Data-Driven Identification and Analysis of the Glass Transition in Polymer Melts

Abstract

Understanding the nature of glass transition, as well as the precise estimation of the glass transition temperature for polymeric materials, remains open questions in both experimental and theoretical polymer sciences. We propose a data-driven approach, which utilizes the high-resolution details accessible through the molecular dynamics simulation and considers the structural information on individual chains. It clearly identifies the glass transition temperature of polymer melts of weakly semiflexible chains. By combining principal component analysis and clustering, we identify the glass transition temperature in the asymptotic limit even from relatively short time trajectories, which just reach into the Rouse-like monomer displacement regime. We demonstrate that fluctuations captured by the principal component analysis reflect the change in a chain’s behavior: from conformational rearrangement above to small fluctuations below the glass transition temperature. Our approach is straightforward to apply and should be applicable to other polymeric glass-forming liquids.

Polymer materials in applications are often in the glassy state. Upon cooling of a rubbery liquid polymer, dynamic properties such as viscosity or relaxation time increase drastically near the glass transition temperature (T_g) in a super-Arrhenius fashion¹⁻⁴ without any remarkable change in structural properties.³ Despite enormous experimental and theoretical efforts,⁵⁻¹⁰ the nature of glass transition as well as the question of a precisely defined T_g still remain unclear.^4,11−14 In computer simulations, T_g is often calculated from characteristic macroscopic properties, e.g., changes in the specific volume, density, or energy.¹⁴⁻¹⁶ The increase in viscosity, equivalent to the terminal relaxation times, is commonly fitted to a Vogel–Fulcher–Tamann behavior that predicts a divergence at T_VFT,¹⁷ typically about 50° below the calorimetric T_g.¹⁸ However, the precise value of the observed T_g depends on the cooling rate and fitting procedures, which can lead to some ambiguities in comparison with experimental values,^19,20 unlike a sharp and distinct change in physical properties. Thus, reliable predictions of T_g are indeed challenging.^12,13,21,22

Attempts to link T_g with the molecular structure of polymeric materials draw more attention. Recent studies predict T_g by quantifying the changes in specific dihedral angles and transitions between states defined by those angles^13,14 or by using averaged intrachain properties.²³ A possibility to specify the structural properties of the glassy systems, which can reflect changes in T_g, is attractive, but it remains challenging and system specific. Machine learning (ML) methods hold great promise to automatize the determination of structural descriptors from molecular simulation data. Recently, the application of ML to nonpolymeric supercooled model liquids allowed to understand the connection between characteristic local structures and the slowing down of dynamical properties.²⁴⁻²⁹ For polymer chains in a melt, the intrachain properties associated with the chain connectivity and flexibility also play an important role in determining T_g. However, the application of ML methods to determine structural changes during the glass transition in polymer chains is limited.^13,30,31

In this Letter, we use unsupervised data-driven methods to identify the glass transition of polymer melts of weakly entangled polymer chains only by employing information about conformational fluctuations at different temperatures. We first analyze the combined data from different temperatures using principal component analysis (PCA),³² followed by clustering and determine a clear signature of glass transition. Considering the simulation data within a finite observation time window up into the Rouse-like regime, our approach allows a very solid extrapolation to infinite times to predict T_g. We then also employ the data-driven methods on individual temperature data separately. The nonmonotonic variation of the magnitudes of leading eigenvalues and the participation ratio derived from PCA captures the signature of the glass transition. It also reflects a change in the nature of the fluctuations in the system. We apply these approaches to the simulation data of a coarse-grained polymer model³³ and compare estimates of T_g obtained from classical fitting of macroscopic properties with the new method. The proposed method has the following advantages: (a) our approach is based on high-resolution microscopic details instead of average macroscopic properties, (b) it does not rely on the fitting protocols, and (c) our analysis focuses on the information about structural fluctuations at the level of individual chains to predict T_g from very moderate simulation trajectories.

