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The atomic force microscope (AFM), as it is able to characterize surface topography as well determine the AFM cantilever tip-surface force, proves effective at estimating the value of the Hamaker constant, A , for a given solid material. Two main AFM-based methods have been proposed for estimating values of A . In the approach-to-contact (AtC) method, Hamaker constants are inferred from the deflections at which the AFM tip first jumps into contact with the substrate. In the pull-off (PO) method, the deflections that arise when the AFM tip is finally pulled away from the substrate are used to estimate values of A . In prior applications of these two methods, the short-range intermolecular repulsive forces that are known to arise between the AFM tip and the substrate were, however, ignored. Upon invoking a physically relevant description of these short-range and steeply repulsive forces, we investigate the effects of repulsive interactions on the values of A estimated from the existing AtC and PO methods when applied to neutrally charged systems under low humidity or vacuum conditions. For experimentally relevant surfaces, we find that repulsive forces have a modest effect on the AtC method, although they still need to be accounted for in order to generate improved estimates of the Hamaker constant. On the other hand, repulsive forces have a significant effect on the Hamaker constants inferred from the PO method, and must be properly accounted for when using this approach. Our analysis also includes an explicit incorporation of surface roughness into the PO method, which is not typically done in most prior applications of the PO method.

Theories of small systems play an important role in the fundamental understanding of finite size effects in statistical mechanics, as well as the validation of molecular simulation results as no computer can simulate fluids in the thermodynamic limit. Previously, a shell particle was included in the isothermal-isobaric ensemble in order to resolve an ambiguity in the resulting partition function. The shell particle removed either redundant volume states or redundant translational degrees of freedom of the system and yielded quantitative differences from traditional simulations in this ensemble. In this work, we investigate the effect of including a shell particle in the canonical, grand canonical, and Gibbs ensembles. For systems comprised of a pure component ideal gas, analytical expressions for various thermodynamic properties are obtained. We also derive the Metropolis Monte Carlo simulation acceptance criteria for these ensembles with shell particles, and the results of the simulations of an ideal gas are in excellent agreement with the theoretical predictions. The system size dependence of various important ensemble averages is also analyzed., (© 2024 Author(s). Published under an exclusive license by AIP Publishing.)

We consider the thermal, mechanical, and chemical contact of two subsystems composed of ideal gases, both of which are not in the thermodynamic limit. After contact, the combined system is isolated, and the entropy is determined through the use of its standard connection to the phase space density (PSD), where only those microstates at a given energy value are counted. The various intensive properties of these small systems that follow from a derivative of the PSD, such as the temperature, pressure, and chemical potential (evaluated via a backward difference), while equal when the two subsystems are in equilibrium are nevertheless found not to behave in accordance with what is expected from macroscopic thermodynamics. Instead, it is the entropy, defined from its connection to the PSD, that still controls the behavior of these small (nonextensive) systems. We also analyze the contact of these two subsystems utilizing an alternative entropy definition, through its proposed connection to the phase space volume (PSV), where all microstates at or below a given energy value are counted. We show that certain key properties of these small systems obtained with the PSV either do not become equal or do not consistently describe the two subsystems when in contact, suggesting that the PSV should not be used for analyzing the behavior of small isolated systems.

Computational predictions of the polymorphic outcomes of a crystallization process, referred to as polymorph selection, can accelerate the process development for manufacturing solid products with targeted properties. Polymorph selection requires understanding the interplay between the thermodynamic and kinetic factors that drive nucleation. Moreover, post-nucleation events, such as crystal growth and polymorphic transformation, can affect the resulting crystal structures. Here, the nucleation kinetics of the Lennard-Jones (LJ) fluid from the melt is investigated with a focus on the competition between FCC and HCP crystal structures. Both molecular dynamics (MD) simulations and 2D free energy calculations reveal that polymorph selection occurs not during nucleation but when the cluster sizes exceed the critical cluster size. This result contrasts with the classical nucleation mechanism, where each polymorph is assumed to nucleate independently as an ideal bulk-like cluster, comprised only of its given structure. Using the 2D free energy surface and the MD simulation-derived diffusion coefficients, a structure-dependent nucleation rate is estimated, which agrees with the rate obtained from brute force MD simulations. Furthermore, a comprehensive population balance modeling (PBM) approach for polymorph selection is proposed. The PBM combines the calculated nucleation rate with post-nucleation kinetics while accounting for the structural changes of the clusters after nucleation. When applied to the LJ system, the PBM predicts with high accuracy the polymorphic distribution found in a population of crystals generated from MD simulations. Due to the non-classical nucleation mechanism of the LJ system, post-nucleation kinetic events are crucial in determining the structures of the grown crystals.

Having a good understanding of nucleation is critical for the control of many important processes, such as polymorph selection during crystallization. However, a complete picture of the molecular-level mechanisms of nucleation remains elusive. In this work, we take an in-depth look at the NaCl homogeneous nucleation mechanism through thermodynamics. Distinguished from the classical nucleation theory, we calculate the free energy of nucleation as a function of two nucleus size coordinates: crystalline and amorphous cluster sizes. The free energy surface reveals a thermodynamic preference for a nonclassical mechanism of nucleation through a composite cluster, where the crystalline nucleus is surrounded by an amorphous layer. The thickness of the amorphous layer increases with an increase in supersaturation. The computed free energy landscape agrees well with the composite cluster-free energy model, through which phase specific thermodynamic properties are evaluated. As the supersaturation increases, there is a change in stability of the amorphous phase relative to the solution phase, resulting in a change from one-step to two-step mechanism, seen clearly from the free energy profile along the minimum free energy path crossing the transition curve. By obtaining phase-specific diffusion coefficients, we construct the full mesoscopic model and present a clear roadmap for NaCl nucleation., (© The Author(s) 2022. Published by Oxford University Press on behalf of the National Academy of Sciences.)

