Origin of ultrastability in vapor-deposited glasses

TL;DR

This study uses molecular simulations to understand how vapor deposition creates ultrastable glasses, revealing that surface relaxation is key and providing a quantitative relation for deposition efficiency.

Contribution

It introduces a scaling relation linking vapor deposition efficiency to surface relaxation, clarifying the mechanism behind ultrastability in vapor-deposited glasses.

Findings

Surface relaxation is crucial for ultrastability.

A scaling relation quantifies vapor deposition efficiency.

Vapor deposition mimics bulk relaxation processes.

Abstract

Glass films created by vapor-depositing molecules onto a substrate can exhibit properties similar to those of ordinary glasses aged for thousands of years. It is believed that enhanced surface mobility is the mechanism that allows vapor deposition to create such exceptional glasses, but it is unclear how this effect is related to the final state of the film. Here we use molecular dynamics simulations to model vapor deposition and an efficient Monte Carlo algorithm to determine the deposition rate needed to create ultra-stable glassy films. We obtain a scaling relation that quantitatively captures the efficiency gain of vapor deposition over bulk annealing, and demonstrates that surface relaxation plays the same role in the formation of vapor-deposited glasses as bulk relaxation does in ordinary glass formation.

Figures

Equations

$$U({\gamma }_c,T)/U_e(T)=\mathcal{C}(x_c),$$

$$U({\gamma }_d,T)/U_e(T)=\mathcal{D}(x_d),$$

Full text

Compared to their liquid-cooled counterparts, vapor-deposited glasses often have a higher density Dalal2012a , a higher kinetic stability Rodriguez2016 ; Swallen2007 ; Leon2010 , and a lower heat capacity Kearns2010 . This makes them promising materials for a wide range of applications, such as drug delivery Hancock1997 , protective coatings Yu2013 ; Chu2012 , and lithography Neuber2014 . Identifying the microscopic process that gives rise to these properties is thus crucial to designing novel amorphous materials today2016 . Vapor deposition indeed does not systematically result in glasses with improved characteristics. It is observed that the substrate ought to be held at a specific temperature (around 85% of the glass transition temperature $T_{g}$ of the liquid Swallen2007 ) and that the deposition rate must be sufficiently slow Chua2015 to get optimal films. A microscopic explanation for the optimality of $0.85T_{g}$, and an estimate of what is a “sufficiently slow” deposition rate are, however, still lacking. Moreover, while simulations and experiments have shown that vapor-deposited glasses may lie lower in the potential energy landscape than liquid-cooled glasses Swallen2007 ; Singh2011 ; Lyubimov2013 ; Lyubimov2015 ; Reid2016 ; Chua2015 ; starr2017 , and sometimes have the same structure as glasses of a comparable energy Reid2016 , it is not known whether vapor deposition can provide truly equilibrium configurations, especially below $T_{g}$.

Here we provide a quantitative test of the role of surface mobility in the creation of vapor deposited glasses. More specifically, we answer two key questions. (i) How much more efficient is vapor deposition than standard cooling in creating a glass or, more precisely, given a substrate temperature and a deposition rate, what is the effective cooling rate that would produce similar configurations? (ii) What is the deposition rate needed to produce fully equilibrated configurations? Answering these questions is a challenging program that requires characterizing equilibrated films at temperatures sufficiently low for a large difference between surface and bulk relaxation to have developed, as well as measuring bulk and surface dynamics in a same material, over the same temperature range, and under the same thermodynamic conditions. We overcome these problems by using, on a properly chosen polydisperse Lennard-Jones model, a swap Monte Carlo algorithm that efficiently samples the energy landscape at very low temperatures, speeding up equilibration by several orders of magnitude over standard molecular dynamics Berthier2016 ; Ninarello2017 . This allows us to compare free standing equilibrated films with those grown on an equilibrated substrate using an algorithm that closely mimics experimental vapor deposition, and to independently determine the low temperature equilibrium energy of the film with no extrapolation.

