Bead-bead Interactions Research Articles

Application of hydrodynamic lubrication in discrete element method (DEM) simulations of wet bead milling chambers

Fine grinding in wet stirred media mills is a key operation in numerous industrial processes. During wet milling, grinding beads (typically millimetre scale) collide with each other under rapid agitation. These collisions transfer energy into the surrounding medium, a feed stream comprising a premixed slurry (a suspension of micron-sized particles), leading to a reduced particle size. Numerical simulation provides an opportunity to gain mechanistic insight by modelling the influences of grinding media, agitator speed and slurry rheology on the breakage. The multiphase nature of wet milling necessitates models that account for both the solid grinding beads and the slurry feed. Established fluid-particle coupled methods are in principle capable of doing this, but the need to resolve the large hydrodynamic lubrication forces between near-contacting beads demands extremely high fluid field resolution, leading to proliferating simulation size and cost. To address this challenge, a Discrete Element Method (DEM) simulation for a lab scale mill is presented in which the bead-bead interaction includes pairwise hydrodynamic lubrication in addition to frictional contact forces. The model accounts for multiphase effects in a simplified yet physically well-motivated way, circumventing the computational cost of a fully fluid-resolved model. A systematic model calibration against the laboratory mill is presented, and thereafter the model provides a good estimation for the empirical power draw across the full range of rotation speeds and considered feed viscosities. The differences in energy dissipation modes and the local distribution of collisions along the mill chamber are examined for the extreme viscosity condition, revealing that most energy dissipation is due to inter-particle forces acting tangentially (shearing and rolling) while highly energetic collisions (impact and torsion), relying on free flows and inter-bead mobility, are relatively unimportant.

Theoretical study on the stability of insulin within poly-isobutyl cyanoacrylate (PIBCA) nanocapsule

ABSTRACTTo study the physical stability of insulin in drug delivery particles, we developed a coarse-grained (CG) model for insulin based on dissipative particle dynamics (DPD). Three insulin modelling schemes were considered: each amino acid as a bead (IM1), each amino acid being separated into one to three beads (IM2), and adding secondary structural information of insulin to IM2 (IM3). The best possible bead-bead interaction parameters were obtained from Hildebrand and Hansen solubility parameters by performing the constant-temperature DPD simulation with insulin models in 20% acetic acid solution. IM3 showed good results in terms of RMSF, RMSD and A1B30 distance compared to those of all-atom models from the literature. Then, the IM3 model was considered in an oil-filled poly (isobutyl cyanoacrylate) (PIBCA) nanocapsule. Two crucial factors were found that mainly influence the stability of insulin in oil: the PIBCA shell thickness and the amount of ethanol in the oil droplet. An appropriate PIBCA shell thickness is necessary to block the interaction between insulin and water outside, and ethanol could stabilise insulin with its good affinity for both insulin and oil.

SAFT-γ Force Field for the Simulation of Molecular Fluids. 5. Hetero-Group Coarse-Grained Models of Linear Alkanes and the Importance of Intramolecular Interactions.

The SAFT-γ Mie group-contribution equation of state [ Papaioannou J. Chem. Phys. 2014 , 140 , 054107 ] is used to develop a transferable coarse-grained (CG) force-field suitable for the molecular simulation of linear alkanes. A heterogroup model is fashioned at the resolution of three carbon atoms per bead in which different Mie (generalized Lennard-Jones) interactions are used to characterize the terminal (CH3-CH2-CH2-) and middle (-CH2-CH2-CH2-) beads. The force field is developed by combining the SAFT-γ CG top-down approach [ Avendaño J. Phys. Chem. B 2011 , 115 , 11154 ], using experimental phase-equilibrium data for n-alkanes ranging from n-nonane to n-pentadecane to parametrize the intermolecular (nonbonded) bead-bead interactions, with a bottom-up approach relying on simulations based on the higher resolution TraPPE united-atom (UA) model [ Martin ; , Siepmann J. Phys. Chem. B 1998 , 102 , 2569 ] to establish the intramolecular (bonded) interactions. The transferability of the SAFT-γ CG model is assessed from a detailed examination of the properties of linear alkanes ranging from n-hexane ( n-C6H14) to n-octadecane ( n-C18H38), including an additional evaluation of the reliability of the description for longer chains such as n-hexacontane ( n-C60H122) and a prototypical linear polyethylene of moderate molecular weight ( n-C900H1802). A variety of structural, thermodynamic, and transport properties are examined, including the pair distribution functions, vapor-liquid equilibria, interfacial tension, viscosity, and diffusivity. Particular focus is placed on the impact of incorporating intramolecular interactions on the accuracy, transferability, and representability of the CG model. The novel SAFT-γ CG force field is shown to provide a reliable description of the thermophysical properties of the n-alkanes, in most cases at a level comparable to the that obtained with higher resolution models.

