Observing classical nucleation theory at work by monitoring phase transitions with molecular precision

Experimental measurement of the diamond nucleation landscape reveals classical and nonclassical features

Nucleation is a core scientific concept that describes the formation of new phases and materials. While classical nucleation theory is applied across wide-ranging fields, nucleation energy landscapes have never been directly measured at the atomic level, and experiments suggest that nucleation rates often greatly exceed the predictions of classical nucleation theory. Multistep nucleation via metastable states could explain unexpectedly rapid nucleation in many contexts, yet experimental energy landscapes supporting such mechanisms are scarce, particularly at nanoscale dimensions. In this work, we measured the nucleation energy landscape of diamond during chemical vapor deposition, using a series of diamondoid molecules as atomically defined protonuclei. We find that 26-carbon atom clusters, which do not contain a single bulk atom, are postcritical nuclei and measure the nucleation barrier to be more than four orders of magnitude smaller than prior bulk estimations. These data support both classical and nonclassical concepts for multistep nucleation and growth during the gas-phase synthesis of diamond and other semiconductors. More broadly, these measurements provide experimental evidence that agrees with recent conceptual proposals of multistep nucleation pathways with metastable molecular precursors in diverse processes, ranging from cloud formation to protein crystallization, and nanoparticle synthesis.

Nucleation of protein crystals - a nanoscopic perspective

Macromolecular phase transitions bear great medical, scientific and industrial relevance, yet the molecular picture of their earliest beginnings is still far from complete. For decades, progress has been hampered by the challenges associated with studying stochastic nucleation phenomena occurring on nanoscopic length scales. In the last 5 years, however, the field has advanced with great strides due to the recent buildout of experimental techniques that allow us to observe details of the nucleation process on the nanoscale. In this review, we present a historical overview and state-of-the-art analysis of protein crystal nucleation from an experimentalist's perspective. After a short introduction of key concepts from classical nucleation theory, we discuss the advancements that have led to the development of alternative models of protein nucleation. We summarize the experimental proof in favour of these various models, but we also focus on some of their shortcomings and experimental blind spots. In our penultimate section we highlight recent works that have provided direct nanoscopic insight into the nucleation of protein crystals. We end with concluding paragraphs discussing outstanding questions and possible strategies to advance the field further in the future.

Solid-to-Solid Crystallization of Organic Thin Films: Classical and Nonclassical Pathways

The solid-to-solid crystallization processes of organic molecules have been poorly understood in view of the complexity and the instability of organic crystals. Here, we studied the crystallization of a π-conjugated small molecular semiconductor, bis-(8-hydroxyquinoline) copper (CuQ₂), by annealing the thin films at different temperatures. We observed a classical film-to-nanorods crystallization at 80 °C, a coexistence of classical and nonclassical nucleation and particle growth at 120 °C, and a nonclassical crystal growth at 150 °C. We found that the growth of the crystals followed the following processes: particle nucleation, particle growth, particle migration, nondirectional particle attachment, and structure reconstruction. We notice that the growth of CuQ₂ particles follows an outside-to-inside process. More interestingly, our experiments suggest that the submicron CuQ₂ particles are able to migrate dozens of micrometers at 150 °C.

Aspects of nucleation on curved and flat surfaces

We investigate the energetics of droplets sourced by the thermal fluctuations in a system undergoing a first-order transition. In particular, we confine our studies to two dimensions with explicit calculations in the plane and on the sphere. Using an isoperimetric inequality from the differential geometry literature and a theorem on the inequality's saturation, we show how geometry informs the critical droplet size and shape. This inequality establishes a "mean field" result for nucleated droplets. We then study the effects of fluctuations on the interfaces of droplets in two dimensions, treating the droplet interface as a fluctuating line. We emphasize that care is needed in deriving the line curvature energy from the Landau-Ginzburg energy functional and in interpreting the scalings of the nucleation rate with the size of the droplet. We end with a comparison of nucleation in the plane and on a sphere.

