Introduction

As a liquid continues to cool below its melting temperature, it will eventually harden through one of two dramatically different processes. The liquid will either form a glass or crystallize. Crystallization is initiated through the process of nucleation, where a small crystallite of a more thermodynamically stable solid phase spontaneously forms. Once this crystallite is of sufficient size to overcome a free energy barrier to growth, the change of phase to the solid will proceed. Alternatively, the liquid might never manage to overcome the free energy barrier to nucleation, but instead undergo a tremendous slow-down in its dynamics that eventually renders it incapable of flowing on an observable timescale.

Both of these transitions, the glass transition and nucleation, have been intensively studied and are important in our understanding of material formation, properties and fabrication. Traditionally, these two processes have been studied as separate phenomena. Recent breakthroughs, however, point towards an intimate connection between the two. It is the goal of my research plan to develop a unified microscopic and thermodynamic understanding of these two processes, connecting glassy dynamics and crystal nucleation, and more generally to quantify the effects of various phenomena in supercooled liquids on nucleation.

The ubiquity of the competition between the glass transition and crystallization, from the physical properties of biological materials to the manufacture of ceramics and optical lenses, underscores the importance of deepening our understanding of the connection between the two phenomena in both bulk liquids and clusters. I will use computer simulation as my primary tool of investigation. It provides the necessary microscopic detail in space and time to capture both the nucleation process and the transport mechanisms inherent in glassy dynamics.

Background

The intriguing nature of the glass transition was pointed out by Kauzmann in 1948 [1]: If one extrapolates the behaviour of a liquid sufficiently below melting, the curve of entropy versus temperature for the liquid crosses that of the solid. In other words, the entropy of the supercooled liquid will be lower than that of the crystal. What prevents this from being observed is the glass transition: the dynamics of the cooling liquid become too slow to ever reach the entropy crossover point in equilibrium. Since there is no thermodynamic transition associated with the glass transition, Kauzmann was left with a paradox, namely that a thermodynamic catastrophe (i.e. a liquid with a lower entropy than that of its underlying crystal) is avoided through a seemingly purely kinetic phenomenon. Combined with the ubiquity of the glass transition, this paradox makes the glass transition an important area of scientific inquiry. Accounting for the tremendous dynamical slowdown in the liquid over a small temperature range near the glass transition is particularly challenging, since there are few if any obvious coincident structural or thermodynamic changes.

Several ideas, concepts and theories have arisen to account for or to characterize the slowdown, or to resolve the paradox. I have experience with many of these including the possibility that the liquid loses thermodynamic stability before the entropy crossover point and that the liquid undergoes an ideal glass transition to an ideal amorphous solid. I have also explored the theory of Adam and Gibbs [2] that relates dynamics to configurational entropy, a thermodynamic quantity, and the inherent structure or potential energy landscape (PEL) formalism, that maps the dynamics to transitions between basins of attraction around local minima in the potential energy of the system. The PEL formalism also allows for the calculation of the configurational entropy. My work has also addressed the classification of glassformers using a spectrum ranging from "fragile" (e.g. orthoterphenyl) to "strong" (e.g. silica), where fragile glassformers undergo a very dramatic dynamic slowdown near the glass transition temperature and are more likely to crystallize, while strong ones exhibit gentler Arrhenius dynamics and are more likely to remain liquids. In my work I have also applied Chandler's theory of dynamic facilitation [3], a coarse-grained description of the dynamics of the system that separates dynamics and thermodynamics.

Another interesting phenomenon associated with the glass transition is dynamical heterogeneity: the coarsening in the size of domains of particles having similar dynamics with decreasing temperature [4]. Recently, researchers have shown that as a liquid is cooled toward the glass transition, particle dynamics become increasingly spatially correlated, with particles of similar mobility tending to form "clumps".

