of values for a salt-water system over the whole water mole fraction range were made on the system at 372 K, by Trudelle, Abraham and Sangster.7 In this abbreviated notation, the number before a chemical formula represents Figure 1 gives the Raoult’s law plot of the data; the system shows positive deviations over almost all water concentration range. This was unusual behavior since the vast majority of electrolytewater systems previously studied, though over limited water concentration ranges, showed strong negative deviations,2, 9 except for such salts as and showing slight positive deviations. From further investigations on the bridging systems 12, 13 where is fixed at 1.06, varying from 0 to 0.125, with M= Cd and Ca, at 372 K, it has been established that three cases of deviations from Raoult’s law may appear, illustrated in Fig. 2:
M. Abraham and M.-C. Abraham
201
202
Bridging Electrolyte-Water Systems
positive deviation from salt to dilute aqueous solutions, negative deviation from salt to dilute aqueous solutions and S-shaped curve with a cross-over point for which and where the deviation changes sign. Positive deviations from Raoult’s law have also been observed with 14-16 solutions of water in the systems where the ratio is fixed at 1.06, varying from 0.025 to 0.100, with M = Na, K, and Cs, at 372 K. Negative deviations from Raoult’ s law have been illustrated with the 17 bridging systems at 18 366 K, at 298 K, and 19 between 360 and 3 90 K, and with sever20 al solutions of water in mixtures of nitrates and/or nitrites, between 383 and 403 K: where varies from
1.00 to 2.33, with M=Na and Cs, and with M = Ca and Mg. 20-31 Some authors have chosen as composition variable the water mole fraction on an ionized basis, that is
in which is the number of ions produced by the total dissociation of the electrolyte i. With this choice of concentration, all kinds of deviations with respect to Raoult’s law have also been observed with bridging systems or solu21 tions of water in electrolytes. Positive deviations have been illustrated , 7 at 372 22, 26, 27, 29, 30 with the systems; 26
K,
at 590 K,
27
between 425 and 492 K, 29 at 492 K, 29 at 492 K,
30
with
26
at 590 K,
29
at 492 K,
M. Abraham and M.-C. Abraham
203 21-23, 25, 26
0.400, 0.600 and 1, at 492 K; negative deviations with the sys9, 23, 32 , between 373 and 436 K, 23 , at 373 K, ,26 at 590 26 K, , at 590 K; alternance of positive and 26 negative deviations with the systems: , 26at 590 K, , 26 at 590 K.
tems:
1. Adsorption Theory of Electrolytes (i) Component Activities
The adsorption theory of electrolytes was initiated, in 1948, by 33 Stokes and Robinson as a conceptual adaptation to electrolyte solutions of the theory of gas adsorption on solid surfaces developed ten years 34 earlier by Brunauer, Emmet, and Teller (BET model). This approach is related to the concept of liquid quasi-lattice. From this point of view, at high electrolyte concentrations, a solution may be pictured as containing ions in various stages of hydration. Some ions have a complete hydration shell forming a monolayer around them, other ions have incomplete shells, and others have several layers, with the second and higher layers much less strongly bound than the first. All the layers are in equilibrium, the relative amounts of each varying with concentration. Since this model bears strong resemblance to that from which the gas adsorption isotherm is derived, Stokes and Robinson33 obtained the following equation giving the water activity as function of the water mole fraction by modification of the notation of the BET isotherm:
where c is an energetic parameter and r, a structural parameter, both independent of The parameter c is given by
204
Bridging Electrolyte-Water Systems
in which
where E is the molar binding energy of water on sites close to the ions, the molar binding energy of water in pure water, R, the gas constant, and T, the Kelvin temperature. The parameter r is expressed by
where is the number of available sites with the molar binding energy E for the water molecules, per mole of electrolyte, and the Avogadro constant. 33 Stokes and Robinson have tested Eq. (7) with 1:1 and 2:1 electrolytes: LiCl, LiBr, HC1, NaOH, and at 298 K, over concentration ranges between 6 and 30 M, by plotting its left-hand side against and obtained good straight lines. For all these electrolyte-water systems, values of lie between 4 and 33 which are reasonable values of energy of adsorption. Values of r for most of these systems lie between 3 and 4, but for and they are 6.7 and 7.1 respectively. The authors were not satisfied by the non-integral values of r which could not, from their point of view, correspond to any physical reality and had arisen as a result of approximations in the BET theory and its application to electrolyte solutions, the most drastic approximation of the theory being that of treating all water molecules beyond the first layer around the ions like pure water. Therefore, taking into account the modification introduced into the BET model by 35 Anderson, they transformed Eq. (7) into
M. Abraham and M.-C. Abraham
205
in which
and
where is the molar binding energy of water in the multilayer hydration sphere of the ions where the nearest neighbors are only water molecules. Fixing r at integral values (4 or 8), Stokes and Robinson33 have tested Eq. (11) and obtained good straight lines as with Eq. (7), values of lying between 0 and – 0.6 and between 5 and Recently, Abraham et al.,19 making a non-linear least-squares fitting of Eq. (11) to the data of the system over the large water mole fraction range at 365, 372, 380, and 388 K, found non-integral values of and observed that is very close to compared to confirming that the parameters c and r of Eq. (7) are sufficient to compute the water activity, as is usually done with other hydrate melts. It must be pointed out that the concept of liquid quasi-lattice would rather be in favor of non-integral values of r. Indeed, Abraham and Abraham,36,37 in discussions of electrolyte-water systems properties, have considered a simplified picture in which quasi-lattices consist of submicroscopic parts representing, in a statistical sense, the most probable groupings of all entities involved: ions, hydrated ions, ion pairs, complex ions, water molecules, structural defects..., by analogy with the cybotactic theory.38 In some submicroscopic parts, salt-like parts, containing ions
206
Bridging Electrolyte-Water Systems
and adsorbed water molecules, these water molecules would be at the average energy level E. Other submicroscopic parts, water-like parts, would have water molecules at the average energy level (or Structure and size of the salt-like parts vary with time while the boundaries between the salt-like and water-like parts of the liquid fluctuate. Even if the picture is simplified so that ions and water molecules are assumed to be the only existing entities in the salt-like parts, the maximum number of available sites close to an ion will vary with time and location so that it is the aver-age of integral values which is measured. Since it is an average value, there is no reason for r to be necessarily an integer. Application of Eq. (7) to various bridging electrolyte-water systems has shown that its validity range extends from anhydrous electrolyte to an upper limit situated between about 0.7 and 0.9, depending upon the
nature of the electrolyte,7, 12, 13, 18, 19, 39, 40 as illustrated in Figs. 2 and 3.
Equation (7) has also been successfully tested with solutions of water in
fused electrolytes.2, 9, 14-16, 41, 42 It is remarkable that whatever the features exhibited by the curves vs (positive or negative deviations with respect to Raoult’s law, cross-over point) satisfactory results are obtained over a large water concentration range with only two parameters r and c.
Abraham43 has derived the equation for the electrolyte activity ae corresponding to Eq. (7) by a treatment based on statistical
thermodynamics44
M. Abraham and M.-C. Abraham
207
with
The parameters r and c are the same as in Eq. (7). In the case of an electrolyte-water system in which the electrolyte is a mixture of several components, represents a sort of average value for the mixed electro-
lyte. Examples of curves vs are given in Fig. 4. Abraham’s treatment attracted criticisms from Voigt45 who proposed an equation apparently different from Eq. (14). These criticisms were refuted by
Braunstein and Ally46 who proved that the Voigt’s equation, considerably more
208
Bridging Electrolyte-Water Systems
cumbersome to apply, is in fact equivalent to the Abraham’s equation and contains no new information. Usefulness of Eqs. (7) and (14) for the prediction of thermodynamic properties of mixtures of water with single or mixed electrolytes is en-
hanced by the existence of additivity rules regarding r and These 12 rules were first proposed by Sangster, Abraham and Abraham in a study systems, where on is fixed at 1.06, varying from 0 to 0.125, at 372 K, and subsequently confirmed with substitution of the salts and to at the same temperature.13–16 In general form, these rules, also called mixing rules40 may be expressed by
M. Abraham and M.-C. Abraham
where r and
209
are the parameters for a mixture of electrolytes, and
the parameters for the electrolyte i.
In Figs. 5 and 6, r and the product r
are plotted against
for all the above-mentioned systems. On the whole, the relative positions of the straight lines are in good correlation with the hydrating power of the cations. At a given mole fraction of the added cations, the stronger the tendency of an added cation to hydrate, the
higher r and r values. The latter property has suggested to express
210
Bridging Electrolyte-Water Systems
hydrating powers by means of the product r .47 According to Abraham,36, 47 the additivity properties of r and r
are
compatible with, though perhaps not uniquely required by, molten
electrolyte characteristics
2, 48–55
and likely to appear when there is no
chemical reaction, complex formation, hydrolysis on mixing the components. They may be the consequences of the preservation of the species, ionic or not, when the electrolytes are mixed, the lack of long range order in conjunction with the presence of structural defects or holes, and the predominance of short range interactions in the phenomenon of water adsorption.
r
Ally and Braunstein40 have compared experimental values of r and with those calculated from the additivity rules, Eqs. (16) and (17),
for solutions containing binary and ternary mixed electrolytes based on
the salts LiBr,
and
at 373 and 393 K.
They concluded that the additivity rules are valid for both, r and r , and apply to common anion mixtures as well as common cation mixtures. It
is interesting to notice that in some instances56–58 the additivity rules have been observed even in solutions studied over limited concentration regions rich in water.
In the framework of statistical mechanics, Abraham43, 47 has shown that in an electrolyte-water system where the electrolyte is a single one or a mixture of electrolytes whose composition is kept constant so that the system is treated as a binary system, the activities of water and electrolyte can also be expressed by
M. Abraham and M.-C. Abraham
211
with the equation
where
is the number of moles of electrolyte,
the total number of
212
Bridging Electrolyte-Water Systems
moles of water, and the number of moles of water with the molar binding energy E in Eq. (9). The derivation of Eq. (18) provides a statistical mechanics support to what is implied in Stokes and Robinson equation, i.e. the water activity 2, 33 represents the fraction of water unbound to electrolyte. In order to express the individual electrolyte activities in electrolytewater systems containing N electrolytes, Ally and Braunstein59 have proposed the following generalization of Eqs. (18)–(20):
with N equations of the form:
where is the number of moles ofwater linked to the electrolyte i with the molar energy In the derivation of Eqs. (21)–(23), two basic assumptions are made. Firstly, that, in a multicomponent system, the parameters and are constants characteristic of an electrolyte i, independent of the other electrolytes and secondly, that, on mixing water and electrolytes, it is sufficient to consider the change in the internal energy due to water adsorption. This latter assumption is a weakness in the theory, since,
M. Abraham and M.-C. Abraham
213
generally there is no ideal mixing of anhydrous electrolytes and an 50, 51 amount of enthalpy is involved. 60 A more rigorous approach proposed by Abraham and Abraham would start with expressing the free energy of mixing all components as:
where is the ideal free energy of mixing the anhydrous electrolytes, the excess (non ideal) free energy of mixing the anhydrous electrolytes, and the free energy of mixing water with the anhydrous electrolytes. Partial differentiation of with respect to gives the activity of the electrolyte i. Calculation of may be carried out with Eqs. (16)–(20). The additivity rules, expressed by Eqs. (16) and (17), which are not used in Ally and Braunstein approach, simplify the calculation procedure and are compatible with a possible slight influence of the electrolyte hydrating powers on one another. The result is:
with
obtained from Eq. (20) written in the form
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Bridging Electrolyte-Water Systems
and where is the excess chemical potential of the electrolyte i in the anhydrous mixture of electrolytes. The excess chemical potential does not obey a simple general equation. Convenient equations have been applied only to a number of anhydrous mixtures of electrolytes, over limited composition ranges, for example, some binary mixtures of molten salts found to conform to equations of the type
where A is a constant introduced by regular solution theories 50, 51, 61 Application of general Margules expansions24, 61 of would perhaps be a line of research worth exploring. With regard to the water activity its calculation is performed by means of Eqs. (7), (8),(16) and (17) since the partial differentiation of with respect to reduces to that of
M. Abraham and M.-C. Abraham
215
(ii) Characteristic Features of Water Activity Coefficient Curves
Figures 7 and 8 illustrate the curves where
at 1.06, with
vs
of the systems is fixed
between 0 and 0.125, M = Na, K, Cs, Cd, and
Ca, at 372 K. Characteristic features of these curves have been dis13, 14, 41, 62 cussed in terms of the structural parameter r and energetic parameter It is seen in Fig. 7 that when the strongly hydrated cations and are added to the relatively weakly hydrated cations and the curves vs are pushed downward in a regular manner and may present a cross-over point where The similarity between the effects of and on the shapes and the situations of these curves is consistent with the similarity of their ionic radii 63 and their Stokes radii in dilute aqueous solution 63
216
Bridging Electrolyte-Water Systems
The experimental observation that there is only one cross-over point on a curve meets a requirement of the form of Eq. (7) which allows only one possible value such that The water mole fraction at this cross-over point is given by12
Equation (29) implies that a necessary condition for the appearance of a
M. Abraham and M.-C. Abraham
217
cross-over point is
In Fig. 8, the curves are slightly raised by addition of the cation to and in contrast to the effect of and which lower the curves, like and but much less markedly. This behavior of is consistent with its weakly hydrating power reflected by the fact that its Stokes radius, 1.2 is smaller than its ionic radius, 1.7 as usually observed with monovalent anions whose ionic radii are larger than 1.33 On the whole, the relative positions of all these curves are in agreement with Eq. (18) from which one expects increase of with decreasing ionic hydrating power, since in this equation re-presents the fraction of unbound water. An equation predicting the existence of an extremum value of has also been proposed.62 The water mole fraction at this extremum is given by
with
Equation (31) implies that a necessary condition for the appearance of an extremum is
The extremum is a maximum if41
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Bridging Electrolyte-Water Systems
which means that the adsorption of water on fused electrolytes releases more energy than does water when it condenses in bulk. This is the case with most of the and based systems in Figs. 7 and 8. None of these systems exhibit a minimum value of Yet, it is noticed that the maximum is removed by the presence of at sufficiently high concentration, for which, from the additivity rules, the sign of the extrapolated value of is negative, meaning that the adsorption of water on releases less energy than does water when it 14 condenses in bulk in contrast with the other electrolytes. 9 Braunstein and Braunstein have suggested application of Eq. (7) to determine the water activity coefficient at infinite dilution in electrolyte In the limit as d 0, Eq. (7) reduces to
Equation (35) leads to Henry’s law constant for dissolution of water in 9 molten electrolyte
Henry’s law is strictly valid only at infinite dilution. There are cases, however, where the ratio varies little over a finite concentration range. Depending on the error which can be tolerated and the concentra62 tion range of interest, a practical Henry’s law constant may be used
which must be distinguished from Since in the region of an extremum a function changes little, a practical Henry’s law constant for water could appear if the extremum
M. Abraham and M.-C. Abraham
219
value of is present in the region of dilute solutions of water in molten electrolyte, which is the case of the systems containing the relatively weakly hydrated cations and as seen in Fig. 8.
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Bridging Electrolyte-Water Systems
In Table 1, values of and calculated by means of Eqs. (35) and (36), are listed for the systems shown in Figs. 2 and 3 and for other 13–15, 19, 41, 42, 62 electrolyte-water systems. Besides, in some instances, values
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221
of these constants have been proposed for dissolution of water in fused 56–58, 66, 67 electrolytes assumed to be supercooled. (iii) Excess Properties
(a) Partial molar excess volumes
From the equation
where the subscript i refers to water or electrolyte and P is the pressure, 40, 68 and with Eqs. (7) and (14), Ally and Braunstein have derived equations expressing the partial molar excess volumes of water and electrolyte, and respectively, assuming arbitrarily that the pressure dependence is entirely in the energy parameter
where
222
Bridging Electrolyte-Water Systems
39
Abraham and Abraham assuming that the pressure dependence is not only in the energy parameter but also in the structural parameter r, have obtained:
where
The functions and are the same as in Eqs. (39) and (40). The partial molar excess volume of water at infinite dilution in molten electrolyte is given by the following simple limiting form of Eq. (45):
The partial molar excess volumes are related to the molar excess
M. Abraham and M.-C. Abraham
volume of the solution
223
by
being defined by
in which
the ideal molar volume, is
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Bridging Electrolyte-Water Systems
and V, the molar volume of the solution, is given by
with the molar volume of the anhydrous electrolyte, the molar volume of pure water, the molar mass of the electrolyte i, the water molar mass, and the density of the solution. The parameters r and c being known, and are determined by fitting Eq. (51) in which and are expressed by Eqs. (45) and (46) to the experimental curve Applying this procedure to bridging electrolyte-water systems, Abraham and Abraham391 have obtained a very good agreement between the experimental data and those predicted from Eq. (51), as shown with three examples in Fig. 9. The same figure shows that dispensing with i.e. when and are expressed by Eqs. (39) and (40), could bring about more or less pronounced deviations of the calculated curves from the experimental data. Utilization of Eqs. (45) and (46) with and values obtained by the preceding procedure is more convenient to evaluate the partial molar excess volumes of electrolyte and water than performing differentiation of the function which could involve non negligible uncer39 tainty. Typical curves of and as functions of are shown in Figs. 10 and 11. Contrary to curves, those of have a common shape, the values of remaining small and quasi-constant over the
*The chemical formula of EAN in Ref. 39 is erroneous. It should be read as and not
M. Abraham and M.-C. Abraham
225
relatively large water concentration range presumably due to the fact that the structure of dilute solutions of water in molten electrolytes is similar to that of the anhydrous electrolytes themselves. The relative positions of the curves are in conformity with the hydrating power of the cations. From the additivity rules, equations involving the parameters and have been proposed39
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Bridging Electrolyte-Water Systems
M. Abraham and M.-C. Abraham
227
Equations (55) and (56) have been tested with the same and based systems12–16 whose study led to the observation of the additivity rules on the parameters r and It is seen in Fig. 12 that obeys a linear relationship and reflects, to a certain extent, the relative cation hy-drating power as does the product r illustrated in Fig 6. Contrary to r, remains quasi-constant at for these systems.
