Excluded Volume Potential Research Articles

Excluded Volume Effect on the Extensional Rheology of Carbon Nanotubes: A Mesoscopic Theory

We present a mesoscopic theory of the extensional rheology of liquid crystalline carbon nanotubes (CNTs). We implement the mesoscopic Leslie–Ericksen theory, considering the repulsive excluded volume potential acting in the direction opposite to the Frank elastic potential. Given the volume fraction, diameter, and length of the CNTs, our model predicts the extensional viscosity. The liquid crystalline phase of the CNTs exhibited extensional thinning behavior. In our theory, the rotational relaxation time is proportional to the reciprocal excluded volume, particularly for long CNTs with waviness. Our model also predicts the minimum draw ratio, below which the nanotube orientation cannot reach the target orientation. The existence of a minimum draw ratio is due to the large excluded volume potential of the long CNTs.

From microscopic theory to macroscopic theory — symmetries and order parameters of rigid molecules

We use density functional theory to describe the phase behaviors of rigid molecules. The construction of kernel function G(x, P, x, P) is discussed. Excluded-volume potential is calculated for two types of molecules with C_{2v} symmetry. Molecular symmetries lead to the symmetries of G and density function f(P), enabling a reduction of configuration space. By approximating G with a polynomial, the system can be fully characterized by some moments corresponding to the form of G. The symmetries of G determine the form of the polynomial, while the coefficients are determined by temperature and molecular parameters. The analysis of the impact of coefficients helps us to choose independent variables in the moments as order parameters. Order parameters for bent-core molecules are predicted.

Recovering the reptation dynamics of polymer melts in dissipative particle dynamics simulations via slip-springs

We report a multi-chain approach for dissipative particle dynamics where the uncrossability constraints of polymer chains are mimicked by temporary cross-links, so-called slip-springs. The conformational statistics of the chains are not affected by the introduction of slip-springs. Dynamical properties such as mean square displacements, diffusion coefficient, and longest relaxation time are in good agreement with the results of reptation theory. According to our analysis, the present formalism is 500 times faster and requires 7 times fewer beads than conventional generic polymer models employing Newtonian dynamics and excluded-volume potentials.

Diffusiophoretic self-propulsion of colloids driven by a surface reaction: The sub-micron particle regime for exponential and van der Waals interactions

Diffusiophoresis is a mechanism for propelling colloid particles in a liquid in which the driving force for the motion derives from intermolecular interactions between solute molecules surrounding the particle and the colloid itself. When solutes are asymmetrically distributed around the particle, the solutal interactions exerted on the colloid are unbalanced, and the particle is propelled. In self-diffusiophoresis, the particle itself creates the asymmetric distribution as a means of autonomous motion (a motor). Experiments implement the asymmetric production of a solutal concentration gradient by functionalizing one side of the colloid with a catalyst, which converts a reactant solute into a product. Previous hydrodynamic models of this design have assumed the length scale L of the intermolecular interaction (typically of order 1−10 nm) to be much smaller than the colloid radius, a (order 1 μm), L/a < 1. In this limit, assuming the catalytic reaction produces a constant flux of solute, and convective effects are negligible, the self-diffusiophoretic velocity is to leading order independent of a. Anticipating future experiments on nanosized motors (a=O(10−100 nm)), numerical solutions are presented for the velocity up to order one in L/a, and an integral asymptotic approximation is constructed accurate for L/a less than 0.1. Three intermolecular interactions are examined, a hard sphere excluded volume potential, an exponential interaction and a long-range van der Waals attraction, which is computed by pairwise additivity and formulated to include the attraction of the solvent with the colloid. For each interaction, the velocity decreases as the colloid radius decreases with the interaction parameters constant. For small L/a, velocity for the exponential potential decreases with an order one correction in L/a while this correction is logarithmic for the van der Waals potential. A curve for velocity as a function of a is constructed for the van der Waals interaction in terms of the pairwise interaction parameters of the colloid with the solute and solvent.

