Shear Viscosity and Diffusion in Supercooled Liquids - ACS Publications

Chapter 10

Shear Viscosity and Diffusion in Supercooled Liquids

Downloaded by UNIV MASSACHUSETTS AMHERST on October 4, 2012 | http://pubs.acs.org Publication Date: September 30, 1997 | doi: 10.1021/bk-1997-0676.ch010

Frank H. Stillinger Bell Laboratories, Lucent Technologies, Inc., 600 Mountain Avenue, Room 1D-359, Murray Hill, NJ 07974

Under normal liquid conditions (at equilibrium or when moderately super cooled), molecular rotational and translational diffusion adhere closely to the sim ple Stokes-Einstein-Debye hydrodynamic model. Deeply supercooled "fragile" liquids present a striking exception, with translational diffusion alone occurring up to 10 — 10 times "too fast". This anomaly rests upon the statistics of fluctuating local viscosity in the fragile glass-forming medium. The simplest version of this picture identifies largefluidizeddomains that spontaneously appear and disappear in a dynamically inert matrix. Plausible and self-consistent values characterizing thesefluctuationsare cited for orthoterphenyl (OTP) at its glass transition. 2

3

Liquids that supercool easily and form glasses have presented many conceptual puz zles, whose resolution has led to deeper understanding of these substances at the molec ular level. This paper focusses on one of those puzzles that mainly concerns the socalled 'fragile" glass formers, identified by the markedly non-Arrhenius temperature de pendences of their shear viscosity rj(T) and various mean relaxation times r ( T ) ( i , 2). Specifically, die need to reconcile separate measurements of translational and of rota tional diffusion rates in fragile liquids just above their glass transition temperature T leads inevitably to a molecular picture of the medium that is strongly inhomogeneous in both space and time. Translational and rotational diffusion constants, A n n * and D , measure rates of increase with time of the mean-square positional and angular displacements for tagged molecules: ((6r) )*£6D 6t, t. g

r o t

2

tTMm

2

K l )

It has been traditional to assess measured values of these diffusion constants by means of the elementary Stokes-Einstein-Debye (SED) hydrodynamic model of a Brownian sphere with radius R embedded in a homogeneous, incompressible, viscous medium (3)-(6). Assuming that "sticking" boundary conditions apply in this model, one has D = k T/§1

Table I: Fluidized domain parameters and related quantities for OTP at T

g

6.10 x 1.70 x 2.11 x 0.426 500 1.53 x 3.5 9.92 x 6.21

%(P) v (A ) ro(cm~ 8~ ) to(s) n 0o R(A) 3

0

z

l

c

0


d

/Z

wmvl R/k T (8) B

a

g

6

10 10 10

5

15

1(T

4

2

10"-

Values taken from ref. 13. T = 240K, rj(T ) = 4 x 10 P. n = mean number of molecules in VQ. o = volume traction of fluidized domains.

h

12

9

c

g

0

d

While it is too much to expect fluidized domain characteristics to follow this behavior precisely, it may be appropriate to suggest that rough conformity obtains, sufficient to justify die simple level of description utilized here. Deeply supercooled glass-forming liquids seem universally to exhibit stretchedexponential relaxation functions (2,19): {t) S e x p { - f t / r ( T ) f } ,

0 < 0 < 1.

(11)

Strong experimental evidence now exists not only that glassy media are dynamically heterogeneous, but that such heterogeneity underlies stretched exponential relaxation (20)-(22). The fluctuating field rj(r, t) is the means for describing this situation in the present context, and its inverse controls the local (in space and time) rate of decay of structure. Consequently, near the glass transition we must have the mathematical con nection: (exp [K j f fo(r, f

= «p{-[t/T(T)]>},

(12)

where K is a positive constant dependent on the property whose relaxation is examined, and where (...) indicates a spatial average over all r in die system volume. Experimen tal determinations of (5 and r thus provide partial information about r;(r, t). Translational diffusion rates for probe molecules of various sizes have been studied experimentally as a means to determine the average size of dynamical heterogeneities — fluidized domains in the present analysis (11,23). Certainly a spherical probe whose volume far exceeds t; would be expected to conform to the SED norm, though probes utilized so far are likely not in this size range. Although this large-probe strategy has merit, considerable caution needs to be exercised in its interpretation. Uncertainties ex ist about how foreign probes in a glass-forming medium might locally perturb the dy namics, and the larger the probe the larger such a perturbation would tend to be. Nev ertheless, the resulting estimates of the mean linear dimension of inhomogeneities is in 0

In Supercooled Liquids; Fourkas, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.

