Homogeneous nucleation in inhomogeneous media. II. Nucleation in a shear flow

Transient nucleation driven by solvent evaporation

We theoretically investigate homogeneous crystal nucleation in a solution containing a solute and a volatile solvent. The solvent evaporates from the solution, thereby continuously increasing the concentration of the solute. We view it as an idealized model for the far-out-of-equilibrium conditions present during the liquid-state manufacturing of organic electronic devices. Our model is based on classical nucleation theory, taking the solvent to be a source of the transient conditions in which the solute drops out of the solution. Other than that, the solvent is not directly involved in the nucleation process itself. We approximately solve the kinetic master equations using a combination of Laplace transforms and singular perturbation theory, providing an analytical expression for the nucleation flux. Our results predict that (i) the nucleation flux lags slightly behind a commonly used quasi-steady-state approximation. This effect is governed by two counteracting effects originating from solvent evaporation: while a faster evaporation rate results in an increasingly larger influence of the lag time on the nucleation flux, this lag time itself is found to decrease with increasing evaporation rate. Moreover, we find that (ii) the nucleation flux and the quasi-steady-state nucleation flux are never identical, except trivially in the stationary limit, and (iii) the initial induction period of the nucleation flux, which we characterize as a generalized induction time, decreases weakly with the evaporation rate. This indicates that the relevant time scale for nucleation also decreases with an increasing evaporation rate. Our analytical theory compares favorably with results from a numerical evaluation of the governing kinetic equations.

Ice nucleation forced by transient electric fields

Icing affects many technical systems, like aircraft or high-voltage power transmission and distribution in cold regions. Ice accretion is often initiated by ice nucleation in sessile supercooled water droplets and is influenced by several influencing factors, of which the impact of electric fields on ice nucleation is still not completely understood. Especially the influence of transient electric fields is rarely or not at all investigated, even though it is of great interest, e.g., for high-voltage transmission lines or for the food industry. In the present study the impact of transient electric fields on ice nucleation in supercooled sessile water droplets is experimentally investigated under well-defined conditions. A set of droplets is cooled down to a certain temperature and is subsequently exposed to electric fields generated from standard lightning or standard switching impulse voltages, which are commonly used for testing of high-voltage equipment. The nucleation behavior of individual droplets is captured using a high-speed camera and the effect of the transient electric field on ice nucleation is analyzed by considering both the singular and the stochastic nature of nucleation. While the singular nature of nucleation is referred to during analysis of the relative number of droplets remaining liquid long times after the impulse voltage, its stochastic nature is accounted for in the analysis of the temporal evolution of the relative number of frozen droplets. It is shown that low electric field strengths (E[over ̂]≤6.52kV/cm) only have a negligible impact on ice nucleation, independent of the supercooling. In contrast, high electric field strengths (E[over ̂]≥9.78kV/cm) promote significantly ice nucleation. It is also shown that depending on the supercooling, the freezing delay of the different droplets in the ensemble may vary over several magnitudes for the same conditions. It is demonstrated that the electric field appears to indirectly affect the nucleation rate by generating droplet oscillations, finally promoting ice nucleation. The experiments clearly demonstrate the possibility to actively force ice nucleation by applying transient electric fields. These results improve the understanding of ice accretion on high-voltage insulators and may also lend insight into freezing processes in food industry. We expect that these results will be a valuable contribution in formulating and/or validating new nucleation models.

A Criterion for the Formation of Nonequilibrium Self-Assembled Structures

Is there a criterion able to determine the type of structures formed in a nonequilibrium self-assembly process? This important question has a clear answer when the process takes place under equilibrium conditions: structures emerge at minimum values of the free energy. Experiments, however, have shown that when self-assembly takes place outside equilibrium, they do not appear at those free energy minima but rather at optimal values of structural parameters. On the basis of these observations, we propose a selection criterion for which structures come up at the minima of a nonequilibrium free energy that takes into account the energy needed to change their configuration. The criterion is able to predict the formation and configuration of structures such as Liesegang rings and patterns in magnetic colloids and could constitute a powerful tool to understand the synthesis of advanced materials, enantiomers, and nanoparticles.