We simulate a polydisperse mixture of Lennard-Jones particles with interaction potential, $V(r_{ij})=\epsilon\left[(\sigma_{ij}/r_{ij})^{12}-(\sigma_{ij}/r_{ij})^{6}\right]$, which is truncated and shifted to zero at $2.5\sigma_{ij}$. The size parameter $\sigma_{ij}$ is given by a non-additive mixing rule $\sigma_{ij}=0.5(\sigma_{i}+\sigma_{j})(1-\Delta|\sigma_{i}-\sigma_{j}|)$, where $\Delta=0.2$. The mixing parameter $\Delta$ is chosen to avoid separation of the particles into small and large components at low pressures and temperatures Ninarello2017 . The particle size parameter $\sigma$ is chosen from the probability distribution $P(\sigma)=A/\sigma^{3}$ for $0.73\leq\sigma\leq 1.62$ and zero otherwise Ninarello2017 . For each temperature we simulate five different realizations of the size distribution. Each particle has the same mass $m$, the unit of energy is $\epsilon$, and the unit of length is the average of the particle diameters $\sigma_{0}$. Our unit of time is $\sqrt{\sigma_{0}^{2}m/\epsilon}$. The freestanding films have box lengths of $11\sigma_{0}$ in the periodic $x$ and $y$ directions. The box length in the $z$ direction is 120$\sigma_{0}$. We simulate systems of $N=4000$ particles, which results in films of around 30$\sigma_{0}$ along the $z$ direction.

To create equilibrium free standing films we used a Monte Carlo swap algorithm. The algorithm consists of two Monte Carlo moves, a standard attempted displacement move and an attempted swap move. For the attempted displacement move we use trial positions within a cube of side $d$, where $d$ is adjusted for each temperature so that the acceptance rate lies between 0.3 to 0.35. For the attempted swap move we consider exchanging the size of two particles chosen at random. The move type is chosen at random, with 20% of the moves being attempted swaps.

For the vapor-deposited films we first create an equilibrated substrate of the same system with $N/2=2000$ particles in a simulation box of the same dimensions as for the free-standing film. We then introduce $N/2$ particles with $x$ and $y$ components of the velocity randomly chosen within the square of side 0.02. The $z$ plane from which particles are introduced moves at a constant velocity in order to remain about $40\sigma_{0}$ above the surface of the substrate. The velocity of the deposited particles in the $z$ direction has a magnitude corresponding to a kinetic temperature of $T=0.1$ (the onset temperature of slow dynamics), directed towards the substrate. The total momentum of the whole system is set to zero at every time step to reduce the drift of the center of mass. The substrate is simulated using a Nosé-Hoover constant $NVT$ algorithm, and the vapor-deposited particles are simulated using a constant $NVE$ algorithm. After all the particles are deposited onto the substrate, the simulation is run for $t=100$ and the following $t=1000$ is used to calculate the average energy for that deposition rate. In all cases the energy changes little over the averaging window.

Figure 1 compares the temperature evolution of the average potential energy, $U$, measured in the exact same film geometry and at the same temperature $T$ for three different protocols. Vapor-deposited films at deposition rate $\gamma_{d}=dz/dt$ and substrate temperature $T$ have energy $U(\gamma_{d},T)$ (triangles), ordinarily-cooled films at cooling rate $\gamma_{c}=dT/dt$ have energy $U(\gamma_{c},T)$ (colored lines), and films equilibrated using swap Monte Carlo have energy $U_{e}(T)$ (black line). In agreement with prior experiments and simulations Lyubimov2015 ; Reid2016 ; starr2017 , the results show that vapor deposition equilibrates our model more efficiently than liquid cooling for a comparable preparation time, especially at low temperatures. The average energy obtained at slower cooling rates indeed deviates from equilibrium below $T\approx 0.085$, while the films grown by vapor deposition have energies much closer to equilibrium. For our slowest deposition rate, the average energy remains equal to the equilibrium energy for all but the lowest temperature studied, $T=0.075$.

We are now in the position to answer the key questions raised above. For liquid-cooled films, the competition between the bulk relaxation time $\tau_{\alpha}^{b}$ and the cooling rate $\gamma_{c}$ controls the distance to equilibrium. This may be captured by the scaling relation

[TABLE]

where $x_{c}=\gamma_{c}\tau_{\alpha}^{b}(T)/T$ represents the (adimensional) ratio between cooling and bulk relaxation timescales. To establish that vapor-deposited films are controlled by the competition between the equilibrium relaxation time at the film surface $\tau_{\alpha}^{s}$ and the deposition rate $\gamma_{d}$ we seek a scaling relation of the same form,

[TABLE]

where $x_{d}=\gamma_{d}\tau_{\alpha}^{s}(T)/\sigma_{0}$ is the (adimensional) ratio between deposition and surface relaxation timescales and $\sigma_{0}$ is the average particle size. By construction, the scaling functions $\mathcal{C}(x)$ and $\mathcal{D}(x)$ should both converge to unity when $x\to 0$.

