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  • 1
    Electronic Resource
    Electronic Resource
    Brownian dynamics simulation of diffusion-limited polymerization of rodlike molecules: Isotropic translational diffusion (1997)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 107 (1997), S. 3289-3294 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Rigid rodlike polymers have considerable technological importance due to their excellent mechanical properties. The polymerization kinetics of such condensation polymers are qualitatively different from flexible polymers, and exhibit significant slowing at the later stages of the reaction. This is due to the slow rotational diffusion of the molecules to an appropriate configuration for reaction. In this work we have carried out Brownian dynamics (BD) simulations to obtain the effective rate constant for reaction between rodlike molecules in the presence of diffusional limitations. The theory of Northrup et al. [J. Chem. Phys. 80, 1517 (1984)] for pairwise BD simulation of reactions is extended to the case of rodlike molecules assuming isotropic translational diffusion. The computed results are compared to exact analytical predictions. Good agreement between computation and theory is obtained over a wide range of parameter values. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.474679
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  • 2
    Electronic Resource
    Electronic Resource
    Brownian dynamics simulation of diffusion-limited polymerization of rodlike molecules: Anisotropic translation diffusion (1998)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 108 (1998), S. 5626-5634 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Step-growth polymerization of rodlike molecules is qualitatively different from flexible molecules, because rotational and translational diffusion limitations result in a rate of reaction which is dependent on the molecular weights of the reacting oligomers. An understanding of the kinetics of polymerization is important for the manufacture of such polymers which have many applications. The theoretical basis and computation details of the pairwise Brownian dynamics method [Northrup et al., J. Chem. Phys. 80, 1517 (1984)] to determine the effective rate constant for reaction between rodlike molecules are presented. In this method the effective rate constant is obtained in terms of the first visit flux (for which an analytical expression is derived) and the reaction probability (which is obtained using Brownian dynamics simulations). A simple derivation for finite domain correction for the simulations is presented, which explicitly accounts for the spatial and orientational variations of the reaction probability. Computations are presented to validate assumptions of the technique. The computed rate constants decrease with decreasing rotational diffusivity to approach an asymptotic value, and decrease with decreasing values of the translational diffusivity perpendicular to the rod axis. The computations show that the effective rate constants obtained for the case of dilute solutions when ratio of the translational diffusivities perpendicular and parallel to the rod axis is equal to 〈fraction SHAPE="CASE"〉12, is close to the case of isotropic translational diffusion. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.475951
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  • 3
    Electronic Resource
    Electronic Resource
    Brownian dynamics simulations of diffusion controlled reactions with finite reactivity (1997)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 107 (1997), S. 1915-1921 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: A new Brownian dynamics simulation technique is presented for the calculation of the effective rate constant for diffusion controlled reactions with a finite intrinsic reactivity. The technique is based on the calculation of the recollision probability of a molecule with a reactive site using a large number of Brownian trajectories, when the probability of reaction upon collision with the reactive site (cursive-phif) is less than unity. The technique is a modification of the earlier work of Northrup et al. [J. Chem. Phys. 80, 1517 (1984)], and is applied to the case of a uniformly reactive target sphere and a target sphere with axially symmetric reactive patches. A theoretical analysis is presented to relate cursive-phif to the intrinsic surface reaction rate constant (k). Computational results for the uniformly reactive sphere are in excellent agreement with theory, and those for the sphere with patches are in very good agreement with the results obtained using a different computational technique [Allison et al., J. Phys. Chem. 94, 7133 (1990)]. The proposed method requires the computation of the recollision probability to a high accuracy; however, this does not result in computational times greater than those of Allison et al. [J. Phys. Chem. 94, 7133 (1990)]. The new method has the advantage that the results of the Brownian dynamics simulation are independent of k and can subsequently be used to calculate the effective rate constant for any given value of k. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.474542
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  • 4
    Electronic Resource
    Electronic Resource
    Diffusion-limited polymerization of rigid rodlike molecules: Semidilute solutions (1993)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 99 (1993), S. 1382-1392 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: The rotational diffusivity and the translational diffusivity perpendicular to the rod axis of rigid rodlike (RRL) molecules decreases rapidly with increasing molecular length, during polymerization in semidilute solutions. This can result in slowing of step-growth polymerizations of RRL molecules with reactive groups at the rod ends and a near-collinearity requirement for reaction. Here a theoretical analysis of the rate of RRL polymerization in semidilute solutions, based on Smoluchowski's approach, and incorporating the rotational and anisotropic translational diffusion of the molecules is presented. The work is an extension of our analysis of polymerization in dilute solutions in which the translational diffusion was assumed to be isotropic [J. Chem. Phys. 96, 7125 (1992)]. The effective second order rate