Nonlinear phenomena of transport at local equilibrium

Reactive Polymers, 2 (1984) 143-151 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

143

N O N L I N E A R P H E N O M E N A OF T R A N S P O R T AT LOCAL EQUILIBRIUM * HEINRICH HOFF

Sektion Kalorimetrie der Universiti~t UIm, Oberer Eselsberg, 7900 Ulm (F. R. G.) (Received August 24, 1982; accepted in revised form November 23, 1982)

Nonlinear constitutive relations do not necessarily' require a new kind of nonequilibrium thermodynamics. There exist a great number of nonlinear transport phenomena which can be treated by well-known thermodynamics. Two classes of nonlinearity are distinguished." that which can be due to linear constitutive equations of a more general system, where the forces are not independent, and that where nonlinearity can be due to "inner polarization". From the latter, information about dependence of transport on reference state is deduced. As examples we treat nonlinear relations observed by reverse osmotic experiments, equisorptic transport and phenomena of asymmetrical transport.

1. INTRODUCTION Transport across membranes includes a great deal of interesting phenomena such as active transport, asymmetrical transport, phenomena of polarization, etc., which can be all grouped under the title "nonlinear phenomena". These phenomena have mostly been investigated with the help of suitable models, e.g., porous membranes and transport by carriers, which apply some kind of "chemical language". Although these models are very useful with respect to a detailed understanding, we need to work out a more general treatment requiring much weaker assumptions. * Paper presented at the NATO Advanced Study Institute "Mass Transfer and Kinetics of Ion Exchange", Maratea (PZ), Italy, May 31-June 11, 1982. 0167-6989/84/$03.00

A general phenomenological treatment of transport is accomplished by nonequilibrium thermodynamics. Transport behaviour is described by constitutive relations between flows and forces, which are often written down as linear relationships. Thus it seems that nonlinear phenomena, such as asymmetrical transport, are excluded by this concept. We shall demonstrate here, however, that wellknown linear constitutive equations are able to describe certain classes of nonlinear phenomena. In the first place, that class which can be due to particular arrangements of more general systems where the forces or flows do not form a set of independent quantities; assuming the constraint relations to be nonlinear, one will obtain nonlinear relations between the sets of independent flows and forces. The second class corresponds to transport where nonlinearity is caused by "inner polari-

© 1984 Elsevier Science Publishers B.V.

144

zation", which can be observed by dependence of transport parameters on reference state [2,4]. As an example of the first class we shall consider a nonlinear correspondence between the flow of volume and pressure occurring for reverse osmosis. As an example of the second class we shall investigate transport of equisorptic solutions observed by the same experimental equipment. By consequence of the concept of inner polarization, it will be demonstrated that a great number of nonlinear transport phenomena can be treated by ordinary nonequilibrium thermodynamics, where the local constitutive equations are linear.

2. NONLINEARITY CAUSED BY CONSTRAINT RELATIONS Throughout this paper we shall discuss unidirectional transport systems, a schematic illustration of which is given in Fig. 1. The so-called outer phases (') and (") are assumed to remain in equilibrium, i.e., transport is assumed to be quasi-steady with respect to

i

n

(')

("1

these phases, which implies that the conductivities of these regions are large compared with those of the region ('"). So all irreversible processes are assumed to happen exclusively in ("'). Transport is described by the well-known constitutive equations [ 1,3,13]: M

J, = Y~. L , k A X k

(i = 1 . . . . . M ) ,

where J, and AX i, (i = 1. . . . . M) denote the flows and forces, respectively. The phenomenological coefficients, L, representing transport behaviour, form a positive definite, symmetrical matrix according to the second law and to the reciprocity relations, provided that flows and forces have been chosen conjugate to each other and that each set consists of independent quantities. There are experimental arrangements, however, where the flows and forces do not form such a set of independent quantities: osmosis for example is a particular case of permeation, where the forces are related to each other by the Gibbs-Duhem equation [4]. It is not surprising that relations between independent flows and forces of those particular systems will be nonlinear if the constraint relations contain nonlinearities. So we assume that there are N flows and forces which are independent, and that the other forces are related to each other by expressions of AX, (AX 1. . . . . AXN) ( i = N + 1. . . . . M). Inserting this into the original constitutive equations of the general system, eqn. (2.1), shows the nonlinearity previously mentioned: N

M

J,= E L,kAXk+ E k=l

_/512

I o

(AX, . . . . . A X N ) . F

(2.1)

k=l

L,k Xk

k=N+l

(2.2)

~,Z

812

Fig. 1. Schematic illustration of an unidirectional steady transport system (e.g., membrane system). The outer phases ('), (") are assumed to be in equilibrium. They are connected with external supplying sources to keep transport steady. ("') is the region where all irreversible processes take place.