To test these scalings, we measure the temperature dependence of the equilibrium relaxation times at the film surface and in its core. To this end, we consider equilibrium films obtained from the swap Monte Carlo algorithm, that we then simulate using ordinary molecular dynamics. The measurements described in the following are thus generic for all films and are not specific to vapor-deposited ones.

Relaxation times are extracted from the decay of the self-intermediate scattering function $F_{s}^{l}(q;t)=(1/N_{l})\sum_{n=1}^{N_{l}}e^{i\mathbf{q}\cdot[\mathbf{r}_{n}(t)-\mathbf{r}_{n}(0)]}$, where the sum is restricted to the $N_{l}$ particles with positions $\mathbf{r}$ either in the bulk, $l=b$, or at the surface, $l=s$, of the film at time $t=0$. The chosen wavevector, $|\mathbf{q}|=7.1$, coincides with the location of the first peak of the static structure factor, and is taken parallel to the surface. The bulk region is defined to be $5\sigma_{0}$ thick at the core of a film approximately $30\sigma_{0}$ thick, while the two surface regions extend $1.5\sigma_{0}$ from the film edge, as illustrated in the inset of Fig. 2.

In Fig. 2(a), $F_{s}^{b}(\mathbf{q};t)$ appears typical of standard glassy dynamics Berthier2011 . A plateau emerges for $T\leq 0.1$, and both its length and height increase with decreasing temperature. Particles are thus localized over an increasingly longer timescale upon increased cooling. The $F_{s}^{s}(\mathbf{q};t)$ in Fig. 2(b) are markedly different. First, there is no distinct plateau at any temperature. Second, a fast initial decay down to $\sim 0.5$ is followed by a slower relaxation. This suggests that surface particles perform vibrational motion with a larger amplitude than bulk ones. More importantly, the long-time dynamics at the surface is much faster than in the core.

Defining the relaxation times as $F_{s}^{l}(q;\tau_{\alpha}^{l})=0.2$ gives a measure of the time needed for a particle to move a distance comparable to its diameter. Empirically, $\tau_{\alpha}^{b}$ is well-described over the whole temperature range by a Vogel-Fulcher form $\tau_{\alpha}^{b}=\tau_{0}^{b}e^{E_{b}/(T-T_{0})}$, with $T_{0}=0.0612\pm 0.001$ (Fig. 3), but other fitting forms work equally well. The surface relaxation time $\tau_{\alpha}^{s}$ also increases with decreasing temperature, although much less than $\tau_{\alpha}^{b}$, and a Vogel-Fulcher form does not fit it well. Instead, $\tau_{\alpha}^{s}$ is better described by an Arrhenius form, $\tau_{\alpha}^{s}=\tau_{0}^{s}e^{E_{s}/T}$, at low temperatures, which is reminiscent of the behavior of surface diffusion observed experimentally Yang2010 ; Zhu2011 ; Daley2012 . As a result, the ratio $\tau_{\alpha}^{b}/\tau_{\alpha}^{s}$ increases dramatically upon supercooling, growing from $\sim 4$ near the onset of localization to $\sim 900$ at the lowest temperature studied (Fig. 4). Under the assumption (proved shortly below) that the surface relaxation time controls the thermalization of the vapor-deposited glass, then the much faster surface relaxation is qualitatively consistent with the enhanced thermalization efficiency of vapor deposition observed in Fig. 1.

To visualize the emergence of a mobile surface layer, we color particles according to their displacement $|\mathbf{r}_{n}(t)-\mathbf{r}_{n}(0)|$ after $t=20\tau_{\alpha}^{s}$ for the highest and lowest temperature examined as insets to Fig. 4. At all temperatures the surface is more mobile than the core, but qualitative differences can be observed. At the highest temperature mobile particles are found throughout the film, while at the lowest temperature only a small layer of mobile particles is observed. This layer is barely thicker than $\sigma_{0}$ in the inset of Fig. 4. (Fitting an exponential decay to the surface relaxation time gradient, see Ref. starr2017 , gives a thickness of order 2-3 $\sigma_{0}$ with a weak temperature dependence.) Such a decoupling between bulk and surface dynamics has been experimentally documented zara2017 , but was not directly connected to the thermalization of ultrastable glasses before.

The mobile surface layer is exploited by the vapor-deposition process to speedup the thermalization of glassy films, as we can directly demonstrate. The combination in Fig. 5 of all our energy measurements in liquid-cooled and vapor-deposited films shows that Eqs. (1, 2) collapse the numerical results very well. Moreover, the simulations indicate that the scaling functions $\mathcal{C}(x)$ and $\mathcal{D}(x)$ are nearly identical. Note that no adjustable parameter is used for these scalings, which combine independent numerical measurements.