constant for the system is obtained for different parameter values using a numerical finite element method. With reduction in rotational diffusivity, for a fixed translational diffusivity, the effective reaction rate constant is found to decrease to a limiting value determined by only the translational flux of the correctly oriented molecules. Similarly, for a given rotational diffusivity, with reduction in translational diffusivity perpendicular to the rod axis, the reaction rate constant is found to decrease to a limiting value determined by the flux only due to translational diffusion parallel to the rod axis, aided by rotational diffusion. An asymptotic analysis for this case is presented. For low rotational diffusivities, reduction in the translational diffusion perpendicular to the rod axis results in a significant decrease in the effective rate constant, even for reactions with relatively slow intrinsic kinetics. A qualitative comparison of the theoretical predictions with experimental results is presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.465382
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  • 5
    Electronic Resource
    Electronic Resource
    Mixing and segregation of granular materials in chute flows (1999)
    Woodbury, NY : American Institute of Physics (AIP)
    Chaos 9 (1999), S. 594-610 
    Details
    ISSN: 1089-7682
    Source: AIP Digital Archive
    Topics: Physics
    Notes: Mixing of granular solids is invariably accompanied by segregation, however, the fundamentals of the process are not well understood. We analyze density and size segregation in a chute flow of cohesionless spherical particles by means of computations and theory based on the transport equations for a mixture of nearly elastic particles. Computations for elastic particles (Monte Carlo simulations), nearly elastic particles, and inelastic, frictional particles (particle dynamics simulations) are carried out. General expressions for the segregation fluxes due to pressure gradients and temperature gradients are derived. Simplified equations are obtained for the limiting cases of low volume fractions (ideal gas limit) and equal sized particles. Theoretical predictions of equilibrium number density profiles are in good agreement with computations for mixtures of equal sized particles with different density for all solids volume fractions, and for mixtures of different sized particles at low volume fractions (ν〈0.2), when the particles are elastic or nearly elastic. In the case of inelastic, frictional particles the theory gives reasonable predictions if an appropriate effective granular temperature is assumed. The relative importance of pressure diffusion and temperature diffusion for the cases considered is discussed. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.166433
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  • 6
    Electronic Resource
    Electronic Resource
    Chaotic mixing of granular materials in two-dimensional tumbling mixers (1999)
    Woodbury, NY : American Institute of Physics (AIP)
    Chaos 9 (1999), S. 195-205 
    Details
    ISSN: 1089-7682
    Source: AIP Digital Archive
    Topics: Physics
    Notes: We consider the mixing of similar, cohesionless granular materials in quasi-two-dimensional rotating containers by means of theory and experiment. A mathematical model is presented for the flow in containers of arbitrary shape but which are symmetric with respect to rotation by 180° and half-filled with solids. The flow comprises a thin cascading layer at the flat free surface, and a fixed bed which rotates as a solid body. The layer thickness and length change slowly with mixer rotation, but the layer geometry remains similar at all orientations. Flow visualization experiments using glass beads in an elliptical mixer show good agreement with model predictions. Studies of mixing are presented for circular, elliptical, and square containers. The flow in circular containers is steady, and computations involving advection alone (no particle diffusion generated by interparticle collisions) show poor mixing. In contrast, the flow in elliptical and square mixers is time periodic and results in chaotic advection and rapid mixing. Computational evidence for chaos in noncircular mixers is presented in terms of Poincaré sections and blob deformation. Poincaré sections show regions of regular and chaotic motion, and blobs deform into homoclinic tendrils with an exponential growth of the perimeter length with time. In contrast, in circular mixers, the motion is regular everywhere and the perimeter length increases linearly with time. Including particle diffusion obliterates the typical chaotic structures formed on mixing; predictions of the mixing model including diffusion are in good qualitative and quantitative (in terms of the intensity of segregation variation with time) agreement with experimental results for mixing of an initially circular blob in elliptical and square mixers. Scaling analysis and computations show that mixing in noncircular mixers is faster than that in circular mixers, and the difference in mixing times increases with mixer size. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.166390
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  • 7
    Electronic Resource
    Electronic Resource
    Simulation of diffusion-limited step-growth polymerization in 2D: Effect of shear flow and chain rigidity (1993)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 99 (1993), S. 3067-3074 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Multimolecule Brownian dynamics simulation results for diffusion controlled polymerization of bead–rod chain molecules in 2D solution are presented. Reaction between any two molecules undergoing Brownian diffusion takes place if the reactive chain ends approach each other to within a certain reaction radius, and if the chain end carrying segments are collinear within certain specified limits. The second order reaction rate constant is found to decrease with time as the molecular lengths increase and the diffusivities decrease. Application of a shear flow is seen to result in alignment of the molecules along the flow direction, thereby enhancing the concentration of molecular pairs with parallel orientation of reactive-end carrying chain segments, and hence the overall reaction rate. This effect is found to be more pronounced in the case of long rigid molecules as compared to flexible molecules because of the slow rotation and high level of orientation under flow of the former. Even the molecular weight distribution (MWD) obtained during polymerization may be affected. For example, longer molecules have lower diffusivities and hence lower reactivities, resulting in a narrower MWD in the absence of flow, as compared to the results with the usual assumption of molecular reactivity being independent of chain length. Furthermore, in the presence of an external flow, the longer molecules orient to a higher degree and hence display a higher enhancement in reactivity. This results in a wider MWD of the polymer. The simulation results are in qualitative agreement with previous experimental data for solution polymerization of rod-like molecules.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.466195
