Supercooled Liquids - American Chemical Society

Chapter 5

Frustration-Limited Domain Theory of Supercooled Liquids and the Glass Transition Downloaded by UNIV MASSACHUSETTS AMHERST on October 4, 2012 | http://pubs.acs.org Publication Date: September 30, 1997 | doi: 10.1021/bk-1997-0676.ch005

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Gilles Tarjus , Daniel Kivelson , and Steven Kivelson 1

Laboratoire de Physique Theorique des Liquides, Université Pierre et Marie Curie, 4 Place Jussleu, 75252 Paris Cedex 05, France Departments of Chemistry and Biochemistry, and Physics, University of California, Los Angeles, CA 90095 2

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We present a novel, and what we believe to be a reasonably successful, way of looking at supercooled liquids. It attributes the salient properties of these latter to the formation of supermolecular domains that, because of inherent structuralfrustration,grow only at a modest rate when the temperature is decreased. The theoretical approach is based on the postulated existence of a narrowly avoided critical point at a temperature T* and makes use of scaling about but below this T*. Application to the description of the structural (α) relaxations and to the explanation of various apparently anomalous properties of supercooled liquids is discussed. Although below the melting point ( T ) the crystal is presumably the stable state, liquids can remain uncrystaJlized over times long compared to the experimentally accessible times and thus exist in a metastable supercooled state. Besides their increasing sluggishness with decreasing temperature (that leads to the dynamic arrest on experimental time scale known as the "glass transition"), supercooled liquids exhibit a rich phenomenology that departs in many ways from that of ordinary liquids above the melting point (7,2). There is no obviously unique way of selecting the salient features in the whole body of thermodynamic and relaxational properties of supercooled liquids. The emphasis one places on specific aspects is usually guided by an underlying view of what is to be explained about supercooled liquids and glass formation. This having been said, we take as particularly significant in the phenomenology of glass-forming liquids the following points (we shall not be concerned here with nonequilibrated systems, i.e. glasses): 1) Strong temperature dependences. The distinctive property of supercooled liquids is of course the stupendous increase of the shear viscosity and the structural (a) relaxation times with decreasing temperature T when approaching the glass transition temperature T . These transport coefficients can increase by 15 orders of magnitude over a T-range of perhaps 150 K (7). Not as spectacular (the entropy difference between the liquid and the crystal often exhibits a factor of 3 to 5 decrease m

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© 1997 American Chemical Society

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

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from T to T ) , but still intriguing, is the marked decrease of the entropy of die liquid as T is lowered towards T ( i ) , a result that leads to the Kauzmann paradox (5) (the vanishing of the difference between the entropy obtained by extrapolation of the liquid curve and the entropy of the crystal at a temperature Tic0, Q>0, and r§ denotes the position of site i . The first term describes the reference system. Short-range ferromagnetic interactions between the spins lead to a continuous phase change at a temperature T* of order J. For simplicity, we consider nearest-neighbor interactions denoted by the symbol , but such short-range details are irrelevant because we only want to study collective, mesoscopic effects. Uniform frustration is described by the second term which introduces long-range competing (anti-ferromagnetic) interactions between the spins. Due to its Coulombic form, it leads to a superextensive growth of strain that opposes the extension of the locally preferred structure (i.e., ferromagnetic order) and macroscopic ferromagnetic ordering is forbidden for any nonzero value of Q. Since we are interested in a narrowly avoided critical point at T*, we always consider Q « J a o , where ao is the lattice spacing that, although ill-defined, we take as being of the order of the intermolecular distance. The model can be completed by introducing Glauber dynamics or other short-range dynamics in order to study relaxational properties. Our claim, so far only partially substantiated, is that this model, which does n include any artificially imposed quenched disorder, provides a minimum theoretica frameworkfor understanding the salientfeatures of the phenomenology ofsupercoole liquids. Despite the apparent simplicity of the frustrated spin model described by equation 1, obtaining numerical, not to mention analytical, solutions still represents a formidable task. The only such problem that has been solved is the mean spherical model (26), in which the spins are taken to be real numbers subject to the global constraint that the thermally averaged norm of a spin be equal to 1. This is not a physically sensible model, but its solution exhibits a number of interesting features. Its phase diagram looks very much like that illustrated in Figure 1 with frustration being measured by the ratio of coupling constants, Q/Jao (26) : the critical point at T* is isolated, the line of transitions (To) to a defect-ordered phase which is characterized by some sort of modulated order (with supermolecular characteristic length) is found for 3-dimensional systems to intersect the zero-frustration (Q=0) axis at a temperature significandy below T*; for Q#0, the critical point at T* is avoided, with no anomaly in the heat capacity, and, although the correlation length of the reference system remains infinite below T* (a peculiarity of the spherical models), there is a second supermolecular characteristic length that increases as T is lowered. These features remain true when 1/n corrections to the spherical model are considered (27). The presence of phases with modulated order has also been shown in zero-T analysis of 2d (28) and 3-d (29) Ising versions of the Hamiltonian in equation 1. In addition to these exact results, Monte Carlo simulations of various spin models with long-range frustration have confirmed the existence of low-T transitions to defect-ordered phases with modulated order as well as the appearance at higher T, but below T*, of mesoscale domains (29,30,31). Despite incomplete information on the relaxational properties, these results are promising since they corroborate the scenario presented in the preceding section.


