Cooperativity at the glass transition: A perspective from facilitation on the analysis of relaxation in modulated calorimetry

The glass transition region in nonconfined polymeric and low-molecular-weight supercooled liquids is probed by temperature-modulated calorimetry at a frequency of 3.3 mHz. From the distribution of relaxation times derived by analyzing the complex heat capacity, the number $N_{\alpha }$ of cooperatively rearranging units is estimated. This is done by resorting to a method in which cooperative motion is viewed as a result of a spontaneous regression of energy fluctuations. After a first, local, structural transition occurs, the energy threshold for the rearrangement of adjacent molecular units decreases progressively. This facilitation process is associated to a corresponding evolution of the density of states in a canonical representation and may be considered as a continuous spanning through different dynamic states toward a condition in which configurational constraints disappear. A good agreement is found with the $N_{\alpha }$ values obtained from the same calorimetric data within the framework of Donth's fluctuation theory. It is shown that, at variance from previous treatments, $N_{\alpha }$ can be estimated from just the relaxation function, without resorting to the knowledge of the configurational entropy. Examples point to a modest dependence of the $N_{\alpha }$ estimates on the experimental method used to derive the relaxation function.

Figure 1

Mobility and metastability regions to which a unit with energy $\epsilon$ may belong when the rearrangement threshold energy is $\zeta$. If the latter is fixed, that is, when no rearrangement occurring in the surroundings of the unit is able to change it, a state characterized by a different threshold can only be reached after a passage through the mobility region; so, the paths labeled with an “$a$” are allowed while the others are not. After a unit has left its frozen state “$F$” to reach the mobility state “$M$” with an energy fluctuation, the subsequent value of the rearrangement threshold is random.

Figure 2

Along the $n=\mathrm{const}.$ line (labeled with $a$), the $z$ units span the available $\zeta$ domain while remaining in the small-scale rearrangement regime. A dynamic transition may only start for $\zeta$ larger than ${\zeta }_0$, that is, the threshold value for which the equality holds in Eq. (8) of the text; ${\zeta }^{*}$ is one such value at which the transition toward a large scale rearrangement happens to start, with the representative point of the local rearranging set of units following the downward curved $\mathrm{\Delta }\mu (n,\zeta )=\mathrm{const}.$ line (labeled with $b$). The curves of the inset represent examples of the normalized integrand of the partition function $Z_{\zeta ,n}$ for different values of $n$, namely 20, 10, 5, and 2, which are labeled as $\alpha ,\beta ,\gamma$, and $\delta$, respectively. The labels on lines $a$ and $b$ highlight a correspondence of different dynamic regimes with the probability profiles of the inset.

Figure 3

Real and imaginary parts of the complex heat capacity (in J/gK) of PVAc on cooling, under the experimental conditions illustrated in the main text. $C_{p,\mathrm{liq}}\left(T\right)$ and $C_{p,\mathrm{glass}}\left(T\right)$ are the equilibrium and unrelaxed heat capacities, respectively. The normalized components of the heat capacity are shown in the inset.

Figure 4

Imaginary part of the heat capacity $C_p^{\prime \prime }$ (in J/gK) for PVAc at different heating/cooling rates. The solid and dashed lines are Gaussian fits (the data points for the cooling rate of 0.5 K/h are not shown for clarity of the figure). The 0.5 K/h heating profile is not reported because it almost completely superposes with the cooling one.

Figure 5

Cole-Cole representations of $C_{p,\mathrm{norm}}^{*}$ for PVAc (upper panel) and PS (lower panel). The solid lines, which best fit the data, correspond to a KWW relaxation in the case of PVAc and to an HN distribution for PS. The dashed lines represent the lower-quality fits obtained with a HN and a KWW distribution for PVAc and PS, respectively. The $C_{p,\mathrm{norm}}^{\prime }$ intervals in which the analyses have been performed are $\left[0.1,0.97\right]$ and $\left[0.08,0.97\right]$ for PVAc and PS, respectively. Insets “a” show, each one, the $T$-dependence of the characteristic relaxation time associated to the solid line of the corresponding main frame (i.e., ${\tau }_{\mathrm{HN}}$ for PS and ${\tau }_{\mathrm{KWW}}$ for PVAc), together with the best VFT approximations. The highlighted temperatures $T_{\mathrm{peak}}$ are found by fitting $C_{p,\mathrm{norm}}^{\prime \prime }\left(T\right)$ with a Gaussian (cf. the main text and Fig. 6). Insets “b” explicitly report the dependencies of $C_{\mathrm{HN}}^{\prime \prime }$ and $C_{\mathrm{KWW}}^{\prime \prime }$ on $\tau$, after Eqs. (20) and (22), respectively, for the value of $\omega$ set by the experimental conditions and the best-fitting shape parameters.

Figure 6

Imaginary part of the normalized heat capacity of PS (upper panel) and PVAc (lower panel) on cooling. The dashed lines are Gaussian fittings for the estimates of the mean temperature fluctuations.

Figure 7

HN relaxation function (dashed line) and fitting curve (solid line) for PS at $T=372.2$ K (the shape parameters are reported in Table 1). The arrows labeled with “$a$” and “$b$” highlight the fitting interval; ${\tau }_0\equiv {\omega }^{-1}\simeq 48$ s is defined by the experimental conditions for a symmetric relaxation. Given $\omega$, the HN shape parameters yield a central relaxation time of $\sim 100$ s (highlighted by the vertical dashed line), consistently with inset “b” of the lower panel in Fig. 5. For a comparison, the dotted line refers to a simple exponential decay $exp\{-t/{\tau }_0\}$, while the dash-dotted one is the relaxation function of a Cole-Cole process, ${[1+{\left(i\omega {\tau }_0\right)}^a]}^{-1}$, with $a=0.46$. The inset compares the PVAc relaxation (dashed) and fitting (solid) at $T=305.6$ K, with the corresponding ones for Ac (dash-dot-dot and solid, respectively) at $T=294.3$ K; again the dotted line indicates the single-time exponential decay of the main frame.