Distribution kinetics of Ostwald ripening at large volume fraction and with coalescence

Condensation phase transitions from metastable fluids occur by nucleation with accompanying particle growth and eventual Ostwald ripening. During ripening the subcritical particles dissolve spontaneously while larger particles grow and possibly coalesce if their volume fraction is large enough. The classical diffusion-influenced rates are also affected by large particle concentrations and are here described by mass-dependent rates. We represent the kinetics of ripening through growth, dissolution, and biparticle coalescence by a new population dynamics equation for the particle size distribution (PSD). Numerical solutions of the scaled governing equations show that coalescence plays a major role in influencing the PSD when the scaled mass concentration (volume fraction) or number concentration is relatively large. The solution describes the time range from initial conditions to the final narrowing of polydispersity. We show that the time dependence of the average particle mass in the asymptotic period of ripening has a power-law increase dependent on rate expressions for particle growth and coalescence at large values of volume fraction.     

Influence of initial conditions on homogeneous nucleation kinetics in a closed system

The formation of nuclei of a new phase from the supersaturated mother phase in a closed system is studied. The depletion of the mother phase due to phase transition is taken into account. Basic kinetic equations describing such process are solved numerically to determine the number density of nuclei of newly forming phase and nucleation rate. It is shown that in contrary to the standard nucleation model, when the depletion of the mother phase is not taken into account, the initial size distribution of the clusters affects considerably the nucleation process at higher supersaturations. Our model starts with the equilibrium size distribution of clusters up to various cluster sizes in the undercritical region. At lower supersaturation the formation of nuclei is similar to the standard model because of the low depletion of the mother phase. At higher supersaturation, the depletion of the mother phase plays an important role and some extremal value appears at the size distribution of nuclei, which is not observed in the standard model. The extremum in the size distribution is not a consequence of the coalescence process itself, but it is caused rather by the depletion of the mother phase during the phase transformation.     

Continuous polydispersity in a self-consistent field theory for diblock copolymers

An efficient algorithm is presented for numerically evaluating a self-consistent field theoretic (SCFT) model of an AB diblock copolymer that incorporates continuous polydispersity in one of the blocks. An interesting segregation effect is found in which chains of intermediate molecular weight are concentrated at domain interfaces. This model of continuous polydispersity is also implemented in the random phase approximation (RPA) to study the order-disorder transition and predicts that the stability of the disordered, homogeneous phase decreases as the polydispersity in one of the blocks increases. The RPA predictions are confirmed by SCFT calculations. Our approach and results are particularly relevant to block copolymers prepared by quasiliving synthesis techniques, where the polymerization of one block is much more controlled than the other block.     

Isotropic-nematic phase transition of nonaqueous suspensions of natural clay rods

A novel model system for studying the behavior of hard colloidal rods is presented, consisting of sterically stabilized particles of natural sepiolite clay. Electron microscopy and scattering results confirmed that the organophilic clay particles were individual, rigid rods when dispersed in organic solvents. With a length-to-diameter ratio of approximately 27, the particles showed nematic ordering for volume fractions phi > 0.06. Polarizing microscopy revealed that the phase separation process involved nucleation, growth, and coalescence of nematic domains. The phase volumes and particle concentrations in the coexisting phases were determined. The dependence of these quantities on the total concentration of the suspension agrees well with Onsager's [Ann. N. Y. Acad. Sci. 51, 627 (1949)] isotropic-nematic phase transition theory extended to bidisperse and polydisperse rod systems, and with previous experimental results for rigid rodlike particles. Particle size distributions were obtained by analyzing transmission electron microscopy images. A significant fractionation with respect to rod length (but not diameter) was observed in the coexisting isotropic and nematic phases. The relative polydispersity of both daughter phases was distinctly smaller than that of the parent suspension. The phase behavior of these daughter fractions agrees well with the predictions for hard spherocylinders of corresponding aspect ratios. An isotropic-nematic-nematic phase equilibrium was seen to develop in phase separated samples after 1 month standing and is ascribed to the effect of polydispersity and possibly gravity. The second nematic phase appearing is dominated by very long rods.     