As inferred from visual observations and turbidity measurements, the average radius of the unilamellar vesicles formed in water from the cationic double-chain surfactant didodecyldimethylammonium bromide (DDAB) varies with the method of preparation, being ∼24 nm after sonication (SS method) and ∼74 nm after extrusion/ultrafiltration (SE method). The radii were larger when the vesicles were produced in 10 mM NaBr, ∼65 nm for the SS method and ∼280 nm for the SE method. The specific turbidity, or turbidity per unit path length divided by the surfactant weight fraction, w , of these vesicular dispersions increased with decreasing w until a constant value was reached at w *, which depends on the preparation method and the dispersion medium. The constant specific turbidities are indicative of single and independent scattering and were used to estimate vesicle radii by solving the specific turbidity equations derived for the Rayleigh-Debye-Gans (RDG) regime. Two turbidity equations were used, one accounting for absorbance errors due to some scattered light reaching the detector and another with no correction. Estimates of the average distances between the vesicles and their corresponding Debye lengths were obtained for evaluating the importance of intervesicle electrostatic interactions, which could lead to dependent scattering at higher weight fractions.

Modifications to the traditional Onsager theory for modeling isotropic-nematic phase transitions in hard prolate spheroidal systems are presented. Pure component systems are used to identify the need to update the Lee-Parsons resummation term. The Lee-Parsons resummation term uses the Carnahan-Starling equation of state to approximate higher-order virial coefficients beyond the second virial coefficient employed in Onsager's original theoretical approach. As more exact ways of calculating the excluded volume of two hard prolate spheroids of a given orientation are used, the division of the excluded volume by eight, which is an empirical correction used in the original Lee-Parsons resummation term, must be replaced by six to yield a better match between the theoretical and simulation results. These modifications are also extended to binary mixtures of hard prolate spheroids using the Boublík-Mansoori-Carnahan-Starling-Leland (BMCSL) equation of state.

Hypothesis: Since the volume fraction of the surfactant bilayer(s), of thickness d b , in a vesicle and liposome is smaller than one, the dependences of the Rayleigh (R) scattering intensity and turbidity on the particle radius a are weaker than those for a homogeneous sphere, which are proportional to a 3 . The dependences of the Rayleigh-Debye-Gans (RDG) scattering intensity and turbidity on a are also weaker. Work done: The dependences of the effective relative refractive index on a, d b , and d w (water layer thickness) were derived. The specific Rayleigh ratio [Formula: see text] and the specific turbidity τ ** for single and independent scattering were derived analytically for R and RDG scattering. Spectroturbidimetry data at 25 ° C for a cationic double-chain surfactant, didodecyldimethylammonium bromide (DDAB) were compared to the turbidity predictions., Findings: For R scattering, [Formula: see text] and τ ** are proportional to a 2 d b for vesicles, and to a 3 d b d w +d b for liposomes. For RDG and particle radii 20-1000 nm, τ ** is proportional to a n , where n is 2 to 0.4 for vesicles and 2 to 1.1 for liposomes. Turbidity data for DDAB vesicles are consistent with the RDG predictions, which are also used to estimate the vesicles' sizes. RDG applies to liposomes < 800 nm and to much larger sizes for vesicles., (Copyright © 2020 Elsevier Inc. All rights reserved.)

A novel method in which vesicular dispersions of the double-chain cationic surfactant DDAB (didodecyldimethylammonium bromide) stabilize suspensions of high density titania particles was recently presented (Yang, Y.-J; Corti, D.S.; Franses, E. I. Langmuir 2015, 31, 8802-8808). At high enough DDAB concentration, the vesicles form a close-packed structure, providing strong resistance to the sedimentation of the titania particles, while the dispersions remain highly shear-thinning with moderate limiting viscosities. Here, to elucidate the key factors of the mechanism by which vesicles or other nonsettling particles stabilize high density particles against sedimentation, we use Brownian dynamics simulations (BDS) to examine the sedimentation behavior of mixtures of "dense particles" that settle rapidly on their own and "light particles" that represent nonsettling "rigid vesicles". BDS confirm that for large enough values of the volume fraction ϕ 2 of the light particles, the dense particles should remain suspended. The rheological behavior of the mixtures is also computed with BDS. The observed shear-thinning behavior of the light particle dispersion suggests that the suspensions of the dense particles are still flowable at high shear stresses. Furthermore, the local viscosity of light particles around the dense particles significantly increases with increasing ϕ 2 , particularly when the same gravitational force applied in the BDS is exerted on a dense particle. The arrangement of light particles around the moving dense particles is an important factor in determining the stability of the dense particles against sedimentation. The BDS results indicate that dispersions of nonsettling particles provide a general method for the stabilization against sedimentation of high density particles.

Hypothesis: Surface effects arising from roughness and deformation can negatively affect the results of AFM contact experiments. Using the non-contact portion of an AFM deflection curve is therefore desirable for estimating the Hamaker constant, A, of a solid material. A previously validated non-contact quasi-dynamic method for estimating A is revisited, in which the cantilever tip is now always represented by an "effective sphere". In addition to simplifying this previous method, accurate estimates of A can still be obtained even though precise knowledge of the nanoscale geometric features of the cantilever tip are no longer required., Experiments: The tip's "effective" radius of curvature, R eff , is determined from a "calibration" step, in which the tip's deflection at first contact with the surface is measured for a substrate with a known Hamaker constant. After R eff is known for a given tip, estimates of A for other surfaces of interest are then determined., Findings: An experimental study was conducted to validate the new method and the obtained results are in good agreement with predictions from the Lifshitz approximation, when available. Since R eff accounts for all geometric uncertainties of the tip through a single fitted parameter, no visual fitting of the tip shape was required., (Copyright © 2018 Elsevier Inc. All rights reserved.)