228
Bridging Electrolyte-Water Systems
(b) Partial molar excess enthalpies From the equation
and with Eqs. (7) and (14), Ally and Braunstein40,68 have derived equations expressing the partial molar excess enthalpies of water and electrolyte, and respectively, assuming that and r are temperature independent:
in which the functions and are the same2 as in Eqs. (39) and (40). These authors have also proposed the following procedure to provide 40,68 crystallization data on hydrates At any point of the liquidus curve where the solid hydrate is in equilibrium with its saturated solution, the temperature is T and the mole fraction of the electrolyte in the liquidus phase is At the melting point of the hydrate, the solid hydrate is in equilibrium with the liquid phase of the same composition in which the mole fraction of the electrolyte is With the approximations that the latent heat of fusion L of the hydrate and the partial molar excess enthalpies and in the hydrate melt are temperature independent, general thermodynamic development leads to the equation
*There is an error in Ref. 40, Eq. (21), p. 228 and in Ref. 68, Eq. (7), p. 6. The parameter in the denominator is missing.
M. Abraham and M.-C. Abraham
229
where and are the activities of electrolyte and water at any point T of the liquidus curve, and at the melting point of the hydrate. The right-hand side and the second term of the left-hand side of Eq. (60) are evaluated knowing L, the parameters r and of the solution derived from vapor pressure data, and using Eqs. (7), (14), (58), and (59). The two values of satisfying Eq. (60) and corresponding to the two branches of the liquidus are found by trial and error using Eqs. (7) and (14), Applying this procedure, the authors have successfully predicted stable and metastable phases of the bridging system Liquidus curves of and systems have also been calculated by Voigt,45 for 0.6 – 0.7, on the basis of the solubility product, the linear relationship between the solubility constant and the reciprocal of the temperature, and the parameters r and obtained from vapor pressure data, taking into account, when available, the influence of temperature on r and (iv) Temperature Dependence of the Adsorption Parameters Some authors19,42,45 have observed linear dependences of r and or r on T. Although reasonable values of a number of thermodynamic parameters have been predicted without taking into account the influence of temperature on r and 40, 68 such simplifications are not always
* In Ref. 40, Ally and Braunstein have noticed that if the hydrate melting points and the heats of fusion are not known, or if the hydrates melt incongruently, the liquidus curves can be calculated if some phase data are available.
230
Bridging Electrolyte-Water Systems
appropriate.42 For example, the partial molar excess entropy and the partial molar excess enthalpy of water at infinite dilution in molten electrolyte may be calculated by the following general equations
If is expressed as a function of the parameters r and (61) and (62) give
Assuming that the parameters r and Eqs. (63) and (64) reduce to
Eqs.
are temperature independent,
For water dissolved in the molten salt mixture 42 data on the influence of temperature on r and and application of Eqs. (63) and (64) give for the values of and and respectively, compared to
M. Abraham and M.-C. Abraham
and
231
with Eqs. (65) and (66). 2. Approaches Related to Regular Solution Theories
(i) Approach with Mole Fractions on an Ionized Basis Pitzer21,22 has proposed the use of equations applicable to regular solutions.61 This author writes for the activities of water and electrolyte the classical equations of regular solution theories in which the ordinary mole fraction x is replaced by the mole fraction on an ionized basis y:
with the parameters
and
expressed by
where v is the number of ions produced by the total dissociation of the electrolyte. The empirical parameters and are non-ideality parameters arising from the difference between the intermolecular attraction of unlike species as compared to the mean of the intermolecular
232
Bridging Electrolyte-Water Systems
attraction of like species, and the ratio is tentatively ascribed to the ratio of the volumes of the molecules or to the ratio of the molar volumes in the liquid. For fused salt-water systems, Pitzer pointed out that it seems best to regard as a freely adjustable parameter to be compared to the ratio of the volumes of species. Equation (67) has been fitted21 to the following bridging systems: 7 at 372 K, with the values 9,32 and and at 373 and 392 K, with the values and The agreement between the experimental data and calculated curves has been found excellent over practically the whole water concentration range. Yet, compar-ison between the ratio and the volume ratio (which is the ratio of molar volume of water to the average volume per ion) of the two systems, 0.82 and 0.87 respectively, is not satisfactory.21, 22 Equation (67) has also been successfully fitted by Tripp20 to water activities of seventeen solutions of water in mixtures of nitrates and/or nitrites containing essentially monovalent cations, and with two of them containing the divalent cations and at mole fraction 0.056. Values of the parameters lie between for a system whose electrolyte has a high hydrating power, 9 at 393 K, and for a system whose electrolyte has a relatively low hydrating power, 14 at 372 K. Tripp has illustrated a near linear correlation between and the weighted average cation charge density Q defined as
where and are the mole fraction, the charge, and the ionic radius of the cation i in the melt, respectively. According to this author, this correlation between Q and strongly supports the concept of preferential hydration previously invoked by Abraham, Abraham, and Sangster62 with regard to the Henry’s law con-
M. Abraham and M.-C. Abraham
233
stant. Besides, Tripp has brought attention to the system 13 where is fixed at 1.06, lying between 0.025 and 0.125, at 372 K. He pointed out the failure of Eq. (67) to correlate the water concentration dependence of water activities for this system, without stating the value(s) at which failure was observed.* Therefore, the writers have tested Eq. (67) noticing that with addition of the ratio decreases gradually, becoming even negative at At this concentration, and this latter value being obviously unrealistic. In order to obtain an accurate representation of the water activity even in very dilute solutions of electrolyte in water, Pitzer21 added to Eq. (67) an extended form of the Debye-Hückel equation previously proposed.73,74 This electrical contribution to the water activity coefficient is given by
where is the usual Debye-Hückel parameter, the ionic strength, and a parameter related to the closest approach of ions. is given by
* There is an error in Ref. 20, p. 855 concerning the evaluation of W and b, respectively designated by and in Eq. (67) in this chapter. The writers have verified that W = 0.69 and 6 = – 98 could not correspond to the data of the system 13 where the ratio is fixed at 1.06, with between 0.025 and 0.125, at 372 K, whatever .Yet, this error does not affect the comments of the author.
234
Bridging Electrolyte-Water Systems
in which is the water density, e, the electronic charge, tric constant of water, and k, the Boltzmann constant. is expressed by
the dielec-
in which and are the mole fraction and charge of the ionic species i, respectively. Addition of the Debye-Hückel term to Eq. (67) by Pitzer gives the following results21: for 7 at at 372 K, and at 373K, 9,32 392 K, for Subsequently, Pitzer and co-workers22–25,31 have proposed more general equations applicable, in principle, to multicomponent systems for calculation of the partial excess free energy, enthalpy and volume of all components. These equations comprising an ideal mixing term, with the assumption of random mixing of all particles, a general Margules term originating from short-range forces, and a third term expressing the Debye-Hückel effect, have been successfully fitted to data of many electrolyte-water systems containing essentially monovalent cations.21,23, 25–31,75 However serious difficulties are encountered with hydrate melts containing appreciable amounts of divalent cations20, 76 and even with dilute aqueous solutions.77 Moreover, this approach has been criticized40, 68,78 for its lack of predicting power, its complexity and the need of too many parameters whose evaluation may pose problems. (ii) Approach with Mole Fractions on an Un-Ionized Basis In a model based on the quasi-lattice concept proposed by Horsák and Sláma,79 the water molecules are introduced into the interpenetrating cationic and anionic sublattices in the ratio of the stoichiometric coefficients of the ions, diluting these two sublattices. The excess free energy of mixing water and electrolyte, and related thermodynamic parameters, are expressed as the sum of two contributions, one arising from short-
M. Abraham and M.-C. Abraham
235
range interactions, analogous to those in regular solutions, the other arising from long-range interactions, analogous to those in ionic crystals. This model has only been tested for 1:1 electrolytes with the following equation:
in which the activity coefficient
is defined as
with
is a short-range force interaction parameter and a Coulombic longrange force interaction parameter. Fitting of Eq. (76) over the whole water concentration range, with for the system 7 at 372 K, and for the system 9,32 at 373 and 392K, is satisfactory.4 3. Surface Properties The first ever determinations of the surface tension of a bridging electrolyte-water system were made by Hadded, Bahri and Letellier80 on
*The values of and in Ref. 79 are inaccurate and, consequently, the remark on the sign of is unfounded, pp. 1678 – 9. The values of and given in this chapter have been determined by the writers.
236
Bridging Electrolyte-Water Systems
the system, at 298 K. Other determinations of were made by Abraham and co-workers on the following bridging sys81 tems, between 350 and 390 K: 81 with M = Cs, Cd, 19 and Ca, and At a fixed water mole fraction, from anhydrous electrolyte to pure water, it was observed19, 81 that the surface tension is a simple linear function of the temperature T, over the explored temperature range:
where and are empirical parameters. Since is the surface free energy per unit area, the surface entropy and the surface enthalpy per unit area, are expressed by the equations:
Thus, and which are respectively represented by and in Eq. (79), appear independent of the temperature for these bridging salt-water systems,19,81 as observed for various pure molten salts, molten salt mixtures, anhydrous or containing small amounts of water.82, 83 As seen in Fig. 13, isotherms vs may exhibit positive, negative and alternate (S-shape curve) deviations from linearity, reminiscent of water activity deviations from Raoult’s law.
M. Abraham and M.-C. Abraham
237
From the surface tension and water activity data, orders of magnitude of the water mole fraction in the surface phase have been obtained81 by application of the Guggenheim and Adam method,84,85 as follows. The Gibbs adsorption equation is written
where is the Gibbs parameter which measures the adsorption of water per unit area, relative to the electrolyte. is related to the numbers of water molecules and electrolyte entities per unit area, and respectively, by
Bridging Electrolyte-Water Systems
238
With the assumption of a monolayer surface phase, and in Eq. (83) are related to the water molecular area and to the electrolyte entity area by
And the water mole fraction in the surface phase
is given by
The areas and were estimated from the particle radii.81 The particle area of water is given by
and taking the nitrate entities as electrolyte entity area is expressed by
an average value of the
in which is the water radius, the radius, the cation i radius, and the cation i valency. As illustrated in Fig. 14 the curves vs show a steady enrichment in water of the surface phase as increases over the whole water concentration range. 19,81 The relative lower values of for the system 19 reflect the strong tendency of the cations to attract water molecules. Computation of the water activity in the surface phase from the water activity in the bulk phase was performed81 by means of the Butler equation86 written in the form
M. Abraham and M.-C. Abraham
239
in which is the surface tension of pure water. In Fig. 15, curves of vs show that the adsorption theory of electrolytes might be applied to the surface phase of some nitrate-water systems. However, the values of r and in the surface phase are different from those in the bulk phase, especially the value of the energetic parameter The related water activity coefficient in the surface phase is obtained by
It has been pointed out81 that, on the whole, the relative position of
Bridging Electrolyte-Water Systems
240
the curves
vs
in the surface phase, for the systems and with M = Cs, Cd, and Ca, reflect, to a certain extent, the hydrating power of the cations as do the curves vs in the bulk phase. Recently, Sonowane, and Kumar87 have proposed the following equation to represent the surface tension of the above-mentioned nitratewater systems, at 373 K, over the whole water concentration range:
where is the surface tension of pure electrolyte and are two empirical constants. It must be underlined that there is a weakness in the approach of these authors. In effect, from the outset of the derivation of Eq. (90), the Butler equation, Eq. (88), is written with substitution of the mole fractions to the activities which is inconsistent with the fact that these systems are not ideal mixtures. 4. Other Approaches and Observations (i) Water Vapor Pressure Tripp and Braunstein1, 2, 20, 32, 88–90 have observed that in a number of hydrate melts the water vapor pressure and the water activity are linear in the water mole ratio (= moles of water / moles of electrolyte), in the region where water is the solute. Tripp20 pointed out that this property holds only in systems which show negative deviations from Raoult’s law and contain highly hydrated cations, such as but does not hold in the and based systems which show positive deviations from Raoult’s law and do not contain highly hydrated cations. Besides, Tripp89 has brought attention to a constant isotopic effect on the Henry’s law constant by substitution of to in the system between 383 and 423 K. The ratio of
M. Abraham and M.-C. Abraham
the Henry’s law constants
241
for water, and
for deuterium oxide, is
(ii) Molar Volume Sacchetto and Kodejš91, 92 have proposed an equation for the dependence of the molar volume of hydrate melts on molar composition which is based on the calculation of the distribution of water molecules between sites near ions and sites near water molecules. This distribution is obtained by a statistical thermodynamic treatment of a quasi-lattice model. Supposing that in an electrolyte-water system the effective molar volume of water in the neighborhood of the ions is the molar volume of water at infinite dilution in the molten electrolyte, and that the effective molar volume of water at sites near the water molecules is the molar volume of pure water, they obtained
242
The parameters
Bridging Electrolyte-Water Systems
and
are given by
and
where is the enthalpy of transfer of water from its pure state to infinite dilution in the molten electrolyte. The parameters and are interpreted as average quasi-lattice coordination numbers of the molten electrolyte and pure liquid water, respectively. Equation (91) was successfully fitted to the data of the bridging systems at 298 K, by Hadded et al.,69 [0.667 93 at 366 K, by Pacák,94 but 95 71 failed with the data of the system at 372 K. (iii) Substitution of an Organic Substance to Water
Kodejš and Sacchetto75, 95 have measured vapor pressures of the system between 360 and 390 K, and compared their results with those obtained with the system 12 at 372 K. They pointed out75 that the substitution of dimethyl sulfoxide (DMSO) to water intensifies the ionsolvent interaction in a manner analogous to the addition of an appreciable amount of divalent cation, such as or to the aqueous sys-
M. Abraham and M.-C. Abraham
243
tem, i.e. positive deviations from Raoult’s lawbecome markedly negative over the whole concentration range. They attributed the difference in behavior of these two systems mainly to the stronger cation-dipole interaction energy in DMSO66, 75, the dipole moments of the two molecules being64 and respectively. It must be noticed that the difference in behavior of these two systems is simply reflected in the adsorption parameters, r=0.49, whence/for water,12 at 372K, and r = 1.13, whence r for DMSO,73 at 383 K, suggesting the use of the product r as a criterion of solvation aptitude of various solvents with reference to a selected electrolyte, by analogy with the use of the same product r as a parameter expressing the hydrating power of various electrolytes.47 Upon substituting DMSO to water in the bridging systems 94 containing the electrolytes 95 at 377 K, and at 383 K, the excess molar volume of the solution becomes more important in both cases while Eq. (91) can still be fitted to the first system experimental data94 but fails with the second system.95
III. TRANSPORT PROPERTIES The first ever determinations of viscosity and electrical conductivity of a bridging electrolyte-water system were made in 1921 by Rabinowitsch96 on the system at 373 K, although there were few measurements over the concentration range where water is the solute. Further measurements of viscosity and electrical conductivity isotherms were carried out, more than 37 years later, on the 97
following systems:
at 405 K,
93
at 366 K, 71. 72, 98 where is fixed at 1.06, with between 0 and 0.125, M=Na, K, Cs, Cd, and Ca, at 372 K, 99 100
at 298 K,
101
at 383 K,
at 358 K. Electrical conductivity
244
Bridging Electrolyte-Water Systems
102 isotherms have been measured on the systems at 495 102 K,and at 453 K. The influence of temperature on both, viscosity and electrical conductivity, have been studied on the following systems: 103 between 413 and 463 K, 104 and 105,106 with M = Cd and Ca, between 350 and 380 K, 19 between 355 and 400K, 107 between 200 and 300 K, and, on the electrical conductivity 101 alone on between 345 and 365 K.
