Introduction to
Non-equilibrium Physical
Chemistry
Towards Complexity and Non-linear Science

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Introduction to
Non-equilibrium Physical
Chemistry
Towards Complexity and Non-linear Science

R.P. Rastogi
Gorakhpur University, Gorakhpur, India

Amsterdam • Boston • Heidelberg • London • New York • Oxford
Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo

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Elsevier
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First edition 2008
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Dedicated to
My Life-long Companion
Mrs Kamla Rastogi

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vii

TABLE OF CONTENTS

PREFACE
ACKNOWLEDGEMENTS
COLOUR PLATE SECTION

xiii
xv
xvii


1 INTRODUCTION

1

1.1 Real systems
1.2 Equilibrium and non-equilibrium states
1.3 Open systems
1.4 Approach to equilibrium
1.5 Non-equilibrium states
1.6 Complex non-equilibrium phenomena
1.7 Scope
References
PART ONE

NON-EQUILIBRIUM STEADY STATES CLOSE
TO EQUILIBRIUM

2 BASIC PRINCIPLES OF NON-EQUILIBRIUM THERMODYNAMICS
2.1 Introduction
2.2 Second law of thermodynamics for open systems
2.3 Law of conservation of mass, charge and energy
2.4 Gibbs equation
2.5 Phenomenological equations for single flows
2.6 Phenomenological equations for coupled flows
2.7 Onsager reciprocity relation
2.8 Entropy production in multi-variable systems
2.9 Basic postulates of non-equilibrium thermodynamics close to equilibrium
2.10 Experimental test of LNT
2.11 Application to other disciplines: sociology, economics and finance

2.12 Concluding remarks
References

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1
1
2
2
3
4
4
7

9
11
11
13
15
16
17
17
19
21
22
22
23
23
24



viii

Table of Contents

3 APPLICATIONS TO TYPICAL STEADY-STATES PHENOMENA
3.1
3.2
3.3

Introduction
Thermodynamic theory of thermo-osmosis
Thermodynamic theory of thermo-osmosis of gaseous
non-reacting mixtures
3.4 Experimental studies
3.5 Thermo-osmosis of gases and gaseous mixtures
3.6 Thermo-osmosis in biological systems
References
4 ELECTRO-OSMOTIC PHENOMENA

27
27
27
35
44
51
55
56
59


4.1 Introduction
4.2 Non-equilibrium thermodynamics of electro-osmotic phenomena
4.3 Theories based on models of membranes
4.4 Experimental test of thermodynamic theory
4.5 Concluding remarks
References

59
59
64
71
76
77

5 NON-EQUILIBRIUM PHENOMENA IN CONTINUOUS SYSTEMS

81

5.1 Introduction
5.2 Theory: thermodynamic considerations
5.3 Experimental studies in gaseous systems
5.4 Dufour effect in liquid mixtures
5.5 Thermal diffusion potential
5.6 Electric potentials generated at crystal interface
References
6 ELECTROPHORESIS AND SEDIMENTATION POTENTIAL
6.1 Introduction
6.2 Thermodynamic theory
6.3 Comparison with Helmholtz double layer theory
6.4 Test of theory: experimental studies

References

81
81
85
86
87
89
91
93
93
93
95
96
98

PART TWO NON-LINEAR STEADY STATES – DISSIPATIVE
STRUCTURE (TIME ORDER AND SPACE ORDER)

99

7

NON-LINEAR STEADY STATES

101

7.1
7.2


101

Introduction
Non-linear flux equations in electro-kinetic
phenomena

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101


Table of Contents
7.3
7.4

8

9

10

ix

Non-linear steady states
Interpretation of second-order coefficients in the light
of double layer theory
7.5 Non-linear transport equations in gaseous medium
7.6 Non-linear flux equations and non-linear steady states
in chemical reactions
7.7 General remarks

References

104

BIFURCATION PHENOMENON AND MULTI-STABILITY

119

8.1 Introduction
8.2 Dynamical non-linear systems
8.3 Typical types of bifurcation
8.4 Bifurcation from steady state to bistability
8.5 Bifurcation from steady state to oscillatory state
8.6 Multi-stability
8.7 A simple mathematical model of bistability
8.8 A simple model for reacting systems
8.9 Bistability in reacting systems
8.10 Spatial bistability
8.11 Bistability in magnetic resonance
8.12 Bistability in electro-kinetic phenomena
8.13 Bistability in biological systems
References