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Frustration-Limited Domain Theory

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Phenomenological scaling approach

Downloaded by UNIV MASSACHUSETTS AMHERST on October 4, 2012 | http://pubs.acs.org Publication Date: September 30, 1997 | doi: 10.1021/bk-1997-0676.ch005

In the absence of exact solutions for realistic models, we have developed a phenomenological scaling description for the system described by equation 1. It is based on the postulate of an isolated, narrowly avoided critical point at T*, a property supported by the calculations discussed above. The critical behavior of die reference system, i.e., the first term in the Hamiltonian of equation 1, is characterized by a correlation length V

£ o c |1-T/T*|- ,

(2)

where, as for usual 3-d systems without quenched disorder, we take v to be close to 2/3. By considering the size at which the adiabatic approximation in the perturbation theory of finite-size systems breaks down, we have shown that a second supermolecular length R appears at temperatures sufficiently below T* in the presence of weak frustration ( Q « J ao) (20). This length scales according to 2

R « ao (Q/J *)'

m

A

%

1/2

~ Q- (1-T/T*)v.

(3)

We interpret R(T) as the mean distance at temperature T over which a nonzero ferromagnetic order parameter can be specified and we thus take R as the mean linear size of the frustration-limited domains. Note that since this applies below T*, the correlation length £ decreases whereas the mean domain size R increases when T is lowered. Before proceeding further in the development of the scaling approach, one has first to address the basic question : why would frustration-limited domains give rise to a dramatic slowing down of die structural relaxation? The answer comes from standard arguments on finite-size systems below their critical temperature. For illustration, consider a simple Ising ferromagnet of volume L , where L is much larger than the lattice spacing. Below the critical point, the system has two low free-energy states characterized by nonzero order parameters of the same magnitude, but different orientation (up or down). Relaxation of the order parameter, i.e., the passage from one state to another, is possible via an activated process which proceeds by creation across die system of a domain wall that separates a region of mostly up spins from a region of mostly down spins. The corresponding activation free energy is proportional to P

Figure 2. Masterplot of Nagel and coworkers (4,5) for frequency-dependent susceptibility data X"((o). (Dp is the frequency at which the maximum of the ocpeak occurs in the suceptibility X"(co), the shape factor W is such that 1.14W is the full width at half-height of the X " vs logio© spectrum, and A X is a normalization factor. The theoretical curves are obtained from the expressions and the values of the parameters that are given in the text and correspond to l i q u i d S a l o l ; 6 curves are incorporated, corresponding to W=l .56,1.61,1.65,1.67,1.70,1.74.


