Thermodynamics: Second Law Analysis - American Chemical Society

14 Nonequilibrium Thermodynamics LUC

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Downloaded by UNIV OF LEEDS on May 17, 2015 | http://pubs.acs.org Publication Date: May 28, 1980 | doi: 10.1021/bk-1980-0122.ch014

Department of Physics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201

Steady states are among the phenomena that thermodynamics studies.

non-equilibrium

When one is not too far from equilibrium

it can be shown that the steady states are stable.

On the other

hand, when far from equilibrium, certain systems can make transitions

to states exhibiting "dissipative structures." The theory

of non-equilibrium developed by I. Prigogine, is quite general and has been applied to a wide range of phenomena.

I tisthe

aim of this lecture to introduce this f i e l d with a few examples. Introduction Self Organization When the temperature of a ferromagnet is brought below the Curie temperature a new order takes place:

the atomic magnetic

moments, which above T were oriented at random, align themselves c

with each other under T , forming magnetic domains. c

The creation or destruction of order in phase transitions is a well known and well studied fact.

What is not so well known

is that self organization also takes place in an entirely d i f f e r ent class of phenomena. Certain systems when far away from equilibrium jump to new states where new structures appear. These structures have been called dissipative structures by Ilya Prigogine, who gave the first general thermodynamic description of these phenomena (1,2). He has been awarded the 1977 Nobel Prize in Chemistry for that work. O-8412-0541-8/80/47-122-227$05.00/0 © 1980 American Chemical Society In Thermodynamics: Second Law Analysis; Gaggioli, R.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

228

THERMODYNAMICS:

B.

SECOND L A W ANALYSIS

Examples of Non Equilibrium Systems

To understand better what will follow it may be helpful to have in mind a few concrete and simple examples.

For instance:

(i) a system consisting of two large heat reservoirs A and Β connected by a small heat conducting system C (Figure l a ) . (ii) a system consisting of two large water tanks connected by a thin pipe (Figure l b ) . ( i i i ) two e l e c t r i c a l l y charged spheres connected by an electric


conductor (Figure l c ) . In each case we are interested in the behavior of C. When T

A

> Τ

β

throughC.,when V

heat flows throughC.,when h > V

A

>

water flows

an electric current flows through C.

The Linear Region Onsager Reciprocity Relations (3) When one is still very close to equilibrium (Τ > Τ , D

A

h

A

> h4, V

A

> V in our three examples) a flow appears in C. To fi

these flows are also associated some generalized forces. For instance in the electric example, an electric f i e l d Ε = appears along the wire which induces the electric current.

In

the thermal example, a temperature gradient is created along the system C which induces the heat flow. At equilibrium both flows and generalized forces vanish. When very close to equilibrium the flows can be written as linear expressions in the generalized forces.

For instance,

J = 0 Έ in

the electric case, which is nothing but ohm's law, or at

= κ 4 VT L

in the thermal case, which is the heat transfer equation. In general there can be several types of flows given systemC.,and several generalized forces

through a

producing them

(for instance, a conducting rod through which flows an electric and a heat current produced by an electric f i e l d and a tempera ture gradient). Close to equilibrium one can thus write the general relation

In Thermodynamics: Second Law Analysis; Gaggioli, R.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

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14.

(1)

J

The

k

Nonequilibrium

229

Thermodynamics

= Τ L . x. h ki ι ι

are called the phenomenological constants.

This region

is called the linear region. Onsager showed in 1931 that with proper choice of fluxes and forces,

=

These relations are now called the Onsager

reciprocity relations. Stability of the Steady State in the Linear Region


If in our three examples A and Β are very large compared to C., the flow through C does not modify A and B.

I f the flow

through C is constant, everything is essentially time indepen dent, including the boundary conditions of the system C.

