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Dynamic Correlation Length Scales under Isochronal Conditions

The origin of the dramatic changes in the behavior of liquids as they approach their vitreous state - increases of many orders of magnitude in transport properties and dynamic time scales - is a major unsolved problem in condensed matter. These changes are accompanied by greater dynamic heterogeneity, which refers to both spatial variation and spatial correlation of molecular mobilities. The question is whether the changing dynamics is coupled to this heterogeneity; that is, does the latter cause the former? To address this we carried out the first nonlinear dielectric experiments at elevated hydrostatic pressures on two liquids, to measure the third-order harmonic component of their susceptibilities. We extract from this the number of dynamically correlated molecules for various state points, and find that the dynamic correlation volume for non-associated liquids depends primarily on the relaxation time, sensibly independent of temperature and pressure. We support this result by molecular dynamic simulations showing that the maximum in the four-point dynamic susceptibility of density fluctuations varies less than 10% for molecules that do not form hydrogen bonds. Our findings are consistent with dynamic heterogeneity serving as the principal control parameter for the slowing down of molecular motions in supercooled materials.

The dynamics of liquids approaching their vitreous state exhibits interesting effects, the most prominent being spectacular changes in viscosity and relaxation times: these quantities may increase several orders of magnitude for a few degrees cooling. Eventually the response becomes so sluggish that the supercooled liquid behaves as a solid over laboratory time scales, and the material is now referred to as a glass. This super-Arrhenius slowing of molecular motions (i.e., the logarithm of the relaxation times increasing faster than linearly with reciprocal temperature) is concurrent with the growth of space-time correlations [1,2,3], as motion of a molecule increasingly requires adjustments of others.
Inherent to spatial correlation of the molecular mobilities is dynamic heterogeneity, the spatial variation of the dynamics reflected in non-exponentiality of the relevant time-correlation function. The growth of transient order manifested as dynamic correlations is commensurate with the size of these dynamic heterogeneities, the latter having been measured experimentally [4,5]. Relatedly, computer simulations of model liquids have found that for a given material, a correlation exists between the dynamic correlation volume and the breadth of the relaxation dispersion [6].
Non-associated liquids, that is, liquids that lack hydrogen bonding, complex formation, etc., exhibit a property referred to as isochronal superpositioning, whereby the shape (breadth) of the relaxation function depends only on the relaxation time, τ [7,8]. Since the relaxation function reflects the distribution of molecular relaxation times (whose average is the τ measured experimentally), isochronal superpositioning implies that dynamic heterogeneity is intimately connected to τ. Of interest herein is whether dynamic correlations and their growth might be the cause of, rather than only accompanying, the slowing down of the dynamics in the supercooled regime. The spatial extent of dynamic correlations exhibits power-law dependences on the relaxation time at constant (low) pressure, with the cooperativity extending to several intermolecular distances at the transition to the glassy state [1]. Such results suggest that dynamic correlations may serve as the control parameter governing vitrification [9,10,11,12,13]. However, to test this idea requires determination of the dynamic length scale in a liquid for different thermodynamic conditions having equivalent molecular mobility. Whether the dynamic correlation volume is invariant for state points with constant τ is a key to solving the glass transition problem [14].