In ref (33), Hsu and Kremer developed a new variant of the bead–spring model^34,35 for studying the glass transition of polymer melts.^36,37 Molecular dynamics simulations of a bulk polymer melt containing n_c = 2000 semiflexible polymer chains of chain length n_m = 50 monomers and a Kuhn length ≈ 2.66σ³⁸ were first performed in the NPT ensemble at P ≈ 0ϵ/σ³ and constant T following a standard stepwise cooling protocol (20 temperatures from 1.0 to 0.05 ϵ/k_B), choosing a fast fixed cooling rate of Γ = 8.3 × 10^–7ϵ/(k_Bτ) (see Supporting Information (SI), Sec. S-I for details). The Rouse time is , and the entanglement time is with the characteristic relaxation time τ₀ ≈ 2.89τ estimated at T = 1.0ϵ/k_B, and the entanglement length N_e = 28 monomers.³⁸ Here τ₀ is the upper limit of time that a monomer can move freely. After the step cooling, subsequent NVT runs up to 3 × 10⁴τ were performed at each T to investigate the monomer mobility characterized by the mean square displacement g₁(t) (for details, see SI, Sec. S-I, Figure S1). In this Letter, we mainly use simulation trajectories from NVT runs stored every 200τ in the time window between 200τ and 3 × 10⁴τ (gray area in Figure S1, SI), resulting in 150 time frames per temperature.

The first estimate of the glass transition temperature at T_g ≈ 0.64ϵ/k_B using a conventional fitting procedure was determined from the volume change (Figure 1a, inset).³³ We here adapt another standard approach to estimate T_g by performing a hyperbolic fit³⁹ on the temperature-dependent density of polymer melt for 0.1 ≤ k_BT/ϵ ≤ 1.0, , where c, T₀, a, b, and f are fitting parameters. T_g is either defined by T_g = T₀ or the intersection point of the two tangents drawn at the high and low temperatures. Both give an identical, more precise, estimate of T_g = 0.660(4)ϵ/k_B, as shown in Figure 1a, and are used as reference values for evaluating the data-driven approach presented below. Note that T_g obtained from the simulation data depends on the cooling rate. We propose here an alternative data-driven approach to gain insight into the glass transition with a minimum a priori knowledge about the system and user input.

Figure 1.

(a) Conventional methods of estimating the glass transition temperature T_g: density (ρ(T)) and logarithm of volume (ln(V/σ³)) (in the inset) plotted versus T. Estimates of T_g via the two tangent (t) fits (dotted lines) at high and low T, hyperbola (h) fit (curve), and two linear fits (dashed lines in the inset) are indicated by vertical lines. (Data are taken from ref (33).). (b–d) Data-driven determination of T_g. (b) Projections of concatenated data from all T for a single chain over multiple time frames in the two first leading principal components (PCs). Each point in the plot corresponds to one chain’s conformation at a given temperature at each time. Projections for T > T_g are colored varying from red to green while they are in purple for T < T_g (data shown for T ≥ 0.45 for clarity). Note that the axis values in the PCA embedding do not correspond to a directly measurable physical quantity, rather could be viewed as a weighted linear combination of scaled input distances. (c) DBSCAN of the PCA projection. The same projection as (b), but it is colored with DBSCAN cluster indices (ID) instead of temperature. DBSCAN assigns the high-temperature liquid state as noise (cluster ID = −1) and the low-temperature glassy state as a cluster (cluster ID = 0). (d) Average cluster ID over all chains versus T. The separation between the liquid and glass state becomes sharper if we use median instead of mean (see SI, Figure S6b).