Free energy landscape and molecular pathways of gas hydrate nucleation

Despite the significance of gas hydrates in diverse areas, a quantitative knowledge of hydrate formation at a molecular level is missing. The impediment to acquiring this understanding is primarily attributed to the stochastic nature and ultra-fine scales of nucleation events, posing a great challenge for both experiment and simulation to explore hydrate nucleation. Here we employ advanced molecular simulation methods, including forward flux sampling (FFS), p_B histogram analysis, and backward flux sampling, to overcome the limit of direct molecular simulation for exploring both the free energy landscape and molecular pathways of hydrate nucleation. First we test the half-cage order parameter (H-COP) which we developed for driving FFS, through conducting the p_B histogram analysis. Our results indeed show that H-COP describes well the reaction coordinates of hydrate nucleation. Through the verified order parameter, we then directly compute the free energy landscape for hydrate nucleation by combining both forward and backward flux sampling. The calculated stationary distribution density, which is obtained independently of nucleation theory, is found to fit well against the classical nucleation theory (CNT). Subsequent analysis of the obtained large ensemble of hydrate nucleation trajectories show that although on average, hydrate formation is facilitated by a two-step like mechanism involving a gradual transition from an amorphous to a crystalline structure, there also exist nucleation pathways where hydrate crystallizes directly, without going through the amorphous stage. The CNT-like free energy profile and the structural diversity suggest the existence of multiple active transition pathways for hydrate nucleation, and possibly also imply the near degeneracy in their free energy profiles among different pathways. Our results thus bring a new perspective to the long standing question of how hydrates crystallize.

Surface-induced crystallization of pharmaceuticals and biopharmaceuticals: A review

Despite the wide occurrence of crystallization in the pharmaceutical industry, deep understanding and fine control of the process remain a tricky issue. Nevertheless, the successful manufacturing of finished pharmaceutical products, as well as the structural determination of biopharmaceuticals, depend on the size, form, shape and purity of the crystals. The ability of substrates with precise chemistry and topological features to induce nucleation has been thoroughly assessed during the recent years. This paper reviews the major advances and discoveries in controlling small molecule drug and protein crystallization by means of engineered surfaces. By designing superficial properties and morphology, it has been possible to tune the polymorph outcome, shorten the nucleation induction time, impose specific crystal shapes, control the crystal size and carry out crystallization at very low supersaturation levels. Such achievements underline the potential of surface-induced crystallization to provide an ideal platform for the study of the nucleation process and gain control over its stochastic nature.

Molecular nucleation mechanisms and control strategies for crystal polymorph selection

The formation of condensed (compacted) protein phases is associated with a wide range of human disorders, such as eye cataracts, amyotrophic lateral sclerosis, sickle cell anaemia and Alzheimer's disease. However, condensed protein phases have their uses: as crystals, they are harnessed by structural biologists to elucidate protein structures, or are used as delivery vehicles for pharmaceutical applications. The physiochemical properties of crystals can vary substantially between different forms or structures ('polymorphs') of the same macromolecule, and dictate their usability in a scientific or industrial context. To gain control over an emerging polymorph, one needs a molecular-level understanding of the pathways that lead to the various macroscopic states and of the mechanisms that govern pathway selection. However, it is still not clear how the embryonic seeds of a macromolecular phase are formed, or how these nuclei affect polymorph selection. Here we use time-resolved cryo-transmission electron microscopy to image the nucleation of crystals of the protein glucose isomerase, and to uncover at molecular resolution the nucleation pathways that lead to two crystalline states and one gelled state. We show that polymorph selection takes place at the earliest stages of structure formation and is based on specific building blocks for each space group. Moreover, we demonstrate control over the system by selectively forming desired polymorphs through site-directed mutagenesis, specifically tuning intermolecular bonding or gel seeding. Our results differ from the present picture of protein nucleation, in that we do not identify a metastable dense liquid as the precursor to the crystalline state. Rather, we observe nucleation events that are driven by oriented attachments between subcritical clusters that already exhibit a degree of crystallinity. These insights suggest ways of controlling macromolecular phase transitions, aiding the development of protein-based drug-delivery systems and macromolecular crystallography.

Influences of Crystal Anisotropy in Pharmaceutical Process Development

Crystalline materials are of crucial importance to the pharmaceutical industry, as a large number of APIs are formulated in crystalline form, occasionally in the presence of crystalline excipients. Owing to their multifaceted character, crystals were found to have strongly anisotropic properties. In fact, anisotropic properties were found to be quite important for a number of processes including milling, granulation and tableting. An understanding of crystal anisotropy and an ability to control and predict crystal anisotropy are mostly subjects of interest for researchers. A number of studies dealing with the aforementioned phenomena are grounded on over-simplistic assumptions, neglecting key attributes of crystalline materials, most importantly the anisotropic nature of a number of their properties. Moreover, concepts such as the influence of interfacial phenomena in the behaviour of crystalline materials during their growth and in vivo, are still poorly understood. The review aims to address concepts from a molecular perspective, focusing on crystal growth and dissolution. It begins with a brief outline of fundamental concepts of intermolecular and interfacial phenomena. The second part discusses their relevance to the field of pharmaceutical crystal growth and dissolution. Particular emphasis is given to works dealing with mechanistic understandings of the influence of solvents and additives on crystal habit. Furthermore, comments and perspectives, highlighting future directions for the implementation of fundamental concepts of interfacial phenomena in the rational understanding of crystal growth and dissolution processes, have been provided.