2.1. Nonlinear phenomena of reverse osmosis

The experimental equipment of reverse osmosis is illustrated schematically in Fig. 2. The general system corresponding to reverse osmosis is a permeation system, where the

145

(see Ref. [4]),

IL

j c = j, _ E ( J , + ./2)

(2.7)

yields the constraint relation between the chemical flow and the flow of volume:

__

..--

--

"////////////////////////,,

permeat e ("}

ZXx -- Jv, 2V"

JC-

I-, t"-

Fig. 2. Schematic illustration of a reverse osmosis system. The retentate solution permeates by a difference of pressure across the membrane.

(2.8)

where V ' is the molar volume of ("). Inserting this into the general constitutive eqns. (2.3) and (2.4) yields the relation we sought:

L22 AX=

--

LuV"

~2 L 1A2 ~ 1P +

¢( .2

L,v)

[

"chemical" force arising from different compositions of (') and (") and the "mechanical" force, Ap, are independent quantities. Constitutive equations of permeation are given by (see Ref. [4]):

J~ = LllA(t + L l z A P

(2.3)

J v = L12A~t + L22 A P '

(2.4)

where the chemical force is related to the difference of mole fractions of (') and (") by the relationship:

a ~ = ( 3 2~ )('~'' 3 X2 ]~ ) T,/TAX=(~JH)TffAx(2"5) where 2 denotes the average mole fraction of the first c o m p o n e n t of (') and ("), fi the average pressure and G the molar free enthalpy. Inspection of Fig. 2, however, suggests that there is only one independent force, k P , whereto the system responds by a flow of volume and by a separation, Ax. Thus there has to be a relation between Ax and k P . Indeed, this relation is found by the stationary condition, which implies that the composition of (") remains constant. So we have

Jl

= 6,'l!J v ,

Or: = c 2 ' J v ,

(2.6)

where c, ( i = 1, 2) denote the molar concentrations and J, the material flows. Insertion into the definition of the chemical flow

+

--

2L,2UJ,~ AP +L~7-12

2 - 2V"G,,AP. (2.9)

The constitutive equation between flow of volume and the difference of pressure is given by the nonlinear relation J v = ½L22 A P --

LllGll V'' --

H

2

+ ¢(½L22AP + L,,G,,V ) - 2L~2UJ,,V"~iP. (2.10) rno[ m2min

1.5

1.0

0.5 //

, 0

20

40

I

I

I

60

80

100

at

~-AP Fig. 3. Total flow of CC14-CH3OH across a membrane made of polyamide versus pressure [6].

146

_j_t m2h .! v .20 20

-Ax

10

•0

/

.4

.6

.8

1.0

~×'

/ / /

Fig. 5. Flow of volume of membrane system toluene-CH3OH with a membrane consisting of polyacrylonitrile versus composition of the retentate solution [ 12].

/ 0

.2

20

40

50

80

100

at

Fig. 4. Selectivity of membrane system CCI4-CH3OH with a membrane consisting of polyamide [6].

Of course, eqn. (2.10) can be linearized by expanding and interrupting the series after the term which is linear in Ap, but the range of validity of eqn. (2.10) is greater than that of the linearized constitutive equation. A detailed discussion shows that Jv (A p ) is monotonous (transport is stable for any Ap), and deviation from linearity agrees qualitatively with Fig. 3. We do not attempt here to give a more precise evaluation of the data presented in Figs. 3 and 4, because this goes beyond the intention of this paper and because deviation from linearity could be caused by other phenomena which would require additional experimental investigation. 2.2. Equisorptic solutions and dependence on reference state Figures 5-7 show diagrams of separation and flow of volume versus composition of the retentate solution. As before, we do not try too precise an evaluation of these data be-

cause this would require additional measurements. We emphasize, however, that eqn. (2.9) shows that separation never can change sign for any composition of the retentate solution if the L parameters are assumed to be constant. The diagrams reveal that the dependence of the transport parameters on reference state

i

-Ax

l

.10

.05

.00

.0

.2

.4

.6

.8

1.0

~--X' Fig. 6. Selectivity of membrane system toluene-CH3OH with a membrane consisting of polyacrylonitrile versus composition of the retentate solution; (+): measured by Hungerhoff [6]; (e): measured by Driessen [12].

147

alone. It is suitable to choose the second set - - t h e so-called reference state--invariant with respect to interchanging (') and (") [4], such as the average:

I -ax

.OE - -

X = ½(X' + X"). l

-S,

.00

-.0£

-

(3.2)

Dependence of transport on reference state is taken into account by considering the L parameters as functions of reference state.