The analysis workflow consists of two different, but related, methods (a sketch is given in SI, Sec. S-III). Both identify the same T_g, but treat the data differently (using combined information from all 20 temperatures or individual information from each temperature). To identify changes in the studied systems, we first define possible descriptors: sets of all pairwise internal distances for a single chain. They are well suited to describe conformational fluctuations of individual polymer chains. Then we apply PCA³² to the high-dimensional descriptor space. PCA has been successfully used to characterize the phase transition in conserved Ising spin systems.^40,41 The method relies on purely structural information without any a priory knowledge of dynamical correlations. A M × L real matrix X_c with elements , 1 ≤ m ≤ M, 1 ≤ l ≤ L is used to represent data for a single chain c. Here c = 1, ..., n_c is a chain index, L is the number of descriptors (e.g., the intrachain distances between any two monomers in a single chain of n_m = 50 monomers: L = n_m × (n_m – 1)/2 = 1225), and M is the number of observations (i.e., M = 150 (time frames) × 20 (temperatures) = 3000 for Method I, and M = 150 (time frames) × 1 (temperature) = 150 for Method II). X_c is standardized column-wise, i.e., each element is converted to , where is the mean value for each column l, and is its corresponding standard deviation for chain c such that the rescaled columns have a mean value of 0 and a variance of 1. PCA is done individually for each chain by first calculating the covariance matrix , where , 1 ≤ j, k ≤ L are elements of C_c, and , are the standardized descriptors. Then the eigenvalues λ_c,i and the corresponding eigenvectors v_c,i of the matrix C_c, for i = 1, 2, 3, ..., min(L, M) are calculated and sorted in decreasing order of λ_c,i. The original data set X_c is converted to by projecting X_c to the new orthogonal basis formed by P-leading eigenvectors v_c,k, where elements of are , k = 1, ..., P, and P ≤ min(L, M) is the reduced number of dimensions (P = 4 in this work).

Due to the correlated motions of neighboring monomers, the intrachain distance space can be reduced by skipping some distances. We discuss this in more detail in SI, Sec. S-IX. All results are similar in nature after reducing the input feature space, and the asymptotic estimate of the glass transition temperature is reported considering every fifth monomer in a chain.

Method I: We perform PCA on a randomly selected single chain using the internal distances over time concatenated for all temperatures. In this way we construct the new basis formed by eigenvectors v_c,i containing information about fluctuations of internal distances at all temperatures. The internal distances of the chain at each simulation snapshot and temperature are projected independently on this new basis. Thus, projections in the new PCA space can be viewed as linear combinations of input distances. Already in the two-dimensional projection one could clearly differentiate between two states (Figure 1b), which occur roughly around the glass transition temperature T_g ≈ 0.65ϵ/k_B (Figure 1a). The scatter of the PCA projection qualitatively changes at and below 0.65ϵ/k_B, indicating the onset of a different state.

To quantify the separation between the liquid and glassy state, we perform such a PCA for each chain separately, followed by clustering. Clustering groups the chains’ conformations at each simulation snapshot and temperature based on similarities in their conformational fluctuations reflected as closeness in the PCA projection space. Thus, each chain conformation is assigned an index corresponding to the group it belongs to. Such an index is called a cluster index (ID). We used density-based spatial clustering of applications with noise (DBSCAN)⁴² for each projection in a four dimensional space of leading principal components (PCs). The cluster ID n_i for a single chain at each time frame is always an integer, i.e., n_i ∈ {−1, 0, 1, 2, ..., n_cluster – 1} for i = 1, 2, ..., n_cM, where the number of chain n_c = 2000, the number of frames M = 150 at each T, n_cluster is a number of clusters found by DBSCAN (max(n_cluster) = 3 in this work). n_i = −1 corresponds to the noise, while n_i ≥ 0 corresponds to the clusters found in the four-dimensional PCA projections using DBSCAN.⁴² The details of clustering, the rationale for choosing four dimensions in PC space, the goodness of clustering are given in the SI (Sec. S-IV, S-V). DBSCAN determines the high-temperature states as sparse or “noise” (and assigns them with cluster ID = −1) and the low-temperature glassy state as a cluster(s) (Cluster IDs ≥ 0), see, e.g., Figure 1b,c. Then, we repeat this clustering on each chain present in the system (2000 chains) to confirm that the separation between liquid and glassy state is consistent for all chains in the melt. To obtain a general estimate of the temperature at which this separation occurs, we calculate the average cluster ID ⟨n(T)⟩. At each temperature T, ⟨n(T)⟩ is given by , where P(n_i, T) is the probability distribution of cluster IDs for all n_c chains over M frames at each T. In Figure 1d ⟨n(T)⟩ shows a sharp transition around T = 0.65ϵ/k_B.