Pre-critical fluctuations and what they disclose about heterogeneous crystal nucleation

Heterogeneous crystal nucleation is ubiquitous in nature and at the heart of many industrial applications. At the molecular scale, however, major gaps in understanding this phenomenon persist. Here we investigate through molecular dynamics simulations how the formation of precritical crystalline clusters is connected to the kinetics of nucleation. Considering heterogeneous water freezing as a prototypical scenario of practical relevance, we find that precritical fluctuations connote which crystalline polymorph will form. The emergence of metastable phases can thus be promoted by templating crystal faces characteristic of specific polymorphs. As a consequence, heterogeneous classical nucleation theory cannot describe our simulation results, because the different substrates lead to the formation of different ice polytypes. We discuss how the issue of polymorphism needs to be incorporated into analysis and comparison of heterogeneous and homogeneous nucleation. Our results will help to interpret and analyze the growing number of experiments and simulations dealing with crystal polymorph selection.

Nonclassical nucleation pathways in protein crystallization

Classical nucleation theory (CNT), which was established about 90 years ago, has been very successful in many research fields, and continues to be the most commonly used theory in describing the nucleation process. For a fluid-to-solid phase transition, CNT states that the solute molecules in a supersaturated solution reversibly form small clusters. Once the cluster size reaches a critical value, it becomes thermodynamically stable and favored for further growth. One of the most important assumptions of CNT is that the nucleation process is described by one reaction coordinate and all order parameters proceed simultaneously. Recent studies in experiments, computer simulations and theory have revealed nonclassical features in the early stage of nucleation. In particular, the decoupling of order parameters involved during a fluid-to-solid transition leads to the so-called two-step nucleation mechanism, in which a metastable intermediate phase (MIP) exists between the initial supersaturated solution and the final crystals. Depending on the exact free energy landscapes, the MIPs can be a high density liquid phase, mesoscopic clusters, or a pre-ordered state. In this review, we focus on the studies of nonclassical pathways in protein crystallization and discuss the applications of the various scenarios of two-step nucleation theory. In particular, we focus on protein solutions in the presence of multivalent salts, which serve as a model protein system to study the nucleation pathways. We wish to point out the unique features of proteins as model systems for further studies.

Kinetics of Vapor-Solid Phase Transitions: Structure, Growth, and Mechanism

The kinetics of the separation between low and high density phases in a single component Lennard-Jones model is studied via molecular dynamics simulations, at very low temperatures, in the space dimension d=2. For densities close to the vapor branch of the coexistence curve, disconnected nanoscale clusters of the high density phase exhibit essentially ballistic motion. Starting from nearly circular shapes, at the time of nucleation, these clusters grow via sticky collisions, gaining filamentlike nonequilibrium structure at a later time, with a very low fractal dimensionality. The origin of the latter is shown to lie in the low mobility of the constituent particles, in the corresponding cluster reference frame, due to the (quasi-long-range) crystalline order. Standard self-similarity in the domain pattern, typically observed in the kinetics of phase transitions, is found to be absent. This invalidates the common method, that provides a growth law comparable to that in solid mixtures, of quantifying growth. An appropriate alternative approach, involving the fractality, quantifies the growth of the characteristic "length" to be a power law with time, the exponent being strongly temperature dependent. The observed growth law is in agreement with the outcome of a nonequilibrium kinetic theory.

Two-Dimensional Nucleation on the Terrace of Colloidal Crystals with Added Polymers

Understanding nucleation dynamics is important both fundamentally and technologically in materials science and other scientific fields. Two-dimensional (2D) nucleation is the predominant growth mechanism in colloidal crystallization, in which the particle interaction is attractive, and has recently been regarded as a promising method to fabricate varieties of complex nanostructures possessing innovative functionality. Here, polymers are added to a colloidal suspension to generate a depletion attractive force, and the detailed 2D nucleation process on the terrace of the colloidal crystals is investigated. In the system, we first measured the nucleation rate at various area fractions of particles on the terrace, ϕ_area. In situ observations at single-particle resolution revealed that nucleation behavior follows the framework of classical nucleation theory (CNT), such as single-step nucleation pathway and existence of critical size. Characteristic nucleation behavior is observed in that the nucleation and growth stage are clearly differentiated. When many nuclei form in a small area of the terrace, a high density of kink sites of once formed islands makes growth more likely to occur than further nucleation because nucleation has a higher energy barrier than growth. The steady-state homogeneous 2D nucleation rate, J, and the critical size of nuclei, r*, are measured by in situ observations based on the CNT, which enable us to obtain the step free energy, γ, which is an important parameter for characterizing the nucleation process. The γ value is found to change according to the strength of attraction, which is tuned by the concentration of the polymer as a depletant.