Influence of molecular weight on scaling exponents and critical concentrations of one soluble 6FDA-TFDB polyimide in solution

Polyimde (PI) samples with different molecular weights were synthesized. Based on SEC coupled with multidetectors measurement and Yamakawa-Fujii-Yoshizaki (YFY) model, eight soluble samples with absolute Mw from 40,600 g/mol to 197,000 g/mol are chosen and applied to investigate the influence of molecular weight on scaling exponents and critical concentrations at 20–45 °C in dilute, semidilute unentangled, and semidilute entangled solutions. Most of the scaling exponents are higher than the theoretical values in three concentration regions, and scaling exponent increases with molecular weight; overlap concentration (C*) increases and entanglement concentration (Ce) decreases with molecular weight. Considering bead-bead interaction, corrected bead-spring model can explain the related results. Finally, the relationship among C*, Ce, and molecular weight is established at different temperatures (from 20 °C to 45 °C), and two linear equations are available at each temperature. Thus, both C* and Ce are calculated at a fixed molecular weight. And from C/C* and C/Ce ratios, the morphology of PI fiber during electrospinning can be controlled. These results are helpful to guide the preparation of polyimide solutions for different processing.

A multiple time step scheme for multiresolved models of Macromolecules

In hybrid particle models where coarse-grained beads and atoms are used simultaneously, two clearly separate time scales are mixed. If such models are used in molecular dynamics simulations, a multiple time step (MTS) scheme can therefore be used. In this manuscript, we propose a simple MTS algorithm which approximates for a specific number of integration steps the slow coarse-grained bead-bead interactions with a Taylor series approximation while the atom-atom ones are integrated every time step. The procedure is applied to a previously developed hybrid model of a melt of atactic polystyrene (di Pasquale, Marchisio, and Carbone, J. Chem. Phys. 2012, 137, 164111). The results show that structure, local dynamics, and free diffusion of the model are not altered by the application of the integration scheme which can confidently be used to simulate multiresolved models of polymer melts.

Brownian dynamics simulations of single polymer chains with and without self-entanglements in theta and good solvents under imposed flow fields

The effects of self-entanglements (spring-spring uncrossability) and solvent quality on the static and dynamic properties of a polymer chain in shear and extensional flows are investigated using Brownian dynamics simulations. We model the polymer chain by a sequence of beads connected by finitely extensible non-linear elastic springs, and spring-spring uncrossability is enforced by applying a spring-spring repulsive potential together with an adaptive time-stepping. Our findings suggest that chain uncrossability has an insignificant effect on the dynamics of a polymer chain. Furthermore, we considered four different combinations of intramolecular interactions: (i) no interactions, (ii) repulsive spring-spring and attractive bead-bead interactions, (iii) only repulsive bead-bead interactions, and (iv) only repulsive spring-spring interactions. The first two cases model “theta” solvents, where the radius-of-gyration of a polymer chain, Rg∼N0.5, where N is the number of beads. For appropriately chosen parameters of interaction potentials, the last two cases model good solvents, where Rg∼N0.59. In the presence of a simple shear flow, the stretching behavior is alike in all the cases except case (ii). In case (ii), the polymer chain forms “pearl-necklace-like” structures at zero Weissenberg number, Wi. With an increase in Wi, Rg first increases due to stretching of bonds between neighboring “pearls,” and then decreases when the decrease in the number of pearls becomes limiting. In the presence of an extensional flow, normal stretching behavior is observed even in case (ii); the number of pearls increases with increasing extension rate because larger pearls break into smaller “pearls.” The results show that even for fixed overall solvent quality for a chain at rest (e.g., a theta solvent), the ability of the chain to stretch in a shear flow is sensitive to the nature of intramolecular interactions. A chain with short-range attraction (between beads) and a long-range repulsion (between springs) tuned to balance each other and so create a theta condition at rest does not stretch much in a shear flow, while a chain for which the theta condition is achieved by applying no interactions at all does stretch in a shear flow.

