Steady-state Cluster Size Distribution Research Articles

Crystallization of Supercooled Liquids: Self-Consistency Correction of the Steady-State Nucleation Rate.

Crystal nucleation can be described by a set of kinetic equations that appropriately account for both the thermodynamic and kinetic factors governing this process. The mathematical analysis of this set of equations allows one to formulate analytical expressions for the basic characteristics of nucleation, i.e., the steady-state nucleation rate and the steady-state cluster-size distribution. These two quantities depend on the work of formation, , of crystal clusters of size n and, in particular, on the work of critical cluster formation, . The first term in the expression for describes changes in the bulk contributions (expressed by the chemical potential difference, ) to the Gibbs free energy caused by cluster formation, whereas the second one reflects surface contributions (expressed by the surface tension, : , , where is a parameter describing the size of the particles in the liquid undergoing crystallization), n is the number of particles (atoms or molecules) in a crystallite, and defines the size of the critical crystallite, corresponding to the maximum (in general, a saddle point) of the Gibbs free energy, G. The work of cluster formation is commonly identified with the difference between the Gibbs free energy of a system containing a cluster with n particles and the homogeneous initial state. For the formation of a “cluster” of size , no work is required. However, the commonly used relation for given above leads to a finite value for . By this reason, for a correct determination of the work of cluster formation, a self-consistency correction should be introduced employing instead of an expression of the form . Such self-consistency correction is usually omitted assuming that the inequality holds. In the present paper, we show that: (i) This inequality is frequently not fulfilled in crystal nucleation processes. (ii) The form and the results of the numerical solution of the set of kinetic equations are not affected by self-consistency corrections. However, (iii) the predictions of the analytical relations for the steady-state nucleation rate and the steady-state cluster-size distribution differ considerably in dependence of whether such correction is introduced or not. In particular, neglecting the self-consistency correction overestimates the work of critical cluster formation and leads, consequently, to far too low theoretical values for the steady-state nucleation rates. For the system studied here as a typical example (lithium disilicate, ), the resulting deviations from the correct values may reach 20 orders of magnitude. Consequently, neglecting self-consistency corrections may result in severe errors in the interpretation of experimental data if, as it is usually done, the analytical relations for the steady-state nucleation rate or the steady-state cluster-size distribution are employed for their determination.

Heating rate effects in time-dependent homogeneous nucleation in glasses

Nucleation kinetics of glass-ceramics is frequently determined using Tammann's double-stage heat-treatment. This method requires a complex deconvolution of the experimentally observed induction time (tind), i.e. the intercept of the linear part of the crystal number density curve with the nucleation time axis, into two components. In this paper, double-stage heat treatments were performed, with heating rates between the nucleation and development temperatures covering two orders of magnitude, in samples of a homogeneously nucleating glass-forming system, lithium disilicate. Our results show that tind increases with increasing heating rates with cubic root dependence. In accordance with the theory, tind was split into the intrinsic time required to establish a steady-state cluster size distribution, τ (time-lag) at the nucleation temperature and an incubation time (ti), which is a size, heating rate and development temperature (Td) dependent growth time. We demonstrate that the Collins-Kashchiev nucleation model performs poorly if ti is approximated by the time needed to experimentally detect the first crystal. In contrast, the Shneidman approach is consistent with theory. We found that at any given nucleation temperature, ti is a strong function of the heating rate, and is proportional to tind, whereas τ is a constant, as expected.

Crystallization of Self-Propelled Hard Discs.

We experimentally study the crystallization of a monolayer of vibrated discs with a built-in polar asymmetry, a model system of active liquids, and contrast it with that of vibrated isotropic discs. Increasing the packing fraction ϕ, the quasicontinuous crystallization reported for isotropic discs is replaced by a transition, or a crossover, towards a "self-melting" crystal. Starting from the liquid phase and increasing the packing fraction, clusters of dense hexagonal-ordered packed discs spontaneously form, melt, split, and merge, leading to a highly intermittent and heterogeneous dynamics. For a packing fraction larger than ϕ^{*}, a few large clusters span the system size. The cluster size distribution is monotonically decreasing for ϕ<ϕ^{*}, nonmonotonic for ϕ>ϕ^{*}, and is a power law at the transition. The system is, however, never dynamically arrested. The clusters permanently melt from place to place, forming droplets of an active liquid which rapidly propagate across the system. This self-melting crystalline state subsists up to the highest possible packing fraction, questioning the stability of the crystal for active discs unless it is at ordered close packing.

Nucleation of charged droplets; an ion-atmosphere model

We have explored the nucleation of charged droplets within the framework of Debye–Huckel ion-atmosphere model. In view of this the nucleation potential energy barrier has been appropriately modified. The Zeldovich–Becker–Doring–Frenkel theory (ZBDFT) and Boundary Layer Theory (BLT) have been exploited to understand the role of droplet charge on the steady state and transient nucleation rates. The time lag associated with the attainment of steady state cluster size distribution and the transient cluster flux have been analyzed in detail. The role played by the critical cluster size and the charge content of the droplets in modulating the transient nucleation rate have been explored. Also, a subtle analysis of the time lag has revealed the importance of diffusion of sub-critical clusters to attain the steady state. As interesting observations we have found complete confinement of droplet charge inside the inner region which is in close proximity with critical cluster size and sudden surge in nucleation rate as the droplet charge surpasses a threshold value. We demonstrate the enhancement of transient nucleation rate involving the charged droplets in comparison with their uncharged counterparts which runs in conformity with the earlier observations. The present work also highlights potential importance in the field of atmospheric physics research where the charged droplets are found as inalienable components.