119
119
121
125
125
126
126

127
128
132
132
133
136
137

TIME ORDER – CHEMICAL OSCILLATIONS

139

9.1
9.2
9.3
9.4
9.5

Introduction
Isothermal chemical dissipative structures
Chemical oscillators
Modelling of oscillatory reactions
Mechanism of B–Z reaction; positive and negative
feedback
9.6 Alternate control mechanism
9.7 Dual control mechanism
9.8 Coupled oscillators
9.9 Logic function
References


139
139
140
148

CHEMICAL WAVES AND STATIONARY PATTERNS

165

10.1 Introduction
10.2 Chemical waves and stationary patterns

165
165

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105
109
111
115
116

149
153
154
160
162
162



x

Table of Contents
10.3 One-dimensional chemical waves
10.4 Mechanism of wave propagation
10.5 Wave formation on membranes
10.6 Wave structures
10.7 Turing instability
10.8 Logic functions
10.9 Precipitation patterns
References

166
167
169
170
171
175
176
184

PART THREE COMPLEX NON-EQUILIBRIUM PHENOMENA
FAR FROM EQUILIBRIUM

187

11

DYNAMIC INSTABILITY AT INTERFACES


189

11.1 Introduction
11.2 Dynamic instability at
11.3 Dynamic instability at
solid–liquid interface
11.4 Dynamic instability at
11.5 Dynamic instability at
References

189
190

12

13

solid–liquid interface
liquid–liquid interface along with
liquid–vapour interface
solid–gas interface [60–68]

199
209
213
215

COMPLEX OSCILLATIONS AND CHAOS


217

12.1 Introduction
12.2 Complex oscillations
12.3 Deterministic chaos
12.4 Routes to chaos
12.5 Characterization of chaos
12.6 Modelling and test of reliability
12.7 Control of chaos
12.8 Noise
12.9 Turbulence
12.10 Future perspectives
References

217
217
223
226
226
229
231
232
233
233
234

COMPLEX PATTERN FORMATION

235


13.1 Introduction
13.2 Experimental studies of complex patterns
13.3 Concluding remarks
References

235
247
266
267

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Table of Contents

xi

PART FOUR NON-EQUILIBRIUM PHENOMENA IN NATURE
AND SOCIETY

271

14

SOCIAL DYNAMICS, ECONOMICS AND FINANCE

273

14.1 Complexity in real systems
14.2 Methodology and strategy for study of complex systems

14.3 Analytical studies of real systems
14.4 Quantification of relationship between cause and effect
14.5 Sociology (social sciences)
14.6 Economics
References

273
273
277
279
283
289
294

LIVING SYSTEMS

297

15.1 Introduction
15.2 Transport through biomembranes
15.3 Biological rhythm
15.4 Concluding remarks
References

297
301
305
312
313


15

EPILOGUE
APPENDIX I
APPENDIX II
APPENDIX III
INDEX

315
321
325
329
333

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xiii

PREFACE

The traditional Physical Chemistry has largely been concerned with equilibrium
phenomena and ideal situations. In the early part of the twentieth century, interest
developed in investigating non-equilibrium phenomena and real situations involving
social dynamics and living state. The obvious tools had been non-equilibrium thermody­

namics, statistical mechanics, molecular dynamics and computer simulation. Significant
advances were made in understanding non-equilibrium processes and discovering new
laws and paradigms for which Onsager was awarded Nobel Prize in 1968 and Prigogine
in 1977. Their researches gave further impetus to the study of more and more complex
phenomena. Complex systems are multi-variable systems involving several processes,
which cannot be understood by using the traditional reductionist approach by dissecting
the phenomenon into parts.
Non-equilibrium phenomena have wide applications in real systems. Living state
is such a typical non-equilibrium state. These have relevance in Physiology, Geology,
Physical Science and Biological Science, Economics and Social Dynamics.
One can have the following type of situations as we move away from equilibrium.
Equilibrium state −→ Linear steady state close to equilibrium −→ Steady state −→
Non-linear steady state −→ Bifurcation phenomena −→ Multi-stability −→ Temporal
and spatio-temporal oscillations −→ More complex situations (chaos, turbulence, pattern
formation, fractal growth). All these stages have been discussed in different chapters of
the book.
During the latter half of the last century and to date, important developments have
taken place and new paradigms have been developed in the field of non-equilibrium
phenomena, giving rise to a new discipline of non-equilibrium Physical Chemistry. Its
importance is further enhanced by the fact that physico-chemical experiments provide
instructive and useful models for Biological and Social Sciences, etc. It may be noted
that several monographs have appeared in the last few decades dealing with these
developments. Lots of experimental studies in these areas have also been reported
during this period which provide a clearer picture. However, so far conventional text
books have laid emphasis on equilibrium phenomena. Undoubtedly there is detailed
reference to chemical kinetics and ion migration, but the above important developments
involving entirely new concepts in the non-equilibrium region have escaped attention.
The purpose of this book is to fill this gap so that it can be used as supplementary
material for teaching as well as for further research. The idea is to present the material
in a sequential, coherent and comprehensive manner with greater emphasis on concepts