As an example of the interdisciplinary crossover of ideas, the PEL has been used extensively by D. Wales [5] and others in describing clusters of atoms. For a small enough cluster, it becomes computationally feasible to enumerate all the states of the system and the connections between them. Liquid clusters are interesting for several reasons. They have a natural heterogeneity in the surface between liquid and vapour. They are small systems in the statistical sense, and therefore fluctuations away from the most probable state become important. The ground states are not bound to be crystalline, as there is no need for translational symmetry. They often have several structures of similar free energy that compete for the ground state. Studying clusters has also been used to determine nucleation rates in bulk systems, at least for the liquid-vapour transition [6]. The many technological uses of clusters on the nanoscale [7], as well as their influence in the atmosphere and environment [8], make understanding their phase behaviour and dynamics urgent.

On the nucleation side of things, Classical Nucleation Theory (CNT) [9] is a well established theory used to describe nucleation in fluids. Only recently, however, have computer simulation methods been developed to directly test CNT and its predictions [10]. Although it is important to test the theory in simpler systems, it is also interesting to check what the effects are of various phenomena on the theory and on nucleation. Some of the cases studied include nucleation near a metastable liquid-gas critical point in a model for globular proteins [11], near a wall [12], near a liquid-liquid phase separation in a mixture [13], near an isostructural phase transition [14] and in the deeply supercooled regime, where the barrier to nucleation might disappear [15]. Furthermore, the connection between crystallinity (i.e. nucleation or crystal growth or crystal-like structures in liquids) and dynamical heterogeneity has only just begun to be explored [16,17].

Through my doctoral and postdoctoral studies, I have extensive experience in simulating model liquids. In particular, I have studied polyamorphism (the existence of multiple amorphous forms of a material), the liquid's approach to the glass transition (in particular the thermodynamic underpinnings of the dynamics of glassforming liquids), the nonequilibrium dynamics associated with glassy aging, the calculation of phase diagrams, colloidal gelation, the calculation of rates of and free energy barriers to crystal nucleation, and nucleation in small clusters. I have therefore studied purely liquid phenomena, as well as crystal nucleation. I have used various free energy and theoretical techniques, as well as different computational approaches. In the case of liquid silica, I have been able to unify the understanding of the seemingly unrelated properties of polyamorphism and strong glass-forming ability [18]. Now I will apply this expertise to the objectives of this research plan.

Objectives

I will build upon my background in both liquid state phenomena and nucleation with the goal of developing a deeper understanding of the connections between the two. This is an emerging direction that a few researchers in the two communities are already taking. In the short-term, I will address specific issues like the effects of dynamical and structural heterogeneities on nucleation, the connection between glassforming fragility and nucleation, and how to use atomic clusters as a tool for understanding bulk behaviour. The results of these investigations will provide building blocks for the construction of the long-term goal of forming a broader view of the supercooled liquid state as a whole, including nucleation. Also for the longer-term is the application of the various theoretical and computational techniques to systems of a biological nature[20], e.g. viral capsid nucleation and understanding growth of artificial bone.

Approach

The research plan relies heavily on classical molecular dynamics (MD) and Monte Carlo (MC) computer simulation. I will use constrained (umbrella sampling) MC in conjunction with parallel tempering methods in determining free energy profiles with respect to one or more order parameters that track the nucleation process, while employing MD to calculate other liquid properties.

The inherent structure formalism has been very useful in the study of liquids, both in terms of linking thermodynamics with dynamics (as I and others have done) and in constructing master equation descriptions [19]. I will continue to use it, this time in combination with nucleation order parameters (like bond orientational order parameters or the size of the largest crystalline embryo in the system) to try to find a coarse-grained master equation description of the supercooled state that includes the nucleation process.

Other computational and theoretical tools that will prove useful for the research plan are the propensity (the mobility of a liquid particle averaged over all momenta for a given atomic configuration), transition path sampling (an indirect way of getting nucleation rates without a good order parameter), and the extended modified liquid drop dynamical nucleation theory (EMLD-DNT) that generates nucleation rates and barriers by first considering small contained clusters of the material [6].