Ice nucleation in high alternating electric fields: Effect of electric field strength and frequency

Icing is a severe problem for many technical systems such as aircraft or systems for high-voltage power transmission and distribution. Ice nucleation in water droplets is affected by several influencing factors like impurities or the liquid temperature, which have been widely investigated. However, although an electric field affects nucleation, this influence has been far less investigated and is still not completely understood. The present work is focused on the influence of high alternating electric fields on ice nucleation in sessile water droplets, which is examined for a systematic variation of the electric field frequency and strength. All experiments used to determine the influence of a single parameter like the electric field strength or frequency are performed with the same set of droplets to ensure well-defined conditions and a high repeatability of the procedure. For each parameter variation a large number of nucleation events is observed and analyzed. Droplet survival curves and the nucleation site density are used to analyze the experiments and to determine the influence of the electric field on ice nucleation. Especially for high electric field strengths, a significant influence on nucleation is observed. Some droplets freeze earlier, which leads to a higher median nucleation temperature. On the other hand, the lowest temperature required to freeze all droplets is almost constant compared to the reference case without an electric field. It is shown that not all droplets are affected by the electric field in the same way, but the influence of the electric field on ice nucleation is rather of singular nature. In addition, the frequency of the applied electric field has an impact on the nucleation behavior. The present experimental data quantitatively demonstrate the effect of an electric field on ice nucleation and improves our understanding of heterogeneous nucleation of supercooled water subjected to high alternating electric fields.

Homogeneous nucleation in a Poiseuille flow

Nucleation in a dynamical environment plays an important role in the synthesis and manufacturing of quantum dots and nanocrystals. In this work, we investigate the effects of fluid flow (low Reynolds number flow) on the homogeneous nucleation in a circular microchannel in the framework of the classical nucleation theory. The contributions of the configuration entropy from the momentum-phase space and the kinetic energy and strain energy of a microcluster are incorporated in the calculation of the change of the Gibbs free energy from a flow state without a microcluster to a flow state with a microcluster. An analytical equation is derived for the determination of the critical nucleus size. Using this analytical equation, an analytical solution of the critical nucleus size for the formation of a critical liquid nucleus is found. For the formation of a critical solid nucleus, the contributions from both the kinetic energy and the strain energy are generally negligible. We perform numerical analysis of the homogeneous nucleation of a sucrose microcluster in a representative volume element of an aqueous solution, which flows through a circular microchannel. The numerical results reveal the decrease of the critical nucleus size and the corresponding work of formation of a critical nucleus with the increase of the distance to axisymmetric axis for the same numbers of solvent atoms and solute atoms/particles.

Classical nucleation theory for the crystallization kinetics in sheared liquids

While statistical mechanics provides a comprehensive framework for the understanding of equilibrium phase behavior, predicting the kinetics of phase transformations remains a challenge. Classical nucleation theory (CNT) provides a thermodynamic framework to relate the nucleation rate to thermodynamic quantities such as pressure difference and interfacial tension through the nucleation work necessary to spawn critical nuclei. However, it remains unclear whether such an approach can be extended to the crystallization of driven melts that are subjected to mechanical stresses and flows. Here, we demonstrate numerically for hard spheres that the impact of simple shear on the crystallization rate can be rationalized within the CNT framework by an additional elastic work proportional to the droplet volume. We extract the local stress and strain inside solid droplets, which yield size-dependent values for the shear modulus that are about half of the bulk value. Finally, we show that for a complete description one also has to take into account the change of interfacial work between the strained droplet and the sheared liquid. From scaling reasons, we expect this extra contribution to dominate the work formation of small nuclei but become negligible compared to the elastic work for droplets composed of a few hundreds of particles.

Self-assembling outside equilibrium: emergence of structures mediated by dissipation

Self-assembly under non-equilibrium conditions may give rise to the formation of structures not available at equilibrium.

Ice Layer Spreading along a Solid Substrate during Solidification of Supercooled Water: Experiments and Modeling

The thermal influence of a solid wall on the solidification of a sessile supercooled water drop is experimentally investigated. The velocity of the initial ice layer propagating along the solid substrate prior to dendritic solidification is determined from videos captured using a high-speed video system. Experiments are performed for varying substrate materials and liquid supercooling. In contrast to recent studies at moderate supercooling, in the case of metallic substrates only a weak influence of the substrate's thermal properties on the ice layer velocity is observed. Using the analytical solution of the two-phase Stefan problem, a semiempirical model for the ice layer velocity is developed. The experimental data are well described for all supercooling levels in the entire diffusion limited solidification regime. For higher supercooling, the model overestimates the freezing velocity due to kinetic effects during molecular attachment at the solid-liquid interface, which are not accounted for in the model. The experimental findings of the present work offer a new perspective on the design of anti-icing systems.

