Monte Carlo simulation of chain length distribution in radical polymerization with transfer reaction

In this paper, the basic principle and a Monte Carlo method are described for numerically simulating the chain-length distribution in radical polymerization with transfer reaction to monomer. The agreement between the simulated and analytical results shows that our algorithm is suitable for systems with transfer reaction. With the simulation algorithm, we confirm that transfer reaction has a similar effect as disproportionation on the molecular weight distribution in radical polymerization with continuous initiation. In the pulsed laser (PL) initiated radical polymerization with transfer reaction, the ‘waves’ on the chain-length distribution profile become weaker as the ratio of transfer reaction rate constant, k_tr, to the propagation rate constant, k_p, is increased in the case with either combination-type or disproportionation-type termination. Moreover, it seems that the combination termination has a broadening effect on the waves. Therefore, k_p can also be determined by precisely locating the inflection point L_o on the chain-length distribution profile for radical polymerization with transfer reaction, unless k_tr is large enough to smear out the waves on the chain-length distribution.     

Monte Carlo simulation of a polymer-analogous reaction in a polymer blend

The autocatalytic polymer-analogous reaction A → B in a blend composed of two contacting layers of compatible homopolymers A and B is studied by numerical simulation using the dynamic continuum Monte Carlo method. The evolution of the numerical density of units A and units initially belonged to the chains of homopolymer A is investigated in the course of the reaction and interdiffusion. Local characteristics of the distribution of the homopolymer with respect to its composition and blocks A and B with respect to their length are calculated at different times. The dispersions of the above distributions are appreciably higher than the corresponding dispersion of the Bernoullian copolymer of the same average composition, despite the random character of the reaction. This effect can be provided by changes in the composition of the blend on the scale of the reacting chain as well as by the diffusive mixing of the above chains. For the products of the polymer-analogous reaction, the broadening of the compositional distribution is predicted also by the theoretical model, which describes interdiffusion in the reacting system on scales that are markedly greater than the size of a polymer chain.     

Efficient implementation of the Hellmann-Feynman theorem in a diffusion Monte Carlo calculation

Kinetic and potential energies of systems of (4)He atoms in the solid phase are computed at T = 0. Results at two densities of the liquid phase are presented as well. Calculations are performed by the multiweight extension to the diffusion Monte Carlo method that allows the application of the Hellmann-Feynman theorem in a robust and efficient way. This is a general method that can be applied in other situations of interest as well.     

Reactive Monte Carlo and grand-canonical Monte Carlo simulations of the propene metathesis reaction system

The influence of silicalite-1 pores on the reaction equilibria and the selectivity of the propene metathesis reaction system in the temperature range between 300 and 600 K and the pressure range from 0.5 to 7 bars has been investigated with molecular simulations. The reactive Monte Carlo (RxMC) technique was applied for bulk-phase simulations in the isobaric-isothermal ensemble and for two phase systems in the Gibbs ensemble. Additionally, Monte Carlo simulations in the grand-canonical ensemble (GCMC) have been carried out with and without using the RxMC technique. The various simulation procedures were combined with the configurational-bias Monte Carlo approach. It was found that the GCMC simulations are superior to the Gibbs ensemble simulations for reactions where the bulk-phase equilibrium can be calculated in advance and does not have to be simulated simultaneously with the molecules inside the pore. The confined environment can increase the conversion significantly. A large change in selectivity between the bulk phase and the pore phase is observed. Pressure and temperature have strong influences on both conversion and selectivity. At low pressure and temperature both conversion and selectivity have the highest values. The effect of confinement decreases as the temperature increases.     

Signal-to-background in EPMA: Measurement and Monte Carlo calculation

By the method of Monte Carlo calculation, the dependence of the signal-to-background ratio of detected X-rays on the energy of electrons as well as on the thickness of the sample was calculated. The range of energy was 40 ÷ 120 keV, the range of thickness was approximately 8 ÷ 80 g/cm² (40 ÷ 400 nm at density = 2 g/cm³). The results were compared with measurements in electron microscope on thin resin standard for biological microanalysis. The measured dependence of signal-to-background ratio on the energy of electrons has the maximum at 80 keV, the calculated one changes at increased thickness from a monotonic form to one with a maximum at a particular thickness. The absolute values (Hall correction procedure was used for measured values) differs mainly at the highest energy used (120 keV); the difference is probably caused by unproper correction of measured value of background at this energy. Simultaneously, the source distribution of emitted X-ray photons is calculated. Its knowledge gives the possibility to estimate simply the interaction volume diameter and, by this way, to determine the spatial resolution of electron probe X-ray microanalysis.     