1. Activation Energy for Viscous Flow
The temperature dependence of viscosity at constant pressure is frequently represented by an Arrhenius type equation52,108:
where the preexponential term and the activation energy for viscous flow are two empirical parameters, often found practically constant over limited temperature ranges. For the bridging system Claes, Michielsen, and Gilbert103 found a linear dependence of the logarithm of the viscosity, In on the reciprocal of the temperature, 1 / T, whatever the water mole fraction, allowing to extract constant values of valid over the explored temperature range, i.e. between 413 and 463 K.
M. Abraham and M.-C. Abraham
245
Abraham and co-workers have also applied Eq. (94) and extracted values of activation energy for the systems: 104
105
with M = Cd and Ca, between 350 and
19 380 K, and between 355 and 400 K, Some curves vs are shown in Fig. 16. Claes, Michielsen, and Glibert,103 observing a minimal value of at concluded that this change in the activation energy suggests a transition from a system exclusively composed of hydrated ions to aqueous solutions where free water is also found. For the five above-mentioned bridging systems, Abraham, Ziogas, and Abraham105 have pointed out the existence of a linear, or quasilinear, relationship between and over the range which
246
Bridging Electrolyte-Water Systems
may be expressed by
where
is the activation energy of the anhydrous electrolyte and an empirical parameter, is the apparent activation energy of water, for viscous flow. The empirical parameter is one among a number of other empirical parameters called apparent parameters by Abraham and co-workers37, 72, 104, 105 used to take into account in the formulations of the properties of the electrolyte-water systems the behavior of water perturbed by the presence of ions, which is linked to the electrolyte in a manner reminiscent of the adsorption theory of electrolytes, over the concentration range In this state of water, there would be perturbations of hydrogen bonds or even the disruption of some of these by the strong electrostatic field of the ions, similar to the effect of temperature and pressure on pure water and aqueous solutions as suggested from Raman and infrared spectral investigations109,110 and recently from neutron diffraction.111 Thus, the apparent energy of activation of water could be seen as the energy of activation pure water would exhibit if it underwent structural modifications analogous to those brought about by the presence of the electrolyte. It is seen in Fig. 16 that may be negative. Concerning these negative values, a similarity between the behavior of perturbed water in these very ionic melts and pure water in its gaseous state has been underlined. 105 As a matter of fact, application of Eq. (94) to water vapor viscosity data64,112 between 373 and 673 K, leads to These results suggest some connection of Eq. (95) with the picture of Eyring significant structure theory of liquids113 which combines solid-like parts with gas-like parts, although this Eyring theory was not meant to deal with hydrates specifically. For the bridging system studied at 358 K by Dumont, Qian, and Conway101 at 358 K, a linear relationship between In and over the very large range has been attributed to a lin-
M. Abraham and M.-C. Abraham
247
ear relationship between and i.e. formally Eq. (95). This has been correlated by the authors with the formation of hydration shells which are supposed to be complete at in accordance with electrical con-ductivity data. 2.Transition State Theory of Viscosity
Eyring114,115 has proposed the following equation for the viscosity of a simple fluid
where h is the Planck constant, , the volume of a hole considered to be close to that of a flow unit, and the molar free energy of activation for viscous flow. Generally, is related to the molar volume of the fluid V by
and Eq. (96) is often written
is related to the enthalpy of activation, of activation, for viscous flow by
and
and the entropy
248
Bridging Electrolyte-Water Systems
In this viscous flow process theory, a particle i, or flow unit, passes from one equilibrium position where it oscillates with a characteristic frequency to another equilibrium position over a free energy barrier This passage requires the availability of a hole, or empty site, in the quasi-crystalline lattice, in which the particle will jump, its vibration being transformed into a translation movement. In molten salts, it has been considered116–118 that originates essentially from the formation of a hole and the jump, or movement, of the flow unit into this hole, whereas the physical meaning of has been less clearly stated. As a matter of fact, holes are parts of liquid structures and their existence has often been invoked in various studies of molten salts.52,53,104, 116–129 Similarities between holes in liquids and defects in solid crystals, such as Schottky (ion pair or single ion vacancy) and Frenkel (dislocated interstitial cation) defects, have been put forward in investigations on molten halides and nitrates. 123–125,128 Besides, studies on water have led to theories making use of such concepts as hole, cavity, defect, interstitial position.129–133 From these theories and other arguments, it has been inferred104 that pure water resembles pure nitrates with respect to hole mechanisms, so that the existence of these mechanisms might also be highly probable in mixtures of water and molten nitrates, and possibly other electrolytes.
M. Abraham and M.-C. Abraham
249
Although Eyring equation was proposed for a simple fluid and other approaches are used when cooperative phenomena have to be considered in liquids of relatively high viscosity,134 interesting results have been obtained by applying Eyring equation to hydrate melts despite the complexity of their liquid structures. In practical situation, in Eq.(96) must be obviously viewed as an average parameter, like and, in hydrates, it will depend on the nature of the flow units originating from the electrolyte and water. Species involved in the viscous flow mechanisms are not known and one might suppose the existence of ion pairs, complex ion clusters... Abraham and co-workers19, 37, 98, 99, 105 have applied Eyring equation with expressed by Eq. (97) to the following 71, 72, 98 systems: where is fixed at 1.06, being between 0 and 0.125, with M = Na, K, Cs, Cd, and Ca, 19, 99
250
Bridging Electrolyte-Water Systems
93
103
and
Curves vs are illustrated in Fig. 17 for some of these systems. Over the range is a linear, or quasi-linear, function of which may be written
of
where is the free energy of activation of the anhydrous electrolyte and the apparent free energy of activation assigned to the perturbed water.* For the and based systems,37, 7l, 72, 98 reflects the perturbing power of the cations through an additivity rule expressed by
where is the parameter for a mixture of electrolytes and the parameter for the component i. As seen in Fig. 18, at any given value ofthe added nitrate mole frac5 tion, the trend of the values
is consistent, on the whole, with the characterization of ions in dilute
*Concerning the system was observed in Ref. 37 that
93
at 366 K, it
does not obey the linear relationship expressed by
Eq. (101). This is all the more surprising that, for the same system, the free energy for electrical conductance obeys a linear relationship analogous to Eq. (101), as will be seen in Section III.6, and that, moreover the writers have verified that the system 101 obeys Eq. (101) although it does not contain any nitrate. In our opinion, the ex perimental data provided by courtesy of Dr Claes on the viscosity of the system [0.667 could be unreliable given the fact that the times of drain of solutions very rich in electrolyte through the capillary tube in the method of measurement are much too long in comparison to those of the solutions rich in water: for example, s at compared to s at
M. Abraham and M.-C. Abraham
251
aqueous solutions as water structure breakers, such as and water 135–139 structure formers, such as If the evaluation of is made assuming the flow units to be the water molecules and the ions, the average volume of a hole in Eq.(96) is then
where and are the numbers of positive and negative ions respec-tively produced by the total dissociation of one mole of the electrolyte i and V is given by Eq. (54). For the system the two alternatives in the evaluation of lead to an appreciable difference in only for the solutions of water in electrolyte, at worst 10% for the anhydrous electrolyte, and the free energy is still linear in when is expressed by Eq. (103).19 In molten carbonates,116 the molar free energy of activation for vis-
252
Bridging Electrolyte-Water Systems
cous flow has been compared to the molar free energy of hole formation , evaluated by means of the following equation used in Fürth hole theory of liquids140
where is the radius of a hole. In the Fürth theory, outside the holes, a liquid is considered as a continuum with normal surface tension and the sizes of the holes obey a statistical distribution law,140 so that they are not comparable to holes in a quasi-lattice.117 Equation (104) has played an important role in the discussion of molten salt properties and the development of their theories52, 53, 116,118,121, 123 in which it was often assumed, or concluded, that the average hole size is close to that of an ion. From this point of view, average values of have been obtained with the following bridging systems 8 1 81
with M=Cs, Cd, 19 and Ca, and at 372 K, taking into account all the species in the solutions, by the equation
where the parameter radii expressed by
For the is close to
is the weighted average of the square particle
and based systems, it was observed that from fused electrolyte to water, and inferred that indeed
253
M. Abraham and M.-C. Abraham
hole formation is an essential step of the viscous flow mechanisms.81 For 19 the system over the range is also close to but their difference is greater for the dilute solutions of water in the molten salt, probably due to the high degree of quasi-crystalline character of the mixture containing an important amount of cations. The presence of divalent cations strengthens the structure of the liquid which will become more crystallike due to intensification of the cationic electrical field, particularly near the pure fused salt end of the concentration range at relatively low temperature, and, therefore, the approximation of a liquid continuum could perhaps be less appropriate, holes resembling more to sort of Schottky or Frenkel defects.19 In the transition state theory, the enthalpy of activation in Eq. (99) is identified to the energy of activation in Eq. (94)
and the preexponential term
in Eq. (94) is interpreted as
Generally and rigorously, like is valid only at a specified temperature. For the bridging system 105 has been plotted against T, at different water mole fractions, showing that varies perceptibly with T, especially at low water concentrations. Nevertheless, this dependence of on T seems sufficiently weak to warrant extraction of meaningful average values of from data on Values of have also been extracted19,37,105 for other bridging 104 nitrate-water systems: 103 105
and
19
Two
254
examples of curves
Bridging Electrolyte-Water Systems
vs
are given in Fig. 19.
The curves vs have been discussed37 considering that the composition range can be divided, at the most, into four regions defined by the algebraic sign of and bounded by the pure components, the two values of at which is zero, and the value of at which is minimum. In region 1, is positive while is negative. In region 2, and are both negative. In region 3, is nega-tive while is positive. In region 4, and are both positive. The changes in the sign of and the existence of a minimum indicate that there is no unique and simple mechanism of passage of a flow unit from one equilibrium position to another over a potential energy barrier. Several simultaneous competing mechanisms occur in
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different regions of the liquid, all accompanied by local structural rearrangements to which different local entropy variations are associated. Some regions would have a relatively high degree of order, close to that of a crystal, where hole formation may obey a sort of Schottky type local mechanism. In this case, one could expect a lowering of the degree of order, and therefore a positive contribution to the entropy of activation. Other regions would have a relatively low degree of order, with interstices of various sizes but too small to be occupied by a flow unit. In these parts, a flow unit may push back and squeeze up other particles yielding a modification of the local structure equivalent to a coalescence of small interstitial holes into a sizeable hole which can accommodate a flow unit. This phenomenon is associated with an increase of the local degree of order and it follows that the passage of a flow unit is accompanied by a negative contribution to the entropy of activation. From this point of view, may be expressed by37
where and are the average positive and negative contributions to the entropy of activation, respectively. and are the probabilities of occurrence of a passage with a positive and negative contribution to the entropy of activation, respectively. These fourfunctions are expected to be generally dependent upon the temperature and the composition of the mixture in an intricate manner. Near the pure electrolyte which has a quasi-lattice structure, processes involving hole formation analogous to that of Schottky defects may predominate. As increases, although the structure of the mixture is similar to that of the molten electrolyte, the electrolyte-water mixture structure becomes more disordered so that the overall entropy of activation decreases. Likewise, near pure water, since the water molecules are connected with each other in a water-like quasi-lattice structure, processes involving hole formation analogous to that of Schottky defects may predominate. As the electrolyte is added to water, the structure becomes more disordered and the overall entropy of activation decreases. Whence the appearance of a minimum in
256
of
Bridging Electrolyte-Water Systems
Over the range expressed by105
where
is a linear, or quasi-linear, function
is the entropy of activation of the anhydrous electrolyte and the apparent entropy of activation assigned to the perturbed water, for viscous flow. In Tables 2 and 3 are reported the values of and for the five nitrate-water systems19, 103–105 whose viscosity has been studied as function oftemperature. The negative values of like indicate that water at low concentration in these ionic systems behaves presumably as a disordered substance in con-trast to ordinary pure water for which and are positive,105 as seenin Figs. 16 and 19. These negative values of are consistent with the remark made in Section III.1 concerning an analogy between the perturbed water and the water vapor, since a gaseous state is characterized by alow degree of order. Although is negative, is so negative that is positive, like for pure water (see Fig. 17).
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3. Free Volume for Viscous Flow
In a study on the viscosity and the diffusivity of molecules in liquids, Hildebrand141 has suggested that the ideas of quasi-lattice structures and activated processes could, in fact, be unrealistic and that the equation of Batschinski,142 based on the available free volume, could be a better approach. This equation, proposed for non-associated liquids, as long ago as 1913, expresses, at a fixed composition, the reciprocal of the viscosity i.e. the fluidity as an indirect function of the temperature, through the difference between the specific volume and a constant similar to the van der Waals co-volume
M. Abraham and M.-C. Abraham
259
where is another constant characteristic of the liquid. Equation (111) may also be written
in which
is the specific free volume defined by
If the specific volume is replaced by the molar volume V, Eqs. (111)–(113) are transformed into141
with the parameters
and
related by
where M is the molar mass, the molar free volume of the liquid, and two empirical constants.6 The constant has been designated by several expressions52, 141: molar co-volume, molar intrinsic volume, molar incompressible volume, molar volume at which the fluidity is zero. It has been found, in some instances, close to the molar volume of the solid at the melting point141,
*There is an error in Ref. 104, Eq. (19), p. 1480. The equation placed by
which is Eq. (117) in the present chapter.
must be re-
260 143
Bridging Electrolyte-Water Systems
or close to the molar volume of the liquid in equilibrium with the solid
at the melting point.104
The other empirical constant may be considered as a measure of the tendency of the fluidity to increase with the free volume so that it could depend primarily upon the interactions between the particles whereas would primarily depend upon the size and form of the particles, reflecting steric effects.104 Abraham and Abraham,98,144 in a derivation of the Batschinski equation from the Eyring equation, have proposed an interpretation of the physical meaning of based on a quantum concept of hole. In the 1930s, Eyring114, 115 considered the holes as playing the same part in a liquid as molecules do in a gas. He thought that a liquid may be regarded as made up of holes moving about in matter just as a gas consists of mole-cules moving about in empty space. In a study published in 1941, Fürth140 viewed the holes in a liquid as the counterparts of the clusters in a dense gas, formed and destroyed by the action of the statistical fluctuations, interacting with each other, and performing a kind of Brownian motion. The phenomenon of viscous flow was then explained by Fürth in a way similar to that in a gas. He assumed that the Brownian motion of the holes produces a transfer of momentum between adjacent layers of the moving liquid. These analogies between holes in matter and particles of matter in space have suggested to Abraham, Chevillot, and Brenet145 that a hole in a liquid exhibits, in connection with its surroundings, wavelike properties. A frequency and a quantum of energy are associated to the hole. The moving hole together with the induced perturbation of the surrounding quasi-lattice is called the lacunon or fluctuating hole.37, 98, 144, 145
In an ideal picture of a simple liquid, in Eq. (96) has been related37 to an equilibrium constant of hole formation
with
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261
N being the number of flow units and the number of lacunons in a given volume of liquid taken as the molar volume. From Eqs. (96), (118) and (119), one can write
The ratio of the total volume of the lacunons to the free volume is assumed to be constant over the temperature range where the Batschinski equation is valid
with the proportionality constant. Equations (115), (120) and (121) lead to the identification of
as
The Bastchinski equation has been applied to various molten electrolytes52,143,146 over large temperature ranges and subsequently, by Abraham and co-workers104, 105, 144 to the following systems: 104
103
105
with M = Cd and Ca, and 19
From fitting of Eq. (114), the constants and are determined as functions of the water mole fraction. From values of are calculated, at a fixed temperature, by means of Eq. (116) in which Vis given by Eq. (54). From values of are determined by means of Eq. (122) where N is taken equal to assuming the flow units to be electrolyte entities and molecules. From the values of
262
and
Bridging Electrolyte-Water Systems
those of are evaluated by means of Eq. (121). It is seen in Fig. 20, showing some curves vs and vs that the fluctuating holes involved in the viscous flow process repre-sent presumably only a small part of the free volume in the fused electro-lytes, contrary to water where about all the free volume would appear in the form of holes which could be one reason why those fused nitrates are much more viscous than water.98, 144 It should be noticed that the preceding development which simultaneously relates Batschinski equation to Eyring equation through the lacunon concept and identify the first Batschinski constant, shows that the two approaches in fact support each other and even merge and there is no reason to give more importance to one approach than to the other.
