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Dynamic scaling for the critical viscosity of a classical fluid
Volume 76A, number 3,4
PHYSICS LETTERS
31 March 1980
DYNAMIC SCALING FOR THE CRITICAL VISCOSITY OF A CLASSICAL FLUID Jayanta K. BHATTACHARJEE Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA and Center for Theoretical Physics, Department of Physics and Astronomy, University of Maryland, College Park, MD 20742, USA 1
and Richard A. FERRELL Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA Rec~ived2 January 1980 Revised manuscript received 4 February 1980
The recently reported deviation from the hydrodynamic critical viscosity of a classical fluid can be attributed to the effect of the finite frequency employed in the measurement. A dynamic scaling function predicts the temperature dependence of the deviation.
It is well established that the hydrodynamic shear viscosity of a classical fluid near its critical point has the critical variation ~271. The critical exponent z~is generally found to be in good agreement with the decoupled-mode value [1] of 8/(l5ir2). ~ is the correlation length and increases without limit as the critical point is approached. The mode-coupling formalisms of Kadanoff and Swift [2] and of Kawasaki [3] are entirely equivalent to the decoupled-mode version of the theory used here, and lead to the same results. An apparent contradiction to the divergence expected for the hydrodynamic shear viscosity is the recent report [4,5] of a fmite limiting value. The purpose of the present note is to point out the important role of frequency dependence near the critical point. We believe that the fact that the viscosity measurements are carned out at a finite frequency may explam the absence of the diverging behavior associated with the strict hydrodynamic zero-fre,quency limit. The characteristic relaxation rate of the fluid has the critical behavior ~ cx ~—3—z 11In terms of y the critical viscosity is i~cx 7~r1 where = zn/(3 + Zn). The basic idea of dynamic scaling is that deviations 1
Permanent address.
290
from the hydrodynamic expression for i~set in when w, the angular frequency of the shear field, becomes as large as y. The decoupled-mode theory [6] is readily generalized [1] to provide the temperature and frequency dependent viscosity 77()f, w). As with the frequency dependence in the non-local limit [71,the local limit which interests us here is described by the singleloop integral 6 d ~(y, w) ~°2 2 ~ (1) (p + ) y(p, ~_1) is the rate at which an order parameter fluctuation of wave number p relaxes when the fluid has correlation length From eq. (1) we have found, as in the non-local case [7], that to a good approximation
~f—~-
.
~—
~.
~(y w)A(y—iw)
—z
~
,
(2)
where A is a non-universal constant and 7 = By(~, ~1). The coefficient of proportionality is B 16\/2/ (3ir). The hydrodynamic limit is recovered by setting 0. For an experiment in which w is kept constant and y is varied by varying the temperature, eq. (2) is conveniently written as =
PHYSICS LEfl’ERS
Volume 76A, number 3,4
~ w)—A~,—~?(ri)~1 ~A —
~_z~(l
+ z~L),
(3)
where L
L1 + iL2 = —ln(I’ i) = —~ln(1+ ~2) + i ~
=
—
r,
(4)
and 71w is a scaled rate The approximation 1’in=the second line of eq.variable. (3) is pei-mitted by the smallness of z~- The point that we wish to make is now expressed quantitatively by the passage to the critical point in eq. (3). With 7 -‘-0 we obtain the finite value n(O, w) =Aw~1exp(iz~ ir/2) and not infinite value expected from the hydrodynamic for-the mula. The detailed temperature dependence of the departure from the hydrodynamic formula is described by L 1(V) and L2(V), the real and imaginary eq. (4). r varies approximately as the square parts of theofreduced temperature along the critical isochore *1 - These functions are shown in fig. 1 by the dashed and dot—dashed temperature dependence of r is more complicated along the non-critical isochores of ref. [2] because the transition is first order. The critical isochore data of ref. [3] are free of this complication.
~ The
2
31 March 1980
curves, respectively. In comparing these results with experimental data it is necessary to take into account the fact that the real and imaginary parts of the viscosity are not normally measured separately, but as explained in the next paragraph, only together as the sum. This is shown by the solid curve in fig. 1, where a point of inflexion wifi be noted at r = 1 2+ determines k, the The diffusion equation iwi~ =is~k complex inverse skin depth. the kinetic viscosity. The momentum flow toward the driving surface (e.g., an oscillating torsional disk, or a vibrating wire) is proportional to2.D i~ times the gradient or i~k = ropping thevelocity factor of w1/2 we conseX (iws~)hI quently have the measured frictional coefficient proportional to j3 = Re(—i~)lI2= Re e~~4(~ 1I2 +1i~2) 4?7I’2(1+ Re e’~ = 2_11217}12(1 + ~fl 2(~~ ~fl2)”2’ (5) 2/fl1)ns2 lI where approximations have been made consistent with fl2 ~ - Hence, as mentioned in the preceding paragraph, experiments which detect the viscous drag on a solid surface necessarily measure the sum of the real and imaginary parts of the viscosity. We note in passing that the complex skin depth gives rise to a non-dissipative inertial drag, observable as a resonance shift proportional to n t2 times w112. —
2)’
The frequency dependence discussed here leads to a frequency dependent critical diffusion [81. Experimental confirmation of this in a binary liquid has been reported by Burstyn et al. [9]. An obvious direct test
-.