so that it may be useful from pedagogical angle. Additional purpose has been to present

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xiv

Preface

an elementary account to provide an insight into non-linear science and complexity for
scientists in other disciplines such as Economics, Physiology and Biological Sciences.
I am extremely grateful to Professors A.C. Chatterji, B.N. Srivastava, M.N. Saha,
K.G. Denbigh, Karl Popper and Nobel laureate Prof. Ilya Prigogine for stimulating my
interest in the study of non-equilibrium phenomena from various angles.
I am indebted to Lucknow University, Punjab University, Gorakhpur University
and Banaras Hindu University and Central Drug Research Institute where most of the
experimental work in different areas could be accomplished. The financial support
of funding agencies like University Grants Commission, Council of Scientific and
Industrial Research and Department of Science and Technology (Government of India)
and Indian National Science Academy for carrying out various projects related to the
theme of the present text are acknowledged. The secretarial assistance provided by Prof.
K.D.S. Yadav, Head, Chemistry Department, Gorakhpur University, is also gratefully
acknowledged.

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xv

ACKNOWLEDGEMENTS


I pay my tribute to Prof. B.N. Srivastava, who was primarily responsible for the
development of my interest in Irreversible Thermodynamics. I am equally grateful to
my former Ph.D. students and other collaborators who had been involved in theoretical
and experimental studies in the areas covered in the book, whose references appear in
the book.
I am particularly thankful to Profs. R.C. Srivastava (Chapters 1–7, 11), Kehar Singh
(3, 4), Ishwar Das (9, 10, 13), Kalanand Prasad (10) and Dr Pankaj Mathur (8, 12, 14)
for collaborating in writing a couple of chapters.
I am also happy to acknowledge the involvement of Dr Ashtabhuja Prasad Mishra,
Sharwan Kumar, Prof. A.K. Jain and Dr Mukul Das for their involvement in the write-up
for specific chapters
I gratefully acknowledge the assistance rendered by Profs. A.K. Dutt and
A.A. Bhalekar related to Appendices II and III.
It is a pleasure to acknowledge the stimulating discussions which I had with
Prof. Raghuveer Singh, Prof. Hem Chandra Joshi, S.C. Mishra, Dr Gopishyam and
Dr Ghanshyam Das in connection with socio-political and financial dynamics.
Further, the help rendered by Prof. N.B. Singh, Dr Vishnu Ji Ram, Dr S.S. Das,
Dr Pankaj Mathur and Mr. Ramendra Pratap in connection with the preparation of the
manuscript is gratefully acknowledged.

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(a)


(b)

(c)

(d)

(e)

Chapter 10, Figure 10.3, p. 177.

Delta wave

Theta wave

Alpha wave
+10 mV
Beta wave
–10 mV
0

1

2

3
4

Seconds



Chapter 12, Figure 12.2, p. 218.

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(b)

x (t + τ)

100 mV

Redox potential

(a)

4 Sec

Time (sec)

x (t )
(d)

(c)

102
101
100

x (t + 2τ)


P(w )

10–1
10–2
10–3
10–4
10–5
10–6

x (t )
x (t + τ)

10–7
0.0

0.1

0.2

0.3

Frequency (Hz)

Chapter 12, Figure 12.9, p. 225.

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0.4


0.5


(a)

(b)

(c)

(d)

(e)

(f)

0.25 mm

Chapter 13, Figure 13.17, p. 254.

(a)

(b)

(c)

(d)

Chapter 13, Figure 13.21, p. 257.

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(e)


(a)
Glass cover
Cobalt(II) nitrate
10% Ammonia solution
(b)

(i)

(ii)

(iii)

(iv)

(v)

Chapter 13, Figure 13.18, p. 255.

Chapter 13, Figure 13.28, p. 266.