Brownian dynamics simulations of single-stranded DNA hairpins

We present a Brownian dynamics model which we use to study the kinetics and thermodynamics of single-stranded DNA hairpins, gaining insights into the role of stem mismatches and the kinetics rates underlying the melting transition. The model is a base-backbone type in which the DNA bases and sugar-phosphate backbone are represented as single units (beads) in the context of the Brownian dynamics simulations. We employ a minimal number of bead-bead interactions, leading to a simple computational scheme. To demonstrate the veracity of our model for DNA hairpins, we show that the model correctly captures the effects of base stacking, hydrogen bonding, and temperature on both the thermodynamics and the kinetics of hairpin formation and melting. When cast in dimensionless form, the thermodynamic results obtained from the present model compare favorably with default predictions of the m-fold server, although the present model is not sufficiently robust to provide dimensional results. The kinetic data at low temperatures indicate frequent but short-lived opening events, consistent with the measured chain end-to-end probability distribution. The model is also used to study the effect of base mismatches in the stem of the hairpin. With the parameters used here, the model overpredicts the relative shift in the melting temperature due to mismatches. The melting transition can be primarily attributed to a rapid increase in the hairpin opening rate rather than an equivalent decrease in the closing rate, in agreement with single-molecule experimental data.

Numerical Study of Extension Rheological Properties of Polymer Solutions using FENE Model

Dilute polymer solutions represent a complex rheological response. A finitely extensible nonlinear elastic (FENE) bead-spring chain model for dilute solutions of polymer molecules in a Newtonian solvent is employed to describe the bead-bead interaction within a molecule. Brownian dynamics simulation is used to capture the most important rheological properties of dilute polymer solutions. A program based on this model is developed to simulate the steady elongational flow and inception of elongational flow, and the flow situations and the rheological material functions are investigated. Semi-implicit predictor-corrector methods are introduced and the results qualitatively agree with rheological theory.

Self-Organization Process of Ordered Structures in Linear and Star Poly(styrene)−Poly(isoprene) Block Copolymers: Gaussian Models and Mesoscopic Parameters of Polymeric Systems

Mesoscopic simulations of linear and 3-arm star poly(styrene)-poly(isoprene) block copolymers was performed using a representation of the polymeric molecular structures by means of Gaussian models. The systems were represented by a group of spherical beads connected by harmonic springs; each bead corresponds to a segment of the block chain. The quantitative estimation for the bead-bead interaction of each system was calculated using a Flory-Huggins modified thermodynamical model. The Gaussian models together with dissipative particle dynamics (DPD) were employed to explore the self-organization process of ordered structures in these polymeric systems. These mesoscopic simulations for linear and 3-arm star block copolymers predict microphase separation, order-disorder transition, and self-assembly of the ordered structures with specific morphologies such as body-centered-cubic (BCC), hexagonal packed cylinders (HPC), hexagonal perforated layers (HPL), alternating lamellar (LAM), and ordered bicontinuous double diamond (OBDD) phases. The agreement between our simulations and experimental results validate the Gaussian chain models and mesoscopic parameters used for these polymers and allow describing complex macromolecular structures of soft condensed matter with large molecular weight at the statistical segment level.

N log N method for hydrodynamic interactions of confined polymer systems: Brownian dynamics

A Brownian dynamics simulation technique is presented where a Fourier-based N log N approach is used to calculate hydrodynamic interactions in confined flowing polymer systems between two parallel walls. A self-consistent coarse-grained Langevin description of the polymer dynamics is adopted in which the polymer beads are treated as point forces. Hydrodynamic interactions are therefore included in the diffusion tensor through a Green's function formalism. The calculation of Green's function is based on a generalization of a method developed for sedimenting particles by Mucha et al. [J. Fluid Mech. 501, 71 (2004)]. A Fourier series representation of the Stokeslet that satisfies no-slip boundary conditions at the walls is adopted; this representation is arranged in such a way that the total O(N2) contribution of bead-bead interactions is calculated in an O(N log N) algorithm. Brownian terms are calculated using the Chebyshev polynomial approximation proposed by Fixman [Macromolecules 19, 1195 (1986); 19, 1204 (1986)] for the square root of the diffusion tensor. The proposed Brownian dynamics simulation methodology scales as O(N1.25 log N). Results for infinitely dilute systems of dumbbells are presented to verify past predictions and to examine the performance and numerical consistency of the proposed method.