In order to minimize the effects of surface roughness and deformation, a new method for estimating the Hamaker constant, A, of solids using the approach-to-contact regime of an atomic force microscope (AFM) is presented. First, a previous "jump-into-contact" quasi-static method for determining A from AFM measurements is analyzed and then extended to include various AFM tip-surface force models of interest. Then, to test the efficacy of the "jump-into-contact" method, a dynamic model of the AFM tip motion is developed. For finite AFM cantilever-surface approach speeds, a true "jump" point, or limit of stability, is found not to appear, and the quasi-static model fails to represent the dynamic tip behavior at close tip-surface separations. Hence, a new "quasi-dynamic" method for estimating A is proposed that uses the dynamically well-defined deflection at which the tip and surface first come into contact, d c , instead of the dynamically ill-defined "jump" point. With the new method, an apparent Hamaker constant, A app , is calculated from d c and a corresponding quasi-static-based equation. Since A app depends on the cantilever's approach speed, v c , and the AFM's sampling resolution, δ, a double extrapolation procedure is used to determine A app in the quasi-static (v c → 0) and continuous sampling (δ → 0) limits, thereby recovering the "true" value of A. The accuracy of the new method is validated using simulated AFM data. To enable the experimental implementation of this method, a new dimensionless parameter τ is introduced to guide cantilever selection and the AFM operating conditions. The value of τ quantifies how close a given cantilever is to its quasi-static limit for a chosen cantilever-surface approach speed. For sufficiently small values of τ (i.e., a cantilever that effectively behaves "quasi-statically"), simulated data indicate that A app will be within ∼3% or less of the inputted value of the Hamaker constant. This implies that Hamaker constants can be reliably estimated using a single measurement taken with an appropriately chosen cantilever and a slow, yet practical, approach speed (with no extrapolation required). This result is confirmed by the very good agreement found between the experimental AFM results obtained using this new method and previously reported predictions of A for amorphous silica, polystyrene, and α-Al 2 O 3 substrates obtained using the Lifshitz method.

Brownian aggregation rates are determined for concentrated dispersions of interacting particles with Brownian dynamics (BD) simulations and various theoretical models. Using simulation results as benchmarks, the predictions of the classical Fuchs-Smoluchowski (FS) model are shown to be quite inaccurate for concentrated dispersions. A new aggregation model is presented which provides significantly improved predictions. This model is developed on the basis of the fundamental measure theory (FMT) which is a rigorous "liquid-state" dynamic density-functional theory (DDFT) approach. It provides a major improvement of the FS model by considering short-range ordering, non-ideal diffusion, and unsteady-state effects. These were recently shown by the authors to play important roles in Brownian aggregation of hard spheres at high concentrations. Two types of interparticle interaction potentials are examined, the purely attractive van der Waals potential and the DLVO potential which includes van der Waals attraction and electrostatic double layer repulsion. For dispersions of particles with purely attractive interactions, the FS model underpredicts the aggregation rates by up to 1000 fold. In the presence of strong interparticle repulsive forces, its predictions are in fair agreement with the BD simulation results for dilute systems with particle volume fractions ϕ < < 0.1. In contrast, the predictions of the new FM-DDFT based model compare favorably with the BD simulation results, in both cases, up to ϕ = 0.3. A new quantitative measure for colloidal dispersion stability, different from the classical FS stability ratio, is proposed on the basis of aggregation half-times. Hence, a better mechanistic understanding of Brownian aggregation is obtained for concentrated dispersions of particles with either attractive or repulsive interactions, or both.

For many applications of colloidal dispersions, the particles must be suspended for a long time. This is often accomplished by preventing agglomeration, which generates aggregates of increasing size. Nevertheless, many colloidal dispersions of dense particles may settle even without agglomeration. Preventing sedimentation without significantly increasing the bulk dispersion viscosity is difficult and has received little attention in the literature. However, settling can be drastically reduced through the novel use of close-packed vesicular dispersions at high enough concentrations, which are non-Newtonian shear-thinning fluids. Such dispersions have much higher viscosities at the low shear stresses "felt" by sedimenting colloidal particles than at the high shear stresses relevant to bulk dispersion flow. In a practical example, dense TiO2 nanoparticles which normally would settle rapidly can remain suspended for at least 6 months without any observable sedimentation when they are introduced into a close-packed vesicular dispersion, while the dispersion retains its flowability. Cryo-TEM images reveal that the vesicles in these dispersions are tightly close-packed. Dynamic light scattering and electrophoretic mobility data also confirm that the vesicles in such dispersions have very low mobilities.

Hypothesis: As more sodium dodecylsulfate (SDS) monomers adsorb at the water/titanium dioxide (TiO2) nanoparticles interface, the particles become more stable against agglomeration and sediment more slowly. SDS micelles are not expected to adsorb on the particles and affect the stability against agglomeration or sedimentation. Since micelles are smaller than the 300 nm TiO2 nanoparticles studied, they may introduce depletion forces which may affect the dispersion stability., Experiments and Models: Sedimentation times were measured in water and in 100 mM NaCl for SDS concentrations from 0.1 to 200 mM. Adsorption densities of SDS and zeta potentials of particles were measured. Dynamic light scattering was used to measure average diameters of particles or particle agglomerates. Modeling of sedimentation/diffusion was done to predict sedimentation times of particles. Modeling of agglomeration rates was done to help predict sedimentation rates of clusters., Findings: At SDS concentrations close to or above the cmc, up to 60 mM in water or 115 mM in 100 mM NaCl, the nanoparticles sediment most slowly without any agglomeration. At higher micelle concentration, SDS micelle depletion forces are very strong, causing fast flocculation, without coagulation. Then sedimentation occurs much faster. The effective micelle depletant size includes about 4 Debye lengths of the charged micelles or particles., (Copyright © 2015 Elsevier Inc. All rights reserved.)

Brownian aggregation in concentrated hard-sphere dispersions is studied using models and Brownian dynamics (BD) simulations. Two new theoretical models are presented and compared to several existing approaches and BD simulation results, which serve as benchmarks. The first new model is an improvement over an existing local density approximation (LDA)-based model. The other is based on the more rigorous Fundamental measure theory (FMT) applied to the "liquid-state" dynamic density-functional theory (DDFT). Both models provide significant improvements over the classical Smoluchowski model. The predictions of the new FM-DDFT-based model for aggregation kinetics are in excellent agreement with BD simulation results for dispersions with initial particle volume fractions, ϕ, up to 0.35 (close to the hard-sphere freezing transition at ϕ = 0.494). In contrast to previous approaches, the nonideal particle diffusion effects and the initial and time-dependent short-range ordering in concentrated dispersions due to entropic packing effects are explicitly considered here, in addition to the unsteady-state effects. The greater accuracy of the FM-DDFT-based model compared to that of the LDA-based models indicates that nonlocal contributions to particle diffusion (only accounted for in the former) play important roles in aggregation. At high concentrations, the FM-DDFT-based model predicts aggregation half-times and gelation times that are up to 2 orders of magnitude shorter than those of the Smoluchowski model. Moreover, the FM-DDFT-based model predicts asymmetric cluster-cluster aggregation rate constants, at least for short times. Overall, a rigorous mechanistic understanding of the enhancement of aggregation kinetics in concentrated dispersions is provided.