138

SUPERCOOLED LIQUIDS

the range of several nanometers for OTP at T (20), which is in rough agreement with Table I. The Adam-Gibbs theory of cooperative relaxation in glass-forming liquids has main tained prominence in the field over the more than thirty years since its inception (14). That approach connects the temperature-dependent mean relaxation time to calorimetric entropy, using the concept of "cooperatively rearranging regions." It is tempting, though probably not correct, to identify these Adam-Gibbs regions with the fluidized domains of the present paper. By themselves, the four parameters used here to charac terize the fluidized domains do not suffice to determine configurational entropy, so at least one further postulate would be required to connect to the Adam-Gibbs viewpoint. One class of theoretical models suggests in fact that the Adam-Gibbs theory cannot be generally valid (24); whether it properly describes a restricted class of theoretical mod els and real substances remains an open issue. For additional perspectives on the phenomena discussed in this paper, the reader may wish to consult the presentations (25) and (26).


g

References [1] C. A. Angell, J. Non-Cryst. Solids 102, 205 (1988). [2] R. Böhmer, K. L. Ngai, C. A. Angell, and D. J. Plazek, J. Chem. Phys. 99, 4201 (1993). [3] A. Einstein, Ann. Phys. (Leipzig) 17, 549 (1905); 19, 371 (1906). [4] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Pergamon, New York, 1987), pp. 63-67. [5] P. Debye, Polar Molecules (Dover, New York, 1929), pp. 77-86. [6] N. E. Hill, W. E. Vaughan, A. H. Price, and M. Davies, Dielectric Properties and Molecular Behavior (Van Nostrand Reinhold, London, 1969), pp. 60-63. [7] J. Jonas, D. Hasha, and S. G. Huang, J. Phys. Chem. 84, 109 (1980). [8] F. Fujara, B. Geil, H. Sillescu, and G. Fleischer, Z. Phys. B 88, 195 (1992). [9] M. T. Cicerone and M. D. Ediger, J. Phys. Chem. 97, 10489 (1993). [10] E. Rössler and P. Eiermann, J. Chem. Phys. 100, 5237 (1994). [11] M. T. Cicerone and M. D. Ediger, J. Chem. Phys. 104, 7210 (1996). [12] F. H. Stillinger and J. A. Hodgdon, Phys. Rev. E 50, 2064 (1994). [13] F. H. Stillinger and J. A. Hodgdon, Phys. Rev. E 53, 2995 (1996). [14] G. Adam and J. H. Gibbs, J. Chem. Phys. 43, 139 (1965). [15] D. J. Plazek, C. A. Bero, and I.-C. Chay, J. Non-Cryst. Solids 172-174, 181 (1994).


10.

STILLINGER

Shear Viscosity & Diffusion in Supercooled Liquids

[16] S. S. Chang and A. B. Bestul, J. Chem. Phys. 56, 503 (1972). [17] R. Zwanzig, Chem. Phys. Letters 164, 639 (1989). [18] J. Jäckle, Rep. Prog. Phys. 49, 171 (1986). [19] K. L. Ngai, Comments Solid State Phys. 9, 127 and 141 (1979).


[20] M. T. Cicerone and M. D. Ediger, J. Chem. Phys. 103, 5684 (1995). [21] A. Heuer, M. Wilhelm, H. Zimmermann, and H. W. Spiess, Phys. Rev. Letters 75, 2851 (1995). [22] B. Scheiner, R. Böhmer, A. Loidl, and R. V. Chamberlin, Science 274, 752 (1996). [23] G. Heuberger and H. Sillescu, J. Phys. Chem. 100, 15255 (1996). [24] F. H. Stillinger, Physica D 000, 0000 (1997). [25] G. Tarjus and D. Kivelson, J. Chem. Phys. 103, 3021 (1995). [26] D. N. Perera and P. Harrowell,J.Chem. Phys. 104, 2369 (1996).


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Shear Viscosity and Diffusion in Supercooled Liquids - ACS Publications

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