Shear gradient dependence of the viscosity of critical mixtures

Volume 50A, number 6

PHYSICS LETTERS

13 January 197.5

S H E A R G R A D I E N T D E P E N D E N C E O F T H E VISCOSITY O F C R I T I C A L M I X T U R E S W. OXTOBY* Department of Chemistry, University of California, Berkeley, California 94720, USA Received 11 November 1974 We present a theory of the nonlinear shear viscosity of fluids near the critical point, and suggest that the experimentally obs6rved leveling off (eusplike behavior) of the excess viscosity may be due to shear gradient effects.

Fluids of small molecules away from the critical p i n t respond linearly to moderate (1-3000 sec -1) shear gradients, became correlations extend only over molecular distances while the local velocity changes on a macroscopic scale. Near the critical point correlation lengths become large, suggesting that nonlinear (non-Newtonian) shear viscosities may be observable. Fixman and co-workers [e.g. 1 ] have developed a phenomenological theory for the dependence of viscosity on shear gradient; in the linear case (small shear gradient) they predict a ( T - Tc)-I/2 divergence of the shear viscosity, which is much stronger than has been found experimentally. More recent microscopic mode-coupling calculations for the linear shear viscosity [2] have predicted a logarithmic divergence, in good agreement with available experimental data. We have developed [3] a microscopic theory for the nonlinear shear viscosity, by extending Mori's [4] projection operator approach to the nonlinear regime. The predicted viscosity is

(and X) very small, this reduces to the linear case [2] and predicts a logarithmic divergence. For X> 1, the viscosity will level off to a constant. We have calculated (by numerical integration) the function A(X) = (n(X = 0) - n(X))/n(X = 0),

which should be a universal function of X, independent of k c. The result is shown in fig. I. For large X, A(X) "" (8/45 ~r2) In X/Xo; Xo ~ 0.45. In capillary viscometer experiments, the shear gradient D is not constant. We have estimated the effective value, Deft , by averaging D over the cross section, weighted by the volume flux in the capillary, assuming a parabolic velocity profile. The results is [3] Deft = 8/15 Dmax where Dmax is the gradient at the capillary walls. 0.06

kc~ ,7(X) = ,7° + (,~(X =0)/8,0) f d3s ~ , , .m, x)

0.05

0.04

where f(s, X) satisfies the equation

k a~xf(S, X) - (Ko(S)/3rrSy)f(s, X) = - 2Sx(1 +s2)-2;

z~(x)

0.03

0.02

Ko(S) = 43-[1 +s 2 + (s 3- s -1) arctan s] and k = ~3D/k B T. r/° is the background shear viscosity, ~ the correlation length, k c a microscopic wave vector cutoff [5], and D the shear gradient. For D * NSF predoetoral fellow and Danforth Foundation fellow.

0.0

i

i

i

i

i llll

0.1

i

I X

i

i

i

i i i

IO

Fig. 1.

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Volume 50A, number 6 0.24

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PHYSICS LETTERS I

to nonlinear effects. In fig. 2 we compare theoretical predictions and experimental results [7] for the isobutyric acid-water mixture. (In the experiment, Dmax ~. 550 sec -1.) The leveling off takes place at approximately the temperature predicted by the theory. The major sources of uncertainty have been discussed elsewhere [3], and further comparisons with experiment have been made. Our conclusion that cusplike behavior in the critical shear viscosity may be caused by nonlinear shear gradient effects, can be tested by varying D i n the large ~, region; care must be taken to exclude impurities, since these also lead to a leveling off of the viscosity and would mask the nonlinear effects discussed here.

I

0.20

\ 0.16

\ \

~/-,r/o

\ \

0.12

\,

0.08

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Theory Experiment (ref. 7)

x

I

-5

I

I

-4 -3 log ( T - Tc/T c)

-2

Fig. 2. Excess shear viscosity of is0butyric acid-water mixture. The experiments [6] that have looked for, and have not found, dependence of r~ on D have involved values of X less than 0.3, for which AQ,) is smaller than the experimental uncertainty. In contrast some experiments that have ignored shear gradient effects have involved values of >, as large as 60. We suggest that the leveling off (cusplike behavior) that has been observed in some of these experiments may be due

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13 January 1975

References

[1] R. Sallavanti and M. Fixman, J. Chem. Phys. 48 (1968) 5326. [2] K. Kawasaki, Ann. Phys. (N.Y.) 61 (1970) 1. [3] D.W. Oxtoby, to be published. [4] H. Mori, Progr. Theor. Phys. 33 (1965) 423. [5] R. Peal and R.A. Ferrell, Phys. Rev. A6 (1972) 2358. [6] D. Woermann and W. Sarholz, Ber. Bunsenges. Phys. Chem. 69 (1965) 319; B.C. Tsai and D. MeIntyre, J. Chem. Phys. 60 (1974) 937. [7] J.C. Allegra, A. Stein and G.F. Allen, J. Chem. Phys. 55 (1971) 1716.