Quantifying dynamic correlation
A number of indirect methods have been proposed to estimate dynamic correlation volumes [15,16,17,18,19], but precise determinations can be obtained from high-order correlation functions [1,20,21,22,23], such as the four-point dynamic susceptibility of the density fluctuations In these equations k is the Boltzmann constant, cP and cV are the respective isobaric and isochoric heat capacities, and κT is the isothermal compressibility. The last term in either eqs. [2] or [3] is not available from measurements, but the insight of Berthier et al. [11] was the notion that both may be sufficiently small near Tg that they can be neglected, with the dynamic susceptibility then approximated as and These expressions, both underestimates of χ4(t), should be nearly equal.
Eqs. [4] and [5] have been applied to many materials [27], but of interest herein are measurements extending to high pressures, which enable assessment of the relationship between Nc and τ. Analyses using eq. [4] of four liquids, salol, polychlorinated biphenyl, propylene carbonate (PC), and a mixture of o-terphyenyl and o-phenyl phenol [28], found that state points having different temperatures and pressures but the same relaxation time had the same correlation volume; that is, Nc is uniquely defined (to within ~10%) by τ, or vice versa. Subsequently, from a similar analysis on dibutylphthalate [29] it was reported that at conditions of constant τ, Nc increased with pressure. A third study [30] used both eq. [4] and [5] and determined that Nc for o-terphenyl, glibenclamide, and phenylphthaleindimethylether decreased with increasing P at fixed τ. Thus, three studies on 8 different liquids concluded that under isochronal conditions the dynamic correlation volume was constant [28], increased [29], or decreased [30] with increasing pressure and temperature.
These discordant results are not because the behavior of Nc is material specific, but rather there are both fundamental and technical issues with the calculation of eqs. [4] and [5]. In simulations χ4 has a dependence both on the dynamics (Newtonian vs. Brownian) and the statistical ensemble (NVT, NPT, etc.) [31,32]. Moreover, it is unclear whether the heat capacity, the excess heat capacity (which isolates the configurational part relevant to structural relaxation), or the change in cp or cV at Tg should be used in applying eqs.
[2] - [5]; furthermore, values for the glassy or non-configurational part of the heat capacity are rarely available at high pressures. The approximation equations also involve derivatives of interpolated data, which can introduce uncertainties into the calculation of Nc. Note that eqs. [2] and [3] are equivalent, and their respective approximations differ only in the neglected last term, presumed to be negligible. However, in ref. [30] Nc from eqs. [4] and [5] differed by as much as 40%; that is, the difference between two small contributions was an appreciable amount of the total χ4. This indicates either a lack of precision in the calculation of the approximations to χ4, or the assumption that 4 NPH χ and 4 NVE χ are small near Tg [33] is incorrect (as has been found in simulations [34]). Notwithstanding the source of the problems in applying eqs. [4] and [5], the obtained results for the behavior of Nc under isochronal conditions are ambiguous, motivating the use of a different method.
Although the linear susceptibility does not detect the transient order associated with dynamic heterogeneity, it has been argued that higher order correlation functions should be manifested in higher order (non-linear) susceptibilities [35]. An example is the non-linear magnetic response of spin glasses, which can detect transitions evident otherwise only in four-point correlation functions [36]. From scaling arguments applied to mode coupling theory, the response of liquids to external perturbations has been shown to be increasingly nonlinear as the glass transition is approached [37]. The inference is that the non-linear susceptibility can be used to measure dynamic heterogeneity; specifically, the amplitude of the nonlinear dielectric susceptibility is proportional to Nc [35] ( ) where ε0 the permittivity of free space, a the molecular volume, Δχ1 the linear dielectric strength, and │χ3│ is the modulus of the third-order susceptibility corresponding to polarization cubic in the applied field. The derivation of eq. [6] is more intuitive than rigorous, and features in the nonlinear susceptibility interpreted in terms of dynamic heterogeneity can be obtained in models that lack dynamic heterogeneity [38,39]. Nevertheless, this equation has been applied by several groups to determine the variation of Nc at ambient pressure with T [40,41,42] and during aging [43]. The main result, that Nc grows on cooling towards Tg in accord with the behavior of the relaxation time, supports the identification of the peak in │χ3│ with the dynamic correlation volume.

Nonlinear dielectric results
We carried out non-linear dielectric measurements under high pressure on two liquids, propylene carbonate and propylene glycol (PG). PC is a non-associated liquid conforming to isochronal superpositioning [7,8], whereas PG is hydrogen-bonded and thus its relaxation spectrum is not constant for constant τ [44]. In Figure 1 are representative │χ3│ spectra obtained at various pressures. For both liquids there is an increase in the peak intensity with decrease in peak frequency, consistent with growth in the correlation volume as the relaxation time becomes longer.
To quantify dynamic heterogeneity requires that the contribution to │χ3│ from saturation of the dipole orientation be removed from the spectra. This saturation effect can be calculated assuming independent, rigid dipoles [40,45] ( ) The distribution of relaxation times (heterogeneous dynamics) are described using the Havriliak-Negami function [46] ( ) ( In the above α and β are constants. We fit the linear dielectric loss peaks to eq.[8], then calculate the saturation effect using eq. [7]; typical results are shown in Figure 2. The difference spectrum can be used to extract Nc via eq. [6]. As seen in Fig. 2, the saturation effect decays rapidly with frequency. This suggests a simpler method to avoid this interference, using a value of │χ3│ at higher frequencies than the peak. Brun et al. [41] have shown that │χ3│ at a frequency 2.5 times fmax, the frequency of the maximum in the linear dielectric loss, provides a measure of Nc unaffected by dipole saturation. In the inset to Fig. 2 we show the results for Nc of PC at ambient pressure obtained after correcting │χ3│ using eq. [7] and by taking the value of │χ3│ at 2.5fmax. Both values have similar temperature dependences, so that either method yields a quantity proportional to the correlation volume. Hereafter we report │χ3(2.5fmax)│.
In Figure 3 are Nc for the two liquids plotted as a function of the linear relaxation frequency. The data for PC show the two regimes expected for dynamic correlations -power-law dependences with a steeper slope at higher frequencies [11]. This supports the interpretation of the peak in the nonlinear susceptibility in terms of dynamic correlations. Within the experimental scatter (ca. 15%), the number of dynamically correlated molecules for PC depends mainly on the relaxation time; there is no systematic  Figure 2. Third-order spectrum of PC as measured (circles) and after subtraction of the saturation effect (squares), the latter indicated by the solid line. Inset shows the magnitude at 2.5fmax in the uncorrected spectrum (circles) and at the peak of the corrected spectrum (squares).
variation in Nc with T or P. We can also arrive at this result without any analysis, by simply comparing │χ3│ for two state points having the same peak frequency in their linear spectra. As seen in Figure 4, these │χ3│ spectra for PC superpose; thus, the isochronal Nc are essentially constant.
PG is an associated liquid and therefore is expected to behave differently. The results in Fig. 3 bear this out: there are substantial variations (>50%) in Nc for a given τ.  this behavior is the change in H-bonding with thermodynamic conditions, whereby the liquid structure is not constant for isochronal conditions. Thus, neither the amplitude of │χ3│ nor the spectrum (Fig. 4) is invariant to changes in T and P, even when τ remains constant.

