Supercooled Liquids

Supercooled liquids are metastable states of glass-forming materials, whose hallmark feature is the dramatic slow down of microscopic dynamics prior to the glass transition. At the mesoscale, supercooled liquids exhibit dynamical heterogeneity, where microscopic reorganization of atoms/molecules are clustered into localized mobile regions and extended “jammed” regions.

Dynamical heterogeneity, as demonstrated through an MD simulation of a binary Lennard-Jones mixture (left) and a polydisperse mixture (right). Red & green color indicates mobile regions (atoms that have moved ~1.5x their own diameter)

My current research focuses on understanding the origin of glassy dynamics and dynamical heterogeneity as well as their impact on phase-ordering phenomena, such as crystallization and grain boundary annealing.

All works are done under a joint effort with Prof. Kranthi Mandadapu. Another close collaborator on this topic is Dimitrios Fraggedakis.

A Theory for Onset Temperature in 2D Supercooled Liquids

The dynamics of glass-forming liquids slow down dramatically below the onset temperature $T_{\mathrm{o}}$. Its microscopic origin has been elusive and is often treated as a fitting parameter to distinguish the supercooled regime from the high-temperature regime. We built upon earlier work on a theory of localized excitations and constructed a theory that allows us to study how inherent states, i.e., the underlying energy-minimizing configurations of the liquid, are stable against excitation fluctuations.

Summary of our work. (Left) The relaxation timescale as a function of temperature, with illustrations delineating the supercooled regime from high temperature. (Top Right) The renormalization group (RG) equations derived from our theory. (Bottom Right) The results of RG calculations on atomistic models of 2D glass formers.

The work was inspired, in particular, by the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory of dislocation mediated melting and the work of Moshe and co-workers. The theory predicts that the supercooled liquid is a state where at intermediate timescales, it behaves like a solid filled with pure-shear localized excitations. As temperature is increased, the excitations destabilize to become a pair of “dipolar” excitations, which are able to freely proliferate in numbers. Such proliferation causes the solid to become unstable and macroscopically behave as a fluid.

By repeating the entire renormalization group (RG) analysis of the KTHNY theory, we are able to derive equations that can predict the onset temperature with reasonable agreement across various 2D glass formers. In addition, the theory is able to explain the finite-size effects of Mermin-Wagner fluctuations that were recently observed in both experiments and also Kob-Anderson-based molecular models of 2D systems, providing a small but important validation of the theory on literature data.

This work is in collaboration with Dimitrios Fraggedakis. Paper can be found in Ref.  [1]

A Theory of Localized Excitations in Supercooled Liquids

To stay mobile at low temperatures, one successful theory of glassy dynamics, known as dynamical facilitation (DF) theory, proposes that microscopic motion is driven by spatially localized excitations, which facilitate the creation and relaxation of nearby excitations in a hierarchical or self-similar manner. Although these excitations are the building blocks of DF theory, their microscopic origin remains unclear.

An illustration of the concept behind dynamical facilitation (DF) theory. Excitations emerge as a coarse-grained description of the liquid, where mobility acts as collective degrees of freedom. These excitations can then facilitate motion in neighboring areas.

On the other hand, it is well-known that glassy dynamics also proceed by hopping between inherent states, i.e., energy-minimizing configurations of the potential energy landscape (PEL). Based on this observation, we constructed a theory which associates excitations of DF theory to hopping events in the PEL.

An illustration of the concept behind our theory. Random energy wells in the PEL are reduced into an equal harmonic energy well picture in a reaction-coordinate space. We then derive the excitation energy barrier by studying barrier-crossing events in this new space.

The theory allows us to derive a formula for the excitation energy barrier, denoted as $J_{\sigma }$ as a function of elastic and structural properties of the inherent states.

$$J_{\sigma }=\underset{T\to 0}{lim}\left[6\pi G^{\mathrm{IS}}{\sigma }^2\frac{{\left({\tilde{u}}^{‡}\right)}^2{(1+{\tilde{u}}^{‡})}^2}{{(1+2{\tilde{u}}^{‡})}^4}\right]$$

where $G^{\mathrm{IS}}$ is the shear modulus averaged over all inherent states, $\sigma$ is the effective particle diameter, and ${\tilde{u}}^{‡}$ is the minimum displacement needed to break an elastic bond. Both $\sigma$ and ${\tilde{u}}^{‡}$ are computed from the peaks of the inherent-state radial distribution function (RDF).

The project has resulted in two plugins for HOOMD-blue to simulate interacting particle systems with continuous poly-dispersity, ParallelSwapMC and PolydisperseMD. In addition, a new Python package PyGlassTools was developed to perform various calculations needed by the theory.

Paper can be found in Ref.  [2]

A Theory for Crystallization vs. Vitrification

We studied how vitrification competes with crystallization using the Arrow-Potts model (a variant of the Potts model) as a coarse-grained model of supercooled liquids. Near the melting temperature, the model produces poly-crystalline microstructures but at low temperatures, fractal & ramified crystalline clusters were observed instead.

A Monte Carlo simulation of supercooled liquid crystallization using the Arrow-Potts model, near the melting temperature (left) and low temperatures (right). Colored sites indicate crystals. Yellow arrows are mobile liquid regions. Blank space indicate glassy regions.

To understand these phenomena, we used the Kolmogorov-Johnson-Mehl-Avrami theory as a framework to combine three essential theories: (1) the field theory of nucleation, (2) a random walk theory for crystal growth, and (3) the dynamical facilitation theory of glassy dynamics.

The theory relates the overall crystallization timescale ${\tau }_{\mathrm{xtl}}$ with the nucleation time ${\tau }_{\mathrm{nuc}}$ and liquid relaxation time ${\tau }_{\mathrm{liq}}$ by the following formula

$${\tau }_{\mathrm{xtl}}\sim {\left({\tau }_{\mathrm{nuc}}{\left({\tau }_{\mathrm{liq}}\right)}^{\frac{\alpha d}{2}}\right)}^{\frac{1}{\frac{\alpha d}{2}+1}}$$

where $\alpha$ is a universal exponent characterizing how mobile regions spread across the dynamically heterogeneous liquid.

Paper can be found in Ref.  [3]

References

[1]

D. Fraggedakis, Hasyim, M. R., and K. K. Mandadapu, Inherent-State Melting and the Onset of Glassy Dynamics in Two-Dimensional Supercooled Liquids, arXiv:2204.07528 (2022).

[2]

Hasyim, M. R. and K. K. Mandadapu, A Theory of Localized Excitations in Supercooled Liquids, J. Chem. Phys. 155, 044504 (2021).

[3]

Hasyim, M. R. and K. K. Mandadapu, Theory of Crystallization Versus Vitrification, arXiv:2007.14968 (2020).