Effects of shear flow on phase nucleation and crystallization

Classical nucleation theory offers a good framework for understanding the common features of new phase formation processes in metastable homogeneous media at rest. However, nucleation processes in liquids are ubiquitously affected by hydrodynamic flow, and there is no satisfactory understanding of whether shear promotes or slows down the nucleation process. We developed a classical nucleation theory for sheared systems starting from the molecular level of the Becker-Doering master kinetic equation and we analytically derived a closed-form expression for the nucleation rate. The theory accounts for the effect of flow-mediated transport of molecules to the nucleus of the new phase, as well as for the mechanical deformation imparted to the nucleus by the flow field. The competition between flow-induced molecular transport, which accelerates nucleation, and flow-induced nucleus straining, which lowers the nucleation rate by increasing the nucleation energy barrier, gives rise to a marked nonmonotonic dependence of the nucleation rate on the shear rate. The theory predicts an optimal shear rate at which the nucleation rate is one order of magnitude larger than in the absence of flow.

Equilibrium fluid-crystal interfacial free energy of bcc-crystallizing aqueous suspensions of polydisperse charged spheres

The interfacial free energy is a central quantity in crystallization from the metastable melt. In suspensions of charged colloidal spheres, nucleation and growth kinetics can be accurately measured from optical experiments. In previous work, from these data effective nonequilibrium values for the interfacial free energy between the emerging bcc nuclei and the adjacent melt in dependence on the chemical potential difference between melt phase and crystal phase were derived using classical nucleation theory (CNT). A strictly linear increase of the interfacial free energy was observed as a function of increased metastability. Here, we further analyze these data for five aqueous suspensions of charged spheres and one binary mixture. We utilize a simple extrapolation scheme and interpret our findings in view of Turnbull's empirical rule. This enables us to present the first systematic experimental estimates for a reduced interfacial free energy, σ(0,bcc), between the bcc-crystal phase and the coexisting equilibrium fluid. Values obtained for σ(0,bcc) are on the order of a few k(B)T. Their values are not correlated to any of the electrostatic interaction parameters but rather show a systematic decrease with increasing size polydispersity and a lower value for the mixture as compared to the pure components. At the same time, σ(0) also shows an approximately linear correlation to the entropy of freezing. The equilibrium interfacial free energy of strictly monodisperse charged spheres may therefore be still greater.

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.

Extension of classical nucleation theory for uniformly sheared systems

Nucleation is an out-of-equilibrium process that can be strongly affected by the presence of external fields. In this paper, we report a simple extension of classical nucleation theory to systems submitted to an homogeneous shear flow. The theory involves accounting for the anisotropy of the critical nucleus formation and introduces a shear-rate-dependent effective temperature. This extended theory is used to analyze the results of extensive molecular dynamics simulations that explore a broad range of shear rates and undercoolings. At fixed temperature, a maximum in the nucleation rate is observed, when the relaxation time of the system is comparable to the inverse shear rate. In contrast to previous studies, our approach does not require a modification of the thermodynamic description, as the effect of shear is mainly embodied into a modification of the kinetic prefactor and of the temperature.

Shear effects on crystal nucleation in colloidal suspensions

Extensive two-dimensional Langevin dynamics simulations are used to determine the effect of steady shear flows on the crystal nucleation kinetics of charge stabilized colloids and colloids whose pair potential possess an attractive shallow well of a few k_{B}T 's (attractive colloids). Results show that in both types of systems small amounts of shear speeds up the crystallization process and enhances the quality of the growing crystal significantly. Moderate shear rates, on the other hand, destroy the ordering in the system. The very high shear rate regime where a reentering transition to the ordered state could exist is not considered in this work. In addition to the crystal nucleation phenomena, the analysis of the transport properties and the characterization of the steady state regime under shear are performed.

The mesoscopic dynamics of thermodynamic systems

Concepts of everyday use such as energy, heat, and temperature have acquired a precise meaning after the development of thermodynamics. Thermodynamics provides the basis for understanding how heat and work are related and the general rules that the macroscopic properties of systems at equilibrium follow. Outside equilibrium and away from macroscopic regimes, most of those rules cannot be applied directly. Here we present recent developments that extend the applicability of thermodynamic concepts deep into mesoscopic and irreversible regimes. We show how the probabilistic interpretation of thermodynamics together with probability conservation laws can be used to obtain Fokker-Planck equations for the relevant degrees of freedom. This approach provides a systematic method to obtain the stochastic dynamics of a system directly from its equilibrium properties. A wide variety of situations can be studied in this way, including many that were thought to be out of reach of thermodynamic theories, such as nonlinear transport in the presence of potential barriers, activated processes, slow relaxation phenomena, and basic processes in biomolecules, such as translocation and stretching.