Phase behavior of weakly polydisperse sticky hard spheres: perturbation theory for the Percus-Yevick solution

We study the effects of size polydispersity on the gas-liquid phase behavior of mixtures of sticky hard spheres. To achieve this, the system of coupled quadratic equations for the contact values of the partial cavity functions of the Percus-Yevick solution [R. J. Baxter, J. Chem. Phys. 49, 2770 (1968)] is solved within a perturbation expansion in the polydispersity, i.e., the normalized width of the size distribution. This allows us to make predictions for various thermodynamic quantities which can be tested against numerical simulations and experiments. In particular, we determine the leading order effects of size polydispersity on the cloud curve delimiting the region of two-phase coexistence and on the associated shadow curve; we also study the extent of size fractionation between the coexisting phases. Different choices for the size dependence of the adhesion strengths are examined carefully; the Asakura-Oosawa model [J. Chem. Phys. 22, 1255 (1954)] of a mixture of polydisperse colloids and small polymers is studied as a specific example.     

Phase coexistence in polydisperse multi-Yukawa hard-sphere fluid: high temperature approximation

High temperature approximation (HTA) is used to describe the phase behavior of polydisperse multi-Yukawa hard-sphere fluid mixtures. It is demonstrated that in the frames of the HTA the model belongs to the class of "truncatable free energy models," i.e., the models with thermodynamical properties (Helmholtz free energy, chemical potential, and pressure) defined by the finite number of generalized moments. Using this property we were able to calculate the complete phase diagram (i.e., cloud and shadow curves as well as binodals) and size distribution functions of the coexisting phases of several different models of polydisperse fluids. In particular, we consider polydisperse one-Yukawa hard-sphere mixture with factorizable Yukawa coefficients and polydisperse Lennard-Jones (LJ) mixture with interaction energy parameter and/or size polydispersity. To validate the accuracy of the HTA we compare theoretical results with previously published results of more advanced mean spherical approximation (MSA) for the one-Yukawa model and with the Monte Carlo (MC) computer simulation results of [Wilding et al. J. Chem. Phys. 121, 6887 (2004); Phys. Rev. Lett. 95, 155701 (2005)] for the LJ model. We find that overall predictions of the HTA are in reasonable agreement with predictions of the MSA and MC, with the accuracy range from semiquantitative (for the phase diagram) to quantitative (for the size distribution functions).     

Effect of quenched size polydispersity on the fluid-solid transition in charged colloidal suspensions

We study the effect of quenched size polydispersity on the phase behavior of charged colloidal suspensions using free-energy calculations in Monte Carlo simulations. The colloids are assumed to interact with a hard-core repulsive Yukawa (screened-Coulomb) interaction with constant surface potential, so that the particles are polydisperse both in size and charge. In addition, we take the size distribution to be fixed in both the fluid and crystal phase (no size fractionation is allowed). We study the fluid-solid transition for various screening lengths and surface potentials, finding that upon increasing the size polydispersity the freezing transition shifts toward higher packing fractions and the density discontinuity between the two coexisting phases diminishes. Our results provide support for a terminal polydispersity above which the freezing transition disappears.     

Effect of emulsion drop-size distribution upon coalescence in simple shear flow: a population balance study