Knowledge-Based Potential for the Polypeptide Backbone

A knowledge-based potential for the polypeptide backbone as a function of the dihedral angles is developed and tested. The potential includes correlations due to the conformations and compositions of adjacent residues. Its purpose is to serve as a major component of a coarse-grained protein potential by including the most relevant local interactions while averaging out nonbonded ones. A probability density estimation algorithm and a multi-resolution probability combination procedure are developed to produce smooth probability distributions and dihedral angle potentials. The potential is described by a set of two-dimensional dihedral angle surfaces involving the various combinations of amino acid triplets and duplets. Several tests are carried out to evaluate the quality of the potential. Monte Carlo simulations, using only the dihedral angle potential and a coarse-grained excluded volume potential, show that the resulting dihedral angle distributions and correlations are consistent with those extracted from protein structures. Additional simulations of unfolded proteins are carried out to measure the NMR residual dipolar coupling (RDC). Significant correlations are obtained between the simulations and the corresponding experiments consistent with other simulations in the literature. Finally, the dihedral angle entropies are calculated for the 20 amino acids. In particular, the entropy difference between alanine and glycine agrees with the ones computed from molecular dynamics simulations ( approximately 0.4 kcal/mol).

Multiple branches of ordered states of polymer ensembles with the Onsager excluded volume potential

We study the branches of equilibrium states of rigid polymer rods with the Onsager excluded volume potential in two-dimensional space. Since the probability density and the potential are related by the Boltzmann relation at equilibrium, we represent an equilibrium state using the Fourier coefficients of the Onsager potential. We derive a non-linear system for the Fourier coefficients of the equilibrium state. We describe a procedure for solving the non-linear system. The procedure yields multiple branches of ordered states. This suggests that the phase diagram of rigid polymer rods with the Onsager potential has a more complex structure than that with the Maier–Saupe potential. A study of free energy indicates that the first branch of ordered states is stable while the subsequent branches are unstable. However, the instability of the subsequent branches does not mean they are not interesting. Each of these unstable branches, under certain external potential, can be made metastable, and thus may be observed.

Effects of strong anchoring on the dynamic moduli of heterogeneous nematic polymers

We focus on the linear viscoelastic response of heterogeneous nematic polymers to small amplitude oscillatory shear, paying special attention to the macroscopic influence of strong plate anchoring conditions. The model consists of the Stokes hydrodynamic equations with viscous and nematic stresses, coupled to orientational dynamics and structure driven by the flow gradient, an excluded-volume potential, and a two-constant distortional elasticity potential. We show that the dynamical response simplifies when plate anchoring is either tangential or homeotropic, recovering explicitly solvable Leslie–Ericksen–Frank behavior together with weakly varying order parameters across the plate gap. With these plate conditions, we establish “model consistency” so that all experimental driving conditions (plate-controlled velocity [strain] or shear stress, imposed oscillatory pressure) yield identical dynamic moduli for the same material parameters and anchoring conditions, eliminating the culpability of device influence in scaling behavior. Two physical predictions emerge that imply significant macroscopic elastic and viscous effects controlled by plate anchoring relative to flow geometry: (1) The storage modulus is enhanced by two to three orders of magnitude for homeotropic relative to parallel anchoring, across all frequencies. (2) The loss modulus exhibits enhancement of a factor of two to three for homeotropic over tangential anchoring, restricted to low frequencies. We further deduce a scaling law for the dynamic moduli versus anisotropy of the distortional elasticity potential.

Using Newton-GMRES for viscoelastic flow time-steppers

Kinetic theory models exhibit dynamics that depend on a few low-order moments of the underlying conformational distribution function. This dependence is exhibited in a compact spectrum of eigenvalues for the Jacobian matrix associated with the dynamical system. We take advantage of this spectrum of eigenvalues through Newton-GMRES iterations to enable dynamic viscoelastic simulators (time-steppers) to obtain stationary states and perform stability/bifurcation analysis. Results are presented for three example problems: (1) the equilibrium behavior of the Doi model with the Onsager excluded volume potential, (2) pressure-driven flow of non-interacting rigid dumbbells in a planar channel, and (3) pressure-driven flow of non-interacting rigid dumbbells through a planar channel with a linear array of equally spaced cylinders.