Bubble formation in lattice Boltzmann immiscible shear flow

Bubble formation and the effect of shear on bubble formation in a van der Waals fluid is investigated by means of lattice Boltzmann mesoscale simulations. In the absence of shear, the maximum number of formed bubbles increases with undercooling but the incubation time before bubble formation decreases dramatically. The results are in agreement with classical phase transition theory. In shear flow, the maximum number of bubbles is not affected by shear but the bubble growth rate is accelerated. The effect of shear on bubble growth rate weakens at large undercoolings. The reasons are twofold. On the one hand the highly undercooled system takes less time to complete phase transition due to the large driving force so that there is less time to accumulate the flow effect. On the other hand the mechanism for bubble growth changes from coarsening to coalescence at large undercoolings.

Nucleation of Protein Crystals under the Influence of Solution Shear Flow

Several recent theories and simulations have predicted that shear flow could enhance, or, conversely, suppress the nucleation of crystals from solution. Such modulations would offer a pathway for nucleation control and provide a novel explanation for numerous mysteries in nucleation research. For experimental tests of the effects of shear flow on protein crystal nucleation, we found that if a protein solution droplet of approximately 5 microL (2-3 mm diameter at base) is held on a hydrophobic substrate in an enclosed environment and in a quasi-uniform constant electric field of 2 to 6 kV cm(-1), a rotational flow with a maximum rate at the droplet top of approximately 10 microm s(-1) is induced. The shear rate varies from 10(-3) to 10(-1) s(-1). The likely mechanism of the rotational flow involves adsorption of the protein and amphiphylic buffer molecules on the air-water interface and their redistribution in the electric field, leading to nonuniform surface tension of the droplet and surface tension-driven flow. Observations of the number of nucleated crystals in 24- and 72-h experiments with the proteins ferritin, apoferritin, and lysozyme revealed that the crystals are typically nucleated at a certain radius of the droplet, that is, at a preferred shear rate. Variations of the rotational flow velocity resulted in suppression or enhancement of the total number of nucleated crystals of ferritin and apoferritin, while all solution flow rates were found to enhance lysozyme crystal nucleation. These observations show that shear flow may strongly affect nucleation, and that for some systems, an optimal flow velocity, leading to fastest nucleation, exists. Comparison with the predictions of theories and simulations suggest that the formation of ordered nuclei in a "normal" protein solution cannot be affected by such low shear rates. We conclude that the flow acts by helping or suppressing the formation of ordered nuclei within mesoscopic metastable dense liquid clusters. Such clusters were recently shown to exist in protein solutions and to constitute the first step in the nucleation mechanism of many protein and nonproteinsystems.

Brownian motion of finite-inertia particles in a simple shear flow

Simultaneous diffusive and inertial motion of Brownian particles in laminar Couette flow is investigated via Lagrangian and Eulerian descriptions to determine the effect of particle inertia on diffusive transport in the long-time limit. The classical fluctuation dissipation theorem is used to calculate the amplitude of random-force correlations, thereby neglecting corrections of the order of the molecular relaxation time to the inverse shear rate. In the diffusive limit (time much greater than the particle relaxation time) the fluctuating particle-velocity autocorrelations functions are found to be stationary in time, the correlation in the streamwise direction being an exponential multiplied by an algebraic function and the cross correlation nonsymmetric in the time difference. The analytic, nonperturbative, evaluation of the particle-phase total pressure, which is calculated to be second order in the Stokes number (a dimensionless measure of particle inertia), shows that the particle phase behaves as a non-Newtonian fluid. The generalized Smoluchowski convective-diffusion equation, determined analytically from a combination of the particle-phase pressure tensor and the inertial acceleration term, contains a shear-dependent cross derivative term and an additional term along the streamwise direction, quadratic in the particle Stokes number. The long-time diffusion coefficients associated with the particle flux relative to the carrier flow are found to depend on particle inertia such that the streamwise diffusion coefficient becomes negative with increasing Stokes number, whereas one of the cross coefficients is always negative. The total diffusion coefficients measuring the rate of change of particle mean-square displacement are always positive as expected from general stability arguments.