In that

case one says that C is in a steady state although it is not in thermodynamic equilibrium. The systems C are open systems:

they can exchange energy

and matter with the exterior (systems A and B). If we assume that the systems C obey local equilibrium thermodynamics (see Ref. 2, page 30) an entropy can be associated to these non equilibrium systems and one can show that the total change of their entropy can be written as (2)

dS = d.S + d S ι e

where d S is the change due to a flow of entropy from A and B, e

whereas d4S is a change of entropy produced by irreversible processes taking place in C. d.S We will c a l l Ρ Ξ -—- the entropy production rate. dt show that (3)

Ρ = [dv il X * i

±

where X4,

One

can

Jl ±

are the flows and generalized forces present in the

system C., and the integration is a volume integral over C. Prigogine has shown that if the boundary conditions of the


THERMODYNAMICS:

230

SECOND L A W

ANALYSIS

systems C are kept constant and if one is in the linear region, then »,

g * . d.S _

As on the other side Ρ Ξ — - > ο (irreversible processes can only at increase the entropy), one concludes that the system C will dP evolve spontaneously to a state where — = 0, where the entropy at


production rate is minimum, this state being a steady state. The behavior of Ρ is i l l u s t r a t e d in Figure 2. In other words if the system C is not too far away from equilibrium, and its boundary conditions are kept constant, it will

eventually reach a steady state.

This steady state is

stable, and corresponds to a minimum entropy production rate. This is called the minimum entropy production theorem (14,2) . We can i l l u s t r a t e the theorem using our thermal example. The graphs (a), (b) and (c) show the evolution of the temperature of C (Figure 3).

In graph (c), which corresponds to the

steady state, we have assumed for simplicity that the coefficient of thermal conduction is independent of temperature. The Non Linear Region S t a b i l i t y Conditions (2) When a system is far away from equilibrium and outside the dP linear region, one cannot say anything about the sign of — : the steady state is not necessarily stable.

In fact, in many

cases it is unstable and the system can jump to new states, the dissipative structure states.

I f we want to determine whether

the steady state is stable, we can perturb the system s l i g h t l y away from the steady state and study its behavior. Let _ ~ 12 AS = S - SQ = OS + — δ S be the change of entropy due to this perturbation, S being the entropy at the steady state. S is a q

functional of the local thermodynamic variables.

In OS we have

collected all the terms linear in the change of these variables


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Nonequilibrium

231

Thermodynamics


(a)

A

R

Β

(c) V Figure 1.

A

C

V

B

Examples of nonequilibrium systems: (a) thermal system; (b) hydrodynamic system; (c) electric system.

Figure 2. Time evolution of the entropy production rate in the linear region


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THERMODYNAMICS:

SECOND L A W ANALYSIS

and in 6 S all the terms q u a d r a t i c in those changes. One can show t h a t if one does not go too f a r from the steady state: OS = 0 6S < 0 2

(5)


(6)

AA-JdïCÏÎI44P

6 Ρ is c a l l e d - the excess entropy p r o d u c t i o n r a t e . I f 4 1 d 2 δ Ρ = Τ Γ δ S > 0, As will e v e n t u a l l y go t o zero: a p e r t u r b a t i o n χ /. a t o f the steady s t a t e goes t o zero w i t h time, and the steady is s t a b l e .

state

I f on the other s i d e δ Ρ < 0, a small p e r t u r b a t i o n is χ

a m p l i f i e d and the steady s t a t e is unstable.

These i n e q u a l i t i e s

are c a l l e d the s t a b i l i t y c o n d i t i o n s . (See F i g u r e 4.) Benard C e l l s as an Example o f D i s s i p a t i v e S t r u c t u r e s A w e l l knowri example o f d i s s i p a t i v e s t r u c t u r e s a r e the Benard c e l l s , a l s o c a l l e d convection c e l l s . f i l l e d w i t h water.

A f l a t tank is

The upper and lower surfaces a r e kept a t

d i f f e r e n t temperatures.

When the lower s u r f a c e is s l i g h t l y

warmer than the upper one, the system is in t h e l i n e a r r e g i o n and one observes a steady and uniform upward flow o f heat. I f the temperature o f the lower s u r f a c e is i n c r e a s e d , suddenly convection c u r r e n t s appear and very r e g u l a r s t r u c t u r e s are formed which when looked a t from above form a hexagonal pattern.

