Volume 144, number 2 PHYSICS LETTERS A 19 February 1990

ON T H E EXTENDED THERMODYNAMICS OF DILUTE DUMBBELL SOLUTIONS

J. CAMACHO and D. JOU Departament de Fisica (Fisica Estadistica), Universitat Autbnoma de Barcelona, 08193 Bellaterra, Catalonia, Spain

Received 18 August 1989; revised manuscript received 15 November 1989; accepted for publication 20 December 1989 Communicated by D.D. Holm

The relation predicted by extended irreversible thermodynamics between the fluctuations of the viscous pressure tensor around an equilibrium state and the high-frequency limit of the imaginary part of the complex viscosity is confirmed from kinetic theory for dilute dumbbell solutions.

The study of the fluctuations of dissipative fluxes bears a deal of interest in the framework of the so- called extended irreversible thermodynamics (EIT), since it provides the coefficients of the nonclassical part of the generalized entropy by using only the equilibrium distribution function and the micro- scopic definitions of the dissipative fluxes [ 1 ]. Mi- croscopic confirmations of this method of evaluating these coefficients have been obtained for the case of ideal [ 2 ] and real gases [ 3 ]. However, polymer so- lutions have not been considered so far, in spite of being of special interest due to the long relaxation times in their viscous pressure tensor. In this article we use more general expressions for the second mo- ments of the fluctuations than the usual ones, since we take into account the existence of additional re- laxation times.

The aim of this article is, on the one hand, to con- firm the mentioned relations and, on the other hand, to show a relation between the second moments of the fluctuations of the viscous pressure tensor and the imaginary part of the complex viscosity (r/" ( to)) in the high-frequency limit.

The Einstein formula for the probability of the fluctuations around the equilibrium state is

Proc exp ( 8 2 S / 2 k ) , ( l )

k being the Boltzmann constant, and 52S the second differential of entropy. Using the nonequilibrium entropy given by extended irreversible thermody- namics [ l ] one obtains for the second moments of

the fluctuations of the viscous pressure tensor the following expression,

( ",, "v krIT~ ~P ij ~P Ja ) e,= = - - ~ "-'Okt , (2)

where I ~ are the components of the viscous pressure tensor, r/is the viscosity coefficient, r the relaxation time associated to I ~v and A is a four-tensor with components

However, this expression has been obtained from a model for I 5v which only contains one relaxation time.

As we have shown [4], the tensor I ~ ~ can be de- composed quite generally as a sum of partial tensors I~} ', each one being an independent variable in the generalized entropy function. Then, we can write the nonequilibrium part of the entropy as [ 4 ]

V oliPi -Pi , (3)

i

where ct~=zi/~l~ and s, u and v are the specific en- tropy, specific internal energy and specific volume, respectively. Obviously we could also consider cross terms like ot~j#~':l~) ', but we ignore them for sim- plicity. The inclusion of such terms leads to com- plicated expressions and more tedious calculations, however, the final results [ 5 ] are the same as those in this paper.

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Volume 144, number 2 PHYSICS LETTERS A 19 February 1990

The second differential of entropy restricted to the #~' variables follows directly from eq. (3):

/')'t'i - ° v G 2 S = -- /~. 2-//TGP~ :GI 5v . (4)

On the one side, using that for a multivariant Gaus- sian distribution of the form Procexp( - ½Gouiuj) the second order moments of the fluctuations are given by (uiuj)=G;f ~ and one easily finds the following expression for the fluctuations of 1~7:

( "v °v kTrh~o. (5) (GP~)xy( GPj )xy )~q = ri V

This macroscopic relation will be compared with the microscopic results derived from kinetic theory for some dilute dumbbell solutions. Nevertheless, this relation has the inconvenience of relating quantities concerning the 1~7 tensors, which sometimes have no direct microscopic interpretation. So it seems suit- able to look for a relation which contains the total viscous pressure tensor #v only. To do that, we may evaluate the second moment of fiv e ,

°v ( $16~xy 8Pxy > eq

. kT rl~ (6) = 2,,j <(GPT)xy(6P;)xy}~q= T ~ r,

On the other side, expression (3) together with the EIT formalism leads [4] to a generalized Maxwell model, with the imaginary part of the complex vis- cosity given by

q" (o9) - 3-" ziqi . (7) oJ "7' 1 + (mzi) 2

For very high frequencies (o9>> z7 ~ Vi) one has

lira oJq"(og)= ~ r h . (8)

Comparison of eqs. (6) and (8) yields

o ~ kT (GPxv G # V y ) e q = - lim mq"(m) (9)

V o ~ a o

This relation only includes global (not partial) quantities. It shows that the fluctuations of the vis- cous pressure tensor around an equilibrium state are related to its high frequency response.

