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THERMODYNAMIC FLUCTUATION THEORY FOR SHEAR FLOW Denis J, EVANS* and H.J.M, HANLEYt Thermophysical Properties Division, National Engineering Laboratory, National Bureau of Standards, Boulder, CO 80303, USA

Received 17 February 1981 A generalization of the Einstein relation for thermodynamic fluctuations is applied to a fluid undergoing Couette flow. It is speculated that if the strain rate fluctuations are to reduce to the known equilibrium results, then the shear dilation effect, that is the variation of the pressure with shear at constant temperature and density, must be a nonanalytic function of strain rate. This conclusion is consistent with computer results obtained previously.

1. Introduction Recently we have used computer simulation ~'2) to show that if a fluid is subjected to a steady shear, characteristic state functions such as a pressure and an internal energy are functions of strain rate at a given temperature and number density. This has prompted us to propose an extension of equilibrium t h e r m o d y n a m i c s to the nonlinear regime~-~). We have made analytical and numerical checks on the validity of some of the predictions of the thermodynamic theory. For instance it has been shown that the thermodynamically predicted relation between shear induced energy and pressure changes is satisfied exactly for the soft sphere fluid3); we have proposed and tested conditions for stability of the fluid under shear. The system discussed in this paper is a macroscopic volume element located within a large surrounding body of fluid undergoing isothermal Couette flow. The viscous heat produced in this dissipative system is removed by an ideal thermostat. [As will be shown later, many of the results should be independent of whether the system is isothermal or adiabatic but the theory is simpler if we consider isothermal systems.] A generalized Einstein relation 6) for f l u c t u a t i o n s in the t h e r m o d y n a m i c variables is postulated. This postulate is additional to and independent of the ones used to construct the generalized thermodynamics reported in refs. 3 and 4. One c o n s e q u e n c e of the postulate is that the shear dilation effect (variation *Fulbright Scholar on leave from the Australian National University, Canberra, Australia. tSupported by the Office of Standard Reference Data. This work was carried out at the National Bureau of Standards and is not subject to copyright. 0378-437118110000-0000/$02.50 O North-Holland Publishing C o m p a n y

568

DENIS J. EVANS AND H.J.M. HANLEY

of pressure with strain rate at constant t e m p e r a t u r e and density) is consistent with a nonanalytic function of the strain rate. This constraint is in fact in a g r e e m e n t with the results of c o m p u t e r simulations in both two and three dimensional systemsl'2).

2. A thermodynamic theory The main p a r a m e t e r s of the s y s t e m are: the density (p) or volume (V) with p = N / V , where N is the n u m b e r of particles; the kinetic t e m p e r a t u r e (T) defined via,

kBT

Smt ,-

1/~ 1

where v~ and u(ri) are the particle velocity and streaming velocity at ri, respectively; an energy (E), where

1

1

with q~ij the pair potential and m the particle mass; and the elements of the pressure tensor P. Defining the element of the strain rate tensor 3' as 7 = ½[(duddy) + (duyldx)] and a pressure p by p -- -~Tr P, we have the relation P = p(7)l

- 2~(7)~,,

(2)

with I the unit tensor and "0 the viscosity coefficient. The simulation procedure evaluates 7'8) the elements of P at constant density and t e m p e r a t u r e and hence a variation of the viscosity and pressure with shear rate can be deduced. R e f e r e n c e should be made to refs. 9-11 for discussions on the c o m p u t e r algorithm and for c o m m e n t s on the statistical mechanical description of a system under shear. Observations for m a n y s y s t e m s in both two and three dimensions for a variety of interaction potentials (both spherical and nonspherical) led us to conclude that the energy of eq. (1) is a state function; E = E(T, V, 3') and we p r o p o s e d 4) a first law incorporating a new state function ~(T, V, 7); d E = d Q - p d V + ~ d7.

(3)

A heuristic t h e r m o d y n a m i c s follows if an entropy is postulated so that T dS ~ d Q . H e n c e one can write a Helmhoitz free energy, A, for a reversible change about a steady state dA= -SdT-pdV+~d3,.

(4)

THERMODYNAMIC FLUCTUATION THEORY FOR SHEAR FLOW

569

Formally ~"= (OE/OT)s,v but using that the hydrostatic pressure of an isothermal system is independent of strain rate in the dilute gas limit (p--> 0), we can derive an expression for ~, namely

V

Numerical culculations of ~ have been reported for a number of model systems3'4'12). Also, with eqs. (1)-(4) and a Maxwell relation an equation for the change of energy with the strain rate, follows 32E _ 3~ 3V07 OV

T 02~ OTOV"

(6)

This equation is satisfied identically for the soft sphere fluid 3) and has been shown, via computer simulation, to be consistent to within statistical uncertainties with data for the L e n n a r d - J o n e s systemJZ).