I -
0.1
-I
—
~
— 0
~
1’
N
-
~
-2
-
-
Fig. 1. Dynamic scaling function for the critical viscosity i~. The temperature dependence of the real and imaginary parts of s~are shown by the dashed and dot—dashed curves, respectively. The solid curve shows the behavior of the experimentally measured sum. The scaled rate r is proportional to the square of the reduced temperature and inversely proportional to the experimental frequency.
out the viscosity measurements at several different frequencies. Dynamic predicts should that such scaled according to scaling the frequency, all data, fall on a single of the scaling scaling curve theory(solid presented curve here of fig. would 1). be to carry Foundation under Grants DMR 79~0l172 and 79We acknowledge support from the National Science 00908 to the University of Maryland and Grant PHY 77-27084, Institute for Theoretical Physics, at the University of California, Santa Barbara. References [1) R. Perl and R.A. Ferrell, Phys. Rev. Lett. 29 (1972) 51; Phys. Rev. A6 (1972) 2358.
291
Volume 76A, number 3,4 [2] [3] [4] [5] [6]
292
PHYSICS LETTERS
L.P. Kadanoff and J.Swift, Phys. Rev. 166 (1968) 89. K. Kawasaki, Ann. Phys. (NY) 61(1970)1. L. Bruschi and M. Santini, Phys. Lett. 73A (1979) 395. A.M. Borghesani, Ph.D. Thesis, Univ. of Padua (1978). R.A. Ferrell, Phys. Rev. Lett. 24 (1970) 1169.
31 March 1980
[71 J.K. Bhattacharjee and R.A. Ferrell, Univ. of Maryland, Dept. Phys. Astron. Tech. Rep. No. 80-031 (Sept. 1979); KINAM, Vol. 2, to be published. [8] J.K. Bhattacharjee and R.A. Ferrell, Univ. of Maryland, Dept. Phys. Astron. Tech. Rep. No. 80-020, to be published. [91 H. Burstyn, RF. Chang and J.V. Sengers, to be published.
PHYSICS LETTERS
31 March 1980
DYNAMIC SCALING FOR THE CRITICAL VISCOSITY OF A CLASSICAL FLUID Jayanta K. BHATTACHARJEE Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA and Center for Theoretical Physics, Department of Physics and Astronomy, University of Maryland, College Park, MD 20742, USA 1
and Richard A. FERRELL Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA Rec~ived2 January 1980 Revised manuscript received 4 February 1980
The recently reported deviation from the hydrodynamic critical viscosity of a classical fluid can be attributed to the effect of the finite frequency employed in the measurement. A dynamic scaling function predicts the temperature dependence of the deviation.
It is well established that the hydrodynamic shear viscosity of a classical fluid near its critical point has the critical variation ~271. The critical exponent z~is generally found to be in good agreement with the decoupled-mode value [1] of 8/(l5ir2). ~ is the correlation length and increases without limit as the critical point is approached. The mode-coupling formalisms of Kadanoff and Swift [2] and of Kawasaki [3] are entirely equivalent to the decoupled-mode version of the theory used here, and lead to the same results. An apparent contradiction to the divergence expected for the hydrodynamic shear viscosity is the recent report [4,5] of a fmite limiting value. The purpose of the present note is to point out the important role of frequency dependence near the critical point. We believe that the fact that the viscosity measurements are carned out at a finite frequency may explam the absence of the diverging behavior associated with the strict hydrodynamic zero-fre,quency limit. The characteristic relaxation rate of the fluid has the critical behavior ~ cx ~—3—z 11In terms of y the critical viscosity is i~cx 7~r1 where = zn/(3 + Zn). The basic idea of dynamic scaling is that deviations 1
Permanent address.
290
from the hydrodynamic expression for i~set in when w, the angular frequency of the shear field, becomes as large as y. The decoupled-mode theory [6] is readily generalized [1] to provide the temperature and frequency dependent viscosity 77()f, w). As with the frequency dependence in the non-local limit [71,the local limit which interests us here is described by the singleloop integral 6 d ~(y, w) ~°2 2 ~ (1) (p + ) y(p, ~_1) is the rate at which an order parameter fluctuation of wave number p relaxes when the fluid has correlation length From eq. (1) we have found, as in the non-local case [7], that to a good approximation
~f—~-
.
~—
~.