-

3.1. Extension to nonlinearity

I .0

.2

.A

.6

.8

1.0

,, × Fig. 7. Selectivity of m e m b r a n e system tolue n e - C 2 H s O H with a membrane consisting of polyamide. ( + ): measured by Hungerhoff [6]; ( t ) : measured by Hensen [5].

Extension to nonlinear transport can be achieved by regarding the L parameters as functions of the forces, which leads to expressions of the form M

J, = E G ( X , AX)aX

(i=1

..... M).

A=I

(3.3)

has to be taken into account [2,4]. The composition of the retentate solution at which separation vanishes is called equisorptic. From the previous paragraph, we conclude that equisorptic composition corresponds to a vanishing coupling parameter, Lt2. Discussion of eqns. (2.9) and (2.10) shows that the equisorptic composition cannot be shifted by changing the difference of p r e s s u r e l a result which has been observed by experimental investigation [7]. In the next section we shall work out that dependence of transport on reference state is closely related to nonlinearity.

3. NONLINEAR TRANSPORT AT LOCAL EQUILIBRIUM Transport across (") is influenced by the sets of intensive variables of both outer phases; so it is easily seen that it cannot be described by dependence on the forces

ax=x'-x"

(3.1)

The tensor function L(,~, AX) may be chosen symmetrical by consequence of the reciprocity relations [2]. In the author's opinion, however, this is not the best possible kind of representation, because the tensor function is not equivalent to the original form of the equations Y( ,,~, /I X ). This is easily understood by the following consideration: the original relations are described by M functions depending each on 2M variables, whereas L(X, AX) is formed by a set of M ( M + 1)/2 independent functions. A detailed investigation [8] shows, indeed, that certain terms describing coupling phenomena of third and higher orders may be chosen arbitrarily without any physical significance, i.e., with respect to the measurable quantities J(X, AX). Therefore, we prefer extending to nonlinearity by simply adding nonlinear terms to the linear correspondence (2.1): M

J,=

M

+ E L,k,(Y)AXkAX, k=l

+ ...

k~l

(i = 1. . . . . M)

(3.4)

148

Instead of (3.4) we shall write the expansion as

Jl E

J,= E

.

.

Jl ~ 1 J2 = 0

JM-, .E . L") J, - J 2 , . . -

_j~,.j,(~')

d.

M d2Xk E ~,, ~ z 2 - 0

JM = 0

M--I

( i = 1 .... ,M)

× FI l=l

(3.5) which is more convenient for further calculation [8]. The laws of linear nonequilibrium thermodynamics have to be included in (3.5), so the matrix of linear parameters L o..... ~.....o is positive definite and symmetrical L(i) ( V ) = I(k) t R ~l 0 ..... 1..... 0 \ ' * ~ 0 . . . . . 1 ..... 0 \ k th p l a c e

(3.6)

i th p l a c e

Transport inside C') can also be treated by the local constitutive equations [1,3]

k=l

(3.7)

whereji and X, denote the local flow densities and the local values of X. X represents the tensor of local conductivities. Sometimes the equivalent formulation M

dX,

Y'. ----=M d2Xk Y'.M d;k~, d X k _ 0 x'* + k = l dz dz k=l

(3.10)

M d2Xk M ~Xik d X l d X , E X,k ~ + E 0X, dz d---Z =0(3"11) k,l=l

k=l

M d2Xk E x , , -+ k= 1 dz2

M

E k,l~l

~)~'ik d X t dXk OXl d z

dz

M ~Xik d X k

0z

d---)-= 0 '

(3.12)

k~l

3.2. Local constitutive equations and determination of transport parameters

( i - 1.... M) ' '

(3.9)

k=l

+ E

( i , k = 1..... M, all .~)

M dX, j, = Y'~ Xi, dz

and explicitly (inhomogeneous polarized membrane). The set of functions X ( z ) is obtained by local balance equations

(3.8)

