Molecular signatures of the glass transition in polymers

The glass transition temperature ($T_g$) is one of the most fundamental properties of polymers. $T_g$ is predicted by some theories as a sudden change in a “macroscopic” quantity (e.g., compressibility). However, for systems with “soft” glass transitions where the change is gradual it becomes hard to pinpoint precisely the transition temperature as well as the set of molecular changes occurring during this transition. Here, we introduce two new molecular signatures for the glass transition of polymers that exhibit clear changes as one approaches $T_g$: (i) differential change of the probability distribution of dihedral angles as a function of temperature and (ii) the distribution of fractional of the time spent in the different torsional states. These new signatures provide insights into the glass transition in polymers by directly exhibiting the concept of spatial heterogeneity and dynamical ergodicity breaking in such systems, as well as provide a key step to quantitatively obtain the transition temperature from molecular characteristics of the polymeric systems.

Figure 1

Glass transition of PEHMA and PMMA from MD simulations using the specific volume ($V_{sp}$) and mobility measures. (a) Chemical structure of EHMA and MMA and corresponding color codings. (b) Glass transition of PEHMA (soft transition) and PMMA (sharp kink transition) and their corresponding $T_g$ as the intercept of extrapolation (dotted line) from the linear regressions at high and low temperature regime (solid line). (c) Uncertainty of fitting a soft curve using linear extrapolation. (d)–(f) Mobility measures of PEHMA and PMMA: (d) Diffusion coefficient ($D$), (d) root-mean-square fluctuations of backbone atoms (RMSF) and (f) Standard deviation (SD) of backbone dihedral angle fluctuations. As can be seen, only in the case of PMMA it is possible to pinpoint the transition using $V_{sp}$, while all the mobility measures do not show a clear transition point and are very similar between both chemistries.

Figure 2

(a) Schematic representation of a potential energy landscape for a glass former. (b) Energy landscape of a double-well potential and the illustration of calculating the pairwise JS divergence (defined in text) for the double well potentials. The pairwise JS divergence for the distributions shown in (b), left panel are JS (red, blue) $=0.07$, JS (red, purple) $=0.99,$ and $\mathrm{JS}$ (blue, purple) $=1.04$. (c) The corresponding dynamical heterogeneity (JS divergence) as a function of $kT$. The error bar is the standard deviation across all pairwise JS divergence.

Figure 3

Probability distribution functions of ten independent trials at four different temperature using Monte Carlo simulation in a double-well potential after mean shift. All temperatures are below the freezing temperature ($kT\approx 1$).

Figure 4

The dynamical heterogeneity (JS divergence) with regard to temperature for (a) PMMA, (b) PEHMA, and (c) PS. The population probability distribution of backbone dihedral angles is also shown as an inset. The piecewise linear regression is shown as solid line.

Figure 5

Average largest standard deviations of all dihedral angles for three homopolymers with increasing temperature. The highest temperature is below the corresponding $T_g$. The reason for choosing the largest standard deviation is because below $T_g$, we expect that the majority of dihedral angles are unimodal. The error bar is the standard error across all the dihedral angles.

Figure 6

Sets of weights of individual dihedral angles for MMA at $300\text{K}, 370\text{K}, 440\text{K}, 510\text{K}, 580\text{K}$, and $650\text{K}$. The three weights in different colors are plotted against the index of all 740 dihedral angles. The percentages of mobile dihedral angles are shown as an inset. A dihedral angles is defined as mobile if the maximum weight is less than 0.95.

Figure 7

The fractional occupation of different torsional states with regard to temperature for: (a) PMMA, (b) PEHMA, and (c) PS. For each dihedral angle, the fractions are ranked and then averaged as blue, purple, and red lines. The error bars are the standard deviation across all the dihedral angles. The definition of fractions can be found in Fig. S2 and S3.