The glass transition is often viewed as the process of falling out of equilibrium during cooling at a given rate or as the onset of ergodicity breaking. Above T_g all states are accessible to the system, while below T_g the system is arrested. Therefore, we expect the dissimilarity between low and high temperature regimes at or around T_g, giving rise to the sharp transition in the average cluster indices. Our result shows a signature of dynamic ergodicity breaking (Figure S1) indicated by a dramatic increase in the equilibration time (see the VFT-plot in ref (33)) for each chain at the same temperature; we report that as T_g. A similar signature of ergodicity breaking has been reported recently using Jensen–Shannon divergence metric for homopolymers.¹³

To extrapolate obtained results to long time limits where the polymer chains are supposed to reach the diffusive regime, we repeated the analysis above for 16 observation time windows Δt ranging from 1000τ to 3 × 10⁴τ. For each Δt, we have used Δt/t_lag consecutive frames with t_lag = 200τ. We see that the transition from the liquid to the glassy state becomes sharper with the increase of the observation time window (Figure 2). To quantify that, we interpolate the data by a hyperbolic tangent function g(T) = C(Δt)(1 – tanh(sT – d))/2 – 1, where s and d are the fitting parameters, C(Δt) is the gap between the two states at T ≫ T_g and T ≪ T_g, respectively. The inflection point of g(T) gives the estimate of T_g(Δt), depending on Δt. The behavior of average cluster ID vs T, as given in Figure 2, is similar to a typical behavior of magnetization vs T for a finite-size 2D Ising model⁴³ and requires further investigation considering the existing discussion in the literature.¹¹ The finite-size (time) effect is often considered for analyzing data obtained from simulations of finite system sizes or limited computing times. Taking into account this finite-time effect, we plot the estimates of T_g(Δt) versus 1/Δt in the inset of Figure 2. We find a remarkable linear dependency, which allows for extrapolation to Δt → ∞ and obtain T_g ≈ 0.6680ϵ/k_B as a best asymptotic estimate of T_g. This is in excellent agreement with the classical analysis of the temperature-dependent density (Figure 1a).

Figure 2.

Average cluster ID ⟨n(T)⟩ for different selected observation time windows Δt, as indicated. The curves give the best hyperbolic fit g(T) going through the data. The inflection point of g(T) shown in the inset gives the estimate of T_g(Δt) at each Δt. Extrapolating to Δt → ∞, we obtain T_g ≈ 0.6680ϵ/k_B.

In order to interpret the obtained projections to the leading PCs, we calculate the correlation between the internal distances and the corresponding projection to PCs (see SI, Sec. S-VII). The mostly correlated distances vary with different chains, with no clear signature of any characteristic distance. Due to standardization of the distances (i.e., see X_c definition for details), PCA accounts for relative changes in the distances rather than the absolute displacement values. As a result, the projections to leading PCs are not dominated by only large distances. However, they are related to physically motivated measures such as R_g, R_e (other physical properties can be also compared) (SI, Figure S8d).

Note that we performed PCA on a single chain, followed by taking an average over all chains in the system. Performing PCA on 2000 chains combined, we only observe the same Gaussian-like distribution within fluctuations, stemming from different chains, which is essentially independent of the temperatures (see SI, Figure S7).