Monte Carlo structure simulations for aqueous 1,4-dioxane solutions

Monte Carlo simulations in the NpT ensembles have been performed for the structure exploration of aqueous 1,4-dioxane solutions. Three different systems with all-atom dioxane:TIP4P water molar compositions of 2:500 (code:D2), 8:465 (D8), and 17:425 (D17) modeled solutions of 0.22, 0.88, and 1.86 mol/dm3 concentrations, respectively, at T = 298 K and p = 1 atm. The calculated solution densities increase from 0.992 to 1.002 g/cm3 with increasing dioxane concentration and approach the experimentally determined densities within 1%. This close agreement was achieved by utilizing RESP charges fitted to the in-solution IEF-PCM/B3LYP/6-31G* electrostatic potential of dioxane taken in its chair conformation and recently developed C, H steric parameters for ethers for calculations with a 12-6-1 all-atom potential. Solution structure analyses pointed out that the dioxane molecules arrange in the solutions with favorable distances of 4-8 angstroms for the ring symmetry centers. Within this range not only pairs of rings but triangular triads and tetrads have also been observed with center-center distances <8 angstroms. For the D8 system, about 25% of the sampled configurations included such a triad. In the case of the D17 model, two simulations starting from different solution configuration predicted different degrees for the dioxane aggregation in aqueous solution. In the more aggregated structure 3-21 triads are consistently maintained and 1-2 tetrads are formed in 58% of the configurations. Each dioxane oxygen forms about one hydrogen bond, on average, to a water molecule in the 0.22-1.86 molar range. The most likely O(dioxane)...H(water) hydrogen bond distance is 1.75-1.80 angstroms compared to the optimal distance of 1.72 angstroms in the isolated dimer. The optimal dioxane-water interaction energy of -5.65 kcal/mol indicates a remarkable hydrogen-bond acceptor character for dioxane.     

Monte Carlo calculation of the thermodynamic properties of water

The Monte Carlo method and parallel computing are used to calculate the thermodynamic properties of water (density, heat capacity, compressibility, thermal expansion coefficient, and static dielectric constant) in a wide range of temperatures (from 70 K to 530 K) at constant (atmospheric) pressure. Four groups of computational experiments are carried out, each for its own model of the water molecule: TIP3P (Jorgensen et al., 1983), SPC/E (Berendsen et al., 1987), TIP4P/2005 (Abascal&Vega, 2005), and TIP5P-E (Rick, 2004). An additional calculation based on the replica exchange method is conducted for the TIP4P/2005 model. A comparison of the calculated properties of water with experimental data suggests that the TIP4P/2005 model can provide highly realistic computer simulation results for water and aqueous solutions.     

Communication: Monte Carlo calculation of the exchange energy

In recent generalized Kohn-Sham (GKS) schemes for density functional theory (DFT) Hartree-Fock type exchange is important. In plane waves and grid approaches the high cost of exchange energy calculations makes these GKS considerably more expensive than Kohn-Sham DFT calculations. We develop a stochastic approach for speeding up the calculation of exchange for large systems. We show that stochastic error per particle does not grow and can even decrease with system size (at a given number of iterations). We discuss several alternative approaches and explain how these ideas can be included in the GKS framework.     

Path integral Monte Carlo method for ab initio calculation

The Feynman path integral method is applied to the many-electron problem. We first give new closure relations in terms of ordinary complex and real numbers, which could be derived from an arbitrary complete set of state vectors. Then, in the path integral form, the partition function of the system and the ensemble average of energy are explicitly expressed in terms of these closure relations. It is impossible to evaluate the path integral by direct numerical integrations because of its huge amount of integration variables. Therefore, we develop an algorithm by the Monte Carlo method with constraints corresponding to the normalization condition of states to calculate the required integral. Finally, the ensemble average of energy for the hydrogen molecule is explicitly evaluated by the quantum Monte Carlo method and results are compared with the result obtained by the ordinary full configuration interaction (CI) method. © 1996 John Wiley & Sons, Inc.     

