Liquid Dynamics - American Chemical Society

Chapter 17

Downloaded by PENNSYLVANIA STATE UNIV on September 14, 2012 | http://pubs.acs.org Publication Date: June 3, 2002 | doi: 10.1021/bk-2002-0820.ch017

Nanoscopic Heterogeneities in the Thermal and Dynamic Properties of Supercooled Liquids Ralph V. Chamberlin Department of Physics and Astronomy, Arizona State University, Tempe, AZ 85287-1504

The theory of small system thermodynamics is applied to the Ising model plus kinetic energy, yielding a partition function for supercooled liquids. Several features near the glass temperature can be attributed to the resulting nanothermodynamic transition. A mean-field-like equilibrium energy exhibits Curie-Weiss-like behavior that provides an explanation for the Vogel-TammanFulcher (VTF) law. Because this energy reduction is intensive, essentially independent of system size, the basic thermodynamic unit(calledaggregate)subdivides into smaller regions (called clusters) lowering the net internal energy. The intensive energy reduction also yields relaxation rates that vary exponentially with inverse size, which when combined with the distribution of aggregate sizes provides an explanation for the KohlrauschWilliams-Watts(KWW)law. Standard fluctuation theory gives a quantitative connection between the spectrum of response and the measured specific heat. Characteristic length scalesfromthe model are related to direct measurements.

Perhaps the two most distinctive features in the behavior of supercooled liquids are the non-Arrhenius activation as a function of temperature T, and the nonexponential relaxation as a function of time t. Near the glass temperature f , g

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© 2002 American Chemical Society

In Liquid Dynamics; Fourkas, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.

229 measurements are usually analyzed empirically in terms of the KohlrauschWilliams-Watts (7,2) (KWW) stretched exponential Φ(0 ~ exp[-(f / τ) ], with the Vogel-Tamman-Fulcher (3,4) (VTF) law for the characteristic relaxation time roc exp[£/(r- Γ )]. Here is the KWW stretching exponent, Bk (with k Boltzmann's constant) gives an energy scale for activation, and T is the Vogel temperature. Simple systems would have single exponential relaxation (β=1) and Arrhenius activation (Γ =0), but most supercooled liquids have β0. Although the KWW and VTF laws have been used to characterize thousands of measurements on hundreds of substances, there is still no widely accepted explanation for either formula. In fact it is still debated whether the behavior near T signifies a true phase transition, or merely a dynamical freezing. Indeed, the search for a fundamental theory of the glass transition has been called the "deepest and most interesting unsolved problem in solid state theory" (5). Our approach (6) is based on the elementary observation that the VTF law from supercooled liquids is suggestively similar to the Curie-Weiss lawfrommagnetism (7), which motivates us to look for a mean-field-like solution for the activation energies in supercooled liquids. This similarity seems never to have been noted before, probably because the Curie-Weiss law is usually applied to susceptibilities not activation energies, and in the mean-field approximation there is no energy reduction above the transition. However, these standard results are valid only in the limit of an infinite thermodynamic system, finite systems have a non-zero CurieWeiss-like energy reduction above the transition, which provides a physical explanation for the VTF law. The theory of small system thermodynamics ("nanothermodynamics") was developed by Terrell Hill about 40 years ago (8-11). Here we add two new features to obtain a partition function for supercooled liquids. First we show that the equilibrium energy of an isolated system is equivalent to the mean-field result, and then we use standard fluctuation theory to obtain the distribution of activation energies in bulk samples. The input parameters for the model include a net interaction energy (ε ) between particles which form strongly interacting clusters, and afractionof nearly-free particles (f) between the clusters. As a function of temperature, the Curie-Weiss-like energy yields the VTF law with T^zJk and B~ Tçjf, butfinite-sizeeffects cause deviationsfromthe VTF law that are observed in the response of several systems (12-15). Similarly, the distribution of activation energies yields KWW-like behavior, but again finite-size effects cause deviations that match the measured response, including the excess wing and other features at very highfrequencies(16-20). Furthermore, the model yields quantitative agreement with measured correlation lengths, and a connection between the response spectrum and the specific heat (21-26). Thus, the model provides a physical basis for both the KWW and VTF laws; while the critical behavior can be attributed to a thermal transition about a midpoint temperature T that is broadened by finite-size effects. β