Research Areas

1) Dynamical heterogeneity and nucleation

Within the last decade, dynamical heterogeneity, i.e. strong spatial correlation in dynamics, has become acknowledged as an important feature of the glass transition [4]. Recently, dynamical heterogeneity has also been shown to affect the dynamics of crystal growth in theoretical models [16]. I will address questions that pertain to its effect on nucleation, such as the following. What effect do these dynamical heterogeneities have on the formation of precritical nuclei, e.g. are precritical nuclei more or less likely to form within a slow or fast domain? Is the length scale of the dynamical heterogeneity important in considering the formation of critical nuclei? Also, it has been recently shown in a 2-dimensional model with tunable global crystallinity that domains with high crystalline order are slow, leading to dynamical heterogeneity [17]. Does this generalize to more realistic models of simple or network-forming liquids? If so, can dynamical heterogeneity arise without crystalline ordering? In 3-d liquids, is this crystalline order the same as the eventual bulk, or is it one that cannot tile space, e.g. icosahedral order? I will quantify the correlation between local structure (using variations on the usual spherical harmonic bond orientation order parameter, for example) and dynamical heterogeneity.

Another interesting aspect of liquid dynamics is the question of relaxation timescales. Typically, liquids equilibrate on a timescale given by the so-called alpha-relaxation time, roughly the time it takes for liquid particles to break the "cage" formed around them by neighbouring particles. However, (and I came across this in my work on nucleation in silica), there could be a separate, longer timescale associated with the evolution of the distribution of precritical nuclei. I will test the implied prediction that liquids with different thermal histories will have different nucleation times by calculating (through direct simulation) nucleation rates as a function of initial and final temperatures, and in so doing determine at what temperatures and timescales thermal history becomes important. I will also use constrained simulations in order to prepare the system at a given temperature but with a cluster size distribution different from the equilibrium one, and then calculate how the distribution approaches equilibrium in time. In general, MD simulation is useful in studying nucleation because it provides detail on timescales and lengthscales difficult to probe otherwise. In this case, it will allow me to uncover the microscopic details of the time evolution of the cluster distribution, in particular how it is connected to dynamical heterogeneity. An additional tool recently developed within simulations that will help me determine this connection is the propensity: the mobility of a liquid particle averaged over all momenta for a given atomic configuration, i.e. the mobility in an isoconfigurational ensemble [21]. The originators of this tool have already used it to show that dynamical heterogeneity has a static, configurational origin.

2) Structural heterogeneity and nucleation

Studies have already shown that a metastable liquid-gas critical point in globular protein solutions affects nucleation [11]. Evidence for a metastable liquid-liquid critical point, in the form of structural heterogeneities in the liquid above the temperature of the would-be critical point, has been demonstrated in my own work on model liquid silica [22], and by others in other network-forming liquids like water and silicon. I will measure the effect these heterogeneities have on nucleation by looking at changes to the free energy profile (along a reaction coordinate to nucleation) as the liquid state point probes deeper into the regions of liquid phase space where the heterogeneities are stronger. I will then determine how these effects can be exaggerated or suppressed, e.g. by preparing the liquid in some nonequilibrium state, perhaps by rapid compression (to manipulate the degree of heterogeneity), followed by rapid cooling (to slow down liquid dynamics).

3) Free energy barrier to nucleation in strong and fragile glassformers

I will differentiate nucleation in fragile and strong glassformers. The interplay between nucleation and the glass transition could be very different for fragile and strong glassformers [23]. In the fragile case, the configurational entropy appears to drop steeply to zero at finite temperature, driving the tremendous superarrhenius slowdown of the dynamics [24]. However, it is often difficult to probe these low temperatures because crystallization occurs quickly. In this case, it could be that at sufficiently low temperature, the barrier to nucleation vanishes. I intend therefore to establish the connection between fragility and nucleation. As a starting point, I already have the tools in place to measure the barriers at low temperatures and to compare them to what the configurational entropy and dynamics are doing. This work will, for the first time, look directly at the competition between the formation of the so-called ideal glass (zero configurational entropy), and barrierless nucleation (the limit of liquid stability). On the other hand, strong liquids like silica resolve the Kauzmann paradox by having a configurational entropy that inflects in such a way as to remain positive at zero temperature [25] (circumventing the possibility of an ideal glass), allowing these systems to have Arrhenius dynamics. I am almost in position to measure the nucleation barriers in the strong regime for silica, and it would be a significant result to show how the behaviour of the barrier in this case differs from the fragile case.