Transient nucleation in isothermal crystallization of poly(3- hydroxybutyrate)

AbstractThe time dependence of nucleation rate in isothermal crystallization of poly(3-hydroxybutyrate) was experimentally shown, both in heterogeneous and homogeneous nucleation. The time dependence of nucleation rate is one of the important limitations for the applicability of the simplified form of Kolmogoroff- Avrami-Evans model with time independent kinetic characteristics. The presented results are interpreted in terms of non-steady-state cluster size distribution underlying transient nature of nucleation. The relaxation time needed for reaching a steady-state cluster size distribution and thus steady-state nucleation rate is relatively long, exceeding the time of exhaustion of heterogeneities. The relaxation time estimated from homogeneous process was tens of seconds in the temperature range between 83 and 120 oC. Application of Arrhenius law allows estimation of relaxation time in broader temperature range, showing an increase of relaxation time with decreasing temperature.

Population balance modeling of aggregation and breakage in turbulent Taylor–Couette flow

An experimental and computational study of aggregation and breakage processes for fully destabilized polystyrene latex particles under turbulent-flow conditions in a Taylor–Couette apparatus is presented. To monitor the aggregation and breakage processes, an in situ optical imaging technique was used. Consequently, a computational study using a population balance model was carried out to test the various parameters in the aggregation and breakage models. Very good agreement was found between the time evolution of the cluster size distribution (CSD) calculated with the model and that obtained from experiment. In order to correctly model the left-hand side of the CSD (small clusters), it was necessary to use a highly unsymmetric fragment-distribution function for breakage. As another test of the model, measurements with different solid volume fractions were performed. Within the range of the solid volume fractions considered here, the steady-state CSD was not significantly affected. In order to correctly capture the right-hand side of the CSD (large aggregates) at the higher solid volume fraction, a modified aggregation rate prefactor was used in the population balance model.

General Becker–Döring equations: effect of dimer interactions

The effect of dimer (two-particle) interactions on the Becker–Doring model of nucleation is investigated. Initially we consider the problem with size-independent aggregation and fragmentation coefficients and a constant monomer concentration. Either an equilibrium or a steady-state solution is found: the former when fragmentation is stronger than aggregation, the latter otherwise. By employing asymptotic techniques, the manner in which the system reaches these states is examined. The dimer interaction is found to accelerate the system towards the equilibrium solution, but has little impact on the relaxation time to the steady-state solution. In cases where aggregation is dominant, the steady-state cluster size distribution can only be determined consistently when the manner of approach to steady state is also known. In the terminology of asymptotic methods, one needs to know the first correction term in order to deduce the leading-order solution. We show how this can be derived and so at steady state we find a flux of matter to larger aggregation numbers due to monomer interactions, with a small and decreasing reverse flux due to dimer interactions. We then consider the case of constant density, that is allowing the monomer concentration to vary, and investigate the effect of a strong dimer interaction on the convergence to equilibrium. Two timescales are present and each one is investigated. We determine the intermediate meta-stable state, the final state and the timescales over which the system relaxes into these states. All results are shown to agree with numerical simulations.

Maxwell model of traffic flows

We investigate traffic flows using the kinetic Boltzmann equations with a Maxwell collision integral. This approach allows analytical determination of the transient behavior and the size distributions. The relaxation of the car and cluster velocity distributions towards steady state is characterized by a wide range of velocity- dependent relaxation scales, ${R}^{1/2}&lt;\ensuremath{\tau}(v)&lt;R,$ with R the ratio of the passing and the collision rates. Furthermore, these relaxation time scales decrease with the velocity, with the smallest scale corresponding to the decay of the overall density. The steady-state cluster size distribution follows an unusual scaling form ${P}_{m}\ensuremath{\sim}〈m{〉}^{\ensuremath{-}4}\ensuremath{\Psi}(m/〈m{〉}^{2}).$ This distribution is primarily algebraic, ${P}_{m}\ensuremath{\sim}{m}^{\ensuremath{-}3/2},$ for $m\ensuremath{\ll}〈m{〉}^{2},$ and is exponential otherwise.

NUCLEATION OF A ONE-COMPONENT NEW PHASE IN A SOLID SOLUTION UNDER DIFFERENT CONDITIONS

The general theory of nucleation and growth of a new phase with a given stoichiometric composition, developed earlier, is reformulated for the case of segregation of a one-component new phase in a solid solution.General expressions for the steady-state cluster size distribution and the steady-state nucleation rate are obtained as well as estimates for the time interval where these expressions may be applied.The general expressions are specified for a description of different particular cases (nucleation in real solutions, nucleation and interfacial kinetics, nucleation under the influence of elastic strains) which are of importance in a large number of technological applications. © 1997 Elsevier Science Ltd

Steady-state cluster size distribution in stirred suspensions

A combinatorial approach for predicting the steady-state agglomerate or drop-size distribution in batch suspensions undergoing intense stirring is proposed. The model accounts for the fact that both coagulation and break-up could occur simultaneously owing to agitation. Overall, considering its simplicity and the fact that it lacks any adjustable parameters, the results of the model compare favourably with experiment.