The excellent data collapse in Fig. 5 indicates that the surface relaxation time and deposition rate determine the distance to equilibrium for vapor-deposited films in the exact same way that bulk relaxation time and cooling rate control the distance from equilibrium for liquid-cooled films. This result suggests that one can convert the deposition rate of a film into an effective cooling rate as $\gamma_{c}^{\rm eff}=\gamma_{d}(\tau_{\alpha}^{s}/\tau_{\alpha}^{b})(T/\sigma_{0})$. Quantitatively, we can fit the scaled data to an empirical power law, $\mathcal{P}(x)=ax^{\nu}+b$ (with $\nu=0.12\pm 0.03$, solid line in Fig. 5) Reid2016 . Solving for $\mathcal{P}(x)=1$ gives the maximal rate, for both preparation processes, at which equilibrium films can be prepared. Our simulations thus provide a simple quantitative criterion, $x_{d}=\gamma_{d}\tau_{\alpha}^{s}/\sigma_{0}\leq 0.005$, to create equilibrium vapor-deposited films.

Consistency with experiments is illustrated by considering the case of indomethacin Swallen2007 ; Zhu2011 , for which we use $\sigma_{0}=1$ nm Zhu2011 as the length unit and $10^{-12}$ s as the time unit Xiang2013 . We further approximate the structural relaxation times using $\tau_{\alpha}^{l}=(q^{2}D^{l})^{-1}$ at $q=2\pi/\sigma_{0}$ with the diffusion coefficients $D$ obtained in Ref. Zhu2011 . For a cooling rate of $40$ K/min, Swallen et al. determined that $T_{g}=315$ K Swallen2007 . We thus estimate that $\gamma_{c}\tau_{\alpha}^{b}(T_{g})/T_{g}\approx 0.0054$, which is in close agreement with the above criterion. Our proposed scaling form, thus captures well the cooling rate dependence of the glass transition. More interestingly, we can compute the lowest temperature at which one can obtain an equilibrium film by vapor deposition for a given deposition rate. For $\gamma_{d}=0.2$ nm/s Dalal2013 , we estimate that the smallest surface diffusion coefficient at which an equilibrium film can be obtained should be $D^{s}\approx 1.0\times 10^{-18}$ m2/s. By extrapolating the surface diffusion coefficients in Ref. (Dalal2013, , Fig. 3), we find that an equilibrium vapor-deposited film should be accessible down to $264~{}{\rm K}\approx 0.84T_{g}$, which is close to the experimental estimate Dalal2013 . Note that for simulations our results imply that an efficiency gain of at most 2-3 orders of magnitude can be expected from vapor deposition over standard bulk annealing, which appears consistent with a recent independent estimate starr2017 . In the regime accessible to experiments, by contrast, that gain can reach eight orders of magnitude.

Our work directly and quantitatively demonstrates how enhanced surface diffusion, as quantified by $\tau_{\alpha}^{s}$, is responsible for the ultrastability of vapor-deposited films. Additional considerations may be needed to account for the full scope of experimental observations. First, additional studies are needed to connect the surface relaxation time measured in this study with the surface mobility inferred from experiments Rodriguez2016 ; Swallen2007 ; Kearns2010 ; Zhu2011 ; Daley2012 ; zara2017 ; Malshe2011 . Second, the shape and chemical nature of vapor-deposited molecules can result in preferential orientation within the vapor-deposited film Lyubimov2015 ; Dalal2012a ; Yokoyama2009 ; Yokoyama2010 ; Dalal2015 ; Jiang2016 , while such alignment bias is not expected for ordinary liquid-cooled films. Although molecular alignment can be used to tailor glassy properties Dalal2015 ; Jiang2016 , it also inherently leads to vapor-deposited films that differ in structure from their liquid-cooled counterpart. For the simple, spherical particles studied in this work, however, we find no evidence in the pair-correlation function (not shown) or the density profile that vapor deposition produces different structures or particle segregation profiles than what is seen in liquid cooling. Our results are thus consistent with those of Reid et al. Reid2016 for a binary Lennard-Jones system. Overall, this work is a first step to obtain a more quantitative understanding of the creation of vapor deposited supercooled liquids and glasses, and additional work is needed to understand the role of the substrate, molecular shape, and other factors in this process.