Reliable radical stabilization energies from diffusion Monte Carlo calculations

We assess the performance of variational (VMC) and diffusion (DMC) quantum Monte Carlo methods for calculating the radical stabilization energies of a set of 43 carbon-centered radical species. Even using simple single-determinant trial wavefunctions, both methods perform exceptionally well, with mean absolute deviations from reference values well under the chemical accuracy standard of 1 kcal/mol. In addition, the use of DMC results in a highly concentrated spread of errors, with all 43 results within chemical accuracy at the 95% confidence level. These results indicate that DMC is an extremely reliable method for calculating radical stabilization energies and could be used as a benchmark method for larger systems in future.     

Monte Carlo simulation of electrodeposition of copper: a multistep free energy calculation

Electrodeposition of copper (Cu) involves length scales of a micrometer or even less. Several theoretical techniques such as continuum Monte Carlo, kinetic Monte Carlo (KMC), and molecular dynamics have been used for simulating this problem. However the multiphenomena characteristics of the problem pose a challenge for an efficient simulation algorithm. Traditional KMC methods are slow, especially when modeling surface diffusion with large number of particles and frequent particle jumps. Parameter estimation involving thousands of KMC runs is very time-consuming. Thus a less time-consuming and novel multistep continuum Monte Carlo simulation is carried out to evaluate the step wise free energy change in the process of electrochemical copper deposition. The procedure involves separate Monte Carlo codes employing different random number criterion (using hydrated radii, bare radii, hydration number of the species, redox potentials, etc.) to obtain the number of species (CuCl(2) or CuSO(4) or Cu as the case may be) and in turn the free energy. The effect of concentration of electrolyte, influence of electric field and presence of chloride ions on the free energy change for the processes is studied. The rate determining step for the process of electrodeposition of copper from CuCl(2) and CuSO(4) is also determined.     

Monte Carlo simulation of kinetics and chain-length distribution in radical polymerization

In this paper, the Monte Carlo method for numerically simulating the kinetics and chain-length distribution in radical polymerization is described. Because the Monte Carlo method is not subject to the assumption of steady-state, it is particularly suitable for studying the kinetic behaviour before the steady-state has been reached and for systems in which the steady-state assumption may be violated. Illustrative applications of the algorithm given in this paper not only demonstrate convincingly both the feasibility and usefulness of the algorithm, but also provide some new insight into the illustrative examples. For the case of pseudostationary radical polymerization such as rotating-sector and pulsed-laser initiations, we have found that the pseudostationary radical concentration can be reached after two or three initiation periods. However, the number-average chain-length x̄_n reaches the pseudostationary value much slower than the radical concentration. It is oscillatively reaching the pseudostationary value, and the amplitudes of the oscillations are decreasing with time. We have also found that the chain-length distribution of the resulting polymer in the case of pseudostationary radical polymerization with termination by combination has stronger periodic modulation. Hence, it should be easier to locate the points of inflection in practice. Therefore, the rate constant of propagation, k_p, can be determined precisely for systems which are dominated by a combination-type of termination.     

Direct calculation of solid-liquid equilibria from density-of-states Monte Carlo simulations

A density-of-states Monte Carlo method is proposed for simulations of solid-liquid phase equilibria. A modified Wang-Landau density-of-states sampling approach is used to perform a random walk in regions of potential energy and volume relevant to solid-liquid equilibrium. The method provides a direct estimate of the relative density of states [Omega(U,V)] and thus the relative free energy within these regions, which is subsequently used to determine portions of the melting curve over wide ranges of pressure and temperature. The validity and usefulness of the method are demonstrated by performing crystallization simulations for the Lennard-Jones fluid and for NaCl.     

Analysis of stress-induced crystallization in high-cis-1,4-polybutadiene

Stress-induced crystallization of a crosslinked polybutadiene with a high 1,4-cis content is investigated, at room temperature. The change in the fraction of crystallinity with deformation is evaluated by two different methods. The first is based on simultaneous measurement of birefringence and stress. The second is based on calorimetric analysis. The data on crystallinity obtained by these two methods are in good agreement in the range of low deformations, while, in the range of higher deformation (λ > 5), the degree of crystallinity obtained by calorimetry is higher than that obtained by the optical method. Qualitative information on the crystallization was also obtained by optical analysis of the hysteresis behavior. The hysteresis data show the presence of ordering phenomena at deformations at which there seems to be no crystallinity according to both quantitative methods.     