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4. Equation for Fluidity with Apparent Parameter
In a study on the systems where is fixed at 1.06, varying from 0 to 0.125, at 372 K, Abraham et al.71 found that the significant decrease which occurs when water is added to the electrolyte, especially for solutions of water in molten electrolytes, may be expressed by a linear relationship between ln or ln and over the range This decrease of the viscosity with increasing water concentration has also been illustrated with other bridging systems72,93,97,100,101,103 and some curves vs are shown in Fig. 21. Abraham and Abraham72 put the relationship between ln and in the following form, containing only one empirical parameter:
264
Bridging Electrolyte-Water Systems
in which is the fluidity of pure electrolyte and the empirical paameter, is the apparent fluidity assigned to the perturbed water which may be seen as the fluidity pure water would exhibit if it underwent structural modifications analogous to those brought about by the presence of electrolytes. The form of Eq. (123) with and appearing as exponent is consistent with the assumption of activation-controlled processes for viscous flow and has been considered as a consequence of the Eyring equation.37
As shown in Fig. 22, for the and based systems,98 reflects the perturbing power of the cations through an additive rule expressed by
where is the parameter for a mixture of electrolytes and the parameter for the component i. Moreover, the fluidity from pure molten electrolyte to pure water may be expressed by the following two empirical parameter equation72:
where and called the excess fluidity of water and the excess fluidity of electrolyte, respectively, are defined by
M. Abraham and M.-C. Abraham
with trolyte.
the fluidity of pure water and
265
the apparent fluidity of elec-
Similarly to the apparent fluidity of water, the apparent fluidity of electrolyte is defined as the fluidity the electrolyte would exhibit if it underwent structural modifications analogous to those brought about by the presence of water at low concentration, for example, the degree of defects or holes. It was found37, 72, 98, 104, 105 that is very close to whereas and are quite different. Examples of values are given in Table 4 together with and for various hydrate melts. 19, 72, 96, 98, 101, 103–105 It must be underlined that Eq. (125) is consistent with the cybotactic theory38 and the significant structure theory113 adapted to feature a picture of electrolyte-water systems comprising submicroscopic parts such as pure electrolyte, electrolyte with small amounts of water, ordinary water, perturbed water, ion pairs, complex ion clusters …whose relative quantities vary with the water mole fraction. The fact that Eq. (125) reduces to Eq. (123) for the solutions of water in electrolyte is in good agreement with a picture comprising above all pure electrolyte and perturbed water
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Bridging Electrolyte-Water Systems
M. Abraham and M.-C. Abraham
267
submicroscopic parts while for solutions rich in water, other submicroscopic parts have to be taken into account. If an analysis is made with the mole fraction on an ionized basis,98 the experimental data cannot be represented over the whole concentration range by an equation similar to Eq. (125). However, in dilute aqueous solutions, i.e. over the range which corresponds to the following equation is obeyed:
with an empirical parameter. The parameter plays formally a role similar to that of in Eq. (123). Arguing from this similarity, has been defined as the fluidity that the electrolyte would exhibit if it underwent structural modifications analogous to those brought about by the presence of water at high concen-tration, in contrast to which is defined in the concentration region rich in electrolyte. It was observed that the perturbed salt behaves in dilute aqueous solutions as a liquid whose fluidity is higher than that of the anhydrous salt whereas, as seen in Table 4, in very concentrated solutions, the perturbed water behaves as a liquid whose fluidity is lower than that of ordinary water This is exemplified with the following sys72 tems: and for which 72 at 372 K, for which and at 372 K, and 19 for which and at 383 K. The shifting of the linear relationship between the logarithm of the fluidity and the concentration towards the dilute aqueous solution range when the ordinary mole fraction is replaced by the mole fraction on an ionized basis and the fact that is greater than may be ascribed, at least partially, to the isolation of the ions at high dilution in the
268
Bridging Electrolyte-Water Systems
solutions. It has been noticed that a relatively high temperature seems required for a simple equation, such as Eq. (128), to be valid. 5. Activation Energy for Electrical Conductance
The temperature dependence of electrical conductance at constant pressure has often been represented by the following Arrhenius type equations52, 108
where the preexponential terms and the activation energies for the electrical conductivity and the equivalent electrical con-ductance are taken as constants over given temperature ranges. The equivalent electrical conductance is defined by
with the electrolyte concentration in equivalent per unit of volume. For a solution in which the electrolyte is a mixture of several components, is related to the molar volume V by
where
is the electrochemical valency defined by
M. Abraham and M.-C. Abraham
269
and being the number of positive and negative ions respectively produced by the total dissociation of one mole of the electrolyte the cation i valency, and the anion i valency. For a mixture of electrolytes, the equivalent electrical conductance in Eq. (131) may be seen as the equivalent conductance of a virtual single electrolyte which would consist of anions and cations carrying the same value of electrical charge and having the same mobility37, 106:
where F is the Faraday constant and all ions, is defined by
the average of the mobilities of
and being the cationic and anionic mobilities, respectively. The activation energies and are related by the following equation148:
where
is the coefficient of thermal expansion of the solution given by
For the bridging system
Claes,
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Bridging Electrolyte-Water Systems
Michielsen, and Gilbert103 found a linear relationship between ln and 1/T, whatever the water mole fraction, and extracted constant values of valid over the explored temperature range, i.e. between 413 and 463 K.
Equation (130) has also been applied to the following bridging systems: 104
106
with M = Cd and Ca, 19
101
and solutions of water in Two curves vs are represented in Fig. 23. Dumont, Qian, and Conway101 have observed for the system 149
M. Abraham and M.-C. Abraham
271
7
a linearity between and over the very large water concentration range By analogy with Eq. (95) for viscosity, this relationship may be expressed by
where is the activation energy for the equivalent electrical conductance of the anhydrous electrolyte and the apparent activation ener-gy for the equivalent electrical conductance of the electrolyte at infinite dilution in the perturbed water. This equation is also valid for nitrate-water systems19, 103, 104, 106 over the concentration range Equation (138) for equivalent electrical conductance, like Eqs. (95) and (123) for viscous flow, is consistent with a picture of hydrate melts inspired by the cybotactic theory38 and the significant structure theory.1l3 Obedience to Eqs. (95), (123) and (138) suggests that is not involved in chemical reactions with ions produced by the electrolyte dissociation. With regard to the salt KF 2HF made up of and ions, Dumont, Qian, and Conway101 concluded from thermodynamic investigations that water added to the electrolyte is ionized to a negligible extent. For solutions of water in Campbell and Williams149 have compared in Eq. (129) and in Eq. (130). Since in Eq. (136) is small and T is relatively low, and were found very close to one another. It can be verified that it is also the case for systems from molten electrolyte to dilute aqueous solution,19, 101, 103, 104, 106 as illustrated
*In Ref. 101, p. 157, in Fig. 9, values of which is in Eq. (130) of this chapter, are erroneous. Calculations made by the writers with the experimental data provided by courtesy of Dr Dumont give values of between 7 and and not between 0.1 and Besides, there is obviously an error concerning the experimental value corresponding to which is far apart from the straight line; the experimental value is
while the extrapolated value is
272
Bridging Electrolyte-Water Systems
in Fig. 23. Besides, the ratio usually close to unity for nitrate melts,52 remains quasi-constant at 1.1 – 0.1 as water is added to the nitrate mixtures over the whole concentration range, suggesting analogous hole mechanisms for both, the electrical conductance and the viscous flow, in dilute aqueous solutions as well as in molten nitrates.104 6. Transition State Theory of Electrical Conductance
The transition state theory of the mobility of the hydrogen ion in hydroxylic solvent, proposed by Steam and Eyring150 has been adapted to the ionic mobility of interstitial ions in liquid silicates by Bockris et al.151 Further applications of this approach were made to molten chlorides152, 153 and nitrates.154 Abraham and Abraham37 have applied the transition state theory to bridging systems, from pure electrolyte to dilute aqueous solutions, extending the equation proposed by Bockris et al.151 to take into account the presence of water and various ions, giving:
where represents the free energy of activation for the conduction process of the electrolyte whose is related to by Eq. (134). The parameter z is an average value of the number of charges carried by all the anions and cations:
The parameter d is something like an average of half the migration distances across the potential barriers taken as equal to the radius that corresponds to the average of all the particle volumes, including water:
M. Abraham and M.-C. Abraham
273
The effective electrical field resulting from a unit applied field strength is expressed by
where and are the dielectric constants of electrolyte and water, respectively. The free energy of activation is related to the enthalpy of activation and the entropy of activation for the electrical conduction process by
and
Values of in Eq. (139) have been determined19, 37, 98, 99, 106 for á 72, l04 the following systems: 72,98,106 with M = K, Cs, Cd, and Ca,
103 101
In Fig. 24,
and, in this chapter, for the system vs curves are given for three
274
Bridging Electrolyte-Water Systems
systems. A linear, or quasi-linear, relationship is observed between for all the above-mentioned systems, over the range may be expressed by the following equation37, 98, 99:
and which
where is the free energy of activation for the equivalent electrical conductance of the anhydrous electrolyte and the apparent free energy of activation for the equivalent electrical conductance of the electrolyte at infinite dilution in the perturbed water. It must be underscored that and appear close to one another for some nitrate systems,19, 37, 98, 99 indicating similarities in viscous flow and electrical conductance mechanisms infused electrolytes. How-ever, as water is added to the anhydrous electrolyte, the difference be-tween the free energies of activation and increases, as seen in Fig. 24. This has been discussed19, 37, 106 in investigations regarding the influence of temperature on viscosity and electrical conductance on the following systems: 105,106 104 with M 19 = Cd and Ca, and 103 Since for these systems (or ) is near (or as seen in Fig. 23, the difference must be looked for in an increase of the difference between and
M. Abraham and M.-C. Abraham
In Fig. 19, it is observed that ( is added to the electrolyte, in particular for
275
) becomes larger as water in connection with
the fact that does not exhibit a marked minimum contrary to For the electrical conductance, the unique participating component is the electrolyte itself over the whole concentration range whereas for viscous flow, electrolyte and water are both participating components. For solutions rich in water, with a water-like quasi-lattice structure, although the ions interact with the water molecules, they are, less than water itself, structural parts of the quasi-lattice. Consequently, they move under the influence of an electric field in obedience to predominant mechanisms in which neighboring water molecules are squeezed up. This is equivalent to coalescence of small holes into larger holes. Whence an overall entropy of activation for the electrical conductance which decreases with increasing water concentration, even in dilute aqueous solutions.
276
Bridging Electrolyte-Water Systems
7. Free Volume for Electrical Conductance
Abraham and Abraham104 have extended utilization of Batschinski type equations to the electrical conductance and, by analogy with the fluidity, wrote the following equations:
and
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277
in which and are two empirical constants called molar co-volumes, and are other constants measuring the tendency of the electrical conductance to increase with the molar free volumes and , for the electrical conductivity and equivalent electrical conductance respectively. Equations (146)–(151) have been applied104 to the systems 104 103 and from fused salts to dilute aqueous solutions. As illustrated in Figs. 25–27 with the system and as well as in the Batschinski equation for fluidity, Eq. (114), behave respectively like the functions and when plotted against
278
Bridging Electrolyte-Water Systems
M. Abraham and M.-C. Abraham
279
The molar free volumes and are close together, over the whole water concentration range for the system 104 as seen in Fig. 28, and, over the range for 103 the system 8. Equation for Equivalent Electrical Conductance with Apparent Parameter
72
For thesystems
72
with M = Cs and Cd, and Abraham and Abraham72 have proposed the following equation, with only one empirical parameter, to represent at fixed temperature experimental data on the equivalent electrical conductance as function of over the range 103
in which is the equivalent electrical conductance of pure electrolyte. The empirical parameter called the apparent equivalent electrical conductance of electrolyte at infinite dilution, represents the equivalent electrical conductance at infinite dilution the electrolyte would exhibit if the properties of pure water were the same as those of the water dissolved in the electrolyte whose fluidity is (see Section III.4). Equation (152) has also been applied to other hydrate melts and various values of and are listed in Table 5. For the and based systems, it must be pointed out that the order of increasing like in Fig. 22, is the same as that of increasing water structure breaking power of the ions in dilute aqueous solutions,135-139 and the order of decreasing in Fig. 18. The common exponential form of the curves vs and vs from pure molten electrolyte to dilute aqueous solutions (see Figs. 26 and 27) has suggested to represent by the following two empirical parameter equation,72 similar to Eq. (125) which expresses :
280
Bridging Electrolyte-Water Systems
where is the excess equivalent electrical conductance of the electrolyte at infinite dilution in water and the excess equivalent electrical conductance of the anhydrous electrolyte defined by
with
the apparent equivalent electrical conductance of the anhydrous electrolyte and the equivalent electrical conductance of the electrolyte at infinite dilution in water obtained by extrapolation of Eq. (153). Since the equivalent electrical conductance of a solution is an homog-eneous function of degree one of the ionic mobilities, the physical meaning of the above-mentioned parameters proceeds from their relation to these ionic mobilities expressed by Eqs. (134) and (135). Equations (154) and (155) correspond to the excess mobility of the ion i at infinite dilution in water and the excess mobility of the ion i in the anhydrous electrolyte defined as follows
where and are the mobilities of the ionic species i in the anhydrous electrolyte and at infinite dilution in water, respectively. The parameter is the mobility at infinite dilution the ionic species i would exhibit if the structure and properties of pure water were the same as that of the perturbed water and, similarly, the parameter is the mobility
M. Abraham and M.-C. Abraham
281
282
Bridging Electrolyte-Water Systems
the ionic species i would exhibit in pure electrolyte if its structure and prop-erties were the same as that of the electrolyte modified by thepresence of water. It has been observed98 that, contrary to the fluidity, the logarithm of the equivalent electrical conductance ln is not a linear function of the water mole fraction on an ionized basis over the dilute aqueous solution concentration range where extensions of the Debye, Hückel and Onsager theories are expected to apply. Consequently, there is no equation similar to Eq. (128) for fluidity. However, a linear relation between ln and is perceptible over an intermediate concentration range. 9. Relation Between Viscosity and Electrical Conductance
Abraham and Abraham104 have found that for the bridging systems 104 and 103 over the range the parameters in Eq. (146) and in Eq. (114) are related in the following simple manner:
where is the proportionality constant. Since over the range the ratio is a quasi-constant at a given temperature and the ratio obeys Eq. (158), the fol-lowing equation can be deduced from Eqs. (115) and (147):
in which and are the electrical conductivity and the viscosity of the pure electrolyte, respectively. This equation is also applicable72, 99, 104, 129, 144 to the bridging systems
M. Abraham and M.-C. Abraham
283 72,129
Cd,andCa, 97
with M=K,Cs, 99 [0.500 96 but fails when tested with the systems
101 and The product of the viscosity and the equivalent electrical conductance, referred to as the Walden product , has often been considered in structural discussions on hydrate melts.21,22,72,93,100,101>129 Some curves
vs
are shown in Fig. 29.
The first determinations of the Walden product over the whole water concentration range were made with the system at 366 K, by Gilbert et al.93 who pointed out that decreases rapidly over the range and remains approximately constant between 0.2 and 1 which is, according to these authors, surprising since at high electrolyte concentration the relaxation and electrophoretic effects should have considerable importance.8 97 For the system at 405 K, the existence of a shallow minimum leading to a quasi-constancy of the Walden product over the range seen in Fig. 29, has suggested to Pitzer21,22 full ionization for hydrate melts as well as for dilute aqueous solutions in which the electrostatic effects are responsible for a decrease in equivalent elec-trical conductance from its limiting value following theoretical equations for fully ionized solute.155,156 For the bridging nitrate-water systems72,96,99,103,129 obeying Eq. (159), Abraham and Abraham72 have found that the Walden product is a linear function of over a large concentration range, as illustrated in Fig. 29 with two systems. Using the concept of the apparent molar volume of water,157 the molar volume of the solution may be written as follows:
*In our opinion, viscosity measurements of the system
should be redone before conducting any discussion implying viscosity data on this system (see note in Section III.2).
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Bridging Electrolyte-Water Systems
Combining Eqs. (131), (132), (159) and (160) gives the following equation to express the Walden product:
For these nitrate-water systems, as a first approximation, the apparent volume being nearly independent of the composition over 17,71,72 the range is a linear function of over this concentration range. It has been numerically tested72 that a minimum value of the Wal-den product may be viewed as a consequence of Eqs. (125), (153) and the values of the empirical parameters in these equations.
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10. Other Approaches and Observations
(i) Logarithmic Equation for Equivalent Electrical Conductance Campbell et al.102 have reported for the systems at 495K, and at 453 K, a linear relationship between and the logarithm of the molarity of the solution, lnC, valid from pure electrolyte to dilute aqueous solutions, expressed by:
where is the molarity of the pure electrolyte and , a proportionality constant. It has been noticed129 that this empirical equation is also valid for the and based systems, at 372 K, but over more limited concentration ranges, especially in presence of divalent cations. For example, for the system Eq. (162) is valid over the limited range (ii) Power-Law Equations for Viscosity and Electrical Conductivity
Concerning the system studied in the subambient 107 region, between 200 and 300 K, Das et al. have found that a power-law type equation158–160 based on the mode coupling theory of relaxation in supercooled liquids161 is better than a Vogel-Tammann-Fulcher type equation108 to describe the temperature dependence of the electrical conductivity and the viscosity . The power-law equation for
and for
160
158, 159
is written
286
Bridging Electrolyte-Water Systems
where and are empirical parameters. For a given composition, the computed value is expected to be very close to since it represents the ideal glass transition
temperature which should be independent of the transport property.162 Das et al.107 found that the values of and are in fact close to one another over the explored concentration range Their investigations have also highlighted correlations between the viscosityconcentration curves and the phase diagram of the system.163 (iii) Substitution of an Organic Substance to Water
Pacák94 has measured the viscosity of the system at 369 K. Comparison of the curve vs with the curve vs for the system 93 at 366 K, shows94 that the viscosity increases considerably when DMSO is substituted to water. Moreover, contrary to the hydrates, the curve with DMSO exhibits a maximum at which, according to Pacák, cannot be described by any currently known equation, but could be related to the solubility curve of the system and to solvate formation. This maximum is not specific to this system but 164 has been also observed with the system between 298 and 333 K, however at Ziogas and Papanastasiou165, 166 have measured the viscosity of the 165 systems and 166 between 376 and 394 K, focusing their discussion on the influence of composition upon the activation parameters and in the transition state theory, Eqs. (98)(100), in connection with the assumption of hole mechanisms in the viscous flow process.