A population balance is used to examine the effect of the shape of the initial drop-size distribution of an emulsion upon its short and long-time evolution in simple shear flow. Initial distributions that are monodisperse, multidisperse, lognormal, bimodal, multimodal, and step functions are considered. At short times, it is shown that the rate of coalescence decreases by up to 25% for step distributions and up to 75% for lognormal distributions as the width of the distribution increases. Bimodal, multidisperse and multimodal distributions show intermediate decreases in the rate of coalescence, between these two values, with increases in the distribution width. Furthermore, it is found that the initial rate of coalescence is strongly dependent upon the presence of large drops. As the number fraction of large droplets within the distribution increases, the rate of coalescence also increases. At long times, all distributions move toward an asymptotic distribution shape in which the frequency of drops decreases algebraically with drop diameter at small drop diameters, and decreases exponentially with drop diameter at large drop diameters. Though portions of each distribution showed the expected asymptotic scaling behavior at long times, each asymptotic distribution nevertheless retains 'fingerprints' of the respective initial distribution. Overall, the rate of coalescence for a system is bounded by the initial rate, which is a function of the initial distribution shape, and the asymptotic rate, which is dependent upon the long-time scaling behavior. Finally, it is shown that the resolution with which the drop-size distribution of an emulsion is experimentally measured can have a significant effect upon predicted rates of coalescence.     

Coalescence in semiconcentrated emulsions in simple shear flow

The coalescence frequency in emulsions containing droplets with a low viscosity (viscosity ratio approximately 0.005) in simple shear flow has been investigated experimentally at several volume fractions of the dispersed phase (2%-14%) and several values of the shear rate (0.1-10 s(-1)). The evolution of the size distribution was monitored to determine the average coalescence probability from the decay of the total number of droplets. Theoretically models for two-droplet coalescence are considered, where the probability is given by P(c)=exp(-tau(dr)tau(int)). Since the drainage time tau(dr) depends on the size of the two colliding droplets, and the collision time tau(int) depends on the initial orientation of the colliding droplets, the calculated coalescence probability was averaged over the initial orientation distribution and the experimental size distribution. This averaged probability was compared to the experimentally obtained coalescence frequency. The experimental results indicate that (1) to predict the average coalescence probability one has to take into account the full size distribution of the droplets; (2) the coalescence process is best described by the "partially mobile deformable interface" model or the "fully immobile deformable interface" model of Chesters [A. K. Chesters, Chem. Eng. Res. Des. 69, 259 (1991)]; and (3) independent of the models used it was concluded that the ratio tau(dr)tau(int) scales with the coalescence radius to a power (2+/-1) and with the rate of shear to a power (1.5+/-1). The critical coalescence radius R(o), above which hardly any coalescence occurs is about 10 microm.     

Model for drop coalescence in a locally isotropic turbulent flow field

The proposed model views drop coalescence in a turbulent flow field as a two-step process consisting of formation of a doublet due to drop collisions followed by coalescence of the individual droplets in a doublet due to the drainage of the intervening film of continuous phase under the action of colloidal (van der Waals and electrostatic) and random turbulent forces. The turbulent flow field was assumed to be locally isotropic. A first-passage-time analysis was employed for the random process of intervening continuous-phase film thickness between the two drops of a doublet in order to evaluate the first two moments of coalescence-time distribution of the doublet. The average drop coalescence time of the doublet was dependent on the barrier for coalescence due to the net repulsive force (net effect of colloidal repulsive and turbulent attractive forces). The predicted average drop coalescence time was found to be smaller for larger turbulent energy dissipation rates, smaller surface potentials, larger drop sizes, larger ionic strengths, and larger drop size ratios of unequal-sized drop pairs. The predicted average drop coalescence time was found to decrease whenever the ratio of average turbulent force to repulsive force barrier became larger. The calculated coalescence-time distribution was broader, with a higher standard deviation, at lower energy dissipation rates, higher surface potentials, smaller drop sizes, and smaller size ratios of unequal drop pairs. The model predictions of average coalescence-rate constants for tetradecane-in-water emulsions stabilized by sodium dodecyl sulfate (SDS) in a high-pressure homogenizer agreed fairly well with the inferred experimental values as reported by Narsimhan and Goel (J. Colloid Interface Sci. 238 (2001) 420-432) at different homogenizer pressures and SDS concentrations.     

