Quantitative description of the self-assembly of patchy particles into chains and rings

We numerically study a simple fluid composed of particles having a hard-core repulsion complemented by two patchy attractive sites on the particle poles. An appropriate choice of the patch angular width allows for the formation of ring structures which, at low temperatures and low densities, compete with the growth of linear aggregates. The simplicity of the model makes it possible to compare simulation results and theoretical predictions based on the Wertheim perturbation theory, specialized to the case in which ring formation is allowed. Such a comparison offers a unique framework for establishing the quality of the analytic predictions. We find that the Wertheim theory describes remarkably well the simulation results.     

A completely analytic equation of state for the square-well chain fluid of variable well width

A completely analytic perturbation theory equation of state for the freely-jointed square-well chain fluid of variable well width (1 ≤ λ ≤ 2) is developed and tested against Monte Carlo simulation data. The equation of state is based on second-order Barker and Henderson perturbation theory to calculate the thermodynamic properties of the reference monomer fluid, and on first-order Wertheim thermodynamic perturbation theory to account for the connectivity of monomers to form chains. By using a recently developed real function expression for the radial distribution function of hard spheres in perturbation theory, we obtain analytic, closed form expressions for the Helmholtz free energy and the radial distribution function of square-well monomers of any well width. This information is used as the reference fluid in the perturbation theory of Wertheim to obtain an analytic equation of state, without adjustable parameters, that leads to good predictions of the compressibility factors and residual internal energies for 4-mer, 8-mer and 16-mer square-well fluids when compared with the simulation results. Further, very good results are obtained when this equation of state with temperature-independent parameters is used to correlate the vapor pressures and critical points of the linear alkanes from methane to n-decane.     

Thermodynamic perturbation theory for fused sphere hard chain fluids using nonadditive interactions

A model is developed for the equation of state of fused chains based on Wertheim thermodynamic perturbation theory and nonadditive size interactions. The model also assumes that the structure (represented by the radial distribution function) of the fused chain fluid is the same as that of the touching hard sphere chain fluid. The model is completely based on spherical additive and nonadditive size interactions. The model has the advantage of offering good agreement with simulation data while at the same time being independent of fitted parameters. The model is most accurate for short chains, small values of Delta (slightly fused spheres) and at intermediate (liquidlike) densities.     

A Thermodynamic Model for Square-well Chain Fluid: Theory and Monte Carlo Simulation

A thermodynamic model for the freely jointed square-well chain fluids was developed based on the thermodynamic perturbation theory of Barker-Henderson, Zhang and Wertheim. In this derivation Zhang's expressions for square-well monomers improved from Barker-Henderson compressibility approximation were adopted as the reference fluid, and Wertheim's polymerization method was used to obtain the free energy term due to the bond connectivity. An analytic expression for the Helmholtz free energy of the square-well chain fluids was obtained. The expression without adjustable parameters leads to the thermodynamic consistent predictions of the compressibility factors, residual internal energy and constant-volume heat capacity for dimer, 4-mer, 8-mer and 16-mer square-well fluids. The results are in good agreement with the Monte Carlo simulation. To obtain the MC data of residual internal energy and the constant-volume heat capacity needed, NVT MC simulations were performed for these square-well chain fluids.     

Dynamic Monte Carlo simulation of chain growth polymerization and its concentration effect

Polymerization kinetics and chain configuration are two of lasting hotspots in Polymer Science. As these problems are concerned, computer experiment affords a very useful method besides theoretical and experi-mental investigations. The conventional Monte …     

Molecular weight distribution in emulsion polymerizations

A mathematical formulation is given which describes the evolution of the number distribution of the molecular weight (MWD) of linear polymer chains that grow in emulsion polymerization systems. The resulting set of coupled ordinary differential equations takes into account the microscopic events of free radical entry, exit, chain annihilation, bimolecular termination (by combination and disproportionation), and chain transfer in a mono- or polydisperse system. Simple analytic solutions are presented for systems in which the number of particles, as well as the average number of free radicals per particle, is constant and in which the rate coefficients are size independent. These solutions indicate that compartmentalization of the free radicals in the latex particles results in a significant increase in the polydispersity of the polymer produced by emulsion polymerization, compared with that in bulk systems. The theory shows that significant mechanistic information may be obtained from experimental MWDs and that, in principle, experimental conditions may be prescribed to grow a desired MWD. The MWDs are presented in a novel manner that facilitates the comparison of theory with experiment.     