M. Abraham and M.-C. Abraham
287
As evidenced in Figs. 30 and 31, for the hydrate melts 104 i.e. replacement of water by DMSO increases significantly and due to stronger cation-dipole interactions in DMSO solutions than in aqueous solutions,165, 166 also invoked in vapor pressure investigations75, 94 (see Section II.4). When water is totally replaced by DMSO, for solutions rich in electrolyte, the simple linear relationships and Eqs. (95) and (110), do not hold and the pronounced minimum in the curve disappears. The low values of in the concentration range rich in DMSO has been attributed by Ziogas and Papanastasiou165 to the existence in DMSO of monomers, dimers and chain-like molecules167 contributing to a relatively low degree of order in contrast to the quasi-lattice structure of nitrate-water systems rich in water as well as systems rich in nitrates.
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Bridging Electrolyte-Water Systems
Claes, Mestdagh, and Gilbert168 have studied the effect of substituting various organic substances to water on the electrical conductivity of the 93 system at 366 K: ethylene glycol (EG), at 403 K, formamide (F), at 363 and 403 K, dimethyl formamide (DMF), at 403 K and dimethyl sulfoxide (DMSO), at 403 K. The most striking difference between the behavior of the electrical conductivity isotherm of the aqueous system and the isotherms of the organic systems is that from pure electrolyte to dilute solutions is less than with the organic substances, while, with water, it increases dramatically from about 10 to the maximum value at in a manner 104 analogous to that of the system illustrated in Fig. 25. The disappearance of the pronounced maximum in the curve vs induced by the substitution of an organic substance to 100 water is also ob-served with the system at 298
M. Abraham and M.-C. Abraham
289
K, when water is replaced by acetonitrile (AN), Occurrence of a pronounced maximum, often observed in aqueous solutions at a mixture composition where the liquid medium is relatively depleted of electrical charge carriers,169 must obviously be ascribed to much higher mobilities of these charge carriers in water compared to the investigated organic substances. In fact, increases much more rapidly with than with the mole fraction of these organic substances.100, 168 From their study, Claes, Mestdagh, and Glibert 168 concluded that the dimensions and shapes on the non-electrolyte molecules are factors more influential than the dielectric constants and acidity properties in the behavior of the isotherms vs
REFERENCES 1
J. Braunstein, Inorg. Chim. Acta Rev. 2 (1968) 19. J. Braunstein, “Statistical Thermodynamics of Molten Salts and Concentrated Aqueous Electrolytes,” in: Ionic Interactions: From Dilute Solutions to Fused Salts, S. Petrucci, ed., Vol. 1, Chapter 4, Academic Press, New York, 1971. 3 R. M. Fuoss, Chem. Rev. 17 (1935) 27. 4 C. A. Kraus, J. Phys. Chem. 58 (1954) 673. 5 R. M. Fuoss and L. Onsager, J. Phys. Chem. 61 (1957) 668. 6 R. H. Stokes and R. Mills, “Concentrated Solutions and Molten Salts,” in: Viscosity of Electrolytes and Related Properties. The International Encyclopedia of Physical Chemistry and Chemical Physics, E. A. Guggenheim, J. E. Mayer, and F. C. Tomp-kins, eds., Vol. 3, Chapter 7, Pergamon Press, New York. 1965. 7 M.-C. Trudelle, M. Abraham, and J. Sangster, Can. J. Chem. 55 (1977) 1713. 8 P. Claes and J. Gilbert, “Water Concentration Dependence of Transport Properties in Ionic Melts,” in: Ionic Liquids, D. Inman and D. G. Lovering, eds., Chapter 14, Plenum Press, New York, 1981. 9 H. Braunstein and J. Braunstein, J. Chem. Thermodynamics 3 (1971) 419. 10 A. N. Campbell, J. B. Fishman, G. Rutherford, T. P. Schaefer, and L. Ross, Can. J. Chem. 34 (1956) 151. 2
11
R. A. Robinson and R. H. Stokes, Electrolyte Solutions, ed., Butterworths, London, 1965. 12 J. M. Sangster, M.-C. Abraham, and M. Abraham, Can. J. Chem. 56 (1978) 348. 13 J. Sangster, M.-C. Abraham, and M. Abraham, J. Chem. Thermodynamics 11 (1979) 619. 14 M.-C. Abraham, M. Abraham, and J. Sangster, J. Solution Chem. 8 (1979) 647. 15 M.-C. Abraham, M. Abraham, and J. Sangster, J. Chem. Eng. Data 25 (1980) 331. 16 J. Sangster, M.-C. Abraham, and M. Abraham, J. Chem. Thermodynamics 14 (1982) 599.
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M. Michielsen, G. Y. Loix, J. Gilbert, and P. Claes, J. Chim. Phys. 79 (1982) 247. M. Biquard, P. Letellier, and M. Fromon, Can. J. Chem. 63 (1985) 3587. 19 M. Abraham, M.-C. Abraham, I. Ziogas, and Z. Kodejš, J. Am. Chem. Soc. 115 (1993) 4300. 20 T. B. Tripp, J. Electrochem. Soc. 134 (1987) 848. 21 K. S. Pitzer, J. Am. Chem. Soc. 102 (1980) 2902. 22 K. S. Pitzer, Phys. Chem. Earth 13-14 (1981) 249. 23 J. M. Simonson, Thermodynamic Properties of Very Concentrated Electrolyte Solutions, Ph. D. Thesis, University of California, Berkeley, 1983. 24 K. S. Pitzer and J. M. Simonson, J. Phys. Chem. 90 (1986) 3005. 25 J. M. Simonson and K. S. Pitzer, J. Phys. Chem. 90 (1986) 3009. 26 O. Weres and L. Tsao, J. Phys. Chem. 90 (1986) 3014. 27 J. C. Barry, J. Richter, and E. Stich, Ber. Bunsenges. Phys. Chem. 92 (1988) 1118. 28 S. L Clegg and P. Brimblecombe, J. Phys. Chem. 94 (1990) 5369. 29 E. Boßmann, J, Richter, and A. Stark, Ber. Bunsenges. Phys. Chem. 97 (1993) 240. 30 J. Geerlings, J. Richter, L. Rørmark, and H. A. Øye, Ber. Bunsenges. Phys. Chem. 101 (1997) 1129. 31 S. V. Petrenko and K. S. Pitzer, J. Phys. Chem. B 101 (1997) 3589. 32 T. B. Tripp and J. Braunstein, J. Phys. Chem. 73 (1969) 1984. 33 R. H. Stokes and R. A. Robinson, J. Am. Chem. Soc. 70 (1948) 1870. 34 S. Brunauer, P. M. Emmet, and E. Teller, J. Am. Chem. Soc. 60 (1938) 309. 35 R. B. Anderson, J. Am. Chem. Soc. 68 (1946) 686. 36 M. Abraham, J. Chim. Phys. 81 (1984) 207. 37 M. Abraham and M.-C. Abraham, Electrochim. Acta 33 (1988) 967. 38 G. W. Stewart and R. M. Morrow, Phys. Rev. 30 (1927) 232. 39 M.-C. Abraham and M. Abraham, Monatsh. Chem. 128 (1997) 805. 40 M. R. Ally and J. Braunstein, Fluid Phase Equilibria 87 (1993) 213. 41 M.-C. Abraham, M. Abraham, and J. Sangster, J. Chim. Phys. 76 (1979) 125. 42 M.-C. Abraham, M. Abraham, and J. Sangster, Can. J. Chem. 58 (1980) 1480. 43 M. Abraham, J. Chim. Phys. 78 (1981) 57. 44 T. L. Hill, J. Chem. Phys. 14 (1946) 263. 45 W. Voigt, Monatsh. Chem. 124 (1993) 839. 46 J. Braunstein and M. R. Ally, Monatsh. Chem. 127 (1996) 269. 47 M. Abraham, Electrochim. Acta 26 (1981) 1397. 48 Y. K. Delimarskii and B. F. Markov, Electrochemistry of Fused Salts, Sigma Press, Washington, 1961. 49 M. Blander, “Thermodynamic Properties of Molten Salt Solutions,” in: Molten Salt Chemistry, M. Blander, ed., Chapter 3, Interscience Publishers, New York, 1964. 50 J. Lumsden, Thermodynamics of Molten Salt Mixtures, Academic Press, New York, 1966. 51 G. J. Janz, Molten Salts Handbook, Academic Press, New York, 1967. 52 H. Bloom, The Chemistry of Molten Salts, W. A. Benjamin Inc., New York, 1967. 53 J. O’M. Bockris and A. K. N. Reddy, “Ionic Liquids,” in: Modern Electrochemistry, Vol.1, Chapter 6, Plenum Press, New York, 1970. 54 J. Lumsden, “Polarization Energy in Ionic Melts,” in: ionic Liquids, D. Inman and D. G. Lovering, eds., Chapter 20, Plenum Press, New York, 1981. 18
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G. N. Papatheodorou, “Structure and Thermodynamics of Molten Salts,”in : Comprehensive Treatise of Electrochemistry, B. E. Conway, J. O’M. Bockris, and E. Yeager, eds., Vol. 5, Chapter 5, Plenum Press, New York, 1983. 36 Z. Kodejš and G. A. Sacchetto, J. Chem. Faraday Trans. 1 82 (1986) 1853. 57 V. Brendler and W. Voigt, J. Solution Chem. 24 (1995) 917. 58 V. Brendler and W. Voigt, J. Solution Chem. 25 (1996) 83. 59 M. R. Ally and J. Braunstein, J. Chem. Thermodynamics 30 (1998) 49. 60 M. Abraham and M.-C. Abraham, Electrochim. Acta 46 (2000) 137. 61 K. S. Pitzer, Thermodynamics, ed., McGraw Hill, New York, 1995. 62 M.-C. Abraham, M. Abraham, and J. M. Sangster, Can. J. Chem. 56 (1978) 635. 63 E. Darmois and G. Darmois, Electrochimie Théorique, Masson et Paris, 1960. 64 Handbook of Chemistry and Physics, ed, C. R. C. Press, Boca Raton, 1980-1981. 65 A. Apelblat, AIChE Journal 39 (1993) 918. 66 G. A. Sacchetto and Z. Kodejš, J. Chem. Soc., Faraday Trans 1 84 (1988) 2885. 67 G. A. Sacchetto and Z. Kodejš, J. Chem. Thermodynamics 21 (1989) 585. 68 M. R. Ally and J. Braunstein, United States Patent, Number: 5,294,357, Mar. 15, 1994. 69 M. Hadded, M. Biquard, P. Letellier, and R. Schaal, Can. J. Chem. 63 (1985) 565. 70 M. Allen, D. F. Evans, and R. Lumry, J. Solution Chem. 14 (1985) 549. 71 M.-C. Abraham, M. Abraham, A. Combey, and J. Sangster, J. Chem. Eng. Data 28 (1983) 259. 72 M. Abraham and M.-C. Abraham, Electrochim. Acta 31 (1986) 821. 73 K. S. Pitzer, J. Phys. Chem. 77 (1973) 268. 74 K. S. Pitzer, Acc. Chem. Res. 10 (1977) 371. 75 Z. Kodejš and G. A. Sacchetto, J. Chem. Soc. Faraday Trans. 88 (1992) 2187. 76 H. H. Emons, W. Voigt, and W. F. Wollny, J. Electroanal. Chem. 180 (1984) 57. 77 K. H. Khoo, K. R. Fernando, R. J. Fereday, and C.-Y. Chan, J. Solution Chem. 24 (1995) 1039. 78 M. R. Ally and J. Braunstein, Fluid Phase Equilibria 120 (1996) 131. 79 I. Horsák and I. Sláma, Collect. Czech. Chem. Commun. 52 (1987) 1672. 80 M. Hadded, H. Bahri, and P. Letellier, J. Chim. Phys. 83 (1986) 419. 81 M. Abraham, M.-C. Abraham, and I. Ziogas, J. Am. Chem. Soc. 113 (1991) 8583. 82 A. N. Campbell and D. F. Williams, Can. J. Chem. 42 (1964) 1778. 83 H. Bloom, F. G. Davis, and D. W. James, Trans. Faraday Soc. 56 (1960) 1179. 84 E. A. Guggenheim and N. K. Adam, Proc. Roy. Soc. A139 (1933) 218. 85 E. A. Guggenheim, Mixtures, Clarendon Press, Oxford, 1952. 86 J. A. V. Butler, Proc. Roy. Soc. A135 (1932) 348. 87 P. D. Sonawane and A. Kumar, Fluid Phase Equilibria 157 (1999) 17. 88 T. B. Tripp and J. Braunstein, Chem. Commun. Com. 1244 (1968) 144. 89 T. B. Tripp, J. Chem. Thermodynamics 7 (1975) 263. 90 T. B. Tripp, “Vapor Pressures of Aqueous Melts at 130 ° Containing ” in: Molten Salts, J. P. Pemsler, J. Braunstein, D. R. Morris, K. Nobe, and N. E. Richards, eds., p. 560, The Electrochemical Society Proceeding Series, Princeton, NJ, 1976. 91 G. A. Sacchetto and Z. Kodejš, J. Chem. Soc., Faraday Trans 1 78 (1982) 3519. 92 Z. Kodejš and G. A. Sacchetto, J. Chem. Soc., Faraday Trans 1 78 (1982) 3529. 93 J. Gilbert, G. Y. Loix, R. Culot, and P. Claes, Electrochim. Acta 23 (1978) 1205.