3. Nonequilibrium thermodynamic fluctuation theory Consider a steady state system for which T, V and y are the independent variables and define a state vector by (X) -- (T, V, 7) = Xo,

(7)

where subscript 0 refers to the steady state. It is then p o s t u l a t e d that the probability of a state variable fluctuation ~X = X - X 0 is related to the minimum reversible work 6Wmin associated with the displacement 6X, P(SX) ~ e x p [

BWmin], k~T J

(8)

where ka is Boltzmann's constant. Eq. (8) represents a generalization of Einstein's equilibrium thermodynamic fluctuation theory6). For our system using eqs. (3) and (4) we find that 6Wmi n

:

~E + p 6 V - T 6 S - ~63,

/a2A~ = ½6X • \oX2/x=xo

.

6 X + 6 T 6 S + ff(,53),

(9)

hence

l

P ( 6X ) c¢ exP 2--~BT [ 6p6 V - ,5S6T - $~/~7].

(1o)

570

DENIS J. EVANS AND H.J.M. HANLEY

Eq. (9) obviously assumes that the Taylor series expansion of the Helmholtz free energy about X0 exists but this is not an expansion about the equilibrium state, rather it is an expansion of the free energy about the steady state. Eq. (10) can be rewritten, again using eqs. (3)-(6), as

P(6X) oc e x p (

1

[C_~3TZ

Op

--

6V

2+

Off

where Cv.~ is the specific heat at constant V and 3'. The expression inside the square brackets must be positive definite for the steady state fluctuations about X0 to be finite. This constraint is identical to the stability conditions derived in ref. 3 without the assumption of (8). An interesting result follows if one considers fluctuations in the strain rate at constant temperature and volume. At constant temperature and volume, eq. (11) is

1 02A,, 2] P(63") oc exp - 2kBT 0y 203' J'

(12)

where the state vector is equated with the strain rate so that, using eq. (12), the quantity (A3' z) can be evaluated around the equilibrium (3' ~ 0 ) state to give (A3'2)~=0 = k , T { k~y2] 02A "~ ' = kBT (0-~1-' ,,u,/ " y=O

(13)

y=O

Further, if one considers the variation of the pressure with 3' as a series lira p(T, V, 3') = p(T, V, O) + Ap(T, 3") I 3" I',

(14)

3,~0

where Ap is a 3'-independent coefficient and m a constant, eq. (13) becomes, using eq. (5),

(15) V

Note that eq. (15) will predict that (A3' z) will only be zero as 3 ' ~ 0 if m < 2 ; otherwise the limiting shear rate fluctuations will be nonzero or infinity. Since the canonical equilibrium ensemble average of (za3'2) is zero for an infinite system ~3) one must speculate that the index m is indeed less than two. If true, an analytical expansion for the pressure around 3, = 0 cannot exist.

THERMODYNAMIC FLUCTUATION THEORY FOR SHEAR FLOW

571

4. Computer simulation We have s h o w n via nonequilibrium molecular dynamics that the pressure variation with strain rate is consistent r4) with the expression

p(T, V, 3') = p(T, V, O) + Ap(T, V) I 3' I 3/2

(16)

for a three dimensional system t'9) and with

p(T, V, 30 = p(T, V, 0 ) + Ap (T, V) 13,11ogl3,'rp(T, V)I

07)

for a two dimensional fluid 2) where A~ and rp are analogous to Ap of eq. (14). Fig. 1 and 2 show the variation of hydrostatic pressure with strain rate as revealed by h o m o g e n e o u s shear nonequilibrium molecular dynamics. Fig. 1 shows the results for the triple point L e n n a r d - J o n e s fluidg), while fig. 2 shows the corresponding results for the two dimensional soft disk fluid close to the freezing density2). Substitution of eqs. (16) and (17) into eq. (15) gives, for three dimensions,

4

(I

(A3,2)v=o = ~ kBTv '12

A~ dV'

]-'

,

(18)

V

1.0 _

0.1

0.01 0.01

J 0.1

1.0

17"1 Fig. 1. The log~0-1og~0plot of the shear induced pressure change zip* as a function of strain rate for the triple point Lennard-Jones system--kT/~ = 0.722, po 3= 0.8442, the straight line is from eq. (16). Note: p* = po'3[~, 3,* = ~,o'(m/~)m. The results.were obtained by Evans9).

572

D E N I S J. E V A N S A N D H.J.M. H A N L E Y 3.0

N= 50

2.0

m

-1.5

-1.0

-0.5

0

loglol~'*l Fig. 2. T h e logl0-1inear plot of the s h e a r induced pressure change as a function of strain rate for the soft disk s y s t e m (4' = ~(~/r)12), close to freezing (ptr 2 = 0.8 (4/3) 1/2, kT/¢ = 1.0). The straight line is from eq. (17). T h e results were obtained for a 50-particle s y s t e m by Evans2).

and for two dimensions (19) v

In both cases the ~, -- 0 limit is zero as one might expect.