Kinetics of vinyl acetate emulsion polymerization

The emulsion polymerization of vinyl acetate has generally been considered a special type of reaction that is not covered by the Smith-Ewart theory. Although the number of particles depends on coalescence rates and can not be predicted by this theory, the polymerization rate data are consistent with the general concepts of Smith and Ewart, including reaction primarily inside swollen polymer particles, escape of radicals from particles, and termination of chains inside the particles. Allowing for rapid exchange of radicals following chain transfer leads to a simple equation which fits much of the published data for cases of both very low and very high values of n , the average number of radicals per particle.      

Interactions between polymer brush-coated spherical nanoparticles: the good solvent case

The interaction between two spherical polymer brushes is studied by molecular dynamics simulation varying both the radius of the spherical particles and their distance, as well as the grafting density and the chain length of the end-grafted flexible polymer chains. A coarse-grained bead-spring model is used to describe the macromolecules, and purely repulsive monomer-monomer interactions are taken throughout, restricting the study to the good solvent limit. Both the potential of mean force between the particles as a function of their distance is computed, for various choices of the parameters mentioned above, and the structural characteristics are discussed (density profiles, average end-to-end distance of the grafted chains, etc.). When the nanoparticles approach very closely, some chains need to be squeezed out into the tangent plane in between the particles, causing a very steep rise of the repulsive interaction energy between the particles. We consider as a complementary method the density functional theory approach. We find that the quantitative accuracy of the density functional theory is limited to large nanoparticle separation and short chain length. A brief comparison to Flory theory and related work on other models also is presented.     

Emulsion polymerization. VII. Effects of instantaneous chain termination during the interval of particle nucleation

In the Smith-Ewart treatment of particle nucleation all particles were assumed to grow as if they contained exactly one radical. Modification of particle growth rate by chain termination in growing particles and reinitiation of nongrowing particles by radicals entering them was neglected in this interval although such effects were taken into account after the particle number became constant. The present theory eliminates this inconsistency for the case where chain termination is instantaneous. This refinement does not change previous predictions for the final number of particles, the steady state rate or the particle radius. Unlike the old theory, the present theory predicts continuous decay of the average number of radicals per particle from the initial value of unity to the steady-state value of one half. It also provides new theoretical predictions for the shape of the conversion-time curve at the initial stages of the reaction. Experimental data are reviewed in the context of the theory. Experimental particle sizes, steady-state conversion rates, and conversions at completion of particle nucleation were often in good quantitative agreement with the theoretical predictions. The predicted maximum in the conversion rate at the time when particle nucleation became completed was observed in a few instances. The theoretically predicted initial shape of the conversion-time curve may not be always observable due to experimental difficulties mainly associated with induction effects.     

Simulation of the latex particle morphology

A model is presented for the simulation of the structuration of polymer particles under conditions in which the new polymer chains are compatible with the polymer previously formed. The model involves the calculation of the monomer concentration gradients within the particles due to discrepancies in thermodynamic interactions between the monomer and the different polymers present in each part of the polymer particle. In addition, the distribution of free radicals in the latex particle is taken into account. This distribution results from the anchoring of the hydrophilic end-group of the growing polymer chain on the surface of the particle. The model was applied to the simulation of the polymerization of vinyl acetate on a butyl acrylate–vinyl acetate copolymer seed. It was found that the development of the particle morphology was mainly due to the profile of concentration of radicals in the particle. On the other hand, the effect of the monomer–polymer thermodynamic interactions on the particle morphology was found to be negligible. However, it has to be pointed out that this is because, for the system studied, the interaction parameters of vinyl acetate with polyvinyl acetate and polybutyl acrylate are nearly identical.     

Phase behavior of self-associating fluids with weaker dispersion interactions between bonded particles

In this study, we explore the global phase behavior of a simple model for self-associating fluids where association reduces the strength of the dispersion interactions between bonded particles. Recent research shows that this type of behavior likely explains the thermodynamic properties of strongly polar fluids and certain micellar solutions. Based on Wertheim's theory of associating liquids [M. S. Wertheim, J. Stat. Phys. 42, 459 (1986); 42, 477 (1986)], our model takes into account the effect that dissimilar particle interactions have on the equilibrium constant for self-association in the system. We find that weaker interactions between bonded molecules tend to favor the dissociation of chains at any temperature and density. This effect stabilizes a monomeric liquid phase at high densities, enriching the global phase behavior of the system. In particular, for systems in which the energy of mixing between bonded and unbonded species is positive, we find a triple point involving a vapor, a dense phase of chain aggregates, and a monomeric liquid. Phase coexistence between the vapor and the monomeric fluid is always more stable at temperatures above the triple point, but a highly associated fluid may exist as a metastable phase under these conditions. The presence of this metastable phase may explain the characteristic nucleation behavior of the liquid phase in strongly dipolar fluids.     

