A structural approach to relaxation in glassy liquids

I Methods

We study a 10,000-particle Kob-Andersen model, a 80:20 binary LJ mixture Kob and Andersen (1994) with parameters: $\sigma_{AA}=1.0$, $\sigma_{AB}=0.8$, $\sigma_{BB}=0.88$, $\epsilon_{AA}=1.0$, $\epsilon_{AB}=1.5$, $\epsilon_{BB}=0.5$. Time is measured in units of $\tau=\sqrt{\epsilon_{AA}/\sigma_{AA}^{2}}$ and the Boltzmann constant is $k_{B}=1$. We cut off the LJ potential at $2.5\sigma_{AA}$ and smooth the potential so that force varies continuously. This mixture has been characterized extensively. In particular, we compare our predictions to the measurements of the onset temperature in Keys et al. Keys et al. (2011). Simulations were done using LAMMPS Plimpton (1995) in an NVT ensemble with a Nosé-Hoover thermostat and a timestep of $0.0025\tau$. We output states every $\tau$ and quench them to their nearest inherent structure using a combination of conjugate gradient and FIRE algorithms. Throughout this study we use inherent structure positions. However, qualitatively similar results can be obtained using time averaged positions. We study this system over the temperatures and number densities listed in Table 1.

I.2 Identifying rearrangements.

We adapt a method first proposed by Candelier et al.Candelier et al. (2010); Smessaert and Rottler (2013). A timescale $t_{R}=10\tau$ is chosen to be commensurate with the amount of time the system takes to complete a rearrangement. Then two time intervals are defined as $A=[t-t_{R}/2,t]$ and $B=[t,t+t_{R}/2]$. An indicator function can then be written as,

where $\langle\rangle_{A}$ and $\langle\rangle_{B}$ are averages over the intervals $A$ and $B$ respectively. $p_{\text{hop}}$ is large when the mean position of a particle changes appreciably. Otherwise, it is similar in magnitude to the variance in particle positions due to noise from the inherent structure calculation.

To find rearrangements we restrict our attention to events in which $p_{\text{hop}}$ exceeds a threshold of $0.05$, that is large compared to the scale of fluctuations in particle positions but small compared to the typical value of $p_{\text{hop}}$ during a rearrangement. As discussed in the supplementary material, we define rearrangements to be those events with $p_{\text{hop}}^{*}>p_{c}=0.2$. Changing this cutoff affects the results only quantitatively and manifests itself primarily as a shift in the energy scale, $\Delta E$ that is approximately logarithmic in the cutoff. This agrees with the observations of Keys et al. Keys et al. (2011) who saw a similar logarithmic shift in the energy scale governing rearrangements with the size of the rearrangements.

Note that rearrangements defined using $p_{\text{hop}}$ result in particle displacements that follow a distribution that depends on the cutoff $p_{c}$ used. This $p_{c}$ dependence needs to be addressed when comparing the probability of rearrangement to the overlap function and its derivative, which are defined in terms of a length scale $a$. To do this, we multiply $P_{R}$ by a temperature-independent constant $c_{a}$, namely the fraction of rearrangements that displace particles by more than $a$.

I.3 Computing softness.

We have made two improvements that greatly increased the prediction accuracy for rearrangements compared to Ref. Cubuk et al. (2015). First, we identified rearrangements more carefully, as detailed above. Second, we defined our training sets more carefully. Each training set contains 6000 particles that rearrange in the next time step, each labeled with $r_{i}=1$, as well as 6000 particles that have not rearranged for a time $\tau_{\alpha}$ before the structure was calculated, each labeled with $r_{i}=0$. These particles were chosen randomly from the set of all particles satisfying these conditions from MD simulations at a low temperature. Then, a training set of N particles can be written as $\left\{\left(\bm{F}_{1},r_{1}\right),...,\left(\bm{F}_{N},r_{N}\right)\right\}$, where $\bm{F}_{i}$ = $\left\{F^{1}_{i},...,F^{M}_{i}\right\}$ are the the $M$ structure functions that describe the local neighborhood of particle $i$ Cubuk et al. (2015). We then use an SVM to find the hyperplane $\bm{w}\cdot\bm{F}-b=0$ that separates the points with $r_{i}=1$ from those with $r_{i}=0$. This hyperplane is used on the rest of the data to reach the results reported.

The SVM is trained, that is, the hyperplane is constructed, on the binary variable $r$ using the LIBSVM package Chang and Lin (2011). It is not possible to find a hyperplane that perfectly separates the two different classes. We use a penalty parameter $C$ and find the optimal hyperplane equation by minimizing

with the constraint $y_{i}\cdot\left(\bm{w}^{T}\cdot\bm{F}_{i}+b\right)\geq 1-\xi_{i}$ and $\xi_{i}\geq 0$. The $C$ parameter was chosen through cross-validation Cubuk et al. (2015). The hyperplane obtained from this training can be used to classify a new particle neighborhood, $\bm{F}_{n}$, as soft or hard. $\bm{F}_{n}$ is soft if $\bm{w}\cdot\bm{F}_{n}-b>0$, and hard otherwise. The continuous variable softness is defined by $S_{n}=\bm{w}\cdot\bm{F}_{n}-b$. Training a neural network to classify soft and hard particles, and using the output from the hidden layer of the neural network as softness, yields similar results. Here we use only the SVM approach.

References