Monte Carlo simulation of the interchain exchange reaction in a blend of incompatible polymers

The interchain exchange reaction in a blend composed of two contacting layers of incompatible A and B homopolymers was simulated by means of the dynamic off-lattice Monte Carlo method. The evolution of local molecular-mass and block mass distributions, depending on the effective temperature and the reaction rate, was studied for the first time. It was shown that the components interpenetrate as the copolymer forms in the interphase layer and the average block length decreases below a certain, temperature-dependent value. The state of dynamic equilibrium, whose characteristics are determined mainly by temperature, is established in the system. The time of establishment of equilibrium and the intensity of compatibilization at the early steps of the process are controlled by the rate of the reaction. The results of the study allow the contribution of the reaction to the interchange processes to be evaluated.     

Stress-optical coefficient of cis-1,4-polybutadiene and cis-1,4-polyisoprene networks. Measurements on cis-1,4-polybutadiene networks and theoretical interpretation

Stress, strain, and birefringence measurements have been carried out on swollen and unswollen networks of ′cis-1,4-polybutadiene polymers. Neither stress-strain nor birefringence-strain relations of unswollen specimens obey the Gaussian network theory, but both can be fitted by the Mooney-Rivlin equation. On the contrary, data on specimens swollen in tetralin, decalin, benzene, and carbon tetrachloride strictly obey the Gaussian network theory. Existing methods for evaluating the temperature coefficient of the unperturbed dimensions, d In 〈r²〉/dT, from the stress-temperature relation are applied to the present data and discussed in some detail. It is concluded that reliable values of d In 〈r²〉/dT are not obtainable from data on unswollen samples because of the pronounced non-Gaussian effect. The value 7.5 ų for the optical anisotropy ų (an alternative to the stress-optical coefficient) for unswollen specimens is markedly larger than values (5.8 ų on the average) for swollen specimens. This is interpreted as due to the shortrange orientational order among polymer segments. The quantities 〈r²〉, ΔΓ, and their temperature coefficients are calculated for both cis-1,4-polybutadiene and cis-1,4-polyisoprene chains, on the basis of the rotational isomeric state approximation for bond rotations. Values of ΔΓ for cis-1,4-polybutadiene calculated using Clément and Bothorel's set of anisotropic bond polarizabilities are in good agreement with observed values for swollen specimens. Those for cis-1,4-polyisoprene obtained using the same set of anisotropic bond polarizabilities are somewhat smaller than observed values for unswollen specimens. This departure is in the direction expected from the behavior of ΔΓ upon swelling (i.e., a decrease in ΔΓ upon swelling).     

Self-assembly of 1,4-benzenedithiolate/tetrahydrofuran on a gold surface: a Monte Carlo simulation study

We report a Monte Carlo simulation study of the self-assembly of 1,4-benzenedithiolate (BDT), tetrahydrofuran (THF), and their mixtures on a Au (111) surface. We use the grand canonical Monte Carlo method to obtain the equilibrium adsorption coverage. Canonical ensemble (NVT) simulation is then used to explore further the structural information of the equilibrated systems. Our results indicate that BDT molecules adsorb onto the Au (111) surface with one of the sulfur atoms bonded to Au atoms. THF molecules form clusters on the surface. For BDT-THF mixtures, BDT can selectively adsorb on Au (111) to form a monolayer, whereas the solvent THF molecules either float above BDT monolayer or occupy vacancies on the surface that are not covered by BDT molecules. BDT molecules adsorb on a Au (111) surface with an average tilt angle of about 18-35 degrees to the surface normal. The tilting angle decreases as the coverage increases. In addition, the BDT monolayer constitutes an ordered herringbone structure on the Au (111) surface, and the ordering pattern is insensitive to the BDT coverage. In comparison, the THF molecules exhibit amorphous structure on the Au surface. Interestingly, simulations indicate that the bonding behavior of BDT molecules on Au (111) is coverage-dependent. BDT bonds preferably on the Au top site when the surface coverage is low. As coverage increases, most BDT molecules bond on the bridge and fcc hollow sites.     

Normal coordinate analysis of cis-1,4-polybutadiene

The fundamental vibrational frequencies of an isolated chain of cis-1,4-polybutadiene have been calculated. Infrared and polarized Raman data are used in the force constant refinement routine. The assignments of the vibrational frequencies are discussed in terms of the potential energy distribution.