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Decoupling between translational and rotational diffusion The preceding discussion of a-relaxations applies when the probed motions are local enough to be completed within a single domain. Then, the a-relaxation time is simply given by (35) oo

oo


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% = Jdt faff) = too J d L L p(L) exp[E(L)/T], o o

(13)

where we have used equation 10. Note that, as required by consistency, the above equation leads back to equations 8 and 9 when the scaling forms for p(L) and E(L) are introduced, provided y(T) is not too small (36). If the relaxation is not completed within a single domain, as for long wavelength concentration fluctuations (8,9), an averaging over domains different than that in equations 10 and 13 must be considered (an intermediate case occurs when the reorientational motions of large solute molecules is studied (14); then, the size of the solute molecule is less than, but comparable to the mean domain size). For instance, translational motion over a scattering wavelength much larger than the mean domain size, as in forced Rayleigh (8) and holographic fluorescence recovery (9) experiments, involves passage through many different domains, which give rise to a "motionally narrowed" relaxation function that is exponential (37). In each domain, translational diffusion, just like rotational diffusion or viscous flow, is controlled by the relaxation of the mesoscopic order parameter and thus involves an activation free energy that is domain size dependent. A simple model of random walk in a random environment formed by independently relaxing domains then leads to the following expression for the translational diffusion constant (37): oo

DT = - (1/6) Jdt < V D ( r ) . VD(r(t))>, o

(14)

where D(r) is a spatially varying diffusion coefficient. In the limiting case where the domains have sharp boundaries, the above expression reduces to oo 2

D « « Doo J d L L p(L) exp[-E(L)/T], o T

(15)

where Doo is a T-independent microscopic diffusion constant. Above T*, E(L)=Eoo and equations 13 and 15 give D x t = constant, as predicted by the Stokes-Einstein-Debye formulas. When domains are present, i.e. at temperatures below T*, the averaging in equation 15 is quite different than that in equation 13 : it is more sensitive to the small domains (with, therefore, smaller activation energies) which, in our picture, are also a


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responsible for the von Schweidler relaxation. As a consequence, translational diffusion is "enhanced" when compared to rotational diffusion or viscosity. This induces a breakdown of the Stokes-Einstein relation at temperatures sufficiently below T* (57), as observed experimentally. In fact, the increase of D T % predicted by equations 13 and 15 when T approaches T is much larger than that observed, presumably because the requirement of sharp domain boundaries needed to derive equation 15 from equation 14 is not met (55). Downloaded by UNIV MASSACHUSETTS AMHERST on October 4, 2012 | http://pubs.acs.org Publication Date: September 30, 1997 | doi: 10.1021/bk-1997-0676.ch005

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Conclusion. In summary, the F L D theory envisages the high-T liquid as a molecular fluid which may not be understood in detail, but which can be described approximately by exponential relaxation functions with relaxation times that exhibit Arrhenius-like behavior. Below a crossover temperature T* that we associate with an avoided critical point, the liquid forms supermolecular frustration-limited domains with a distribution of sizes, each domain being specified by a nonzero mesoscopic order parameter. As a result, the a-relaxation becomes heterogeneous and strongly nonexponential with relaxation times that show strong superArrhenius behavior. The F L D theory is based on a well-defined statistical-mechanical model for which we have proposed a phenomenological scaling description. As discussed in this article, this allows us to reproduce the main features of the a-relaxations in glass-forming liquids, including temperature and frequency dependences, at a quantitative level with a limited number of species-dependent, but T-independent adjustable parameters. Although not yet worked out in all details, the theory also provides a consistent framework to explain most of the apparent anomalies observed in supercooled liquids. We have already discussed the decoupling between translational and rotational diffusion. It is tempting to interpret the bifurcation between a- and (3relaxations as a consequence of the existence of two supermolecular characteristic lengths below T* : whereas a-relaxations occur on a lengthscale corresponding to the mean domain size R, additional faster relaxations presumably take place on the smaller lengthscale associated with the correlation length £ of the reference system. Although we have not yet studied these shorter-range processes, it seems natural to take them as the fast (J-relaxations. Likewise, the rapid decrease of the entropy difference between the supercooled liquid and the crystal can be understood as resulting from formation and growth of frustration-limited domains. Elsewhere (55), we show that the scaling description, when extrapolated to low temperatures, leads to an entropy crisis as described by Kauzmann, but that this crisis is avoided in a more exact description because of the breakdown far from T* of the scaling formulas and the interposition of a transition to a defect-ordered phase. Such a transition has been described in the above sections, and it may indeed have been observed experimentally (25,38). Clearly, additional work is needed in order to obtain a complete (at least numerical) solution of a realistic frustrated spin model and thus to put the scaling approach on firmer ground. A fully satisfying description of supercooled liquids and


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glass formation would also require a molecular interpretation of the order variable associated with the locally preferred liquid structure. However, even at the present stage, the physical picture and die comparison to experiment drawn from the F L D theory are quite encouraging.