In Vitro Characterization of the Two-Stage Non-Classical Reassembly Pathway of S-Layers

The recombinant bacterial surface layer (S-layer) protein rSbpA of Lysinibacillus sphaericus CCM 2177 is an ideal model system to study non-classical nucleation and growth of protein crystals at surfaces since the recrystallization process may be separated into two distinct steps: (i) adsorption of S-layer protein monomers on silicon surfaces is completed within 5 min and the amount of bound S-layer protein sufficient for the subsequent formation of a closed crystalline monolayer; (ii) the recrystallization process is triggered—after washing away the unbound S-layer protein—by the addition of a CaCl₂ containing buffer solution, and completed after approximately 2 h. The entire self-assembly process including the formation of amorphous clusters, the subsequent transformation into crystalline monomolecular arrays, and finally crystal growth into extended lattices was investigated by quartz crystal microbalance with dissipation (QCM-D) and atomic force microscopy (AFM). Moreover, contact angle measurements showed that the surface properties of S-layers change from hydrophilic to hydrophobic as the crystallization proceeds. This two-step approach is new in basic and application driven S-layer research and, most likely, will have advantages for functionalizing surfaces (e.g., by spray-coating) with tailor-made biological sensing layers.

Crystallization of supercooled fenofibrate studied at ambient and elevated pressures

Investigation of the thermodynamic history in the T,p-plane of the crystallization process of a supercooled liquid.

Birth of NaCl Crystals: Insights from Molecular Simulations

Molecular dynamics simulations are used to investigate the factors that influence the nucleation of NaCl crystals in a supersaturated aqueous solution. We describe a methodology for detecting solidlike NaCl clusters (potential nuclei) and following their evolution in time until they achieve nucleation (which is very rare) or dissolve back into solution. Through an analysis of cluster lifetimes and multiple nucleation events, we demonstrate that cluster size is not the only property that influences cluster stability and the probability of achieving nucleation. We introduce a parameter called cluster crystallinity, which is a measure of the solidlike order in a particular cluster. We show that cluster order (as measured by this parameter) has a strong influence on the lifetime and nucleation probability of clusters of equal sizes, with the lifetime and probability of nucleation increasing with increasing crystallinity. These observations remain true for clusters as small as six ions, showing that the structural factors are important even at the earliest stages of crystal birth.

Mechanism for the stabilization of protein clusters above the solubility curve: the role of non-ideal chemical reactions

Dense protein clusters are known to play an important role in nucleation of protein crystals from dilute solutions. While these have generally been thought to be formed from a metastable phase, the observation of similar, if not identical, clusters above the critical point for the dilute-solution/strong-solution phase transition has thrown this into doubt. Furthermore, the observed clusters are stable for relatively long times. Because protein aggregation plays a central role in some pathologies, understanding the nature of such clusters is an important problem. One mechanism for the stabilization of such structures was proposed by Pan, Vekilov and Lubchenko and was investigated using a dynamical density functional theory model which confirmed the viability of the model. Here, we revisit that model and incorporate additional physics in the form of state-dependent reaction rates. We show by a combination of numerical results and general arguments that the state-dependent rates disrupt the stability mechanism. Finally, we argue that the state-dependent reactions correct unphysical aspects of the model with ideal (state-independent) reactions and that this necessarily leads to the failure of the proposed mechanism.

Crystal Nucleation in Liquids: Open Questions and Future Challenges in Molecular Dynamics Simulations

The nucleation of crystals in liquids is one of nature's most ubiquitous phenomena, playing an important role in areas such as climate change and the production of drugs. As the early stages of nucleation involve exceedingly small time and length scales, atomistic computer simulations can provide unique insights into the microscopic aspects of crystallization. In this review, we take stock of the numerous molecular dynamics simulations that, in the past few decades, have unraveled crucial aspects of crystal nucleation in liquids. We put into context the theoretical framework of classical nucleation theory and the state-of-the-art computational methods by reviewing simulations of such processes as ice nucleation and the crystallization of molecules in solutions. We shall see that molecular dynamics simulations have provided key insights into diverse nucleation scenarios, ranging from colloidal particles to natural gas hydrates, and that, as a result, the general applicability of classical nucleation theory has been repeatedly called into question. We have attempted to identify the most pressing open questions in the field. We believe that, by improving (i) existing interatomic potentials and (ii) currently available enhanced sampling methods, the community can move toward accurate investigations of realistic systems of practical interest, thus bringing simulations a step closer to experiments.