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P. Pacák, J. Solution Chem. 22 (1993) 839. G. A. Sacchetto and Z. Kodejš, “Studies on the Behaviour of Hydrous Melts: the NonAqueous Liquid System from Thermodynamic and Volumetric Measurements,” in: Molten Salt Chemistry and Technology, M. Chemla and D. Devilliers, eds., Vol. 73-75, p. 227, Trans Tech Publications, Materials Science Forum, Switzerland. 1991. 96 A. J. Rabinowitsch, Z. Phys. Chem. 99 (1921) 417. 97 A. N. Campbell and W. G. Paterson, Can. J. Chem. 36 (1958) 1004. 98 M. Abraham and M.-C. Abraham, J. Phys. Chem. 94 (1990) 900. 99 I. Ziogas, M.-C. Abraham, and M. Abraham, Electrochim. Acta 37 (1992) 349. 100 G. Perron, A. Hardy, J.-C. Justice, and J. E. Desnoyers, J. Solution Chem. 22 (1993) 1159. 101 H. Dumont, S. V. Qian, and B. E. Conway, J. Molec. Liq. 73 (1997) 147. 102 A. N. Campbell, E. M. Kartzmark, M. E. Bednas, and J. T. Herron, Can. J. Chem. 32 (1954) 1051. 103 P. Claes, M. Michielsen, and J. Glibert, Electrochim. Acta 28 (1983) 429. 104 M. Abraham and M.-C. Abraham, Electrochim. Acta 32 (1987) 1475. 105 M. Abraham, I. Ziogas, and M.-C. Abraham, J. Solution Chem. 19 (1990) 693. 106 M. Abraham, M.-C. Abraham, and I. Ziogas, Electrochim. Acta 41 (1996) 903. 107 A. Das, S. Dev, H. Shangpliang, K. L. Nonglait, and K. Ismail, J. Phys. Chem. B 101 (1997) 4166. 108 C. T. Moynihan, “Mass Transport in Fused Salts,” in: Ionic Interactions: From Dilute Solutions to Fused Salts, S. Petrucci, ed., Vol. 1, Chapter 5, Academic Press, New York, 1971. 109 G. E. Walrafen, “Raman and Infrared Spectral Investigations of Water Structure,” in: Water, a Comprehensive Treatise, F. Franks, ed., Vol. 1, Chapter 5, Plenum Press, New York, 1972. 110 W. A. P. Luck, Disc. Faraday Soc. 43 (1967) 115. 111 R. Leberman and A. K. Soper, Nature 378 (1995) 364. 112 K. Sigwart, in: Properties of Ordinary Water-Substance, Compiled by N. E.Dorsey, p. 64, Reinold, New-York, 1940. 113 M. S. Jhon and H. Eyring,“The Significant Structure Theory of Liquids,”in: Physical Chemistry, an Advance Treatise, D. Henderson, ed., Vol. VIIIA, Chapter 5, Academic Press, New York, 1971. 114 H. Eyring, J. Chem. Phys. 4 (1936) 283. 115 S. Glasstone, K. J. Laidler, and H. Eyring, The Theory of Rate Processes, McGraw-Hill, New York, 1941. 116 G. J. Janz and F. Saegusa, J. Electrochem. Soc. 110 (1963) 452. 117 J. O’M. Bockris and S. R. Richards, J. Phys. Chem. 69 (1965) 671. 118 T. Emi and J. O’M. Bockris, J. Phys. Chem. 74 (1970) 159. 119 J. Zarzycki, J. Phys. Rad. 18 (1957) 65. 120 J. O’M. Bockris and N. E. Richards, Proc. Roy. Soc. A241 (1957) 44. 121 J. O’M. Bockris, E. H. Crook, H. Bloom, and N. E. Richards, Proc. Roy. Soc. A255 (1960) 558. 122 M. Abraham and J. Brenet, C. R. Acad. Sci. 251 (1960) 2921. 95
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G. J. Janz and M. R. Lorenz, J. Electrochem. Soc. 108 (1961) 1052. M. Abraham, Etude sur les Potentiels d’Elctrodes dans les Sels Fondus, Thèse de Doctorat, Université de Strasbourg, France, 1963. 125 B. Cleaver, Nature 207 (1965) 1291. 126 P. Cerisier, Rev. Int. Hautes Tempér, et Réfract. 8 (1971) 133. 127 P. Cerisier and B. Blin, Ultrasonics 5 (1982) 130. 128 M. Abraham and M.-C. Abraham, J. Chim. Phys. 83 (1986) 115. 129 M.-C. Abraham, Pression de Vapeur d’Eau, Densité, Viscosité et Conductivité Elec-trique de Solutions Electrolytiques Fortement Ioniques, Thèse de Doctorat, Université de Montréal, Canada, 1987. 130 T. L. Hill, J. Chem.Phys. 28 (1958) 1179. 131 A. H. Narten and H. A. Levy, “Liquid Water: Scattering of X-Rays,” in: Water, a Comprehensive Treatise, F. Franks, ed., Vol. 1, Chapter 8, Plenum Press, New York, 1972. 132 A. Ben-Naim, “Application of Statistical Mechanics in the Study of Liquid Water,” in: Water, a Comprehensive Treatise, F. Franks, ed., Vol. 1, Chapter 11, Plenum Press, New York, 1972. 133 H. S. Frank, “Structural Models,” in: Water, a Comprehensive Treatise, F. Franks, ed., Vol. 1, Chapter 14, Plenum Press, New York, 1972. 134 M. Spiro and F. King, “Transport Properties in Concentrated Aqueous Electrolyte Solutions,” in: Ionic Liquids, D. Inman and D. G. Lovering, eds., Chapter 5, Plenum Press, New York, 1981. 135 H. S. Frank and M. W. Evans J. Chem. Phys. 13 (1945) 507. 136 H. S. Frank and W. Y. Wen, Discuss. Faraday Soc. 24 (1957) 133. 137 M. Kaminsky, Discuss. Faraday Soc. 24 (1957) 171. 138 R. L. Kay and D. F. Evans, J. Phys. Chem. 70 (1966) 2325. 139 R. E. Verrall, “Infrared Spectroscopy of Aqueous Electrolyte Solutions,” in: Water, a Comprehensive Treatise, F. Franks, ed., Vol. 3, Chapter 5, Plenum Press, New York, 1973. 140 R. Fürth, Proc. R. Soc. Camb. Phil. Soc. 27 (1941) 252. 141 J. H. Hildebrand, Science 174 (1971) 490. 142 A. J. Batschinski, Z. Phys. Chem. 84 (1913) 643. 143 R. P. Chhabra and R. J. Hunter, Rheol. Acta 20 (1981) 203. 144 M. Abraham, “Some Recent Contributions of Molten Salts Studies to the Under-standing of Very Concentrated Aqueous Solutions,” in: Molten Salts: Ninth Inter-national Symposium. Proceedings, C. L. Hussey, D. S. Newman, G. Mamantov, and Y. Ito, eds., Volume 94-13, p. 83, The Electrochemical Society, Pennington, NJ, 1994. 145 M. Abraham, J. P. Chevillot, and J. Brenet, C. R. Acad. Sci. 256 (1963) 2129. 146 S. Zuca, Rev. Roum. Chim. 15 (1970) 1277. 147 J. Kestin, M. Sokolov, and W. A. Wakeham, J. Phys. Chem. Ref. Data 7 (1978) 941. 148 R. L. Martin, J. Chem. Soc. 3 (1954) 3246. 149 A. N. Campbell and D. F. Williams, Can. J. Chem. 42 (1964) 1984. 150 A. E. Stearn and H. Eyring, J. Chem. Phys. 5 (1937) 113. 151 J. O’M. Bockris, J. A. Kitchener, S. Ignatowicz, and J. W.Tomlinson, Trans. Faraday Soc. 48 (1952) 75. 124
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5
Factors Limiting Applications of the Historically Significant Born Equation: a Critical Review B.E. Conway Chemistry Department, University of Ottawa, 10 Marie Curie Street, Ottawa, ON KIN 6N5, Canada
1. GENERAL INTRODUCTION The state of ions in solution depends in a major way on their solvation, i.e. the interaction of the ions originating from dissolution of a salt or dissociation of an acid, with solvents. The standard Gibbs energies of solvation of ions determine solubilities of salts and dissociation constants of acids. Solvation also plays a major role in determining conductance of electrolytes and activity coefficients at high concentrations. Such energies and solvated states of ions also play a central role in kinetics of electron-transfer processes ("reorganization energies" in the activation process) and in the structure and properties of the double-layer at electrode interfaces. Also, the calculation of solvation energies and entropies, coupled with modeling of the solvated state of ions in solution, as in the early Modern Aspects of Electrochemistry, Number 35, Edited by B.E. Conway and Ralph E. White, Kluwer Academic/Plenum Publishers, 2002 295
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treatment of Born1, has constituted a principal area in the field of Ionics in Electrochemistry where Born's equation has played an important part in the early development of the subject. That the Born equation1 for calculations of ionic solvation energies, based on dielectric polarization, has a principal limitation due to assumption of a continuum dielectric having an homogeneous value of its permittivity, is well known. However, there are a number of other, more subtle, difficulties that require discussion. These are the subject of the present review which examines the bases of the Born treatment and its relation to more satisfactory molecular-level calculations. Notwithstanding such limitations, the Born-type model has been involved in a central but implicit way in treatments of the activation process in electrochemical "outer-sphere" electron-transfer reactions according to the "polaron" concept in works by Marcus and by Levich, while Born solvent polarization calculations have appeared, even until recent times, for the purpose of evaluation of dielectric polarization energies, and their derivatives with respect to temperature and pressure as are involved, for example, in entropies and volumes of acid dissociation processes. Hence the justification for further examination of the Born treatment as used in more recent contexts. As will be seen from the text which follows, a number of significant and interesting problems arise in the Born treatment in addition to the well known, unrealistic assumption of a continuum dielectric model. Some new aspects are treated here.
II. HISTORICAL INTRODUCTION At the time of Faraday's experiments on electrolytic conduction of solutions of salts and acids2, the origin of this phenomenon remained controversial and poorly understood for some years. Several principal reasons for that situation, seen in hindsight,
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were as follows: a) heats of formation of salts from their elements were known to be large, on the order of yet when such salts were dissolved in water their respective heats of solution were only on the order of zero to ± i.e. ca. 10% of the corresponding energies of formation; b) the dissolution process from the solid crystalline state was usually rapid and spontaneous, as was the generation of conducting solutions by dissolution of molecular acids such as HC1 gas and almost anhydrous although for the latter cases the process of dissolution into water was more exothermic; c) it was not understood, at the time of Faraday's works, that solid salts were in most cases (with the exception e.g. of already in a fully ionized state as a 3-dimensional lattice of ionized atoms and finally, d) that there was a "driving force" for dissolution arising from a negative, and usually large, Gibbs energy of solvation [hydration] of the ions of the solid salt lattice or of the ions of acids resulting from spontaneous solvolytic dissociation of the parent molecular acids. That the resulting exothermic solvation energies of the dissolving ions largely compensate the endothermic energy of splitting up the salt lattice or forming cations and anions from an acid, "HA", by solvolytic dissociation, was not understood until much later when Born-Haber cycles for the component processes became to be written. Another complicating feature, again not initially understood, was the generation of alkalinity and acidity in the electrolysis of neutral salts, e.g in parallel with respective generation of and in a cell with separated cathode and anode compartments, while, with acids, only formation of and appeared to be the result of electrolysis2,3,4. The interpretation of the result of the latter type of experiment with acids was, of course, faulty since the pH near the cathode would actually have been increased and that near the anode decreased under corresponding conditions. In a stirred solution of salt, the net result will, of course, always be removal of water according to the complementary processes and corresponding to a net process of
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formation of 2 with recombination of and ions in solution. In the case of aqueous the resulting process is simply that the concentration of the acid increases. (Note, however, the result that, with HC1, the anion species is continuously removed as causing dilution of the acid solution.)
III. BASIS OF THE DERIVATION OF THE BORN EQUATION FOR EVALUATION OF ENERGIES OF ION SOLVATION 1. Principles of the Calculation Ion/solvent interaction is a major feature of solution chemistry, especially electrochemistry. The strength of such interactions remained unappreciated through Faraday’s time and it was not until 1920 that an elegantly simple calculation of energies of solvation (hydration) was given by Born1 in terms of dielectric polarization and the self-energy of a charged particle in a solvent compared with that in the gas-phase. The Born calculation provided values of standard Gibbs solvation energies that were of the correct order of magnitude required to account for salt solubilities and acid dissociation constants, and predicted meaningful dependences of such energies on ionic radius and charge number. Additionally, in the earlier years and, in fact, continuing in some cases to more recent times, the Born equation has been applied to some organic reaction mechanisms where ionic intermediates are involved. Depending on the solvent, stabilization of such intermediates can arise through dielectric* polarization the energy of which can be estimated be means of the Born calculation; the
* It is of historical interest that the word dielectric was suggested to Faraday by the polymath scholar, Whewell, of the University of Cambridge in correspondence about electrical terminology. Other suggested words were cathode, anode and electrode.
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Menchutkin reaction, quaternization, and solvolysis of esters in various media, are examples5. Other applications are to solvent effects on dissociation constants of acids5,6,7 and derivative quantities such as partial molar entropies (from or volumes (from In order to be able to discuss limitations of and problems with the Born model in following sections of this article, it is first necessary to present the formal derivation of Bern's equation for calculation of ionic solvation (or hydration) energies, as follows. The Born equation is based on comparison between the energy change associated with the notional process of charging a neutral particle having the radius of the ion concerned in the solvent phase (dielectric permittivity ) and that for the process conducted in the gas phase (Fig. 1, p. 253 in ref. 37). Both charging processes lead to a Gibbs energy so the difference corresponds to a Gibbs energy of solvation of the ion in the solvent medium. The conventional derivation of the Born equation involves the following steps: suppose the initial particle has acquired a transitional charge, being a fraction of its final, full charge q, where The electric potential is in the gas-phase or in the dielectric solvent. Continuation of the charging process by addition of a further element of charge involves an energy increase of or dq can be represented as giving the energy increment as or for the solvent phase. Integration w,r.t. then gives the overall charging energies respectively as or Upon performing the integrations, the energy difference corresponding to a Gibbs energy of solvation of the resulting ion is found as
For an ion, q is identified with its formal ionic charge. Note that this resulting energy is a Gibbs energy, the entropy component of
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which arises from the T-dependence of which, for water, is appreciable, arising from the latter’s substantial H-bonded structure. Also, it is seen from Eq. (1) that is little dependent on for most polar solvents where A conceptionally analogous relation follows from the earlier paper by Drude and Nernst8. An alternative derivation (not previously made, according to the author’s knowledge) can be given as follows. The capacitance, C, of a sphere is identical with its radius, here The self-energy, G, of a capacitance is where V is the electric potential (difference) to which the plates of C are charged. Here, hence, with
for a vacuum or for a dielectric of permittivity in the notional capacitor. Thus Bern's relation can be recovered by a different route. It should be remarked that the Born process of charging an initially neutral atom or molecule to generate a corresponding ionic species does not correspond to a real chemical process (see also Section VII for the case of since formation of an ion from an atom or molecule is a discontinuous and quantized event, also producing an electron (in cation formation – the ionization energy) or receiving an electron (in anion formation – the electron affinity). It is for this reason that we refer to the Born charging process as "notional". Note that in reality, if the conceptual gas-phase to solution ion-transfer process (Fig. 1, p. 253 in ref. 37) were actually to take place, an additional interfacial transfer Gibbs energy, would arise, corresponding to transfer of the ion charge, ze across the interfacial surface potential difference, of the solvent. The total transfer energy is then called the “real” solvation energy, as it would correspond to the energy change for supposed real transfer of an ion from the gas to the solution phase, across the boundary of the latter.
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In practice, such a situation does not normally arise; for example, for homogeneous ion formation, Section X, as arises in acid dissociation (process II later), there is no requirement for the charged ions actually to be transferred across any interphasial boundary at which a surface potential difference may exist. An interesting aspect of the condition of a polarized dielectric surrounding an ionic charge was noted by Gurney13: if the charge was adiabatically neutralized, like in adiabatic demagnetization, a substantial cooling effect should arise, mainly an account of gain of entropy of the dielectric. A similar situation could arise in discharge of a charged capacitor but, in that case, ohmic heating could obscure the effect due to the required electrolytic passage of discharging current. In some ionic processes, homogeneous extinction of charges does take place, e.g. in the neutralization of an acid by a base the heat of neutralization includes the loss of polarization energy of the solvent around the ions coupled with the energy of formation of an OH- bond in the production of 2. Comparison with Experimental Data Comparison of ionic hydration energies, calculated by means of Born's equation, with individual "experimental" values is restricted by the fact that individual ionic values of hydration energies can only be derived from data for salt pairs of ions (cation + anion values) by application of some extra-thermodynamic principle for dividing up the experimentally accessible salt values. Notwithstanding this limitation, comparisons can be attempted, using appropriate ionic radii and the best values for "experimental" ionic components of salt values. However, the above restriction and associated problems can be avoided if calculated cation + anion values are directly compared with the overall total salt hydration energy. Oddly enough, this approach does not seem to have been made in the literature, so we perform such a comparison here, giving the results shown in Table 1 for the alkali
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halides. The calculated values are based on the Pauling scale of crystal ionic radii.
It is seen that the calculated values differ by a large factor from the experimental ones, the difference decreasing with increasing radii of the cations or anions of the salt. This is to be expected as the effect of finite size of the solvent molecules (Section VI) becomes relatively smaller at larger ions. However, the ratio of calculated to experimental values diminishes somewhat from LIF (1.68) to RbF (1.26) or from LiI (1.90) to RbI (1.36), indicating large discrepancies.
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Mainly but not entirely (because of dielectric saturation effects which tend to decrease with ion size), the discrepancies revealed in Table 1 arise because the Born model allows strong dielectric polarization right up to the peripheries of ions since the solvent size factor (Section VI) is not brought into consideration. As noted there, this effect is analogous to the neglect of finite ion size in the GouyChapman theory of the double-layer, which leads to values of interfacial capacitance that are much too large.
IV. CHANGE OF RADIUS OF AN ION UPON ENTRY INTO SOLUTION The Born equation refers to the difference between the charging energy of an initially neutral particle in solution compared with that in the gas phase, with a common value of its radius being assumed for both the gas-phase and solution-phase environments. However, this assumption requires comment. In solution, the ion, initially in the gas-phase, suffers intrinsic electrostriction due to the derivative of the large electrostatic energy force that arises between the ion and its coordinating solvent dipoles, giving rise to the ion’s solvation energy. Correspondingly, the ion-solvent interactions also give rise to electrostriction of the solvent, especially of the solvent's structural free volume which is particularly large (ca. in the case of water. In formal terms of dielectric polarization, these electrostriction effects would be calculated (see chapter 25 in ref. 37) in terms of the pressure dependence of the dielectric constant. The intrinsic electrostriction of ions can be estimated from the crystal ionic volumes, as listed in Table 2. In solution, ions in solvated states experience interaction energies with solvent dipoles (especially for high polarity solvents containing a strongly electronegative atom, e.g. F or O, as in or HF) comparable with the electrostatic energies experienced by ions in salt crystal lattices
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with neighboring ions of opposite charge. Hence, the ion "selfelectrostriction" in solution10 can be anticipated to be similar to that in a crytalline salt but dependent on the identity and charges of the coions of the salt lattice.
a
Units,
Numbers in brackets are corresponding Pauling crystal ionic volumes.
b
From Pauling univalent radius for cation of noble gas.
c
Values derived from corresponding gas-phase radii derived by Stokes using the
"quantum-mechanical scaling principle" (Ref. 9).