We propose an equilibrium embryo definition for homogeneous bubble nucleation within the pure component superheated Lennard-Jones liquid. The suggested embryo definition serves as an improvement to a previous (n,v) embryo model (Uline and Corti 2007). In that model, the constrained equilibrium between the bubble and the surrounding superheated liquid was maintained by placing n particles within a spherical volume v, without concern for the redundant counting of configurations and the relevance of the model to the dynamics of a nucleation process. To eliminate this redundancy, while only considering those embryos that should most likely appear at a transitional state, we now define the volume of the embryo via the use of a shell particle and only include n particles inside the volume that are deemed to be "vapor-like". The underlying free energy surface of formation of the new (n,v) embryo model is generated via Monte Carlo simulation and also approximately by a suggested density functional theory method. The resulting surface implies that small and locally confined regions of near-zero density serve as precursors initiating homogeneous bubble nucleation. Furthermore, the nonredundant counting of the embryo configurations yields a well-defined and sharp conduit in the free energy surface that directs the initially formed embryos toward the critical nucleus. We discuss how the suggested equilibrium embryo model aids in both the identification of the transitional configurations and calculation of the average number density of the critical nuclei within the superheated liquid phase, which is the focus of the companion paper (DOI 10.1021/jp404151h).

In the present paper, we develop a method to calculate the rate of homogeneous bubble nucleation within a superheated L-J liquid based on the (n,v) equilibrium embryo free energy surface introduced in the first paper (DOI: 10.1021/jp404149n). We express the nucleation rate as the product of the concentration of critical nuclei within the metastable liquid phase and the relevant forward rate coefficient. We calculate the forward rate coefficient of the critical nuclei from their average lifetime as determined from MD simulations of a large number of embryo trajectories initiated from the transitional region of the metastable liquid configuration space. Therefore, the proposed rate coefficient does not rely on any predefined reaction coordinate. In our model, the critical nuclei belong to the region of the configuration space where the committor probability is about one-half, guaranteeing the dynamical relevance of the proposed embryos. One novel characteristic of our approach is that we define a limit for the configuration space of the equilibrium metastable phase and do not include the configurations that have zero committor probability in the nucleation free energy surface. Furthermore, in order to take into account the transitional degrees of freedom of the critical nuclei, we develop a simulation-based approach for rigorously mapping the free energy of the (n,v) equilibrium embryos to the concentration of the critical nuclei within the bulk metastable liquid phase.

The classical steady-state Smoluchowski model for Brownian coagulation is evaluated using Brownian Dynamics Simulations (BDS) as a benchmark. The predictions of this approach compare favorably with the results of BDS only in the dilute limit, that is, for volume fractions of φ≤5×10(-4). From the solution of the more general unsteady-state diffusion equation, a new model for coagulation is developed. The resulting coagulation rate constant is time-dependent and approaches the steady-state limit only at large times. Moreover, in contrast to the Smoluchowski model, this rate constant depends on the particle size, with the transient effects becoming more significant at larger sizes. The predictions of the unsteady-state model agree well with the BDS results up to volume fractions of about φ=0.1, at which the aggregation half-time predicted by the Smoluchowski model is five times that of the BDS. A new procedure to extract the aggregation rate constant from simulation results based on this model is presented. The choice of the rate constant kernel used in the population balance equations for complete aggregation is also evaluated. Extension of the new model to a variable rate constant kernel leads to increased accuracy of the predictions, especially for φ≤5×10(-3). This size-dependence of the rate constant kernel affects particularly the predictions for initially polydisperse sphere systems. In addition, the model is extended to account in a novel way for both short-range viscous two-particle interactions and long-range many-particle Hydrodynamic Interactions (HI). Predictions including HI agree best with the BDS results. The new models presented here offer accurate and computationally less-intensive predictions of the coagulation dynamics while also accounting for hydrodynamic coupling., (Copyright © 2012 Elsevier Inc. All rights reserved.)

The effect of Triton X-100 on the colloidal dispersion stability of CuPc-U (unsulfonated and hydrophobic) and CuPc-S (surface sulfonated and hydrophilic) particles in aqueous solutions (water and NaNO(3)) was investigated at 25 °C. Its adsorption density was determined from surfactant concentrations analyzed by an HPLC method with a UV detector. The experimental dispersion stability ratios of the particles were determined from dynamic light scattering (DLS) data, with the Rayleigh-Debye-Gans (RDG) light scattering theory. The adsorption densities of Triton X-100 on both the CuPc-U and CuPc-S increase with increasing concentration of surfactant up to the critical micelle concentration (cmc), and then reach a plateau. The maximum adsorption density Γ(m) is higher for the CuPc-U (d(h)=160 nm) than that for the CuPc-S (d(h)=90 nm). The hydrophobic chains are inferred to be adsorbed onto the surfaces, and the hydrophilic ethylene oxide chains are in a coil conformation. The W(app)-values for the CuPc-U dispersions are affected mainly by the surfactant fractional surface coverage θ. Adding NaNO(3) has no significant effect on the dispersion stability. The stabilization mechanism for the CuPc-U is inferred to be primarily steric, as expected. The stability ratios for the CuPc-S in solutions with NaNO(3) are higher than those for CuPc-U, and decrease with increasing concentration of NaNO(3), indicating that the stabilization is affected by the screening of electrostatic repulsive forces. The zeta potential is not a good predictor of the electrostatic stabilization, pointing to the need for new and improved theories., (Copyright © 2011. Published by Elsevier Inc.)