These s t r u c t u r e s have been observed by Benard f o r the

f i r s t time in 1901, and a r e very w e l l known by m e t e o r o l o g i s t s . Systems I n v o l v i n g Chemical Reactions (2_) The B a s i c

Equations

Up t o now the argument has been very g e n e r a l .

We saw t h a t

some systems when f a r from e q u i l i b r i u m jump from the steady

state

to d i s s i p a t i v e s t a t e s , and we saw t h a t the s t a b i l i t y depends on


14.

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Nonequilibrium

A

Thermodynamics

Β C

(a)


(b)

(c)

t(a) (b)

(c)

Figure 3. Time evolution of a nonequi librium system towards a steady state in the linear region

Stable

Figure 4. The time evolution of the excess entropy production 8S in the non linear region gives a criterion for the stability of the steady state 2

Unstable


234

THERMODYNAMICS:

SECOND L A W

ANALYSIS

the sign of δ Ρ, the excess entropy production rate. χ

If we want to learn more about those dissipative structures, we have to study precise examples.

The type of systems that

Prigogine and his school have decided to study in detail are composed of a medium (a l i q u i d solvent or an inert gas) in which are diluted η chemical components.

The temperature of the system

is assumed to be constant, and the system is assumed to be at mechanical equilibrium (no mass flow) and is not subject to external f i e l d s . 1


If we c a l l

the density of the i * " * chemical component, one

can show that the following equation is true:

(7)

= D. V

2 P ±

+ F.({p }) k

The D. are diffusion coefficients and the F. are non linear ι ι functions of the p. . This last term comes from the reaction k rates. These equations have been called the basic equations. As one can see we are dealing with non linear d i f f e r e n t i a l equations.

There exists no general method for solving this type

of equation.

In order to be able to solve them one has to

simplify the model as much as possible. The Tri-molecular Model This model consists of:

(1) two i n i t i a l chemical components

A and Β which are absorbed by the system from outside.

Their

densities are kept constant throughout the system by a continuous supply from outside.

(2) Two f i n a l chemical components D and Ε

which are rejected by the system.

Their densities are kept

constant a r t i f i c i a l l y too, by extracting them continuously. (3) Two intermediate components X and Y, the densities of which are variable. Prigogine and co-workers have chosen the simplest reactions among those components for which i n s t a b i l i t y can take place


14.

Nonequilibrium

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(8)

A ;

235

Thermodynamics

X

B +

X Î Y + D

2X + Y J

3X

X Î E and found the f o l l o w i n g system o f equations. 4

= A -

(B+1)X +

2

X Y + DjV

2

X


(9) 3Y 2 4 = B x - X Y

2

+ D ?Y 2

where X , A , e t c . are p r o p o r t i o n a l to the d e n s i t i e s o f the corresponding components, and where D c o e f f i c i e n t s o f components X and Y,

and

1

are the d i f f u s i o n

respectively.

Under t h i s form the equations are

still

too hard to s o l v e .

One has to make a f i n a l s i m p l i f i c a t i o n , and assume t h a t the system is e s s e n t i a l l y a one dimensional system o f l e n g t h £. Depending on the boundary c o n d i t i o n s

( i . e . the v a l u e s o f D4,

D4 and A ) , the d i s s i p a t i v e s t r u c t u r e s are time independent o r time dependent.

For the time independent case one f i n d s the

following r e s u l t s .

(10)

Χ = Χ

where X

q

ο

B-B ± (—4

h

m ïïr sin

+ ...

corresponds to the steady s t a t e .

be represented g r a p h i c a l l y

(Figure

These s o l u t i o n s can

5).

The d e n s i t y o f A being kept constant, the system is brought away from e q u i l i b r i u m by v a r y i n g B.

B

c

is the c r i t i c a l v a l u e a t

which the system b i f u r c a t e s from the steady branch (a) t o one o f the d i s s i p a t i v e s t r u c t u r e branches (b) o r ( c ) . The expressions f o r X , Y show t h a t these c o n c e n t r a t i o n s vary in space as s i n — 4 — where m values o f D4,

c

is an i n t e g e r which depends on the

and A .