Estimates of the self-electrostriction effect are given in Table 3 based on data given in refs. 10 and 11. The above electrostriction effects have not previously been included in evaluations of ionic hydration Gibbs energies by means of the Born equation or of corresponding derivative quantities.
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However, it can be usefully noted that for processes involving direct solution of a solid salt with corresponding solvation of the dissolved ions, the relevant radii of the actual ions before and after dissolution from the solid salt will fortuitously be quite comparable, within a few percent. This point has not been previously discussed but has been implicity assumed.
Units,
Based on the respective differences of the data in Table 2, for gas-
phase volumes and the crystal tonic volumes derived from the cubes of the corresponding Pauling radii, multiplied by
V.
CHANGE OF RADIUS OF A PARTICLE UPON CHARGING
Another problem with the notional process of charging an initially uncharged atomic particle, used in the Born treatment for evaluating ionic solvation energies, is that a change of value of the particle's electronic radius takes place upon charging: a decrease, in the case of cation formation and an increase in the case of anion
306
Born Dielectric Polarization
formation. This is readily seen by comparison between the radii of isoelectronic cations, anions and a corresponding noble gas, e.g. in the series and (see Table 2 above). By means of the so-called “quantum – mechanical scaling procedure,” Stokes9 has evaluated such effects, as illustrated in Table 2.
VI. COMPARISON OF THE GAS-PHASE AND SOLUTION-PHASE CHARGING ENERGIES IN THE BORN EQUATION As discussed above, in Section III, the Born equation is composed of two different charging energy components: and For most solvents in which salts are significantly or apprecially soluble , Hence the principal origin of the relatively large, negative, Gibbs energies of solvation of ions (on the order of –200 to is the loss of the gas-phase energy of charging, which should have been separately evaluated in the term using the gas-phase value of ri: In fact, as referred to in Section III, the Born equation predicts little specificity of the Gibbs energy of ionic solvation to the dielectric properties of the solvent, through its value, until values of arise (cf. Fig. 2). However, for such solvents, salts solubilities are very low (except for some organic salt having hydrophobic groups of ca. 6 C atoms, as in aromatic salts) and ionic processes in such media are only of interest in regard to certain organic-reaction mechanisms in non-aqueous media12 and to ion-pairing. The charging process envisaged in Bern's calculation for the solute particle in the dielectric medium leads to a decrease of the Gibbs energy of the solvent corresponding to a lowering of the energy of the dissolved ion, relative to that in the gas phase or vacuum leading to a large negative Gibbs energy of solvation. As noted above, this mainly originates from the loss of the charging "self-energy" the ion has in vacuum (the term in the factor in the Born equation). The
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lost Gibbs energy is stored in the dielectric as electric polarization of the solvent molecules and as energy associated with local changes of the solvent's structure, especially in the case of H-bonded solvents, alcohols, HF, etc.
VII. STRUCTURE AND VOLUME FACTORS IN THE SOLVENT CO-SPHERE AROUND AN ION The Born model treats a structureless continuum of the solvent up to the electronic boundary of the ion. In this sense, it allows a continuity of dielectric polarization right up to the ion's geometrical surface (its electronic boundary) which is physically totally unrealistic owing to the finite volume of solvent molecules and the discrete nature of the ion-solvation complex (Fig. 3), Gurney's cosphere13. This difficulty is analogous to that of the Gouy-Chapman 14,15 model of the diffuse layer at electrodes where the distribution of
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Born Dielectric Polarization
charge in the double-layer is treated as that of point charges (Fig. 4a) which can closely approach the electronic surface of the metal and thus generate an high space-charge density, corresponding to an unrealistically high specific interfacial capacitance. This anomaly is only resolved by taking account of the finite sizes of the ions adsorbed at the electrode giving rise to an inner contact limit for the diffuse layer determined by the radii of solvated cations or of the usually less solvated anions, as recognized in the model of Grahame16 and earlier of Stern17 (see Fig. 4b). The above situation for an ion of radius being solvated by, say, n solvent molecules of radii leads to a primary (cf. refs. 18 and 19) solvation cosphere having an annular structure in which a freespace arises (which is unpolarizable) determined by the quantity As a fraction, f, of the total apparent polarizable volume of the cosphere, due to the ion's field, f is obviously
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The above situation for an ion of radius being solvated by, say, n solvent molecules of radii leads to a primary (cf. refs. 18 and 19) solvation cosphere having an annular structure in which a freespace arises (which is unpolarizable) determined by the quantity As a fraction, f, of the total apparent polarizable volume of the cosphere, due to the ion's field, f is obviously
in terms of the ratio of solvent radius to ion radius (Fig. 5). Then, of course, as continuum model, and Thus, as the result in Eq. 6 corresponds to the Born model in the limit of a continuum dielectric where the solvent is a continuum of
310
Born Dielectric Polarization
point-dipoles hence giving rise to a continuum dielectric. The non-polarizable, free-space fraction is obviously smaller when the accommodation of coordinating solvating solvent molecules, n per ion, is larger. This will be allowed as increases down the sequence of elements of a given group of the Periodic Table, for a given charge value for the ion and a given value.
This discreteness of size factor is, in some ways, indirectly allowed for in attempts to include the effect of dielectric saturation determined by the distance dependence of the ion’s electrostatic field across the annular solvation cosphere, as illustrated in Fig. 3 and treated in the paper of Laidler and Pegis20 for monatomic ions and of Conway, Desnoyers and Smith21 for poly-ions (polyelectrolytes). (Note that analogous treatments for solvent dielectric saturation within the double-layer at electrodes were given by Grahame22 and by Conway, Bockris and Ammar23, based on the polarization theory of Booth24.)
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VIII. CASES OF HYDRATION OF THE PROTON AND THE ELECTRON For the cases of the proton and the electron application of the Born equation to evaluation of their respective hydration energies leads to a virtual reductio ad absurdum. For the gas-phase proton, the ionic radius is on the order of and ze is The Gibbs energy for its hydration would then be on the order of owing to the assumed, very small value of its ionic radius. However, strictly, the Born charging process would notionally be carried out on the atom H but, to make from it, involves diminution of the initial H particle's radius by the order of 5 magnitudes, ca. to Hence, obviously, the formal Born charging process is inapplicable here since the ionization of H involves the complete removal of the 1s electron with an energy of and also generates a free electron charge having itself a large self-energy. In the case of it is well known25,26,27 that its hydration involves first the ion-molecule reaction.
the energy of which process is the proton affinity of water, analogous to the ionization of giving (cf. ref. 26). The energy change in reaction I is ca. while the further interaction energy of with bulk water is ca. corresponding to an hydrated state of having a formal stoichiometric formula of (cf. ref. 28), as identified by means of gas-phase mass spectrometry29. While the above figure is large, it is much less than that which would formally arise from application of the Born charging process to not recognizing the then quantum-mechanical aspect of ion formation from H, referred to earlier. Such a value is on the
312
Born Dielectric Polarization
order of or ca. starting from an H particle having the Bohr 1s electron radius. Note that the gas-phase ionization energy of H is accurately known to be Clearly, application of the Born charging process for leads to a reductio ad absurdum. Rather, the gas-phase process for treating the charging process for generation of should start with a neutron, formally. Similar problems arise for ionization of Li. Considering the hydration of the electron runs into apparently similar problems: first, what is the gas-phase particle at which notional negative charging to form takes place? As for the formal Born charging to form would presumably be very large. For the solution-phase process of hydrating (or solvating) the electron, a good deal of information is available in the literature30,31,32. However, uncertainty arises in ascribing a radius to the electron in solution owing to its wave character, so that its effective radius is believed to be quite large, on the order of 0.2 run and its hydration energy is ca. In fact, the situation of in a solvent, e.g. or approaches the well known quantum-mechanical case of a "particle in a box". However, in water, it is a transient species, leading to as known from radiation chemistry of water. In liquid ammonia or solutions of dissolved metallic sodium, it is, however, much more stable, a situation originating from ion-pairing with species. However, such solutions have electronic conductivity. Also, solvated electrons can be directly injected into solvents by cathodic electrolysis at a metal, e.g. Hg. It may seem that the case of the solvated electron in solution is least likely to be treatable by means of the Born model and calculation. Nominally, the electron would be treated as a particle having an extremely small radius: however, as referred to above, the wave-character of the electron allows it to have relatively large effective radius related to the size of the solvent structural cavity in which it finds itself. In water, the average cavity volume is ca. 30% of the molar volume of liquid water at 298K.
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In polar solvents, the energy levels of the solvated electron have been calculated in an early paper by Landau30 and treated by Jortner31 on the basis of a continuum model (like that in Bern's calculation) in which trapping of the electron is considered in terms of the polarization it induces in the dielectric medium (cf. the Born model) and with which it interacts. The charge distribution and energy levels depend on the interaction of the electron with the polarized dielectric, treated on a "long-range polaron" interaction basis, an assumption of doubtful validity. For example, in water or ammonia, structural fluctuations of solvent dipoles (vibrational and librational modes) complicate the electron's local environment and arise at infra-red frequencies. The calculations lead to results in only fair agreement with experiment; for example, for the enthalpy of solution, the data compare as follows: theoretical, 1.32 eV experimentally derived, 1.6 eV However, the experimental value is itself not without significant uncertainty. Another complicating factor is that the electron arising from dissolution of Na metal in liquid is principally in an ion-paired state with the ion. However the solvated electron can also be injected directly into appropriate solvents such as from a cathodically polarized electrode. In such a system, its state is then different from that of an electron in and more appropriate for calculations of the solvation of the actual free electron. The final conclusion here is that any attempt to apply the Born charging treatment to solvation of or e would be far from realistic and fraught with many uncertainties.
314
Born Dielectric Polarization
IX. RELATION TO MOLECULAR MODELING THROUGH ION-DIPOLE INTERACTIONS In terms of polarizable volume of solvent (water) near to an ion in its Gurney cosphere (Section VI), it is, of course, much more preferable to treat the ion-solvent interaction in that region locally, on a molecular basis, i.e. in terms of ion/solvent-dipole and ion/solventquadrupole interactions, as in the works of Bernal and Fowler27, Verwey33, Eley and Evans34, and Buckingham35, for the case of the latter ion/quadrupole component of ion/solvent interaction (especially for aqueous ion solutions). In relation to the Born calculation, differentiation of the polarization in terms of local components (6 in the case of octahedral coordination) in the Gurney cosphere10 is equivalent conceptually to introducing dielectric saturation effects within that region, thus reducing in a field-dependent way, over that volume. Introduction of the effect of dielectric saturation in the Gurney cosphere represents the first and elementary step towards molecular-level modeling of the important Gurney solvation cosphere of ions. Such calculations have been made by means of electrostatic modeling in the paper by Laidler and Pegis20, and for polyions by Conway, Desnoyers and Smith21. Taking account of dielectric saturation over the cosphere "primary hydration"18 region has the effect of diminishing the difference between the term and unity in the Born equation, thus tending to improve agreement with Gibbs energies of hydration determined for individual cations and anions36, based on experimental data for "salt pairs" of ions, by means of extrathermodynamic procedures as reviewed in refs 36 and 37.
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X. BORN EQUATION AS A BASIS FOR PLOTTING PROCEDURES FOR EVALUATION OF IONIC SOLVATION ENERGIES Gibbs energies or enthalpies of hydration of salt-pairs of anions and cations are easily and reliably evaluated from thermodynamic information on salt solubilities or heats of dissolution of salts, using an appropriate Born-Haber cycle36. However, often of greater interest is the evaluation of the individual ionic hydration energies of cations and anions. Such values can only be derived by employment of some extra-thermodynamic procedure for division of the salt values into ionic components. Various methods have been proposed (see review in ref. 36) amongst which is that of Latimer, Pitzer and Slansky38 based on a linear plotting procedure applied to differences of Gibbs energies of hydration of salts in either a series of salts having a common cation or one having a common anion, e.g. as for alkali halides. The values of such differences fall on a line when plotted versus the reciprocal, • of the ionic radius, based on the form of the Born equation (using the macroscopic dielectric constant of water). The best fit to the Born relation was obtained by empirically adding 0.01 nm to the crystallographic radii of the anions and 0.085 nm to those of cations. Such plots based on these "effective radii" are then linear and parallel. The salt value for the Gibbs energy of hydration of CsI was then divided into ionic components in such proportions that the Gibbs energies of hydration of all the alkali-metal and halide ions fell on one common line when the ionic components, thus derived, were plotted versus the reciprocals of the adjusted respective ionic radii, i.e. on the basis of a Bora polarization relation. Criticism may be raised against this procedure since, on the basis of the preferred ion-dipole interaction model (refs 27, 33 and 34), the relation of individual ion hydration energies to ionic radii is more complex: it depends, in the first analysis, on the reciprocal of
316
Born Dielectric Polarization
squared where is the radius of the solvent dipole; also, beyond the first Gurney coordination shell10, a residual Born charging energy arises, reciprocal in according to Bernal and Fowler27, thus further complicating the basis of a plotting procedure involving ion size, Results derived from the procedure of Latimer et al38. markedly disagree with those of Bernal and Fowler27 but are more consistent with those of Verwey33 based on an extrapolation procedure in again based on Bern's relation, for enthalpies of ionic hydration. It must be concluded that neither of these plotting procedures give reliable or consistent results since they are based on a substantially incorrect model of ion/solvent interaction. However, in their time, they did give important directions for improved modeling and more exact calculations, e.g. those of Eley and Evans34 and later by molecular dynamics methods. One of the principal difficulties concerned with application of the Born equation to "experimental" results36 for ionic hydration energies is that the well established19,36 differences of hydration energies of anions and cations of comparable (or interpolated) ionic radii (data for anions being consistently greater than those for cations, on this basis) are not accounted for through the Born equation since no ion-specific parameter (except is involved. In fact, only by empirical, different, additions to ionic radii of cations compared with anions could Latimer, Pitzer and Slansky38 usefully apply the Born equation to evaluation of individual ionic energies of hydration.
XI. RELATION TO IONIZATION PROCESSES IN SOLUTION It should be noted that the notional process of transfer of an ion into solution from the gas-phase, treated by Born, rarely takes
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place as a chemical process except with gas-phase electric discharges across a vapor/conducting-liquid interface at high voltages. However, direct heterogeneous electron transfer into solution takes place by photo-emission when the incident radiation photon energy exceeds the electron work formation of the emitter. A similar situation arises at cathode interfaces with certain non-aqueous media at sufficiently negative electrode potentials, forming solvated electrons, as mentioned in Section VII. Direct heterogeneous formation (or removal) of ions into (from) solution occurs, of course, in Faradaic oxidation (or reduction) of metals but requires a complementary electron transfer process for maintenance of "electro-neutrality". Many ion-forming processes in chemistry take place homogeneously without involvement of an interfacial charge transfer. Such reactions as acid ionization and transition-metal complexation are in this category, and involve a solvolytic process in which the required bond-breaking energy is almost matched by the solvation (hydration) energy associated with interaction of the resulting cation and anion with polar solvent (s) molecules. Thus
Such processes, unlike the notional Born model, always involve both cations and anions, solvated together in the bulk solvent. However, the chemical reaction II can be broken down into component steps by means of a Born-Haber cycle:
318
Born Dielectric Polarization
where is the enthalpy of solvolytic acid ionization of HA, is the H-to-A bond strength, the ionization energy of and the electron affinity of quantities are the solvation energies of and in the solvent, S. A similar Born-Haber cycle can be written in terms of standard Gibbs energy quantities, giving the of the acid HA. The Born-Haber cycle serves to illustrate the involvement of the notional Born gas-phase ion to solution-phase ion transfer processes for the cation and the anion that are involved implicitly in the homogeneous ionization reaction of HA in solvent S. The case of quaternization of tertiary organic N-bases by alkyl halides (RX) provides another example of creation of a pair of ions in solution, in this case according to a reaction of the type
in which the solvation of the resulting pair of ions is a major component of the "driving force" (standard Gibbs energy) for the above type of reaction to proceed to the right. In this example, however, again application of the Born equation is unreliable but for the additional reason that, in the water solvent, the hydration of the quaternary cation involves significant hydrophobic solvent structural interactions related to changes of solvent H-bonding effects around the ion; these are essentially non-electrostatic in nature and thus are superimposed in a complex way on the Born electrostatic polarization beyond the coordinating alkyl groups. In fact, the positively charged N-center is fully coordinated by low polarizability alkyl groups which completely block or screen the electrostatic interaction of the with polar solvent water molecules which would otherwise directly interact with the - center as at primary, secondary or tertiary However, beyond the hydrophobic screened coordination shell around the center, the ionic field is still strong enough for weak electrostatic polarization to
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remain, depending on the size of the coordinating alkyl (or aryl) groups which locally cause the structural changes when water (and to a lesser extent, methanol) is the solvent. However, in this latter region, a continuum, structureless dielectric medium is still inapplicable. Analogous effects arise in ionization of sterically hindered acids and protonation of sterically hindered N-bases, as studied by Everett and Wynne-Jones45 with respect to molar heat capacities and entropies of ionization.