We present a new self-consistent thermodynamic formalism for the interfacial properties of nanoscale embryos whose interiors do not exhibit bulklike behavior and are in complete equilibrium with the surrounding mother phase. In contrast to the standard Gibbsian analysis, whereby a bulk reference pressure based on the same temperature and chemical potentials of the mother phase is introduced, our approach naturally incorporates the normal pressure at the center of the embryo as an appropriate reference pressure. While the interfacial properties of small embryos that follow from the use of these two reference pressures are different, both methods yield by construction the same reversible work of embryo formation as well as consistency between their respective thermodynamic and mechanical routes to the surface tension. Hence, there is no a priori reason to select one method over another. Nevertheless, we argue, and demonstrate via a density-functional theory (with the local density approximation) analysis of embryo formation in the pure component Lennard-Jones fluid, that our new method generates more physically appealing trends. For example, within the new approach the surface tension at all locations of the dividing surface vanishes at the spinodal where the density profile spanning the embryo and mother phase becomes completely uniform (only the surface tension at the Gibbs surface of tension vanishes in the Gibbsian method at this same limit). Also, for bubbles, the location of the surface of tension now diverges at the spinodal, similar to the divergent behavior exhibited by the equimolar dividing surface (in the Gibbsian method, the location of the surface of tension vanishes instead). For droplets, the new method allows for the appearance of negative surface tensions (the Gibbsian method always yields positive tensions) when the normal pressures within the interior of the embryo become less than the bulk pressure of the surrounding vapor phase. Such a prediction, which is allowed by thermodynamics, is consistent with the interpretation that the mother phase's attempted compression of the droplet is counterbalanced by the negative surface tension, or free energy cost to decrease the interfacial area. Furthermore, for these same droplets, the surface of tension can no longer be meaningfully defined (the surface of tension always remains well defined in the Gibbsian method). Within the new method, the dividing surface at which the surface tension equals zero emerges as a new lengthscale, which has various thermodynamic analogs to and similar behavior as the surface of tension.

A completely generalized version of an inhomogeneous scaled particle theory (I-SPT) for hard particle fluids confined by hard walls is presented, whereby the reversible work of cavity insertion can be determined for a cavity of any radius located at any distance from the hard wall. New exact and approximate conditions on the central function Ḡ of I-SPT are developed, where Ḡ is related to the average value of the anisotropic density of hard-sphere centers at the surface of the cavity. The predictions of the work of insertion and the form of Ḡ are quite accurate up to moderate bulk densities as compared to molecular simulation results. The accuracy of I-SPT begins to decline at high densities, due to limitations of certain needed approximations required for a complete description of Ḡ. Finally, interesting insights into the origin of depletion effects between a hard-sphere solute and the hard wall are generated via this version of I-SPT. The oscillatory nature of depletion forces, exhibiting both attractive and repulsive domains, is found to arise from the interplay between bulk SPT and I-SPT relations.

The free energy of forming a droplet and a bubble with a given particle number n and volume v within the pure-component Lennard-Jones supercooled vapor and superheated liquid, respectively, are further explored using density-functional theory. Similar to what was found previously [M. J. Uline and D. S. Corti, Phys. Rev. Lett. 99, 076102 (2007); M. J. Uline and D. S. Corti, J. Chem. Phys. 129, 234507 (2008)], the limits of stability again appear within both free energy surfaces evaluated at two other metastability conditions, one closer to the binodal and one closer to the spinodal. Furthermore, an ad hoc bond connectivity criterion is also applied in an attempt, however approximately, to eliminate certain configurational redundancies that arise from the chosen droplet and bubble definitions. What results are free energy surfaces describing the formation of equilibrium embryos that should be an improved representation of the fluctuations that are relevant to those nonequilibrium embryos seen in an actual nucleation event. Finally, we discuss in some detail the use of the (n,v) reaction coordinate within the framework of an equilibrium-based theory and its relation to other descriptions of nucleation.

The free energy of forming a droplet and a bubble with a given number of particles n inside a volume v within the pure component Lennard-Jones supercooled vapor and superheated liquid, respectively, is further explored using density-functional theory. Certain key aspects of the free energy surface for bubble formation, such as the radius of the bubble at a stability limit, are found to scale in a nearly temperature independent manner when plotted versus a parameter that quantifies the location of the given state point in the metastable region. The corresponding work at this stability limit exhibits scaling for small values of n, but shows a strong temperature dependence for large n. No aspect of the free energy surface for droplet formation shows scaling over the full range of metastability conditions, including the work of forming the critical droplet and the radius of a droplet at its stability limit. Hence, there is no "universal" surface for embryo formation in metastable fluids. We also generate by thermodynamic arguments alone droplet and bubble trajectories along the corresponding free energy surfaces that avoid by construction the locus of instabilities, which match quite well the results obtained from other approaches. We also discuss in greater detail the use of the (n,v) order parameter within an equilibrium-based description of embryo formation, focusing on why the density profile of the embryo is found to be discontinuous at the embryo surface and why stability limits are expected to develop at certain bubble radii.

Previous equilibrium-based density-functional theory (DFT) analyses of cavity formation in the pure component superheated Lennard-Jones (LJ) liquid [S. Punnathanam and D. S. Corti, J. Chem. Phys. 119, 10224 (2003); M. J. Uline and D. S. Corti, Phys. Rev. Lett. 99, 076102 (2007)] revealed that a thermodynamic limit of stability appears in which no liquidlike density profile can develop for cavity radii greater than some critical size (being a function of temperature and bulk density). The existence of these stability limits was also verified using isothermal-isobaric Monte Carlo (MC) simulations. To test the possible relevance of these limits of stability to a dynamically evolving system, one that may be important for homogeneous bubble nucleation, we perform isothermal-isobaric molecular dynamics (MD) simulations in which cavities of different sizes are placed within the superheated LJ liquid. When the impermeable boundary utilized to generate a cavity is removed, the MD simulations show that the cavity collapses and the overall density of the system remains liquidlike, i.e., the system is stable, when the initial cavity radius is below some certain value. On the other hand, when the initial radius is large enough, the cavity expands and the overall density of the system rapidly decreases toward vaporlike densities, i.e., the system is unstable. Unlike the DFT predictions, however, the transition between stability and instability is not infinitely sharp. The fraction of initial configurations that generate an instability (or a phase separation) increases from zero to unity as the initial cavity radius increases over a relatively narrow range of values, which spans the predicted stability limit obtained from equilibrium MC simulations. The simulation results presented here provide initial evidence that the equilibrium-based stability limits predicted in the previous DFT and MC simulation studies may play some role, yet to be fully determined, in the homogeneous nucleation and growth of embryos within metastable fluids.