Our aim is to compare the macroscopic results, eqs.

(5) and (9), with microscopic ones for some dilute dumbbell solutions. In the next paragraph we start with rigid dumbbell solutions.

From Kramers' expression for the viscous pres- sure tensor [ 6,ch. 14 ] we can distinguish two partial tensors for the rigid dumbbell contribution,

Py = - 3 n k T ( ( u u ) - ~O), (10)

P ~ = - 6 n k T 2 V v : ( u u u u > , ( 11 )

n is the number of dumbbells per unit volume, ;t is a time parameter, O is the unit tensor, u the unit vec- tor in the dumbbell direction and ( ) is the average over all configurations. Let us begin with Py. To check eq. (5) we must calculate the microscopic sec- ond moments of the fluctuations of Py,

( (GP'i')xy( GP~')xy )eq

=(3nkT)2 I du' u'~u'y I d , u u, < >o,.

(12)

To do this, we need to evaluate the equilibrium av erage of the fluctuations of the distribution function. This can be done by introducing the Boltzmann expression for the entropy (S= - Nkfdu f l n f ) into the Einstein formula for the fluctuations, eq. ( 1 ); this yields

1 < Gf(u) Gf(u') )~q = ~f~qS(u-u ) , (13)

where N is the total number of dumbbells. With the help of this expression and f~q= l /4n, we can write eq. (12) as

( (GP~')xy( GP'i')xy ) ~

(3nkT)2 I 2 2 3 (nkT) 2 -- ~ d l l feqUxUy: 5 N (14)

After manipulating the microscopic expression for Py and the diffusion equation for the distribution function one finds [ 4 ] r/z = ]nkT2, z~ =2, so the right- hand side in eq. (5) becomes

kTql 3 (nkT) 2 - - _ ( 1 5 ) Vzl 5 N

Comparison with eq. (14) shows that expression ( 5 ) in correct in this case.

The situation is different for the tensor P[ since,

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Volume 144, number 2 PHYSICS LETTERS A 19 February 1990

as can be easily seen [4 ], r2 = 0. This means that there is no contribution of P~ to the generalized entropy up to second order, hence expression ( 1 ) cannot be used to evaluate the fluctuations of P~.

Kramers ' expression for the dumbbell contribu- tion to the viscous pressure tensor in a Hookean dumbbell solution gives

P~y= - n n ( QxQy ) , (16)

H is the elastic constant and Qi are the components of the vector which joins the beads. Following the same procedure as earlier one finds for the fluctuations

( (~pV)xy( ~pV)xy )eq

(nH)Z ~ ( n k T ) 2 2 2 (17) - dQf~qQxQy - N

Here we have used the Hookean dumbbell equilib- rium distribution function

f~q = (H/2rckT) 3/2 e x p ( - H Q 2 / 2 k T ) .

Introducing q= n k T z - as stands for dilute solutions [ 6 ] - in the second member of eq. (5), one obtains the same result as in eq. (17).

The extension to the Rouse-Zimm model for polymers is straightforward since in this model pv is a sum of tensors P~', each one having a microscopic expression like (16). The only difference is that the elastic constant H must be replaced by an effective constant H* which combines H with the ith eigven- value of the modified Rouse matrix. So eq. (17) re- mains valid for these P~', since it does not contain the elastic constant. Moreover, as qi= nkTzi, eq. (5) is also verified for each tensor P~.

Let us finally consider FENE dumbbells, for which we do not know the microscopic definitions of the I~ ', so we try to verify eq. (9) and not eq. (5). (No- tice that eq. (9) is also verified in the rigid, the Hoo- kean and the Rouse-Zimm cases.)