Microfluidic capturing-dynamics of paramagnetic bead suspensions

We study theoretically the capturing of paramagnetic beads by a magnetic field gradient in a microfluidic channel treating the beads as a continuum. Bead motion is affected by both fluidic and magnetic forces. The transfer of momentum from beads to the fluid creates an effective bead-bead interaction that greatly aids capturing. We demonstrate that for a given inlet flow speed a critical density of beads exists above which complete capturing takes place.

Effects of the bead-bead potential on the restricted rotational diffusion of nonrigid macromolecules.

The influence of the bead-bead interaction on the rotational dynamics of macromolecules which are immersed into a solution has been investigated by starting from the microscopic theory of the macromolecular motion, i.e., from a Fokker-Planck equation for the phase-space distribution function. From this equation, we then derived an explicit expression for the configuration-space distribution function of a nonrigid molecule which is immobilized on a surface. This function contains all the information about the interaction among the beads as well as the effects from the surrounding solvent particles and from the surface. For the restricted rotational motion, the dynamics of the macromolecules can now be characterized in terms of a rotational diffusion coefficient as well as a radial distribution functions. Detailed computations for the rotational diffusion coefficient and the distribution functions have been carried out for HOOKEAN, finitely extensible nonlinear elastic, and a DNA type bead-bead interaction.

Effects of bead-bead interactions on the static and dynamical properties of model polymer solutions.

The effects of segment-segment interactions on the static and dynamical properties of model polymer solutions are examined by Brownian dynamics simulations in the free-draining limit over a wide concentration range. A bead-and-spring model is used to describe the polymer chains at a coarse-grained level, in which segment-segment interactions are represented by a bead-bead pair potential with a Gaussian analytic form, beta u(ev)(r)=A exp(-r(2)/2 sigma(2)), where beta=1/k(B)T and A and sigma are characteristic energy and distance scales, respectively. The chain dimensions, self-diffusion coefficient, and viscosity of the systems are studied as functions of number density of beads of the system, rho, at given excluded-volume potential parameters, A and sigma. Our results show that in the limit of infinite dilution even for short chains (N approximately 10) there is statistically significant scaling behavior in the static and dynamical properties. For a system with given values of A and sigma the change in polymer coil size shows a realistic trend as the concentration of the system increases. In the dilute and concentrated regions the coil size decreases as a result of increasing interchain repulsions, while in the highly concentrated region the coil size increases again, showing a return to Rouse-like behavior because the intrapolymer and interpolymer segment-segment interactions become effectively indistinguishable for an arbitrary bead and to a large extent are "balanced out." In the limit of infinite dilution, the self-diffusion coefficient of the center of mass, D(cm), depends on N only and not on the potential parameter A, while in contrast the specific viscosity eta(sp) depends on both N and A. As the concentration increases D(cm) decreases and eta(sp) increases consistent with the behavior of real polymers. When the system becomes highly concentrated, however, both D(cm) and eta(sp) unrealistically return to the Rouse limit. This suggests that from the concentrated region upward in concentration, the entanglement or the topological constraints caused by the physical connectivity of the chains significantly influence their dynamical behavior. The mean-field segment-segment interactions or excluded-volume effects incorporated in the current coarse-grained bead-spring approach cannot capture this entanglement effect, and therefore give rise to unrealistic dynamical behavior in the concentrated regime.

Molecular dynamics of model liquid crystals composed of semiflexible molecules.

Results of molecular-dynamics computer simulations are presented for a simple microscopic model of thermotropic liquid crystals. The system is composed of short multibead semiflexible chains where the beads are connected by an anharmonic spring. The intermolecular bead-bead interactions are modeled by Lennard-Jones potentials, the attractive part is taken into account only between the stiff parts. Heating the system, solid, smectic-A, and liquid phases are found. For the symmetric molecules and anisotropic potentials studied first, the smectic-A phase is clearly defined over a wide range of temperatures, whereas the nematic phase is not present or too narrow in temperature to be seen clearly. The evolution of the systems has also been studied as a function of the length of flexible parts and the strength of the stiff-stiff potential. \textcopyright{} 1996 The American Physical Society.

Multicomponent diffusion in polymeric liquids.

It is shown how the phase-space kinetic theory of polymeric liquid mixtures leads to a set of extended Maxwell-Stefan equations describing multicomponent diffusion. This expression reduces to standard results for dilute solutions and for undiluted polymers. The polymer molecules are modeled as flexible bead-spring structures. To obtain the Maxwell-Stefan equations, the usual expression for the hydrodynamic drag force on a bead, used in previous kinetic theories, must be replaced by a new expression that accounts explicitly for bead-bead interactions between different molecules.