The dispersion stability and the zeta potentials of nonspherical crystalline (beta-form) copper phthalocyanine (CuPc) particles of hydrodynamic diameter d(h) approximately = 90 nm were investigated at 25 degrees C in water and in aqueous solutions of NaNO(3). The electrolyte concentrations c ranged from 1 to 500 mM and the particle concentrations ranged from 50 to 10,000 ppm (0.005 to 1 wt %). In each case, the Fuchs-Smoluchowski stability ratio W was determined from dynamic light scattering (DLS) data and the Rayleigh-Debye-Gans (RDG) scattering theory. The data suggest that electrostatic effects play a major role in the stability of CuPc-based dispersions. The calculated particle charge z per CuPc particle based on the zeta potential data and the area of the particles (assumed to be cubical) suggests that there is preferential adsorption of NO(3)(-) ions on the uncharged CuPc surface, and the surface charge increases with increasing electrolyte concentration. Furthermore, two new models of the DLVO theory, for spheres and for parallel face-to-face cubes, were reformulated in dimensionless form. The Hamaker constant of CuPc particles was calculated by the same authors on the basis of theoretical models in J. Chem. Theory Comput. (2010, in press). The key dimensionless group is the ratio N of the electrostatic double layer energy to the Hamaker constant. The two DLVO models were used to predict the value of a dimensionless maximum potential energy Phi(max), the conditions when it may exist, and then the value of W. In water, the DLVO model for spheres overpredicted the stability, while the model for cubes underpredicted the stability. At c = 1 mM, both models overpredicted the stability. At c = 10 mM, the model for spheres underpredicted the stability, whereas the model for cubes overpredicted the stability. Hence, there seems to be some significant shape effects on the electrostatic stabilization of CuPc particle dispersions. At c = 100 and 500 mM, both models underpredicted the stability substantially, suggesting the existence of additional short-range repulsive forces, which may primarily control the stability. Simple sensitivity analysis on the calculations supported these conclusions.

The colloidal dispersion stability at 25 degrees C of aqueous dispersions of sonicated DPPC (dipalmitoylphosphatidylcholine) vesicles was quantitatively evaluated from the Fuchs-Smoluchowski stability ratio W. Data of average particle size vs. time were obtained with dynamic light scattering measurements. Dispersions in water, 1, 10, or 150mM aqueous sodium chloride, and phosphate buffer saline (PBS) solutions were tested. The W-values ranged from about 8x10(6) to 6x10(8). The dispersed particles had small but significant zeta-potentials zeta, implying that the vesicles had some significant charge due to preferential adsorption of negative hydroxyl ions over hydronium ions in water, and chloride ions over sodium ions in the electrolyte solutions. Hence, there was some contribution of the double-layer electrostatic forces to the dispersion stability. The initial values of zeta ranged from -30 to 8mV. The vesicle non-retarded Hamaker constants were estimated from the published values of the Hamaker constants of DPPC and water in vacuum, and they varied with the vesicle size. A new dimensionless model of the DLVO theory for spherical particles was formulated, focusing on the conditions for the existence of a positive interaction potential energy maximum, Phi(max), which is linked to W. The dimensionless number N, which is defined as the ratio of the electric double layer energy to the Hamaker constant A, was found to be the key determinant of Phi(max) and W. DLVO calculations of Phi(max), and W with error analysis show that the charged DPPC vesicles tested are quite more stable than predicted. This discrepancy highlights some significant weaknesses in the premises and approximations of the DLVO theory, and the need for improved theories, possibly considering other repulsive forces., (Copyright 2009 Elsevier Inc. All rights reserved.)

A time-dependent density functional theory (TDDFT) scheme has been validated for predictions of the dispersion coefficients of five molecules (H2O, NH3, CO2, C6H6, and pentane) and for predictions of the static dipole polarizabilities of three organometallic compounds (TiCl4, OsO4, and Ge(CH3)4). The convergence of grid spacing has been examined, and two types of pseudopotentials and 13 density functionals have been tested. The nonretarded Hamaker constants A11 are calculated by employing a semiempirical parameter a along with the standard Hamaker constant equation. The parameter a is optimized against six accurate Hamaker constants obtained from the full Lifshitz theory. The dispersion coefficients of copper phthalocyanine CuPc and CuPc-SO3H are then computed. Using the theoretical densities of ρ1 = 1.63 and 1.62 g/cm(3), the Hamaker constants A11 of crystalline α-CuPc and β-CuPc are found to be 14.73 × 10(-20) and 14.66 × 10(-20) J, respectively. Using the experimentally derived density of ρ1 = 1.56 g/cm(3) for a commercially available β-CuPc (nanoparticles of ∼90 nm hydrodynamic diameter), A11 = 13.52 × 10(-20) J is found. Its corresponding effective Hamaker constant in water (A121) is calculated to be 3.07 × 10(-20) J. All computed A11 values for CuPc are noted to be higher than those reported previously.

We present a formally exact thermodynamic treatment of curved boundary layers. Specifically, we extend the boundary layer analysis of Mandell and Reiss [ J. Stat. Phys. 1975 , 13 , 107 ] for a spherical cavity located within a uniform bulk fluid to the case of a cavity intersecting a hard, structureless wall. We derive various expressions for the line tension of an intersecting cavity, all of which can be evaluated for a hard-sphere fluid using existing versions of scaled particle theory. Since the analysis is similar to the standard approach for describing curved interfacial layers, several boundary analogues to conventional interfacial relations appear. In some instances, we obtain results that apparently have not yet been derived either for boundary layers or for their parallel relations in interfacial thermodynamics. Several results offer interesting insights into the behavior of the line tension of a cavity when the cavity approaches macroscopic sizes or in the specific limit where the cavity no longer intersects the wall.

For the pure-component supercooled Lennard-Jones vapor, the free energy of forming a droplet with a given particle number and volume is calculated using density-functional theory. In contrast to what was noted in previous studies, the free energy surface beyond the pseudosaddle point no longer exhibits a valley but rather channels the nuclei toward a locus of instabilities, initiating an unstable growth phase. Similar to a previous study of bubble formation in superheated liquids [M. J. Uline and D. S. Corti, Phys. Rev. Lett. 99, 076102 (2007)], a new picture of homogeneous droplet nucleation and growth emerges.