Molecular weight distribution in emulsion polymerization. I. The homopolymer case

A model for evaluating the instantaneous degree of polymerization distribution of homopolymers produced in emulsion, based on the mathematics of the Markov chains, is developed. The model accounts for any number of active chains per particle, as well as for the two fundamental mechanisms of chain termination: mono- and bi-molecular, both by combination and by disproportionation. The core of the model is the so called subprocessmain process treatment, which allows us to correctly evaluate the degree of polymerization of the chains growing in the polymer particles, by distinguishing between the events experienced by the polymer chain which imply a change of its degree of polymerization (subject transitions) and those which imply only a change in the particle state (environment transitions). This is obtained by properly defining the one-step transition probability matrix of the relevant Markov process. Once this is done, the evaluation of the distribution of the degrees of polymerization reduces to a few simple operations among matrices. Explicit expressions for the instantaneous probability density functions and the relative cumulative distributions are obtained. The application of such relationships is facilitated by the numerical procedures reported in the Appendices. The results of the model developed in this work are in agreement with those of earlier models in the range of parameter values of practical interest. In the limit of very low molecular weights, only the model developed in this work provides the correct answer. Moreover, a much more significant result is its applicability to the case of emulsion copolymerization, as it is shown in Part II.     

Emulsion polymerization. I. Recalculation and extension of the Smith-Ewart theory

The most important assumptions underlying the Smith-Ewart theory are that the locus of chain propagation is the monomer-swollen latex particle, polymeric chains are initiated by radicals entering from the water phase into the particles, chain termination is an instantaneous reaction between two radicals within one particle, and particles are nucleated by radicals absorbed into monomer-swollen soap micelles. Right or wrong, these and other assumptions used by Smith and Ewart are retained in this paper. The newly derived and experimentally verifiable equations contain only such parameters which can be determined by experiments not involving emulsion polymerization. The proportionality constant between the particle number and the appropriate powers of soap and initiator concentrations is defined in terms of these independent parameters. Absolute rate equations are presented for the intervals before and after the completion of particle nucleation. To calculate these rates it is not necessary to have prior knowledge of the experimental particle number. The conversion at which particle nucleation is complete is calculated. The molecular weight is defined in terms of independent parameters. Predictions are made for the particle size distribution. It is shown that the validity of the theory is confined to specifiable intervals of conversion, to a certain range of monomer/water ratio, and to soap concentrations whose upper and lower limits are given.     

Effect of friction between particles in the dynamic response of model magnetic structures

A simple particle-level simulation model that takes into account interparticle friction forces is developed to describe the dynamic response of magneto-rheological fluids. The results obtained for single-width particle chains are found to be in good agreement with slender body theory predictions [J. de Vicente, M.T. López-López, J.D.G. Durán, G. Bossis, J. Colloid Interface Sci. 282 (2005) 193]. The addition of side chains to a single-width one results in one order of magnitude increase of storage modulus and relaxation. The double logarithmic plot of storage and loss moduli vs frequency gives a limiting slope of one when including friction forces between particles. Simulation results are found to be in agreement with experimental measurements on an iron/kerosene model MR-fluid.     

Emulsion polymerization of styrene in a single continuous stirred-tank reactor

The effects of mean residence time, initiation rate, and emulsifier concentration on particle formation, particle growth, and polymerization rate are examined for the emulsion polymerization of styrene in a completely mixed continuous stirred-tank reactor. Experimental measurements of number of particles, particle size distribution, polymerization rate, and molecular weights are compared with theoretical predictions. A theoretical model which incorporates Stockmayer's modification of the Smith-Ewart theory into the particle growth equation allows reasonably accurate prediction of polymerization rate, particle formation rate, and particle size distribution. Agreement between experimental measurements of number-average and weight-average molecular weights and a theory based on Smith-Ewart case 2 kinetics is also reasonable.     

Filler-induced deformations of amorphous polyethylene chains. The effects of the deformations on elastomeric properties, and some comparisons with experiments

Monte Carlo simulations based on rotational isomeric state models were carried out to determine how amorphous polyethylene chains are deformed by the presence of spherical filler particles. The resulting distributions of the chain end-to-end distances were then employed to calculate mechanical properties of the cross-linked chains. The basic goals were (i) to refine previous simulations of this type carried out on this system to remove possible sampling biases, and (ii) to document the circumstances under which the particles stretch the chains to larger end-to-end distances, rather than compress them to smaller distances, and (iii) to study in more detail the flattening of the chains near particle surfaces. Comparisons with experiments include filler-induced changes in chain dimensions measured by neutron scattering, and general features of the stress-strain isotherms widely observed for reinforced elastomers in simple elongation.     