XII. DIELECTRIC POLARIZATION EFFECTS IN KINETICS OF REACTIONS INVOLVING CHARGED TRANSITION STATES The kinetics of many chemical processes, including some organic reactions, e.g. the Menschutkin reaction, (quaternization of tertiary N-bases), and solvolysis of esters and alkyl halides, depend on the properties of the solvent in which such chemistry takes place. Such effects are rationalized in terms of transition states in which charges, or a distribution of charges, are developed which experience interactions with polar solvents through electrostatic dielectric polarization. The resulting effects on the kinetics of the reaction are treated in terms of a Born-type solvation of the transition state involving the dielectric constant of the solvent. A variety of works have been addressed to the subject of solvent effects on reaction kinetics. For example, Menschutkin studied quaternization in many solvents39 and Tommila examined various reactions, the kinetics of which are solvent-dependent40,41. In relation to Tommila's results, Laidler42 gave a comprehensive treatment of the kinetics in terms of the principles presented in the important paper of Laidler and Landskroener43, involving application of the Born equation.
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Born Dielectric Polarization
The simplest case is when two ions having charges and react via a double-sphere activated complex. The electrostatic (Coulombic) work of bringing the two ions together in an activated complex where they are distant d apart is This energy is the electrostatic part of the Gibbs energy of activation, complementary to a non-electrostatic, usually structure-dependent, component, Then the rate constant, k, can be written as
If the reacting particles are regarded as merging into an activated complex comprising a single sphere of charge the Born equation can be applied for evaluating the change of Gibbs energy, of the reacting ions as they form the activated complex, giving
where is the radius of the charged activated complex. Then the electrostatic component of ln k is
A more realistic treatment of the activated complex for a reaction involving ionic charges was given by Laidler and Landskroener43 in which the complex was regarded as a sphere in which the charges were embedded in a particular manner to represent the charge distribution in the complex ‡. Such a model can be treated by means of an expression derived by Kirkwood44 for the electrostatic change of energy, when a sphere of radius r is transferred from a medium where to one where the dielectric constant is i.e. a process corresponding to that in the treatment of Born. is then
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Note that since Eq. (10) gives a negative value for This result is simply a Born polarization energy term plus a second term arising from the additional effect of the charges embedded in a sphere. The resulting electrostatic component of the rate constant is then
noting that the factor for sufficiently large values of e.g. 78 for water. The above equation predicts that the ln of the rate constant will vary as and gives an explicit dependence of the slope of such a relation on the charges, radii and dipole moments of the species involved. The electrostatic treatments were applied by Laidler42 to reactions in relation to those following kinetics. Experimental results in the literature could be well accounted for.42
REFERENCES 1
M. Born, Zeit. Phys. 1 (1920) 45.
2
L. Pearce Williams, Michael Faraday: a biography, Chapman and Hall, London (1965).
3
M. Faraday (On Electrical Decomposition), Phil. Trans. Roy. Soc., London, 1834.
4
W. Nicholson, Nicholson's J. 4 (1800) 183.
5
S.J. Paddison, R. Pratt, T. Zawodzinski and D.W. Reagor, Fluid Phase Equilibric, 151 (1998) 235.
6
L.R. Pratt, G.J. Tawa, G. Hummer, A.E. Garcia and S.A. Corcelli, Intl. J. Quant. Chem., 64(1997) 121.
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7
S.B. Rempe, L.R. Pratt, G. Hummer, J.D. Kress, R.L. Martin and A. Redondo, J. Amer. Chem. Soc., 122 (2000) 966.
8
A, Drude and W. Nernst, Zeit. phys. Chem. 15 (1894) 79.
9
R.H. Stokes, J. Amer. Chem. Soc. 86 (1964) 979 and 982.
10
B.E. Conway and E. Ayranci, J. Soln. Chem. 28 (1999) 163.
11
J.E. Desnoyers, R.E. Verrall and B.E. Conway, J. Chem. Phys. 43 (1965) 243.
12
C.A. Winkler and C.N. Hinshelwood, J. Chem. Soc., London (1935) 1117.
13
R.W. Gurney, Ionic Processes in Solution, Dover Publications, New York (1953).
14
G. Gouy, J. Phys. 9 (1910) 457.
15
D. Chapman, Phil. Mag. 25 (1913) 475.
16
D.C. Grahame, Chem. Rev., 47 (1947) 441.
17
O. Stern, Zeit. Elektrochem. 30 (1924) 508.
18
J. O'M. Bockris, Quarterly Rev. Chem. Soc. London, 3 (1949) 173.
19
B.E. Conway and J.O'M. Bockris, Modern Aspects of Electrochemistry, 1 (1954) 47, Butterworths Sci. Publications, London, UK.
20
K.J. Laidler and G. Pegis, Proc. Roy. Soc., London, A241 (1957) 80.
21
B.E. Conway, J.E. Desnoyers and C. Smith, Phil. Trans. Roy. Soc., London, A256 (1964) 389.
22
D.C. Grahame, J. Chem. Phys. 18 (1950) 903.
23
B.E. Conway, J.O'M. Bockris and I. A. Ammar, Trans. Faraday Soc., 47 (1951) 756.
24
F. Booth, J. Chem. Phys. 19 (1951) 391 and 1327, and 1451.
25
G. Sherman, Chem. Rev. 11 (1932) 98.
26
S. Grimm, Handbuch der Physik, 27 (1924) 518.
27
J.D. Bernal and R.H. Fowler, J. Chem. Phys., 1 (1933) 515.
28
M. Eigen and L. de Maeyer, Proc. Roy. Soc., London, A247 (1933) 515.
29
P. Kebarle and E.W. Godbole, J. Chem. Phys. 39 (1963) 1131.
30
L. Landau, Phys. Zeit. Sowjetunion, 3, (1933) 664.
31
J. Jortner and S.A. Rice in Solvated Electrons, Chapter 2, Adv. in Chemistry Series, vol. 55, Ed. R.F. Gould, Amer. Chem. Soc., Washington, DC (1965).
32
B.E. Conway, Modern Aspects of Electrochemistry, Eds B.E. Conway and J.O'M. Bockris, 7 (1972) 83, Plenum Publ. Co., New York.
33
E.J. W. Verwey, Rec. Trav. Chim., Pays Bas 61 (1942) 127.
34
D.D. Eley and M.G. Evans, Trans. Faraday Soc. 34 (1938) 1093.
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35
A.D. Buckingham, Disc. Faraday Soc. 24 (1957) 151.
36
B.E. Conway, J. Soln. Chemistry, 7 (1978) 721.
37
B.E. Conway, chapter 16 in Ionic Hydration in Chemistry and Biophysics, Elsevier Publ. Co., Amsterdam (1981).
38
W.M. Latimer, K. Pitzer and Slansky, J. Chem. Phys. 7 (1939) 108.
39
N. Menschutkin, Zeit. physick, Chem., 5 (1890) 589 and 6 (1890) 41.
40
E. Tommila, Suomen Kemistilehti, B25 (1952) 37.
41
E. Tommila, E. Paakkala, U.K. Virtanen, A. Erva and S. Varila, Ann. Sci. Fenn., All, 91 (1959).
42
K.J. Laidler, Suomen Kemistilehti, A33 (1960) 44.
43
K.J. Laidler and P.A. Landskroener, Trans. Faraday Soc., 52 (1956) 200.
44
J.G. Kirkwood, J. Chem. Phys., 2 (1934) 351.
45
D.H. Everett and W.F.K. Wynne-Jones, Proc. Roy. Soc. London, A177 (1941) 499.
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Index
Boranes, amine, reducing agents, 61 Born equation, 295 Born equation from capacitance of a sphere, 300 dielectric constant factor in, 307 dielectric saturation factor, 310 historical, 296 limitations of, 295 plotting procedures from, 314 references on, 321-323 theoretrical basis, 298, 299 Born-Haber cycle, 315, 317 Bornhydration energies, comparison with experiment, 301 Borohydride reducing agents, 301 Bridging electrolyte/water systems, 198
Absorption, of H into metals, 22 Absorption of H, at Pd, 36 Absorption, from UPD H, 31 AC admittance, 4, 6 AC impedance, 1 AC modulation of diffusion of H, 24 Activation energy, for conductance, 268 Activation parameters for viscous flow, 257, 258 Activity coefficient curves, for water, 201, 216 Activity of water, in concentrated solutions, 201, 211 Admittance, AC, 4, 6 Adsorption capacitance, 10 Adsorption capacitance, potential dependence of, 10 Adsorption model, parameters, 229 Adsorption pseudocapacitance, AC results for, 14 Algorithms, finite difference simulation, 178 Algorithms, for kinetic simulations, 104, 166 Atomic H, in electroless deosition, 87 Autocatalysis, in electroless deposition, 55 Automation, in laboratories, 144
Capacitance adsorption, 10 double-layer, 9 Catalysis, in electroless deposition, 51 Ceramics, metallization of, 111 Charge-transfer resistance, 11 Charging effect on ionic radii, 306 Chemistry, computational, 140 Chemometrics, 144 Cobalt deposit, SEM of, 104 Codeposition, electroless, 53 Complex-plane plots
ion, 64 Bieniaz computational procedures by, 159 325
326
for H at Pt,12 for H sorption and evolution, 30, 34 Complex-plane plot and transfer function, 46 Complexing agents, in electroless deposition, 58, 81 table of, 59 Composite coatings, electroless deposition of, 113 Computer experiments, 140 Computer procedures, in electrochemistry, 135 Computers, in electrochemical kinetics, 151 Computers, in natural sciences, 137 Computational electrochemistry, 135 Computational electrochemistry, references, 187-195 Computational science, 145 future of, 149 Conductance electrical, 268 and free volume, 276 Conductance log equation for, 285 transition-state theory, 272 Constant phase element, 9 CPE, 9 Crank-Nicholson method, in simulations, 169, 171 Coverages at electrodes, 2 in kinetic equations, 2, 17
Deposition contact, 55 of metals, displacement, 54 of metals, electroless, 51 Determinants, for impedance evaluation, 5
Index
Dielectric constant, in Born equation, 299 Dielectric constant effect, in Born equation,307 Dielectric, origin of term, 298 Dielectric permittivity, 300 Dielectroc saturation, and Born equation,310 Diffusion fluxes, in impedance, 3 Diffusion in H absorption into metals, 23 of H, under AC modulation, 24 Diffusion-length, finite, 29 Diffusion, spherical, 33 Dipole-ion interaction, 313 Double-layer capacitance, 9 Electrochemical impedance spectroscopy, 1 Electrochemical instrumentation, control of, 155 Electrochemical kinetics, computers in, 151 Electrochemical mechanism, in electroless deposition, 91 Electrochemical mechanisms, and equations, 160 Electrochemistry, computational, 135, 156 Electroless codeposition, 53 Electroless deposition, 51 autocatalysis in, 51, 55, 56 basic research in, 121 electrochemical mechanism, 91 hydride ion in, 90 kinetics of 80 mechanisms of 87 of composites, 112 of gold, 114 recent develoopments, 107 reducing agents for, 59
Index
327
references on, 124-133 solutions for, 56 stabilizers in, 76 “universal mechanism”, 105 table of, 53 Electrolyte solutions, structural parameter for, 209 Electrolyte/water systems, 197 references on, 289-294 transport properties of, 243 thermodynamics of, 199 viscous flow, 244 Electrolytes adsorption theory of, 203 concentrated, 197 Electron hydration of, 311 hydration energy of, 312 Electrostriction, for ionic volumes, 306 Enthalpy of activation, for viscosity and evolution, 29, 32, 33 Equations for cathodic evolution, 13 kinetic, for despotion, 80 Equivalent conductance data, table, 281 equation, 279 and viscosity, 282 Eyring hole theory, 260 Faradaic admittance, 4 involving diffusion of H, 27 Fluidity in electrolyte water systems, 265 equation, 263
Gas-phase ionic volumes, 305 Gibbs activation energies, components of, 319, 320 Gold, electroless deposition of, 114 Grid strategy, for equation solving, 174 Gurney cosphere, 308 H, in electroless deposition, 87 H sorption equivalent circuit for, 28 from UPD H, 31 into metals, 22 Hydration cosphere finite volume factor, 309, 310 volume factor, 308 Hydration effects, historical, 299, 296 Hydration energies, Born calculation of, 295 Hydration energy calculations, comparison with experiment, 301 Hydration energy of electron, 312 Hydration energies extrapolation procedures for, 314 tabulated, 302 Hydration, molecular modeling of, 313 and molten-salt transition, 197 of proton and electron, 311 Hydrogen evolution equivalent circuit, 29 equations for, 13 evolution reaction, 15 mass transfer of, 19 Hydrogen UPD, 6 kinetic equations, 7 at noble metals, 6
328
Hydrogen, at Pt, complex plane plots for, 12 Hydrogen processes, impedance of, l Hydride ion mechanism, in electroless deposition, 90 Hypophosphite, for electroless deposition, 52, 60, 81 Impedance AC, l applications of, 1 determinations of, 2 diffusion fluxes in 3 Impedance equations, matrix forms, 4, 18 Impedance mass-transfer factor in, 19 at single-crystal surfaces, 12 spectroscopy, 1 transfer formation, 39 Warburg component, 35 Informatis, chemical, 142 Integral equations, solving, 172 Intercalation of relation to H sorption, 47 Interfacial potential, 300 Ion-dipole interactions, 313 Ionic radii charging effect on, 306 in solution, 304 Ionic reactions dielectric affects in, 319 kinetics of, 319 Ionic volumes electrostriction in, 306 in gas phase, 305 Ionization processes in solution, 316 Ionization, quantized nature of, 300
Index
Kinetic algorithms, 164 Kinetis, in computational electrochemistry, 135 Kinetic equations and coverage, 2 for H UPD, 7 for evolution, 16 for reaction steps, 16 Kinetic simulations, algorithms of, 102 Kinetics of electroless deposition, 80, 95 Kinetics, of ionic reactions, 319 Kinetics, modeling of, 151 Laboratory automation, 144 Laplace transform operator, 39 ion, intercalation in relation to H sorption into metals, 47 Liebig, description of electroless deposition, 52 Mass-transfer, of 19 Matrix form, of impedance equations, 4, 18 Mechanisms of electroless deposition, 89 Metal deposition, electroless, 51 Metal hydroxide mechanism, 100 Metals, H sortion into, 22 Metals, various, deposition of, 117 Mixed potentials, 97, 99 Modeling, in science, 148 Modeling, of kinetic experiments, 151 Mole fraction approach, for electrolyte/water systems, 234 Molten-salt/solution transition, 197
Index
Non-conductive surfaces, metallization of, 107 Nyquist plots; see complex-plane plots, 12 Organic/water mixtures, 286 Polaron, 296 Polymers, metallization of, 112 Proton, hydration of, 311 Pseudocapacitance, AC results for, 14 Publication numbers, in computational electrochemistry, 136 Reducing agents amine boranes, 67 B-containing, 64 formaldehyde, 69, 86 hydrazine, 72 in electroless deposition, 59-76 Regular solution theories, 231 Resistance, charge-transfer type, 11 Rotating disc electrode, fluxes at, 20 Simulations kinetic, 162 methods, new, 172 Single-crystal surfaces, impedance at, 12 Solution, ionization in, 316 Solution/molten-salt transition, 197 Solvent cosphere, volume factor, 308 Sphericl diffusion, 33
329
Surface properties, of electrolyte/water systems, 236 Surface potential, at liquid interface, 300 Stabilizers, in electroless deposition, 76 Stokes and Robinson, theories of, 203 Surface tension, of electrolyte solutions, 237 Thermodynamics, of electrolyte/water systems, 199 Transfer function applications, 39 and complex-plane plot, 44 in impedance, 39, 44 concept of, 37 Transients, electrochemical, 179 Transition-state theory of conductance, 272 of viscosity, 247 Transport properties, of electrolyte/water systems, 243 Underpotential deposition, of H, 6 UPD, of H, 6 UPD H, sorption into metals, 31 Viscosity enthalpy of activation for, 287 entropy of activation for, 254, 288 and equivalent conductance, 282 transition-state theory, 247 Viscous flow
330
activation energy for, 244, 251 Eyring hole model, 260 of solutions, 244 Volume excess, in water solutions, 223 excess of water, curves for, 225, 226 Volumes, of ions, in gas phase, 305 Walden product, 283 Warburg impedance, 35, 38 Water activity, 201, 211 Water, activity coefficient curves, 215, 216 Water activity, in molten electrolytes table, 219, 220 Water/electrolyte systems, 197 Water, mixtures with organics, 280 Water vapor pressure, 240
Index