4) Free energy and potential energy landscapes of freezing clusters

Having the expertise to study the nucleation pathway using one or simultaneously two order parameters (such as embryo size and crystallinity) [26], I will continue to systematically study differences in the nucleation process in different types of clusters. I will also build on the work by others that attempts to develop a detailed unified understanding of all dynamical properties of a cluster. For example, Berry has been developing a master equation description (based on the potential energy landscapes of small clusters) of the dynamics and crystallization in very small clusters [27]. Recently, he has proposed a way of extending his ideas to larger systems. I intend to combine the order parameter and potential energy surface approaches in order to obtain a tractable course-grained description.

I will also apply the EMLD-DNT theory mentioned above to deriving the behaviour of the bulk liquid by looking at the behaviour of clusters. The intriguing part is that some structures to which the clusters freeze are not compatible with the bulk, and so the resulting bulk nucleation rates and barrier heights may be connected with some frustrated phase that has more to do with glassy dynamics than it does with crystal nucleation. To gain experience with the EMLD-DNT theory, I am currently working with it on a simpler problem, namely that of bubble nucleation in stretched liquids.

5) Nucleation of viral capsids

I will apply the machinery built up for studying nucleation to the problem of viral capsid formation. A classical-type theory for viral capsids from constituent subunits, or capsomeres, has been recently introduced [28]. Using a suitable, simplified model of capsomere interactions, I will be able to quantitatively test the predictions of the theory. From this starting point, the project can proceed by incorporating details into the model that are relevant to the problem such as internal structural change within the capsomeres that may be responsible for the icosahedral symmetry of the capsids [29]. Further, the project can continue by looking at various physical phenomena associated with viruses, or move onto self-assembly problems in other biological systems. I have been in contact with a neurologist, Dr. Daniel Mendonca, currently a Clinical Fellow in the Movement Disorders Program at the London Health Sciences Centre, London, Ontario, who is interested in what simulations could reveal about misfolded proteins implicated in Alzheimer's disease [30]. We will be exploring possibilities for collaboration.

6) Colloidal self-Assembly controlled by external fields (with Dr. Anand Yethiraj)

In addition to their widespread industrial uses, colloids are of great scientific interest because colloidal particles can be thought of as model atoms, yet they are large enough to directly visualize with light microscopes. Thus, colloidal systems are used to experimentally study liquid state phenomena like crystal nucleation and vitrification on a level of detail impossible to obtain for atomic systems. Journals such as Nature, Science and Physical Review Letters avidly publish works in which colloids are used to model atomic systems. Furthermore, there are phenomena unique to colloids, such as gelation.

References

1. Kauzmann, W. Chem. Rev. 43, 218 (1948).

2. G. Adam and J.H. Gibbs, J. Chem. Phys. 43, 139-146 (1965).

3. J.P. Garrahan and D. Chandler, Proc. Natl. Acad. Sci. USA 100, 9710-9714 (2003).

4. W. Kob, C. Donati, S.J. Plimpton, P.H. Poole, and S.C. Glotzer, Phys. Rev. Lett. 79, 2827-2830 (1997).

5. D. J. Wales, Energy Landscapes, With Applications to Clusters, Biomolecules and Glasses (Cambridge University Press, Cambridge, 2003).

6. D. Reguera and H. Reiss, J. Phys. Chem. B 108, 19831 (2004); D. Reguera and H. Reiss, Phys. Rev. Lett. 93, 165701 (2004); R. Zandi, D. Reguera, and H. Reiss, J. Phys. Chem. B, ASAP Article (2006).

7. http://www.innovation.gc.ca/gol/innovation/site.nsf/en/in02362.html

8. National Science Foundation (USA) workshop report on âœEmerging Issues in Nanoparticle Aerosol Science and Technologyâ, Editors S. K. Friedlander and D. Y. H. Pui (2003).