Molecular dynamics simulations
As discussed above, the four-point dynamic susceptibility enables determination of Nc.
Molecular dynamics (MD) simulations of Lennard-Jones particles found that the Nc from χ4 was constant for constant τ [25]. This result is also predicted for simple liquids in the NVT ensemble [32], "simple" defined as liquids displaying certain properties, such as isochronal superpositioning of the relaxation function. Accordingly, to compare with the dielectric results herein, we carried out MD simulations on two rigid, polar united-atom structures, modelling a three-site hydrogen-bonded molecule and a twosite structure with the same dipole moment but no H atom (and thus non-associating). Details of the simulations can be found in ref. [47]; the difference herein is a larger system size (10,000-16,000 molecules).
To compare to the dielectric relaxation results, we calculated χ4 for dipole reorientation from the variance of the rotational autocorrelation function, C1(t) (rather than eq.[1], the 4-point correlation function for density fluctuations)   [11] was also computed to assess the susceptibility loss spectra at various state points with constant relaxation time, as done for the experimental dielectric spectra.
χ4 determined for the NVT ensemble for the non-associated liquid at four isochronal state points are shown in Figure 5. The peak heights, and thus Nc, are within 4% for density variations of as much as 16%. For the H-bonded structure (Figure 6), this variation under isochronal conditions is much larger, more than a factor of two. Similarly, there is a breakdown of isochronal superpositioning, as shown in the inset to Fig. 6. The very different behavior of the non-associated and the H-bonded materials, as summarized in Figure 7, corroborate the nonlinear dielectric determination that for propylene carbonate, but not for the H-bonded propylene glycol, Nc is sensibly constant at constant τ, its variation being a few percent. This behavior is consistent with isochronal superpositioning of the linear relaxation function for PC, but not for PG. χ'' f* Figure 5. χ4 for the dipole autocorrelation function for a polar, non-associating liquid at three state points having equal reduced relaxation times (t*=ρ 1/3 (kT) 1/2 t). The density change between the highest and lowest pressure data is 16%. The inset shows the linear susceptibility for the same state points versus reduced 1/t*.

Summary
For non-associated liquids, the third-order dielectric susceptibility and four-point dynamic susceptibility from MD simulations both show that there is only a small (~10%) variation in Nc for state points having  time. This is important for development of a theory of the glass transition, since viable models that predict τ implicitly make simultaneous predictions for Nc and the distribution of relaxation times.

Materials and Methods
Samples for nonlinear dielectric experiments were contained within Teflon gaskets, 10 -25 µm diameter, and between polished steel parallel plates that served as the electrodes. This capacitor assembly was placed in a Manganin cell (Harwood Engineering) inside a Tenney environmental chamber.
Pressure was applied via using an Enerpac pump in combination with a Harwood pressure intensifier; ?? was used as the hydraulic fluid. The voltage for the nonlinear experiments was as large as 200 V (rms), with a Novocontrol HVB 4000 dielectric analyzer used for th measurements.
The molecular volumes, a(T,P) in eq. [6], were obtained from the published equation of states for PC [49] and PG [ 50 ].
MD simulations were carried out using GROMACS [46]. For details of the system see ref. [48].
Larger system sizes were used herein (N=10000 for the H-bonded and N=16000 for the non-associated liquid).