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1

Chapter 1

INTRODUCTION

1.1. Real systems
Traditional Physical Chemistry has largely been concerned with state of matter in
equilibrium. Non-equilibrium aspects come up in the case of kinetic theory, chemical
kinetics and ion transport. Real systems like living state and socio-economic systems are
always in non-equilibrium and invoke a number of complicated phenomena. Science is
now getting oriented to the study of such complex systems. From the philosophical angle
the question whether science deals with real world [1a] has been raised. The concept of
evolution in social systems in contrast to biology is moving towards an interdisciplinary
theory of change of state [1b]. This is provoking interest in experimental and theoretical
studies of analogous non-equilibrium phenomena in physico-chemical systems which
can serve as a model for economists and biologists. Joint effort in understanding complex
phenomena with the help of various disciplines is leading to the growth of Synergetics
meaning, a new discipline.

1.2. Equilibrium and non-equilibrium states
The important feature of the equilibrium state is that variables such as temperature T ,
pressure P, chemical potential and electric potential everywhere are the same in the
system. There can be two types of equilibrium states, viz. (a) dynamic equilibrium and
(b) static equilibrium. Vapour–liquid equilibrium and chemical equilibrium are typical
examples of dynamic equilibrium, where in the first case, the rate of condensation
and rate of vaporization are equal while in the second case, the rates of forward and
backward reactions are equal. Simple crystals belong to the class of static equilibrium.
In non-equilibrium state, the thermodynamic variables are not the same everywhere
in the system, on account of which gradients of variables (e.g. grad P, grad T , etc.)
develop which act as the cause (force) for generating effects (flows) such as volume
flow or heat flow (fluxes). Non-equilibrium thermodynamics has the advantage of being
used for identifying cause and effect, i.e. forces and fluxes, and also coupling between
the fluxes without a detailed knowledge of the systems. However, for real systems

problem arises in identifying variables, fluxes and forces, involved in processes and

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2

Introduction to Non-equilibrium Physical Chemistry

cross-phenomena. Further in many systems, additional difficulty arises on account of
lack of knowledge about the nature and magnitude of the coupling coefficients between
fluxes and forces [1].
The systems can be of three types:
Isolated systems in which there is no exchange of matter or energy with the
surroundings;
closed systems in which there is no exchange of matter with the surroundings but
exchange of energy can occur; and
open systems, which exchange both matter and energy with the surroundings.

1.3. Open systems
Open system is always in non-equilibrium. A closed system can be in non-equilibrium
depending on the circumstances. It may have subsystems between which exchange of
matter and energy can take place or in the system itself, thermodynamic variables may
not be constant in space. A typical example of the former type is thermo-osmosis, which
is discussed in Chapter 3, where the two subsystems are separated by a membrane.
Example of the latter type is thermal diffusion, which has been discussed in Chapter 5.
When the flows and counter-flows in opposite directions are generated by corresponding
gradients, steady state is obtained. Both equilibrium and non-equilibrium steady states
are time-invariant states, but in the latter case both flows and gradients are present.
Real systems are open systems and may consist of numerous subsystems; global

system, human society and human body are typical examples. The nature of subsystems,
variables, fluxes and forces, their coupling leading to cross-phenomena, temporal and
spatio-temporal changes, pattern formation and self-organization would be discussed in
the subsequent chapters.

1.4. Approach to equilibrium
For taking a comprehensive view, it is also desirable to keep in mind the process
of approach to equilibrium. Chemical kinetics and kinetic theory of gases have been
the traditional tools. Simple reactions have been studied by Monte-Carlo technique or
stochastic approach by monitoring random picks of molecules represented by digits on
the computer and employing a criterion that accepts or discards potential conversions.
The methodology adopted for the study of simple set of simultaneous reactions has been
received by Gupta, which involves comparison of experimental results with postulated
mechanism [2]. For complex reactions, numeric integration techniques are employed to
abstract concentration profiles. The methodology involved is essentially linear kinetics.
Chemical kinetics is now moving towards the study of more and more complex reaction

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Chapter 1. Introduction

3

network which may simultaneously involve (i) electron transfer reaction, (ii) free-radical
reaction, (iii) organic reaction, (iv) inorganic reaction and (v) reaction between organic
and inorganic species. For such type of systems, a new methodology called non-linear
kinetics involving non-linear differential equations is emerging.
There is considerable error in thermodynamic prediction if true equilibrium is not
maintained, a condition never maintained in industry due to time factor. Rastogi and