Phase transitions of a single polyelectrolyte in a poor solvent with explicit counterions

Conformational properties of a single flexible polyelectrolyte chain in a poor solvent are studied using constant temperature molecular dynamics simulation. The effects of counterions are explicitly taken in to account. Structural properties of various phases and the transition between these phases are studied by tracking the values of asphericity, radius of gyration, fraction of condensed counterions, number of non-bonded neighbours, and Coulomb interaction energies. From our simulations, we find strong evidence for a first-order phase transition from extended to collapsed phase consistent with earlier theoretical predictions. We also identify a continuous phase transition associated with the condensation of counterions and estimate the critical exponents associated with the transition. Finally, we argue that previous suggestions of existence of an independent intermediate phase between extended and collapsed phases is only a finite size effect.     

Population Balance Model for Miniemulsion Polymerization, 2

Summary: The predictions of the model developed in Part 1 of this series are compared with experimental values taken from literature. Initially, the method of solution of the population balance equation and the simulation algorithm are given. Various radical entry mechanisms are discussed in adequate detail. Plausible arguments are given to identify the correct radical entry mechanism. An expression to evaluate the radical exit coefficient is given. Model predictions of a number of variables are discussed. These include average number of radicals per particle, particle phase monomer volume fraction, average number of radicals averaged over all particles, monomer volume fraction averaged over all particles, variation of nucleation rate, variation of fraction of droplets nucleated, variation of average diameter, variation of standard deviation, variation of polydispersity index, and development of particle size distribution with time. Finally, model predictions for the variation of conversion with time for five different initiator concentrations, number average diameter, standard deviation and full distribution are compared with experimental values.

Variation in the average number of radicals per particle with time, at different collocation points.     

Emulsion liquid membranes: Role of internal droplet size distribution on toluene/n-heptane separation

Using oil/water/oil-type emulsion liquid membranes, batch wise extraction experiments are carried out to separate toluene from a mixture of toluene and n-heptane. In the separation process using emulsion liquid membranes, the internal phase polydispersity affects mass transport of a solute because under steady operating conditions, internal droplet size and size distribution are proportional to the interfacial area. The present study aims to assess the polydispersity character of the internal droplets of emulsion globules. In this paper, the important variables affecting dispersed drop sizes as well as internal droplets mean diameter and size distribution of the emulsion globule, including impeller speed during emulsification, surfactant concentration, volume ratio of surfactant solution, carrier concentration and composition of feed phase are systematically investigated.     

Emulsification in turbulent flow 1. Mean and maximum drop diameters in inertial and viscous regimes

Systematic experimental study of the effects of several factors on the mean and maximum drop sizes during emulsification in turbulent flow is performed. These factors include: (1) rate of energy dissipation, epsilon; (2) interfacial tension, sigma; (3) viscosity of the oil phase, eta(D); (4) viscosity of the aqueous phase, eta(C); and (5) oil volume fraction, Phi. The emulsions are prepared by using the so-called "narrow-gap homogenizer" working in turbulent regime of emulsification. The experiments are performed at high surfactant concentration to avoid the effect of drop-drop coalescence. For emulsions prepared in the inertial turbulent regime, the mean and the maximum drop sizes increase with the increase of eta(D) and sigma, and with the decrease of epsilon. In contrast, Phi and eta(C) affect only slightly the mean and the maximum drop sizes in this regime of emulsification. These results are described very well by a theoretical expression proposed by Davies [Chem. Eng. Sci. 40 (1985) 839], which accounts for the effects of the drop capillary pressure and the viscous dissipation inside the breaking drops. The polydispersity of the emulsions prepared in the inertial regime of emulsification does not depend significantly on sigma and epsilon. However, the emulsion polydispersity increases significantly with the increase of oil viscosity, eta(D). The experiments showed also that the inertial turbulent regime is inappropriate for emulsification of oils with viscosity above ca. 500 mPa s, if drops of micrometer size are to be obtained. The transition from inertial to viscous turbulent regime of emulsification was accomplished by a moderate increase of the viscosity of the aqueous phase (above 5 mPa s in the studied systems) and/or by increase of the oil volume fraction, Phi>0.6. Remarkably, emulsions with drops of micrometer size are easily formed in the viscous turbulent regime of emulsification, even for oils with viscosity as high as 10,000 mPa s. In this regime, the mean drop size rapidly decreases with the increase of eta(C) and Phi (along with the effects of epsilon, sigma, and eta(D), which are qualitatively similar in the inertial and viscous regimes of emulsification). The experimental results are theoretically described and discussed by using expressions from the literature and their modifications (proposed in the current study).     