β

0

B

B

0


0

0

g

0

B

iV/


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Thermodynamic heterogeneity Most recent experimental evidence favors heterogeneity as the main source of complexity in the dynamics of condensed matter (27). Techniques which demonstrate the heterogeneous nature of the slow response include: nuclear magnetic resonance (28-30% chemical-probe bleaching (31) and solvation dynamics (32). Detailed analysis of several techniques (33,34) indicates that the intrinsic local relaxation is consistent with simple-exponential response, so that the entire spectral width can be attributed to a heterogeneous distribution of local relaxation times. From this width, dynamical correlation lengths of 1-5 nmnear the glass transition have been deduced (35-37). Indeed, direct measurements (38-41) have confirmed that different relaxation times coincide with distinct spatial regions having an average radius of 1 -5 nm. Dynamic heterogeneity has also been found in computer simulations (42), interacting hard spheres (43), and colloidal suspensions (44,45). Here we focus on evidence indicating that the primary response in condensed matter involves thermodynamic heterogeneity. Several specific heat measurements have shown that energy flows slowly from the thermal bath into the slow degrees offreedom.One technique involves monitoring the temperature change of the sample (AT) as a function of time after applying a heat pulse (46-50). At short times, AT rises sharply as the heat diffuses rapidlyfromthe heater through the sample to the thermometer. At long times, Δ Γ returns exponentially to zero as heat flows out of the sample via a weak link to the cryostat. At intermediate times, AT relaxes non-exponentially as heat flows slowly from the thermal bath into the sample's slow degrees offreedom.Indeed, when Δ Γ is inverted to give the excess heat in the thermal bath, this enthalpy relaxation is similar to the dielectric and mechanical response in the sample. Likewise, if the amplitude and phase of AT is monitored as a function of heaterfrequency,the specific heat loss spectrum is similar to other loss spectrafromthe sample (51-53). A technique that simultaneously establishes the weak thermal coupling and the heterogeneity in the primary response is nonresonant spectral hole burning, NSHB, Fig. 1. NSHB utilizes a large-amplitude pump oscillation to add energy to the slow degrees offreedom,then a probe step to measure the modification. Observation of NSHB requires that the excess energyfromthe pump oscillation mustflowslowly out of the selected slow degrees offreedom,consistent with the inverse process of dynamic specific heat where energy added to the thermal bath flows slowly into the slow degrees offreedom.The intrinsic behavior of the response is deducedfromthe measured modification. For homogeneous response [Fig. 1(a)] the entire spectrum is shifted uniformly toward shorter times, whereas for heterogeneous response [Fig. 1(b)] a spectral hole develops. Because NSHB uses the responding degrees of freedom as their own probe, it is the most direct technique for investigating the nature of the net responsefroma bulk sample, but it cannot determine the length scale of the heterogeneity. The inset of Fig. 1 (b) showing a cartoon sketch of



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Figure 1. NSHB characteristics for (a) homogeneous and (b) heterogeneous response after a pump oscillation at frequency Ω/2π. spatially-localized spectrally-distinct degrees of freedom is established by other types of measurements that utilize local-probes. NSHB was first applied to the nonexponential dielectric relaxation of supercooled liquids (54), Fig. 2. A simple model (55, inset of Fig. 2) consists of a "box" for each set of slow degrees of freedom, with a local temperature 7], response rate 1/T and corresponding recovery rate 1/(ξ+γ) via a set of thermal resistors coupled to the thermal bath. Here l/γ is a uniform limit to how fast heat can flow out of the slow degrees of freedom, which can be attributed to a thermal bottleneck. Deduced values of l/γ are within experimental error of the average alpha response rate, consistent with the nearly uniform rate seen for aging of the alpha wing after a thermal quench (56). The model provides good agreement with all prominent features in the NSHB behavior (solid curves in Fig. 2). Dielectric NSHB has also been observed in a relaxor ferroelectric (57) and an ionic glass (58); while magnetic NSHB has been measured in a spin glass (59) and two crystalline ferromagnets (60). In every case, the observation of spectral holes indicates that the nonexponential primary response involves thermodynamic heterogeneity. This picture of spectrally-distinct regions that are weakly coupled to a thermal bath is the basis of the nanothermodynamic model presented here. h