9. P.G. Debenedetti, Metastable Liquids: Concepts and Principles Princeton University Press, Princeton, New Jersey, 1996.

10. P.R. ten Wolde, M.J. Ruiz-Montero, and D. Frenkel, J. Chem. Phys. 104, 9932 (1996); P.R. ten Wolde and D. Frenkel, J. Chem. Phys. 109, 9901 (1998); P.R. ten Wolde, M.J. Ruiz-Montero, and D. Frenkel, J. Chem. Phys. 110, 1591 (1999); S. Auer and D. Frenkel, Nature London 409, 1020 (2001); S. Auer and D. Frenkel, J. Chem. Phys. 120, 3015 (2004).

11. P.R. ted Wolde and D. Frenkel, Science 277, 1975 (1997).

12. S. Auer and D. Frenkel, Phys. Rev. Lett. 91, 015703 (2003).

13. X. Zhang, Z. Wang, X. Dong, D. Wang, and C.C. Han, J. Chem. Phys. 125, 024907 (2006).

14. A. Cacciuto, S. Auer, and D. Frenkel, Phys. Rev. Lett. 93, 166105 (2004).

15. F. Trudu, D. Donadio, and M. Parrinello, Phys. Rev. Lett. 97, 105701 (2006).

16. L. Granasy, T. Pusztai, T. Borzonyi, J.A. Warren, and J.F. Douglas, Nat. Mater. 3, 645 (2004)

17. H. Shintani and H. Tanaka, Nature Physics 2, 200 (2006).

18. I. Saika-Voivod, P.H. Poole and F. Sciortino, Nature 412, 514-517 (2001).

19. 7. J. C. Dyre, Phys. Rev. Lett. 58, 792 (1987); Phys. Rev. B 51, 12276 (1995); T. Keyes, Phys. Rev. E 66. 051110 (2002); E.M. Bertin, J.-P. Bouchaud, J.M. Drouffe, and C. Godreche, J. Phys, A 36, 10701-10719 (2003).

20. K. Keough, Biophys. J. 85, 2785-2786 (2003).

21. A. Widmer-Cooper, P. Harrowell, and H. Fynewever, Phys. Rev. Lett. 93, 135701 (2004).

22. I. Saika-Voivod, F. Sciortino and P.H. Poole, Phys. Rev. E. 63, 011202 (2001).

23. C.A. Angell, J. Non-Crsyt. Solids 131-133, 13-31 (1991).

24. S. Sastry, Nature 410, 663-667 (2001).

25. A.J. Moreno, S.V. Buldyrev, E. La Nave, I. Saika-Voivod, F. Sciortino, P. Tartaglia, and E. Zaccarelli, Phys. Rev. Lett. 95, 157802 (2005).

26. I. Saika-Voivod and R.K. Bowles, submitted (2006).

27. K.D. Ball and R.S. Berry, J. Chem. Phys. 109, 8541 (1998); K.D. Ball and R.S. Berry, J. Chem. Phys. 111, 2060 (1999); J. Lu, C. Zhang, and R.S. Berry, Phys. Chem. Chem. Phys. 7, 3443 (2005).

28. R. Zandi, P. van der Soot, D. Reguera, W. Kegel, and H. Reiss, Biophys. J. 90, 1939-1948 (2006).

29. R.F. Bruisma, W.M. Gelbart, D. Reguera, J. Rudnick, and R. Zandi, Phys. Rev. Lett. 90, 248101 (2003).

30. K. Blennow, M.J. de Leon, and H. Zetterberg, Lancet 368, 387-403 (2006); A. Melquiond, N. Mousseau, and P. Derreumaux, Proteins 65, 180-91 (2006); A. Melquiond, G. Boucher, N. Mousseau, and P. Derreumaux, J. Chem. Phys. 22, 174904 (2005); M.M. Pallitto and R.M. Murphy, Biophys. J. 81, 1805 (2001) Erratum in: Biophys. J. 2002 82, 2826 (2002).