Modeling of composite latex particle morphology by off-lattice Monte Carlo simulation

Composite latex particles have shown a great range of applications such as paint resins, varnishes, water borne adhesives, impact modifiers, etc. The high-performance properties of this kind of materials may be explained in terms of a synergistical combination of two different polymers (usually a rubber and a thermoplastic). A great variety of composite latex particles with very different morphologies may be obtained by two-step emulsion polymerization processes. The formation of specific particle morphology depends on the chemical and physical nature of the monomers used during the synthesis, the process temperature, the reaction initiator, the surfactants, etc. Only a few models have been proposed to explain the appearance of the composite particle morphologies. These models have been based on the change of the interfacial energies during the synthesis. In this work, we present a new three-component model: Polymer blend (flexible and rigid chain particles) is dispersed in water by forming spherical cavities. Monte Carlo simulations of the model in two dimensions are used to determine the density distribution of chains and water molecules inside the suspended particle. This approach allows us to study the dependence of the morphology of the composite latex particles on the relative hydrophilicity and flexibility of the chain molecules as well as on their density and composition. It has been shown that our simple model is capable of reproducing the main features of the various morphologies observed in synthesis experiments.     

Application of a renormalization-group treatment to the statistical associating fluid theory for potentials of variable range (SAFT-VR)

An accurate prediction of phase behavior at conditions far and close to criticality cannot be accomplished by mean-field based theories that do not incorporate long-range density fluctuations. A treatment based on renormalization-group (RG) theory as developed by White and co-workers has proven to be very successful in improving the predictions of the critical region with different equations of state. The basis of the method is an iterative procedure to account for contributions to the free energy of density fluctuations of increasing wavelengths. The RG method has been combined with a number of versions of the statistical associating fluid theory (SAFT), by implementing White's earliest ideas with the improvements of Prausnitz and co-workers. Typically, this treatment involves two adjustable parameters: a cutoff wavelength L for density fluctuations and an average gradient of the wavelet function Φ. In this work, the SAFT-VR (variable range) equation of state is extended with a similar crossover treatment which, however, follows closely the most recent improvements introduced by White. The interpretation of White's latter developments allows us to establish a straightforward method which enables Φ to be evaluated; only the cutoff wavelength L then needs to be adjusted. The approach used here begins with an initial free energy incorporating only contributions from short-wavelength fluctuations, which are treated locally. The contribution from long-wavelength fluctuations is incorporated through an iterative procedure based on attractive interactions which incorporate the structure of the fluid following the ideas of perturbation theories and using a mapping that allows integration of the radial distribution function. Good agreement close and far from the critical region is obtained using a unique fitted parameter L that can be easily related to the range of the potential. In this way the thermodynamic properties of a square-well (SW) fluid are given by the same number of independent intermolecular model parameters as in the classical equation. Far from the critical region the approach provides the correct limiting behavior reducing to the classical equation (SAFT-VR). In the critical region the β critical exponent is calculated and is found to take values close to the universal value. In SAFT-VR the free energy of an associating chain fluid is obtained following the thermodynamic perturbation theory of Wertheim from the knowledge of the free energy and radial distribution function of a reference monomer fluid. By determining L for SW fluids of varying well width a unique equation of state is obtained for chain and associating systems without further adjustment of critical parameters. We use computer simulation data of the phase behavior of chain and associating SW fluids to test the accuracy of the new equation.     

A single particle model to simulate the dynamics of entangled polymer melts

We present a computer simulation model of polymer melts representing each chain as one single particle. Besides the position coordinate of each particle, we introduce a parameter n(ij) for each pair of particles i and j within a specified distance from each other. These numbers, called entanglement numbers, describe the deviation of the system of ignored coordinates from its equilibrium state for the given configuration of the centers of mass of the polymers. The deviations of the entanglement numbers from their equilibrium values give rise to transient forces, which, together with the conservative forces derived from the potential of mean force, govern the displacements of the particles. We have applied our model to a melt of C(800)H(1602) chains at 450 K and have found good agreement with experiments and more detailed simulations. Properties addressed in this paper are radial distribution functions, dynamic structure factors, and linear as well as nonlinear rheological properties.