Y'~ Pi, J* = d z k=l

will be more convenient for calculation, p denotes the tensor of local resitivities, which exists by consequence of local validity of the second law. We shall distinguish the following four cases: (a) ~k is constant (homogeneous membrane); (b) Mz) depends on space explicitly (inhomogeneous membrane); (c) ~ , ( X ( z ) ) depends on space implicitly (homogeneous polarized membrane); (d) ~ . ( X ( z ) , z ) depends on space implicitly

referring to the cases (a), (b), (c), (d), respectively. These balance equations are valid for the steady state. Obviously, polarization is described by an additional term, which makes the solution of the differential equations difficult as it is nonlinear. Equations corresponding to case (d) are given in Refs. [9-11] for conduction of heat across inhomogeneous polarized media. Methods of numerical solution can be found in Ref. [10]. Fortunately, determination of the global transport parameters does not require knowledge of the total solution X ( z ) " we are only interested in the dependence of the flows on the boundary conditions X ' = X(~/2), X " = X ( - 8/2). The method of finding the global transport parameters consists of integrating the local constitutive eqns. (3.7) and (3.8) over the volume of ('"). 3.2.1. Unpolarized medium

For case (a), integration of eqn. (3.7) yields M F

4 = E

X,,AX,

k=l M

= Z Li, A X , k=l

( i = 1..... M)

(3.13)

149 Case (b) is resolved by integrating eqn. (3.8). We obtain M

1 1"8/2

A X, = ,,,: j~-fi j _ +/2p,~ d z

out the integration is avoided, and an expression is obtained, which can be compared with eqns. (3.4) and (3.5). After a long and tedious calculation it is found [8]

au,(x,o)

M

= ~_, J~R,k

(i = 1 .....

M)

(3.14)

Asymmetrical transport does not occur, although the structure of ("') is permitted to be asymmetrical. Additionally there is no dependence of transport on reference state--this result had to be expected by inspection of the differential equations (3.9) and (3.10). As these equations are linear, any relations derived from these equations have to be invariant with respect to a constant shift .,~ of X.

3.2.2. Homogeneous polarized medium Integration of eqn. (3.7) yields M

which is a line integral, the calculation of which requires knowledge of the path X ( z ) (for - 6 / 2 <~z <~6/2) from (") to ('), i.e. the solution of the differential equation (3.11). The path can be chosen arbitrarily if ~ is assumed to be a function of state of all variables. So we choose a path first parallel to the X~ axis, then parallel to the X 2 axis and so on. Finally we obtain

o(,~x~) .....

a(Ax,)a(ax~)

F O+-lh,j(~)

1

2 +.-, 8 0 x r . . . . .

axo

( i , j , r = 1. . . . . M; s = 1,3 .... odd)

a'J,( x,o)

(3.18)

=0

aCAXr) . . . . . O(AX.)a(AXj)

(s = 2,4 .... even) (3.19) Derivatives of even orders vanish, as it should be by consequence of symmetry: the structure of ( ' " ) is homogeneous, its response has to be symmetrical with respect to interchanging (') and C). The parameters of transport depend on the reference state by consequence of inner polarization, as we expected by inspection of eqn. (3.11). Comparison with the series (3.5) yields the transport parameters [8]: M

L~,I +,_+,:.... ~.... _ , , , . , . , ( R ) =

1

H t=l

1 (Jr-Jr+,)!

a~,4(2,o)

x

0 ( a x , ) j ' - * , .... 0(AXM_,) j" ' ' ' 0 ( ~ X ~ , ) j," M-I

=1-I

1

/=+ ( j / - j / + 1)!2 j ' - t

J i = - S k=,

~}j+- Ii(i)

x x

t

- 0 ..... , ..... o ( . f )

o.+( ,+`'-+. . . . . aR[,-,~. ] - ' . . . . . a , % -

it

(X~ . . . . . X'k_,,Xk,X~,+, . . . . . X ~ ) d X k

(3.16) Instead of X' and X", we insert the forces and the reference state: M F x~+ AX~/2

X ( X I + ½A X, . . . . . X,,,..... , 2 M - ½ A X M ) d X 1 , (3.17) and expand into powers of AX. Thus carrying

(3.2o)

( i = 1 . . . . . M; j~ = 1,3 .... odd; J M ~ J M _ , ~ ... ~ j , ) Parameters of even orders vanish as we stated above. 3.2.3. Inhomogeneous polarized medium Dependence of transport on the boundary conditions cannot be determined by integra-

150

tion of eqns. (3.7) and (3.8), respectively, since the integrand can never be represented by a complete differential--consequently knowledge of the path of integration, i.e., of the solution of eqn. (3.12) is required. Thus we try to calculate X(z) by perturbation theory from eqn. (3.8). The resistivity, O, is assumed to depend weakly on X. Expanding around reference state yields:

O,k ( X,z ) = oik (

)

M

+ IE= 1 (x,- Z) ~ o

Pik

(X,z)

M

g (x,-g)(xo, l,m=

1

2z

ax, axm

...