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log (t/s) Figure 2. Spectral holes in supercooled propylene carbonate after pump oscillations at three different frequencies (54). The bath temperature was held constant at 157.4 K. Changes in local relaxation time are plotted as changes in local temperature using the measured activation energy of the alpha response peak. The inset shows the "box" model (55) used to calculate the solid curves. Nanothei niodynamic M o d e l We characterize the local interactions between slow degrees of freedom using the Ising model. In the Ising model, the i degree of freedom may be represented by a "particle" with a state variable (σ ) that is either "up" (+1), or "down" (-1). For magnetic systems, σ, represents the up or down orientation of each Ising spin. Other systems with two degrees of freedom, such as a lattice gas or binary alloy, map directly onto the Ising model. For supercooled liquids, the binary degrees of freedom may correspond to the two lowest energy levels in the interaction between molecules. Often the energy levels depend on the electron alignments in the molecules, as in the case of bonding and anti-bonding states, which would also map to the Ising model. Otherwise the Ising model is just the first step towards characterizing the actual interaction between complex molecules, for which we assume that each molecule with multiple degrees of freedom may be represented by multiple Ising-like "particles." Consider a cluster containing a total of m particles, with a uniform interaction energy of tjlz between each particle and its ζ nearest neighbors in the cluster. The net interaction energy per particle is V^-^Jlzrn^ôpj, where denotes the sum over all pairs of interacting particles. In terms of the number of up particles ($) th

4


233 and the number of pairs of interacting up particles the energy may be written as (61) V (lJU^(tJ4)(MJzm - 41/m+l). Most of the difficulty in evaluating ^(i,{++) comesfromthe pair-interaction term In the mean-field approximation, this difficulty is avoided by using the average value £ 1m « (z/2)(i/m) , leaving V (i) * -(8o/4)(2i/m - l) . m

2

++

2

m


Equilibrium energy of finite clusters Although the mean-field approximation is usually associated with long-ranged interactions, in fact an identical expression is obtained for nearest-neighbor interactions in an isolated cluster (microcanonical ensemble) with two basic assumptions. 1) Net momentum is conserved; thus changes in ? occur only through contact to neighboring clusters, whereas ί-eonserving changes in the local configuration ({++) occur internally. 2) Net energy per particle [ε (ί)] is conserved by a balance between potential and kinetic energies; thus configurations with low potential energy have high kinetic energy and vice versa, so that the virial theorem for any realistic potential near a stable equilibrium yields t (i)-2 V ({). In the Ising model, kinetic-energy-carrying "demons" have been used to maintain the microcanonical ensemble (62,63). Using the fundamental postulates of statistical mechanics - that all allowed states are equally likely and that the equilibrium of a finite system is foundfromits time-averaged properties - the net energy per particle is eÔ^-CeîîCi-iymCm-l) - mim + 1]. One way to demonstrate the validity of ε (ί) is to examine the interaction energies of some specific examples. Consider four Ising particles (m=4) arranged in a squareringso that each particle interacts with only its two nearest neighbors. For two up particles (i=2), one set of four-fold degenerate configurations has both aligned particles on the same side of the square (i++=l) giving F (2,l)=0; and a doubly-degenerate set has the aligned particles at diagonal corners (i++=0) giving F (2,0) = +εt /k . However, this standard result is valid only for infinite systems. We now show that afinitesystem can lower its energy by subdividing into small clusters. 2

+

B

B

Q

B


OC

235 Continuing with the integral representation of the partition function, the average energy for a cluster of size m (from k T d\n A IdT) is 2

B

+

m

2

K = J_ Wiô^ )exp[-mf(I)/^71/A

(2)

m

For large but finite m, the integrals may be approximated by a steepest-descents procedure, where the argument in the exponent of each integrand is expanded about its maximum. Specifically for the partition function in the denominator, at T>tJk where 1^=0, f(Z)*f(0)+f \0)Υι1 gives k *2 ^2π/ηι(\-ε ί k T). However the extra factor of L in the numerator yields a different transcendental equation Z,-tanh[Z (8 /^ 7)+2/(/72LJ], with a non-zero order parameter |ZJ~

Liquid Dynamics - American Chemical Society

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