Based on the approach of Gruhn and Monson [Phys. Rev. E 63, 061106 (2001)], we present a new method for deriving the collisions dynamics for particles that interact via discontinuous potentials. By invoking the conservation of the extended Hamiltonian, we generate molecular dynamics (MD) algorithms for simulating the hard-sphere and square-well fluids within the isothermal-isobaric (NpT) ensemble. Consistent with the recent rigorous reformulation of the NpT ensemble partition function, the equations of motion impose a constant external pressure via the introduction of a shell particle of known mass [M. J. Uline and D. S. Corti, J. Chem. Phys. 123, 164101 (2005); 123, 164102 (2005)], which serves to define uniquely the volume of the system. The particles are also connected to a temperature reservoir through the use of a chain of Nose-Hoover thermostats, the properties of which are not affected by a hard-sphere or square-well collision. By using the Liouville operator formalism and the Trotter expansion theorem to integrate the equations of motion, the update of the thermostat variables can be decoupled from the update of the positions of the particles and the momentum changes upon a collision. Hence, once the appropriate collision dynamics for the isobaric-isenthalpic (NpH) equations of motion is known, the adaptation of the algorithm to the NpT ensemble is straightforward. Results of MD simulations for the pure component square-well fluid are presented and serve to validate our algorithm. Finally, since the mass of the shell particle is known, the system itself, and not a piston of arbitrary mass, controls the time scales for internal pressure and volume fluctuations. We therefore consider the influence of the shell particle algorithm on the dynamics of the square-well fluid.

We consider various ensemble averages within the molecular dynamics (MD) ensemble, corresponding to those states sampled during a MD simulation in which the application of periodic boundary conditions imposes a constraint on the momentum of the center of mass. As noted by Shirts et al. [J. Chem. Phys. 125, 164102 (2006)] for an isolated system, we find that the principle of equipartition is not satisfied within such simulations, i.e., the total kinetic energy of the system is not shared equally among all the translational degrees of freedom. Nevertheless, we derive two different versions of Tolman's generalized equipartition theorem, one appropriate for the canonical ensemble and the other relevant to the microcanonical ensemble. In both cases, the breakdown of the principle of equipartition immediately follows from Tolman's result. The translational degrees of freedom are, however, still equivalent, being coupled to the same bulk property in an identical manner. We also show that the temperature of an isolated system is not directly proportional to the average of the total kinetic energy (in contrast to the direct proportionality that arises between the temperature of the external bath and the kinetic energy within the canonical ensemble). Consequently, the system temperature does not appear within Tolman's generalized equipartition theorem for the microcanonical ensemble (unlike the immediate appearance of the temperature of the external bath within the canonical ensemble). Both of these results serve to highlight the flaws in the argument put forth by Hertz [Ann. Phys. 33, 225 (1910); 33, 537 (1910)] for defining the entropy of an isolated system via the integral of the phase space volume. Only the Boltzmann-Planck entropy definition, which connects entropy to the integral of the phase space density, leads to the correct description of the properties of a finite, isolated system. We demonstrate that the use of the integral of the phase space volume leads to unphysical results, indicating that the property of adiabatic invariance has little to do with the behavior of small systems.

We present a modification to a previously proposed method of adapting scaled particle theory (SPT) to an arbitrary hard-sphere equation of state that satisfies a large number of exact SPT conditions, including thermodynamic consistency. By introducing a set of functions to interpolate the density of hard-spheres centers at the cavity surface, a broad range of hard-sphere properties, in particular the planar surface tension and related properties, are predicted with high accuracy as compared to simulation data. Similarly accurate results are obtained when this modified interpolation scheme is incorporated into a self-consistent version of SPT, i.e., an equation of state is a predicted output of the method. Hence, SPT is now able to closely match the surface thermodynamic properties of the hard-sphere fluid either without using any adjustable parameters or by simply setting the pressure and chemical potential via a reliable equation of state. We also consider other interpolation schemes, some of which better represent certain exact relations that can be derived within SPT. The limited success of these more rigorous approaches provides insights into the various trade-offs between the simplicity and rigor of the chosen interpolation method, as well as the accuracy of the results, that arise in any (inexact) version of SPT.

For the superheated Lennard-Jones liquid, the free energy of forming a bubble with a given particle number and volume is calculated using density-functional theory. As conjectured, a consequence of known properties of the critical cavity [S. N. Punnathanam and D. S. Corti, J. Chem. Phys. 119, 10 224 (2003), the free energy surface terminates at a locus of instability. These stability limits reside, however, unexpectedly close to the saddle point. A new picture of homogeneous bubble nucleation and growth emerges from our study, being more appropriately described as an "activated instability."

We present a further development of an inhomogeneous scaled particle theory (I-SPT) for hard particle fluids confined by hard walls, such that the reversible work of cavity insertion can now be determined for all cavities that intersect one of the walls. Building upon a previous version of I-SPT [D. W. Siderius and D. S. Corti, Phys. Rev. E, 71, 036141 (2005)], a new function, F[over ] , is introduced, which is proportional to the net force on the surface of the cavity in the direction normal to the wall. The reversible work of cavity insertion is then determined by an integral over the force required to "push" the cavity of fixed size into the fluid starting from a position behind the wall. An exact relation for F[over ] at certain cavity locations and radii is derived and an accurate interpolation scheme is proposed for the computation of F[over ] beyond these exact limits. The chosen interpolation incorporates a large number of exact and nearly exact conditions, several of which follow from the surface thermodynamics of macroscopic cavities. Work predictions using F[over ] are highly accurate as compared to simulation results at low to moderate fluid densities. Good agreement still persists at densities near the hard-sphere freezing transition. The interpolation of F[over ] is also used to estimate the depletion force between a hard sphere solute and the wall. The I-SPT entropic force predictions are in good agreement with simulation results presented in the literature. Due to its reliance upon physical and geometric arguments, I-SPT provides important insights into the origin of various depletion effects such as how the interplay between geometry and the varying local density at the cavity surface gives rise to the appearance of multiple attractive regions at intermediate solute sizes and a universal repulsive region, both within solute to wall separations that are less than the diameter of a solvent particle. Finally, all of the scaled particle theory-based methods presented here can, in principle, be extended to describe hard particle fluids confined by nonplanar surfaces, thereby providing estimates of the depletion force between a solute and a variety of surfaces of interest.