(3.21) After some calculation [8], a series of AX(X, J) is obtained, the parameters of which are given by (we only write terms up to third order): 1

f~/2

R,~(X) = - - . p,k(X,z) dz F J_ s/2

(3.22)

,=,

×

pl,(X,z') dz' [Jo: 6/2

fo

-p,,(,~,-y))dy]dz

(3.23)

----5 2F ,. =, -~/2 0X,~0Z pik ('~'z)

x [ fo:o,.(X,z') dz' I

1"8/2,

,-

Ip,°(x,y)

-Pl,(X,-y))dY]

× [foZpmr(A',z') dz' -½ fo~/Z(p,r(X,y) - O , r ( . ~ , - y ) ) dy] dz

(3.24)

The transport parameters depend on reference state by consequence of inner polarization. The second-order parameters (3.23) describe asymmetrical transport since the reference state is chosen invariant with respect to interchanging (') and ("). The convolution integral on the right hand side of (3.23) vanishes for a symmetrical structure of ("') as it should because p is an even function of z under this condition. Formula (3.23) reveals necessary and sufficient conditions with respect to asymmetrical transport: asymmetrical structure and inner polarization of ("), which can be observed by dependence of the linear parameters on reference state.

3. 3. Relationship between dependence of transport on reference state and that on the forces The formulae (3.20), (3.23) and (3.24) demonstrate that the dependence of transport on forces is related to that on reference state. Inner polarization is a necessary and sufficient condition of nonlinear transport phenomena. Thus the nonlinear parameters can be calculated from the linear ones by (3.20) if ('") is homogeneous. Nonlinear transport is represented by the M ( M + 1)/2 functions Lik(X ), each depending on M variables, in this case. This drastic reduction of the number of transport parameters could not be concluded by symmetry considerations, although it is a consequence of spatial symmetry. The latter is proved by eqns. (3.23) and (3.24): nonlinear parameters are related to the linear ones, but their calculation requires knowledge of the explicit dependence of the resistivity on space, i.e., of the structure of ("'). On the other hand,

151

knowledge of linear and nonlinear parameters could be used to get some information about structure. We do not discuss this problem here.

It should be interesting to investigate what information about dependence on reference state can be concluded from conditions of stability.

4. FINAL REMARKS

ACKNOWLEDGEMENT

The present paper has demonstrated that certain classes of nonlinear transport phenomena can arise from well-known nonequilibrium thermodynamics, where linear constitutive equations are applied. Nonlinearity is introduced by nonlinear constraint relations between the forces and the flows, respectively, or by inner polarization, where the transport parameters depend on reference state. Although analytical solutions determining the intensive variables as functions of space inside the membrane region cannot be obtained for polarized systems, certain properties describing global transport behaviour have been calculated. The result gives a relationship between the dependence of transport on reference state and on forces. In the case of a homogeneous membrane, the nonlinear parameters can be deduced from the dependence of Onsager's tensor on reference state, which is a drastic reduction of the number of independent parameters. For the case of an inhomogeneous membrane, necessary and sufficient conditions with respect to asymmetrical transport have been deduced. Additionally, it is proved that certain information about structure of the membrane is hidden in the nonlinear parameters.

I wish to thank Dipl. Chem. H. Hensen for helpful discussions and for asking the questions which led to this work.

REFERENCES 1 R. Haase, Thermodynamics of Irreversible Processes, Addison-Wesley, Reading, MA, 1969. 2 F. Sauer, Handbook of Physiology-Renal Physiology, American Society of Physiology, Washington, DC, 1972. 3 S.R. de Groot and P. Mazur, Non-equilibrium Thermodynamics, North-Holland, Amsterdam, 1962. 4 H. Hoff, J. Chem. Soc., Faraday. Trans. 1, 77 (1981) 2325-2340. 5 H. Hensen, Thesis, RWTH Aachen, 1977. 6 J. Hungerhoff, Ph.D. Thesis, RWTH Aachen, 1975. 7 J. Kopecek and S. Sourirajan, Reverse Osmosis Separation of Mixtures of Organic Liquids, Reverse Osmosis, Academic Press, New York, 1970. 8 H. Hoff, Representation of Nonlinear Constitutive Equations, to be published. 9 H.S. Carlslaw and J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, Oxford, 1959. 10 P.I. Schneider, Conduction Heat Transfer, Addison-Wesley, Reading, MA, 1957. 11 U, Grigull and H. Sander, W~irmeleitung, Springer Verlag, Berlin, Heidelberg, New York, 1979. 12 H. Driessen, Ph.D. Thesis, RWTH Aachen, 1976. 13 S. de Groot, Thermodynamik irreversibler Prozesse, B.I., Mannheim, 1960.