Method II: In the following, we change our approach and perform PCA for individual chains, but at different temperatures independently. In this way the new basis formed by eigenvectors v_c,i(T) differ for each temperature (see SI, Sec. S-IX, showing examples of first eigenvectors for Methods I and II) and no information on individual chain conformations from other temperatures is accessible to Method II. The resulting projections are shown in SI, Figure S9. Notably, for the majority of chains in the melt, we could observe the change from a completely random distribution of points in the projection to more “clustered” with the decrease of T. This behavior can be quantified by the magnitude of the eigenvalues of PCA. In general, this magnitude is not a uniform value for independently projected data, but in our case all distances are standardized. Thus, we could average over the first eigenvalue for all projections (see Figure 3a), which shows a (weak) maximum close to T_g. This suggests that above T_g large scale fluctuations dominate, while below T_g fluctuations are dominated by many contributions from different, but short length scales. As a more general criterion, we use the participation ratio (PR) defined at each temperature over 150 frames as , where λ_c,i are eigenvalues sorted in the descending order (see Figure 3b). PR reflects decay rate of eigenvalues: the steeper is the change the smaller PR will be, if all λ_c,i are equal then PR = k. A typical spectrum of eigenvalues λ_c,i with different decay rates are plotted in SI, Figure S4c. The leading k = 25 eigenvalues from min(L, M) eigenvalues are counted to preserve at least 80% data fluctuations in PCs. Results are averaged over all chains, deviations are shown as error bars. The increase in magnitude of the first eigenvalue (or the decrease in PR) on approaching T_g can be related to an appearance of state separation in the system and change in a local structure as some recent studies suggest.^13,14 We argue that a prominent change in the monotonic behavior of PR (or the first eigenvalue) is connected with a change in the nature of the fluctuations in the system: from local configurational rearrangements (the rearrangement of parts of chain conformations) above T_g to only localized fluctuations along the chain below T_g (similar to observations in metallic glasses⁴⁴). As a result, more dimensions are needed to describe the random motion below T_g. To test the hypothesis about local structural changes above T_g, we perform the same analysis on simulation trajectories within a relatively short time window between 0.2τ and 20τ (blue area in Figure S1). Results are shown in the inset in Figure 3. We no longer see the nonmonotonic signature around T_g since chains remain in their initial conformations within 1σ fluctuation in such a small time window. Projections of short-time data from individual temperatures are given in SI, Figure S10.

Figure 3.

Analysis of each temperature independently. Mean values including deviations of the magnitude of first eigenvalues (a) and the participation ratio (b). Data taken from the time window between 200τ and 3 × 10⁴τ (gray area in Figure S1). The results for a shorter time up to 20τ (blue area in Figure S1) are shown in the insets.

In general, with method II, one can perform PCA on simulation trajectories at each temperature and monitor the eigenvalues and PR. Once we observe the nonmonotonic change in both quantities around T_g, further simulations at lower temperatures are not required to localize T_g.

In summary, we propose a new approach for determining the glass transition temperature from molecular dynamics simulation data with a fixed stepwise cooling protocol. The proposed data-driven protocol requires minimum input parameters and defines T_g in a robust and transferable fashion. Our analysis focuses on the information about structural fluctuations at the level of individual chains to identify the glass transition temperature and predict T_g for infinite simulation time from moderate simulation trajectories. We hypothesize that the relative distance fluctuations measured by the PCA may be directly correlated with the configurational entropy in the space of a single chain.³⁰ The method can be applied to a wide range of systems with microscopic/atomistic information. The generality of our approach could be tested with different dimensionality reduction and clustering methods. Further work in this direction is in progress.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsmacrolett.2c00749.

• Simulation details (S-I), static properties of polymer melts (S-II), additional information on the data-driven approach (S-III), variance explained ratio of pca projections (S-IV), detailed description of clustering (S-V), principal component analysis on combined chains (S-VI), interpretation of leading principal components (S-VII), projections of a chain after performing PCA independently at each temperature (S-VIII), and results with reduced number of descriptors (S-IX) (PDF)

Author Contributions

CRediT: Atreyee Banerjee data curation (lead), formal analysis (equal), investigation (equal), software (equal), validation (equal), visualization (lead), writing-original draft (lead), writing-review & editing (equal); Hsiao-Ping Hsu data curation (lead), formal analysis (equal), investigation (equal), software (equal), validation (equal), writing-review & editing (equal); Kurt Kremer conceptualization (equal), methodology (supporting), resources (lead), supervision (equal), writing-review & editing (equal); Oleksandra Kukharenko conceptualization (lead), formal analysis (equal), methodology (lead), software (supporting), supervision (lead), writing-original draft (supporting), writing-review & editing (equal).

Open access funded by Max Planck Society.

The authors declare no competing financial interest.