5. Conclusion An expression for the strain rate fluctuations of a fluid subjected to a shear in the a s y m p t o t i c limit of zero shear has been derived using t h e r m o d y n a m i c arguments. It is then suggested that the pressure should be a nonanalytic function of the strain rate at constant t e m p e r a t u r e and volume or density. This conclusion is reinforced by our c o m p u t e r simulation results for a three and a two dimensional systemt'2'9"]2). For example, our t h e r m o d y n a m i c argument leads to the result that the index m of eq. (14) must be less than two. We have two remarks: 1. For sufficiently small strain rates the t e m p e r a t u r e rise in an adiabatic

THERMODYNAMIC FLUCTUATION THEORY FOR SHEAR FLOW

573

s y s t e m u n d e r g o i n g C o u e t t e flow is p r o p o r t i o n a l to 3, 2. S i n c e as w e h a v e j u s t s e e n , t h e i n d e x m, o f eq. (14) g o v e r n i n g t h e s h e a r d i l a t i o n e f f e c t a p p e a r s to b e less t h a n t w o w e c a n , b y c h o o s i n g a sufficiently small s t r a i n r a t e , m a k e the r a t i o o f v i s c o u s h e a t i n g to s h e a r d i l a t i o n as s m a l l as w e p l e a s e . T h i s s u g g e s t s that although our previous computer results have been derived for isothermal s y s t e m s t h e y m a y also a p p l y to a d i a b a t i c s y s t e m s in the 3' = 0 limit. S o m e a m b i g u i t i e s as to t h e d e f i n i t i o n a n d c o n t r o l o f the t e m p e r a t u r e c o u l d t h e n b e removed. 2. I n the f u t u r e it m a y b e p o s s i b l e to c a l c u l a t e

Acknowledgments W e a r e g r a t e f u l to Dr. J.R. D o r f m a n f o r c o m m e n t s in this w o r k a n d to M r s . Karen Bowie for preparing the paper.

References l) 2) 3) 4. 5)

D.J. Evans and H.J.M. Hanley, Physica 103A (1980) 343. D.J. Evans, Phys. Rev. 22A (1980) 290. D.J. Evans and H.J.M. Hanley, Phys. Lett. 80A (1980) 175. D.J. Evans and H.J.M. Hanley, Phys. Lett. 79A (1980) 178. See the book: G. Astorita, An Introduction to Nonlinear Continuum Thermodynamics (Societa Editrice di Chimica, Milan, 1975) and the review J. Keizer, Accounts of Chemical Research (July 1979). 6) A. Einstein, Ann. der Phys. 33 (1910) 1275. See also A. Einstein, Investigations on the Theory of the Brownian Motion (Dover, 1956). 7) W.H. Bauer and E.A. Collins, Rheology, F.R. Eirich, ed. (Academic Press, New York, 1967) vol. 4. 8) J.H. Irving and J.G. Kirkwood, J. Chem. Phys. 60 (1950) 3567. 9) D.J. Evans, Phys. Rev. 23A (1981) 1988. 10) D.J. Evans, Mol. Phys. 37 (1979) 1745; J. Stat. Phys. 22 (1980) 81. 11) D.J. Evans, W.G. Hoover and A.J.C. Ladd, Phys. Rev. Lett. 45 (1980) 124. 12) H.J.M. Hanley and D.J. Evans, in preparation. 13) J.P. Hansen and I.R. McDonald, Theory of Simple Liquids (Academic Press, New York, 1976). See especially their equation 7.115.

574

DENIS J. EVANS AND H.J.M. HANLEY

14) These functional forms have also been predicted theoretically although, as we have discussed previously, the predicted values of the coefficients are several orders of magnitude smaller than the values found in simulation: K. Kawasaki and J.D. Gunton, Phys. Rev. A8 (1973) 2048. See also M.H. Ernst, B. Cichocki, J.R. Dorfman, J. Sharma and H. van Beijeren, J. Stat. Phys. 18 (1978) 237.

Note added in proof If we define a w a v e v e c t o r d e p e n d e n t strain rate y ( k ) in the u s u a l mannerS3), t h e n e q u i l i b r i u m statistical m e c h a n i c s s h o w s lim lim (Ay(k)2)~gk) = 0. k~0 ~o-~0

T h e b r a c k e t s d e n o t e a n o n e q u i l i b r i u m s t e a d y state a v e r a g e in a k - d e p e n d e n t s t r a i n field of a m p l i t u d e ~/o. N o w if (Ay(k)2)vo(k) is a continuous f u n c t i o n of 7o a n d k then lim lim (Ay(k)2)vo~k) = 0, ~'0~0 k-~O

that is lira (Ay2)~o -- 0. yO~0

I n this c i r c u m s t a n c e eq. (15) i m p l i e s m < 2.

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