For FENE dumbbells the elastic force reads FC= H Q / [ 1 - ( Q/ Qo ) 2 ] . Qo is the maximum dumb- bell extension. Kramers ' expression for the dumbbell contribution to the viscous pressure tensor is [ 6 ]

P ~ = - n H I _ ( Q / Q o ) 2 . (18)

Following the same procedure as in the previous cases one gets

( (SP").~,( 8PV)~y )~q

(nil)2 f Q2Q2 - --ff . jdQf~q [ I _ ( Q / Q o ) 2 1 2 . (19)

With the use of the equilibrium distribution function

1 ~eq = ~q [ 1 - (Q/Qo) 2 ] b/2

( J~ is the normalization constant, and b - H Q ~ / k T is a dimensionless parameter) and after some long though straightforward calculations one obtains

( n k T ) 2 b ( (SpV)xy( 8PV)xY)~- N b - 2 " (20)

The solution of the distribution function equation for an oscillating velocity gradient can be introduced in the microscopic definition, eq. (18), to give q" (o9). For high frequencies one finds [6, oh. 13]

q"(og)/o9 b 1 + .... (21)

n k T - b - 2 o92

A direct comparison of eqs. (20) and (21) reveals that relation (9) holds, as we intended to show.

Relations (5) and (9) obtained from EIT have been confirmed by kinetic theory for dilute rigid, Hookean and FENE dumbbell solutions. Straight- forward extensions can be made for the rod model and the Rouse-Zimm model.

The results of this paper are not restricted to the generalized Maxwell model. As we mentioned ear- lier, one could consider cross terms in the general- ized entropy; this would lead, through the formalism of extended thermodynamics, to a more general dy- namics for the viscous pressure tensor than the gen- eralized Maxwell model, which is recovered in the particular case when all the coefficients ot o vanish. It can be seen [5 ] that the inclusion of these cross terms in the entropy modifies the second moments of the fluctuations of I b v and the complex viscosity in such a way that relation (9) remains valid. Note that the presence of the coefficients ot o provides the model with some flexibility to adapt itself to specific models, although it may fail for some cases, as Jeffreys'. This is not surprising because such a model leads to in- finite speed for transverse velocity waves in the high-

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Volume 144, number 2 PHYSICS LETTERS A 19 February 1990

frequency limit, which, in our opinion, is an un- physical feature. EIT, on the other hand, provides constitutive equations leading to finite wave speeds.

Relation (9) shows that the fluctuations of the viscous pressure tensor around an equilibrium state are related to the high frequency response of pv. This result admits the following interpretation. When a system is at equilibrium fluctuations take it out of equilibrium so we can think that fluctuations pro- vide the system with a storage of free energy. Since ~/" (oJ) is also connected to the stored energy it is not surprising that the imaginary part of the complex viscosity is linked to the fluctuations and that the real part, related to dissipation, is not. In fact, according to the EIT formalism, dissipation vanishes in the high frequency limit. Moreover, the existence of this high frequency limit in eq. (9) could reside in the fact that fluctuations take place in a period of time much shorter than the relaxation times of the viscous pres- sure tensor.

From an operational point of view, expression (9) allows one to obtain a nonstationary property, as limo~ootor/"(to), without solving the evolution

equation for the distribution function. All we need is the equilibrium distribution function, and the mi- croscopic definition of the viscous pressure tensor.

Financial support of the Direcci6n General de In- vestigaci6n Cientifica y T6cnica of the Spanish Min- istry of Education under the grant PB/86-0287 is acknowledged.

References

[ 1 ] D. Jou, J. Casas-VD.quez and G. Lebon, Rep. Prog. Phys. 51 (1988) 1105.

[2] D. Jou and T. Careta, J. Phys. A 15 (1982) 3195. [3] D. Jou, Physica A 155 (1989) 221. [4] J. Camacho and D. Jou, On the thermodynamics of dilute

dumbbell solutions under shear, J. Chem. Phys. (1989), to be published.

[ 5 ] J. Camacho, Trabajo de investigaci6n, Universitat Autbnoma de Barcelona (1989).

[6] R.B. Bird, O. Hassager, R.C. Armstrong and C.F. Curtiss, Dynamics of polymeric liquids, Vol. 2 (Wiley, New York, 1987).

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