I f we imagine our one dimensional system as a long t e s t tube


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THERMODYNAMICS:

in which the r e a c t i o n s take p l a c e , the branch

SECOND L A W

ANALYSIS

(a) o f F i g u r e 5

would correspond to the case where the c o n c e n t r a t i o n s X and Y are constant along the tube, whereas branch

(b) o r (c) would c o r r e s -

pond to the case where the c o n c e n t r a t i o n s X and Y vary p e r i o d i c a l l y along the tube forming an a r r a y o f l a y e r s which can be observed.

I l l u s t r a t i o n s corresponding to t h i s type o f e x p e r i -

ments are g i v e n in Refs. 1 and

2.

The Belousov-Zhabotinski Reactions


Experimentally d i s s i p a t i v e s t r u c t u r e s in chemical systems have been observed.

The b e s t known are probably the Belousov-

Zhabotinski reactions.

Already two a r t i c l e s d e s c r i b i n g them have

appeared in S c i e n t i f i c American, in the i s s u e s o f June 1974 July

and

1978. The Josephson

Effects

Introduction We have seen in the general i n t r o d u c t i o n t h a t s e l f o r d e r i n g can occur in two very d i f f e r e n t s i t u a t i o n s : a system goes through a phase t r a n s i t i o n

a t e q u i l i b r i u m , when

(example o f ferromagne-

tism) o r f a r away from e q u i l i b r i u m when d i s s i p a t i v e s t r u c t u r e s appear in the system We

will

(example o f Benard

cells).

see in t h i s s e c t i o n t h a t there e x i s t s a t h i r d type

o f phenomena, the Josephson

e f f e c t s , which occupy a p o s i t i o n

between phase t r a n s i t i o n s and d i s s i p a t i v e s t r u c t u r e s . The analogy between phase t r a n s i t i o n and b i f u r c a t i o n s to d i s s i p a t i v e s t r u c t u r e s has been noted by s e v e r a l authors. authors

(4)

Some

a r e even l o o k i n g f o r a u n i f i e d formalism t h a t c o u l d

d e s c r i b e both o f these phenomena.

The study o f the

Josephson

e f f e c t s can thus be a u s e f u l t o o l in t h i s l i n e o f thought. We

will

d e s c r i b e two d i f f e r e n t types o f Josephson

junctions:

(i) The Anderson-Dayem b r i d g e s , the behavior o f which can be i n t e r p r e t e d very n i c e l y in terms o f d i s s i p a t i v e s t r u c t u r e s , but f o r which the t h e o r e t i c a l d e s c r i p t i o n is not so simple.


14.

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(ii)

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237

Thermodynamics

The oxide barrier Josephson junctions, which can be des cribed by rather simple theoretical models. The Anderson-Dayem Bridges (5) An Tinderson-Dayem bridge consists of a thin superconducting

film having the shape represented in Figure 6. The narrowed part of the bridge, NP, is typically of the order 3y. The poten t i a l difference V is progressively increased starting from zero. q

The current going through the bridge and the potential drop


across the bridge are measured. (Figure 7) At f i r s t a current is observed but no potential drop. The current is a supercurrent, and the system is said to beinthe pure superconducting state (P.S. state).

This situation contin

ues u n t i l a c r i t i c a l current I is reached. c

When V is increased ο

some more, I increases and a voltage drop appears.

Along ab one

says that the system is in the resistive superconducting state (R.S. state). The interpretation of the behavior of the bridge along ab is that vortices are created along the line NP. The structure of these vortices consists of a nucleus in which is trapped a quantum of magnetic flux, surrounded by concentric super currents.

This magnetic f i e l d is perpendicular to the film.

The Lorentz force between the magnetic fluxes and the current induces the vortices to move sideways along the line NP. (Figure 8) It can be shown that there exists a simple relation between the rate dn/dt at which the vortices cross the junction, and the potential drop (6):

Thermodynamics: Second Law Analysis - American Chemical Society

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