Kinetics of phase separation in polymer blends. Calculations based on nonlinear theory

Suzuki's scaling theory for transient phenomena is applied to the calculation of the kinetics of phase separation in the early-to-intermediate stage based on a nonlinear theory proposed by Langer, Bar-on, and Miller (LBM). Calculated results are compared with experimental data on light scattering from a polymer blend system. Deviations from predictions of Cahn's linearized theory in the early time range of phase separation can be explained well by the proposed method of calculation. Nonlinear effects are found to play an essential role in characterizing the light scattering behavior of phase separation in the intermediate stage. Time evolutions of the single-point distribution function of composition are calculated, and the results are in good agreement with those reported in digital imaging analysis experiments and computer simulations of the time-dependent Ginzburg-Landau equation. The influence of asymmetry of free-energy on the single-point distribution function is also investigated in this study. © 1993 John Wiley & Sons, Inc.     

分散聚合与水热处理相结合制备单分散聚苯乙烯微球的研究

将分散聚合与水热处理相结合,以聚乙烯醇为稳定剂,以乙醇和水为分散介质,三羟甲基丙烷三丙烯酸酯为交联剂,一步法成功制备得到不同粒径的单分散交联聚苯乙烯微球.以乙醇/水的比例为50/50的反应体系为基础,研究了聚乙烯醇类型和含量,有机相含量,引发剂浓度,以及水热釜填充量等对所制备的微球形貌的影响,发现聚乙烯醇类稳定剂的分子量的降低和含量的增多倾向于生成黏连的微球;在有交联剂的条件下,不含稳定剂的体系仍能够得到单分散的交联PS微球;有机相含量的增加会导致微球呈现多分散性;而体系中引发剂的含量和反应液在水热釜中的填充量对微球的形貌影响不大.进一步针对水热法的特点分析探讨了一步法成功制备单分散的交联聚苯乙烯微球的原因及其机理.     

Efficient emulsification of viscous oils at high drop volume fraction

It is shown experimentally in this study that the increase of drop volume fraction can be used as an efficient tool for emulsification of viscous oils in turbulent flow. In a systematic series of experiments, the effects of drop volume fraction and viscosity of the dispersed phase on the mean, d(32), and maximum, d(V95), diameters of the drops, formed during emulsification, are quantified. The volume fraction, Φ, of the dispersed oily phase is varied between 1% and 90%, and oils with viscosity varying between 3 and 10,000 mPa.s are studied. All experiments are performed at sufficiently high surfactant concentration, as to avoid possible drop-drop coalescence during emulsification. The analysis of the experimental data shows that there is a threshold drop volume fraction, Φ(TR), at which a transition from inertial turbulent regime into viscous turbulent regime of emulsification occurs, due to the increased overall viscosity of the emulsion. At Φ < Φ(TR), d(32) and d(V95) depend weakly on Φ and are well described by known theoretical expression for emulsification in inertial turbulent regime (Davies, Chem. Eng. Sci. 1985, 40, 839), which accounts for the effects of oil viscosity and interfacial tension. At Φ > Φ(TR), both d(32) and polydispersity of the formed emulsions decrease very significantly with the increase of Φ (for the oils with η(D) > 10 mPa.s). Thus, very efficient emulsification of the viscous oils is realized. Very surprisingly, a third regime of emulsification is observed in the range of concentrated emulsions with Φ > 75%, where the mean drop size and emulsion polydispersity are found experimentally to be very similar for all oils and surfactants studied-an experimental fact that does not comply with any of the existing models of drop breakup during emulsification. Possible mechanistic explanations of this result are discussed. The experimental data for semiconcentrated and concentrated emulsions with Φ > Φ(TR) are described by a simple scaling expression, which accounts for the effects of all main factors studied.     