~(y w)A(y—iw)
—z
~
,
(2)
where A is a non-universal constant and 7 = By(~, ~1). The coefficient of proportionality is B 16\/2/ (3ir). The hydrodynamic limit is recovered by setting 0. For an experiment in which w is kept constant and y is varied by varying the temperature, eq. (2) is conveniently written as =
PHYSICS LEfl’ERS
Volume 76A, number 3,4
~ w)—A~,—~?(ri)~1 ~A —
~_z~(l
+ z~L),
(3)
where L
L1 + iL2 = —ln(I’ i) = —~ln(1+ ~2) + i ~
=
—
r,
(4)
and 71w is a scaled rate The approximation 1’in=the second line of eq.variable. (3) is pei-mitted by the smallness of z~- The point that we wish to make is now expressed quantitatively by the passage to the critical point in eq. (3). With 7 -‘-0 we obtain the finite value n(O, w) =Aw~1exp(iz~ ir/2) and not infinite value expected from the hydrodynamic for-the mula. The detailed temperature dependence of the departure from the hydrodynamic formula is described by L 1(V) and L2(V), the real and imaginary eq. (4). r varies approximately as the square parts of theofreduced temperature along the critical isochore *1 - These functions are shown in fig. 1 by the dashed and dot—dashed temperature dependence of r is more complicated along the non-critical isochores of ref. [2] because the transition is first order. The critical isochore data of ref. [3] are free of this complication.
~ The
2
31 March 1980
curves, respectively. In comparing these results with experimental data it is necessary to take into account the fact that the real and imaginary parts of the viscosity are not normally measured separately, but as explained in the next paragraph, only together as the sum. This is shown by the solid curve in fig. 1, where a point of inflexion wifi be noted at r = 1 2+ determines k, the The diffusion equation iwi~ =is~k complex inverse skin depth. the kinetic viscosity. The momentum flow toward the driving surface (e.g., an oscillating torsional disk, or a vibrating wire) is proportional to2.D i~ times the gradient or i~k = ropping thevelocity factor of w1/2 we conseX (iws~)hI quently have the measured frictional coefficient proportional to j3 = Re(—i~)lI2= Re e~~4(~ 1I2 +1i~2) 4?7I’2(1+ Re e’~ = 2_11217}12(1 + ~fl 2(~~ ~fl2)”2’ (5) 2/fl1)ns2 lI where approximations have been made consistent with fl2 ~ - Hence, as mentioned in the preceding paragraph, experiments which detect the viscous drag on a solid surface necessarily measure the sum of the real and imaginary parts of the viscosity. We note in passing that the complex skin depth gives rise to a non-dissipative inertial drag, observable as a resonance shift proportional to n t2 times w112. —
2)’
The frequency dependence discussed here leads to a frequency dependent critical diffusion [81. Experimental confirmation of this in a binary liquid has been reported by Burstyn et al. [9]. An obvious direct test
-.
I -
0.1
-I
—
~
— 0
~
1’
N
-
~
-2
-
-
Fig. 1. Dynamic scaling function for the critical viscosity i~. The temperature dependence of the real and imaginary parts of s~are shown by the dashed and dot—dashed curves, respectively. The solid curve shows the behavior of the experimentally measured sum. The scaled rate r is proportional to the square of the reduced temperature and inversely proportional to the experimental frequency.
out the viscosity measurements at several different frequencies. Dynamic predicts should that such scaled according to scaling the frequency, all data, fall on a single of the scaling scaling curve theory(solid presented curve here of fig. would 1). be to carry Foundation under Grants DMR 79~0l172 and 79We acknowledge support from the National Science 00908 to the University of Maryland and Grant PHY 77-27084, Institute for Theoretical Physics, at the University of California, Santa Barbara. References [1) R. Perl and R.A. Ferrell, Phys. Rev. Lett. 29 (1972) 51; Phys. Rev. A6 (1972) 2358.
291
Volume 76A, number 3,4 [2] [3] [4] [5] [6]
292
PHYSICS LETTERS
L.P. Kadanoff and J.Swift, Phys. Rev. 166 (1968) 89. K. Kawasaki, Ann. Phys. (NY) 61(1970)1. L. Bruschi and M. Santini, Phys. Lett. 73A (1979) 395. A.M. Borghesani, Ph.D. Thesis, Univ. of Padua (1978). R.A. Ferrell, Phys. Rev. Lett. 24 (1970) 1169.
31 March 1980
[71 J.K. Bhattacharjee and R.A. Ferrell, Univ. of Maryland, Dept. Phys. Astron. Tech. Rep. No. 80-031 (Sept. 1979); KINAM, Vol. 2, to be published. [8] J.K. Bhattacharjee and R.A. Ferrell, Univ. of Maryland, Dept. Phys. Astron. Tech. Rep. No. 80-020, to be published. [91 H. Burstyn, RF. Chang and J.V. Sengers, to be published.