The results of a series of constant pressure and temperature molecular-dynamics (MD) simulation studies based on the rigorous shell particle formulation of the isothermal-isobaric (NpT) ensemble are presented. These MD simulations validate the newly proposed constant pressure equations of motion in which a "shell" particle is used to define uniquely the volume of the system [M. J. Uline and D. S. Corti, J. Chem. Phys. (to be published), preceding paper]. Ensemble averages obtained with the new MD NpT algorithm match the ensemble averages obtained using the previously derived shell particle Monte Carlo NpT method [D. S. Corti, Mol. Phys. 100, 1887 (2002)]. In addition, we also verify that the Hoover NpT MD algorithm [W. G. Hoover, Phys. Rev. A 31, 1695 (1985); 34, 2499 (1986)] generates the correct ensemble averages, though only when periodic boundary conditions are employed. The extension of the shell particle MD algorithm to multicomponent systems is also discussed, in which we show for equilibrium properties that the identity of the shell particle is completely arbitrary when periodic boundary conditions are applied. Self-diffusion coefficients determined with the shell particle equations of motion are also identical to those obtained in other ensembles. Finally, since the mass of the shell particle is known, the system itself, and not a piston of arbitrary mass, controls the time scales for internal pressure and volume fluctuations. We therefore consider the effects of the shell particle on the dynamics of the system. Overall, the shell particle MD algorithm is an effective simulation method for studying systems exposed to a constant external pressure and may provide an advantage over other existing constant pressure approaches when developing nonequilibrium MD methods.

Current constant pressure molecular-dynamics (MD) algorithms are not consistent with the recent reformulation of the isothermal-isobaric (NpT) ensemble. The NpT ensemble partition function requires the use of a "shell" molecule to identify uniquely the volume of the system, thereby avoiding the redundant counting of configurations [e.g., G. J. M. Koper and H. Reiss, J. Phys. Chem. 100, 422 (1996); D. S. Corti, Phys. Rev. E, 64, 016128 (2001)]. So far, only the NpT Monte Carlo method has been updated to allow the system volume to be defined by a shell particle [D. S. Corti, Mol. Phys. 100, 1887 (2002)]. A shell particle has yet to be incorporated into MD simulations. The proper modification of the NpT MD algorithm is therefore the subject of this paper. Unlike Andersen's method [H. C. Andersen, J. Chem. Phys. 72, 2384 (1980)] where a piston of unknown mass serves to control the response time of volume fluctuations, the newly proposed equations of motion impose a constant external pressure via the introduction of a shell particle of known mass. Hence, the system itself sets the time scales for pressure and volume fluctuations. The new algorithm is subject to a number of fundamentally rigorous tests to ensure that the equations of motion sample phase space correctly. We also show that the Hoover NpT algorithm [W. G. Hoover, Phys. Rev. A. 31, 1695 (1985); 34, 2499 (1986)] does sample phase correctly, but only when periodic boundary conditions are employed.

Integral equations describing the structure of a hard sphere fluid surrounding a cavity that intersects a hard wall are derived from scaled particle theory (SPT). The new expressions are solved exactly for specific cavity radii and the predictions are compared to simulation-generated results, showing excellent agreement. Additional simulation studies are conducted for cavity radii that fall outside the range of exact solution. For all cavity sizes, an enhancement of the local density of hard spheres over that of the hard wall contact value is seen for positions near the point of intersection of the cavity and the hard wall. The local density in front of the cavity and away from the hard wall is depleted at small cavity sizes, but eventually approaches the density profile created by a cavity placed within a bulk hard sphere fluid at larger cavity radii. The exact solutions and simulation results are also used to understand why a minimum appears in the inhomogeneous SPT function G (lambda,h) [D. W. Siderius and D. S. Corti, preceding paper, Phys. Rev. E 71, 036141 (2005)].

The methods of traditional scaled particle theory (SPT) are used to develop an extended scaled particle theory that is now applicable to hard particle fluids confined by hard walls. The new theory, labeled inhomogeneous SPT (I-SPT), introduces the function G that describes the average value of the anisotropic density of hard particle centers contacting a cavity placed at or behind a hard wall. We present an exact relation describing G for certain cavity sizes and, similar to the original SPT, an accurate interpolation scheme for larger cavity radii. Given G , the reversible work of inserting a cavity centered at or behind the hard wall can be estimated. The work predictions at low to moderate packing fractions are extremely accurate, though small deviations from simulation results become apparent at packing fractions close to the bulk hard sphere freezing transition. I-SPT also reveals the importance of the line tension in determining the free energy of cavity formation for cavities intersecting a hard wall, a term which has been previously neglected. Furthermore, this paper provides the initial groundwork needed to develop a more complete SPT-based theory that can accurately predict the depletion force between a hard particle and a hard structureless wall.

We present density-functional theory (DFT) calculations for critical cavities inside model superheated liquids with varying intermolecular potentials. Our calculations show that the radius of the critical cavity and the ratio of the work of formation of the critical cavity to the work of formation of the critical bubble as predicted by the classical nucleation theory exhibit universal scaling across similar intermolecular potentials. We then utilize this observed scaling behavior by proposing two new criteria for the kinetic spinodal of superheated liquids. These criteria are based on various properties of the critical cavity as obtained from our DFT studies of the superheated Lenanrd-Jones liquid. Our predictions of the kinetic spinodal compare favorably with experimental data of the limits of superheating for various organic liquids., ((c) 2004 American Institute of Physics.)

A semiempirical approach, based on both the scaled particle theory of hard particle fluids and free-energy perturbation methods, that predicts the work of formation of a cavity inside a model fluid is presented. The method is tested using the pure component Lennard-Jones fluid. A good agreement between values obtained via the theory and molecular simulation is observed even as the size of the cavity becomes larger than the effective diameter of Lennard-Jones particles. The method also yields reasonably accurate estimates when the pressure of the liquid is quite low, which is the case for liquids close to the coexistence line, or when the pressure is negative, which is the case when the liquid is metastable.

The application of the isothermal-isobaric (N-P-T) ensemble to small systems is considered. In the small system limit, which is currently gaining in scientific and technological significance, a volume scale must be introduced in order to obtain a partition function that is dimensionless. The volume scale, however, must be carefully chosen since it depends upon the nature of the boundary separating the system from the surroundings. If the incorrect volume scale is used, the resulting N-P-T ensemble partition function will not rigorously describe the small system of interest. Although volume scales become inconsequential in the thermodynamic limit, care must be exercised in formulating the ensembles used to study small systems.