Behaviour of formula emulsions containing hydrolysed whey protein and various lecithins

Formula emulsion systems are used as enteral, sports and health products. In some formulas addition of hydrolysed protein is necessary to guarantee ease of digestion and hypoallergenicity. In the low fat emulsion model an increase in the content of lecithin (phospholipid mixture) was required, in consideration of the advice of the Food and Nutrition Board (USA) for choline supplementation. The individual and interactive effects of whey protein isolate (WPI) or hydrolysate (WPH) (3.7 and 4.9% w/w), unmodified deoiled or hydrolysed lecithin (0.48 or 0.7% w/w) and carbohydrate in the form of maltodextrin with dextrose equivalent (DE) 18.5 or glucose syrup with DE 34 (11% w/w) on the properties of formula emulsions with 4% v/w sunflower oil, were investigated using a full factorial design. The emulsions were characterised by particle size distribution, coalescence stability, creaming rate, and also surface protein and lecithin concentration. WPI-containing emulsions proved to be stable against coalescence and showed only little creaming after 1 and 7 days standing. There was a significant increase in the mean droplet size and a significant deterioration of coalescence and creaming stability when WPH instead of WPI was used as the protein source, due to the lower number of large peptides and lower surface activity of the WPH. Increasing the WPH concentration led to an increase in oil droplet size and further deterioration of the stability of the emulsions. The starch hydrolysate and lecithin also significantly influenced the emulsion properties. Their influence was less strong when the emulsion contained WPI. Under the conditions used WPH-based emulsions were more stable, in terms of creaming and coalescence, when a low level of protein was used in conjunction with hydrolysed lecithin and glucose syrup. Oil droplets in emulsions containing unmodified lecithin in either the continuous or disperse phase and WPH in the continuous phase were very sensitive to coalescence. The addition of starch hydrolysates (DE 18.5) induced intensive flocculation and phase separation in these emulsions.     

分子量多分散性对AB二嵌段共聚物相行为的影响

采用Monte Carlo方法研究了分子量多分散性对AB型嵌段共聚物相行为的影响. 通过调整嵌段共聚物中组分含量, 考察了整体多分散性和单嵌段多分散性对嵌段共聚物共混物的有序-无序转变(Order-disorder transition, ODT)、 形貌及链尺寸的影响. 研究结果表明, 多分散度的增大使无序相向较大χN区域略微移动, 形成的片层结构厚度增加. 在形成微观有序形貌后, 较大分散度时各亚组分的链会得到更大的伸展, 表明分子链堆积受挫的程度减小, 因此, 涨落作用受到的抑制作用减小, 无序相区向更低温度区域移动.     

Direct measurement of depletion and structural forces in polydisperse, charged systems

The effect of polydispersity in macromolecule size or surface potential on the depletion interaction between a spherical silica particle and a silica flat in solutions containing two different types of nonadsorbing charged spherical macromolecules was studied with an atomic force microscope (AFM). The macromolecules used here were negatively charged nanospheres of either polystyrene or silica. To investigate the effect of size polydispersity, experiments were performed under the condition of either constant macromolecule number density or constant volume fraction as the relative proportions of smaller and larger polystyrene nanospheres in the suspension were varied. Similarly, for the experiments with surface potential polydispersity, the suspensions contained varying fractions of more highly charged (polystyrene) and less highly charged (Ludox silica) nanospheres at constant number density. The experimental results were compared to the predictions of the modified force-balance model of Piech and Walz and semiquantitative agreement was found. In particular, the maximum attraction and repulsion observed in the measured force profiles were found to agree with the predicted trends as the makeup of the macromolecules was varied. The trend in the maximum attraction was also consistent with predictions made using a simple "scaling" analysis derived using the equation for hard-sphere interactions.