Micro-structural Change During Nucleation: From Nucleus To Bicontinuous Morphology

Although the microstructure of coexistence phase provides direct insights of the nucleation mechanism and their change is substantial in the phase transition, their study is limited due to the lack of suitable tools capturing the thermodynamically unstable transient states. We resolve this problem in computational study by introducing a generalized canonical ensemble simulation and investigate the morphological change of the nucleus during the water evaporation and condensation. We find that at very low pressure, where the transition is first order, classical nucleation theory holds approximately. A main nucleus is formed in the supersaturation near spinodal, and the overall shape of the nucleus is finite and compact. On increasing the pressure of the system, more nuclei are formed even before spinodal. They merge into a larger nuclei with a smaller free energy penalty to form ramified shapes. We suggest order parameters to describe the extent of fluctuation, and their relation to the free energy profile.

Unification of classical nucleation theories via a unified Itô-Stratonovich stochastic equation

Classical nucleation theory (CNT) is the most widely used framework to describe the early stage of first-order phase transitions. Unfortunately, the different points of view adopted to derive it yield different kinetic equations for the probability density function, e.g., Zeldovich-Frenkel or Becker-Döring-Tunitskii equations. Starting from a phenomenological stochastic differential equation, a unified equation is obtained in this work. In other words, CNT expressions are recovered by selecting one or another stochastic calculus. Moreover, it is shown that the unified CNT thus obtained produces the same Fokker-Planck equation as that from a recent update of CNT [J. F. Lutsko and M. A. Durán-Olivencia, J. Chem. Phys. 138, 244908 (2013)10.1063/1.4811490] when mass transport is governed by diffusion. Finally, we derive a general induction-time expression along with specific approximations of it to be used under different scenarios, in particular, when the mass-transport mechanism is governed by direct impingement, volume diffusion, surface diffusion, or interface transfer.

Do protein crystals nucleate within dense liquid clusters?

Protein-dense liquid clusters are regions of high protein concentration that have been observed in solutions of several proteins. The typical cluster size varies from several tens to several hundreds of nanometres and their volume fraction remains below 10(-3) of the solution. According to the two-step mechanism of nucleation, the protein-rich clusters serve as locations for and precursors to the nucleation of protein crystals. While the two-step mechanism explained several unusual features of protein crystal nucleation kinetics, a direct observation of its validity for protein crystals has been lacking. Here, two independent observations of crystal nucleation with the proteins lysozyme and glucose isomerase are discussed. Firstly, the evolutions of the protein-rich clusters and nucleating crystals were characterized simultaneously by dynamic light scattering (DLS) and confocal depolarized dynamic light scattering (cDDLS), respectively. It is demonstrated that protein crystals appear following a significant delay after cluster formation. The cDDLS correlation functions follow a Gaussian decay, indicative of nondiffusive motion. A possible explanation is that the crystals are contained inside large clusters and are driven by the elasticity of the cluster surface. Secondly, depolarized oblique illumination dark-field microscopy reveals the evolution from liquid clusters without crystals to newly nucleated crystals contained in the clusters to grown crystals freely diffusing in the solution. Collectively, the observations indicate that the protein-rich clusters in lysozyme and glucose isomerase solutions are locations for crystal nucleation.

Real-time observation of nonclassical protein crystallization kinetics

We present a real-time study of protein crystallization of bovine β-lactoglobulin in the presence of CdCl(2) using small-angle X-ray scattering and optical microscopy. From observing the crystallization kinetics, we propose the following multistep crystallization mechanism that is consistent with our data. In the first step, an intermediate phase is formed, followed by the nucleation of crystals within the intermediate phase. During this period, the number of crystals increases with time, but the crystal growth is slowed down by the surrounding dense intermediate phase due to the low mobility. In the next step, the intermediate phase is consumed by nucleation and slow growth, and the crystals are exposed to the dilute phase. In this stage, the number of crystals becomes nearly constant, whereas the crystals grow rapidly due to access to the free protein molecules in the dilute phase. This real-time study not only provides evidence for a two-step nucleation process for protein crystallization but also elucidates the role and the structural signature of the metastable intermediate phase in this process.

Nanostructure formation via post growth of particles

Post growth of nanoparticles enables new nanostructure formation and blurs the boundary between crystals and molecules.