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  • 8
    Electronic Resource
    Electronic Resource
    Competition effects in surface diffusion controlled reactions: Theory and Brownian dynamics simulations (1993)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 99 (1993), S. 9237-9247 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Surface diffusion controlled reactions on a heterogeneous catalyst surface comprising randomly placed circular reactive sites are considered. The diffusing species adsorbs onto the surface following Langmuir–Hinshelwood kinetics and reacts instantaneously on contact with a reactive site. Approximate theories are formulated to describe the process for high concentrations of the reactive sites, when competition between the sites is significant, following three different approaches: (i) modification of the single sink theory; (ii) using a cell model; and (iii) using an effective medium theory. The predictions of the theories are compared with the results of multiparticle Brownian dynamics simulations for the overall reaction rate, the bulk concentration of the reactive species, and the ensemble averaged concentration profile around a reactive site. The effective medium theory is found to give the best results among the theories considered, and the predictions are in good agreement with the computational results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.465540
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  • 9
    Electronic Resource
    Electronic Resource
    Diffusion-limited polymerization of rigid rodlike molecules: Dilute solutions (1992)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 96 (1992), S. 7125-7134 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: A theoretical investigation of rates of diffusion-limited polymerization between rigid rodlike molecules in dilute solutions is presented, incorporating both translational and rotational diffusion of the molecules. Simplified criteria for reaction are proposed according to which molecules must approach within a specified reaction radius, and have a relative orientation angle in a specified range to react. The translational diffusion of the molecules is assumed to be isotropic. The results of the exact analytical solution, finite element computations, as well as approximate solutions for low rotational diffusivity, are presented. The match between the solutions is satisfactory in the appropriate limits, although the exact solution is found to be computationally intensive for low rotational diffusivities and rapid intrinsic kinetics. The reaction rate is found to decrease to a limiting value with decreasing rotational diffusivity, and the magnitude of the decrease in rate increases with intrinsic rate of reaction. The limiting value of rate is entirely due to the translational diffusive flux of the correctly oriented molecules from the far field, and an asymptotic expression for the rate in this limit is given. Comparison with reported experimental data is qualitative, since the data is only available for the semidilute regime for such systems, where the anisotropic translational diffusion is expected to play a prominent role.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.462546
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  • 10
    Electronic Resource
    Electronic Resource
    Radial segregation of granular mixtures in rotating cylinders (1997)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Fluids 9 (1997), S. 3600-3614 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: Simultaneous mixing and segregation of granular materials is of considerable practical importance; the interplay among both processes is, however, poorly understood from a fundamental viewpoint. The focus of this work is radial segregation—core formation—due to density in a rotating cylinder. The flow regime considered is the cascading or continuous flow regime where a thin layer of solids flows along a nearly flat free surface, while the remaining particles rotate as a fixed bed along with the cylinder. The essence of the formation of a central segregated core of the more dense particles lies in the flow, mixing, and segregation in the cascading layer. The work involves experiments and analysis. A constitutive model for the segregation flux in cascading layers is proposed and validated by particle dynamics and Monte Carlo simulations for steady flow down an inclined plane. The model contains a single parameter, the dimensionless segregation velocity (β), which is treated as a fitting parameter here. Experimental results for the equilibrium segregation of steel balls and glass beads are presented for different fractions and different extents of filling. There is a good match between theoretical predictions and all experimental results when the value of dimensionless segregation velocity is taken to be β=2. The extent of segregation is found to increase with increase in the dimensionless segregation velocity and dimensionless diffusivity but is independent of the level of filling. Lagrangian simulations based on the theory and experiments demonstrate the competition between segregation and mixing. In the case of slow mixing, the intensity of segregation monotonically decreases to an equilibrium value; for fast mixing, however, there exists an optimal mixing time at which the best mixing is obtained. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.869498
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