Denbigh [3] investigated this aspect theoretically. As an illustration, they examined the
reaction
2HI

H2 + I2

for which the equilibrium constant K is of the order of 2 18 × 10−2 at 763.8 K. The
energy of activation of the forward reaction and H, the enthalpy change of the
reaction, has values of 44 000 cal mol−1 and 3000 cal mol−1 , respectively. Let r denote
the ratio of the true temperature coefficient of the yield to the temperature coefficient
of the yield, which would be predicted on the assumption that the system is at equili­
brium. For f = 0.9, the true temperature coefficient of the yield is 10.6 times the value
predicted thermodynamically on the supposition of equilibrium. Further when f = 0 99,
the corresponding factor is as much as 2.8.
In a similar manner, cooling rate at the rocket nozzle throat used to be computed by
assuming isentropic flow [4]. However, it has been shown that the cooling rate at the
throat is likely to increase when departure from equilibrium becomes significant [5].

1.5. Non-equilibrium states
There was tremendous interest in mid-twentieth century in exploring general prin­
ciples for understanding non-equilibrium phenomena along with the development of
non-equilibrium thermodynamics and non-equilibrium statistical mechanics. Pioneering
work of Professor Prigogine [6] and his school in Brussels stimulated a good deal
of interest in the field of non-equilibrium statistical mechanics. Formal solutions of
Liouville equation [7] in terms of a Greenian and complete internal propagator leads to
a theoretical expression for electrical conductivity tensor, which easily leads to classical
formula for electrical conductivity of metals based on the free-electron model [8].
Kinetic theory, non-equilibrium statistical mechanics and non-equilibrium molecular
dynamics (NEMD) have proved to be useful in estimating both straight and crosscoefficients such as thermal conductivity, viscosity and electrical conductivity. In a
typical case, cross-coefficient in case of electro-osmosis has also been estimated by

NEMD. Experimental data on thermo-electric power has been analysed in terms of free
electron gas theory and non-equilibrium thermodynamic theory [9]. It is found that phe­
nomenological coefficients are temperature dependent. Free electron gas theory has been
used for estimating the coefficients in homogeneous conductors and thermo-couples.

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4

Introduction to Non-equilibrium Physical Chemistry

Onsager relations are satisfied, showing that free electron gas theory is consistent
with thermodynamic theory. The free electron theory correctly predicts the temperature
dependence of thermo-electric power. Similarly, the interpretation of the phenomenon
of thermo-osmosis of gases on the basis of non-equilibrium thermodynamics and kinetic
theory of gases is mutually consistent.

1.6. Complex non-equilibrium phenomena
The departure from equilibrium occurs primarily on account of appearance of gra­
dients such as temperature and concentration leading to flow of heat or of some species
and subsequently leading to a specific non-equilibrium state. Earlier in the first instance,
uncoupled flows, e.g. heat conduction, Poisseuille flow and electrical conduction, were
the subject of investigation. Discussion of such processes has been given due attention
in conventional Physical Chemistry texts. However, complex and exotic phenomena
in the non-equilibrium thermodynamics provide a good tool for understanding such
phenomena.
Far from equilibrium, one comes across exotic phenomena as pointed out earlier.
The study of these involved novel theoretical approaches and novel experimental studies
on a variety of phenomena to check the validity of phenomenological relations, Onsager

reciprocity relations and thermodynamic predictions regarding steady state along with
analysis of data in the non-equilibrium region on the basis of non-equilibrium thermody­
namics. These developments led to the study of non-linear steady states, non-linear flux
equations along with the attempts to understand the origin of non-linear terms. Further
away, bifurcation phenomena are encountered involving multi-stability and oscillatory
behaviour, followed by spatio-temporal oscillations, chaos and noise. All these phenom­
ena attracted good deal of attention from theoretical and experimental angle in view of
practical interest in Physiology and other disciplines.

1.7. Scope
The great importance of thermodynamics and hydrodynamic methods lies in the
fact that these provide us with a reduced description in simplified language to describe
macroscopic systems as stated by Glansdroff and Prigogine. The present contribution is
intended to present a coherent account of developments, both theoretical and experimen­
tal, in the advancing field of knowledge related to complex phenomena from equilibrium
to far from equilibrium region. From this angle, it is reasonable to expect that the
concepts and thought methodology would be useful for taking a synergetic view of
real systems in nature and social surroundings. Recent developments in non-equilibrium
Physical Chemistry have also been examined and discussed in this context.

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