We wish to thank the C.N.R.S., the N.S.F., the Research Corporation, and N A T O for their support. We are indebted to Pascal Viot for many useful discussions.

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16. T . R. Kirkpatrick, D. Thirumalai, and P. G. Wolynes, Phys. Rev. A 40, 1045 (1989). 17. J. P. Sethna, J. D. Shore, and M. Huang, Phys. Rev. B 44,4943 (1991). 18. R. M. Ernst, S. R. Nagel, and G. S. Grest, Phys. Rev. B 43, 8070 (1991). 19. R. L. Leheny, N. Menon, S. R. Nagel, D. L. Price, K. Suzuya, and P. Thiyagarjan, J. Chem. Phys. 105, 7783 (1996). 20. D. Kivelson, S. A. Kivelson, X.-L. Zhao, Z. Nussinov, and G. Tarjus, Physica A 219, 27 (1995). 21. K. J. Standburg, Rev. Mod. Phys. 60, 161 (1988). 22. F.C. Frank, Proc. Royal Soc. London 215A, 43 (1952). 23. D. R. Nelson and F. Spaepen, Solid State Phys. 42, 1 (1989). 24. M. Kleman and J. F. Sadoc, J. Phys. Lett. (Paris) 40, L569 (1979). J. P. Sethna, Phys. Rev. Lett. 51, 2198 (1983). S. Sachdev and D. R. Nelson, Phys. Rev. Lett. 53, 1947 (1984); Phys. Rev. B 32, 1480 (1985). 25. D. Kivelson, M. Lee, J.-C. Pereda, K. Luu, H. Sakai, A. Ha, I. Cohen, and G. Tarjus, these Proceedings. 26. L. Chayes, V. J. Emery, S.A. Kivelson, Z. Nussinov, and G. Tarjus, Physica A 225, 129 (1996). 27. Z. Nussinov, private communication. 28. U. Low, V. J. Emery, K. Fabricus, and S. A. Kivelson, Phys. Rev. Lett. 72, 1918 (1994). 29. P. Viot, private communication. 30. D. Wu, D. Chandler, and B. Smit, J. Phys. Chem. 96, 4077 (1992). 31. I. Booth, A. B. MacIsaac, J. P. Whitehead, and K. De'Bell, Phys. Rev. Lett. 75, 950 (1995). 32. S. A. Kivelson, X-L Zhao, D. Kivelson, C. M. Knobler and T. Fischer, J. Chem. Phys. 101, 2391 (1994). D. Kivelson, S. A. Kivelson, X-L Zhao, C. M. Knobler and T. Fischer, in "Lectures on Thermodynamics and Statistical Mechanics" Ed. M. Costas, R. Rodrigues, and A. L. Benavides (World Scientific, London, 1994), p. 32. 33. D. Kivelson, G. Tarjus, and S. A. Kivelson, Prog. Theoret. Physics (Japan), to appear. 34. D. Kivelson, G. Tarjus, X-L Zhao, and S. A. Kivelson, Phys. Rev. E 53, 751 (1996). 35. G. Tarjus and D. Kivelson, in preparation. 36. Experimentally, there may be different ways to determine the characteristic α-time for a given motion; see for instance the discussion of retardation vs relaxationtimesin I. Chang and H. Sillescu, preprint (1996). This may require slight modifications of equation 13 and may induce differences in the T-dependences of the various times. However, these differences seem to remain typically less than an order of magnitude around T and will not be considered here. 37. G. Tarjus and D. Kivelson, J. Chem. Phys. 103, 3071 (1995). 38. I. Cohen, A. Ha, X-L. Zhao, M. Lee, T. Fischer, M. J. Strouse, and D. Kivelson, J. Phys. Chem. 100, 8518 (1996). g


Supercooled Liquids - American Chemical Society

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