Study of phase transition in homogeneous, rigid extended nematics and magnetic suspensions using an order-reduction method

We study the phase transition in rigid extended nematics and magnetic suspensions by solving the Smoluchowski equation for magnetically polarized rigid nematic polymers and suspensions in equilibrium, in which the molecular interaction is modeled by a dipolar and excluded volume potential. The equilibrium solution (or the probability distribution of the molecular distribution) is given by a Boltzmann distribution parametrized by the (first-order) polarity vector and the (second-order) nematic order tensor along with material parameters. We show that the polarity vector coincides with one of the principal axes of the nematic order tensor so that the equilibrium distribution can be reduced to a Boltzmann distribution parametrized by three scalar order parameters, i.e., a polar order parameter and two nematic order parameters, governed by three nonlinear algebraic-integral equations. This reduction in the degree of freedom from 8 (3 in the polarity vector and 5 in the nematic order tensor) to 3 significantly simplifies the solution procedure and allows one to conduct a complete analysis on bifurcation diagrams of the order parameters with respect to the material parameters. The stability of the equilibria is inferred from the second variation of the free energy density.

Colloidal brushes in complex solutions: Existence of a weak midrange attraction due to excluded-volume effects

Towards the prediction of thermodynamic properties for complex systems and the design of self-assembly techniques for the production of nanocomposite materials, we employ Monte Carlo simulations to calculate the effective interactions between colloidal brushes dissolved in nonadsorbing polymer solutions. For simplicity, all interactions are reduced to excluded-volume potentials. Our results indicate that (1) due to excluded-volume effects and the absence of attractive van der Waals forces in our calculations, the effective interactions between colloidal brushes in a continuum medium are always repulsive; and (2) that the short-ranged repulsion between the colloidal brushes may be coupled to a midrange attraction when nonadsorbing polymers are in solution. We prove that a depletion mechanism is responsible for the midranged attraction and we observe that the strength of the induced attraction is weaker compared to the depletion attraction computed for hard-sphere colloids in nonadsorbing polymer solutions. More interestingly, however, our results indicate that, even for the simple interaction potentials employed here, it is not possible to superimpose depletion attraction and steric repulsion to predict the correct pair potential between colloidal brushes in nonadsorbing polymer solutions.

Monodomain dynamics for rigid rod and platelet suspensions in strongly coupled coplanar linear flow and magnetic fields. II. Kinetic theory

We establish reciprocity relations of the Doi-Hess kinetic theory for rigid rod macromolecular suspensions governed by the strong coupling among an excluded volume potential, linear flow, and a magnetic field. The relation provides a reduction of the flow and field driven Smoluchowski equation: from five parameters for coplanar linear flows and magnetic field, to two field parameters. The reduced model distinguishes flows with a rotational component, which map to simple shear (with rate parameter) subject to a transverse magnetic field (with strength parameter), and irrotational flows, for which the reduced model consists of a triaxial extensional flow (with two extensional rate parameters). We solve the Smoluchowski equation of the reduced model to explore: (i) the effect of introducing a coplanar magnetic field on each sheared monodomain attractor of the Doi-Hess kinetic theory and (ii) the coupling of coplanar extensional flow and magnetic fields. For (i), we show each sheared attractor (steady and unsteady, with peak axis in and out of the shearing plane, periodic and chaotic orbits) undergoes its own transition sequence versus magnetic field strength. Nonetheless, robust predictions emerge: out-of-plane degrees of freedom are arrested with increasing field strength, and a unique flow-aligning or tumbling/wagging limit cycle emerges above a threshold magnetic field strength or modified geometry parameter value. For (ii), irrotational flows coupled with a coplanar magnetic field yield only steady states. We characterize all (generically biaxial) equilibria in terms of an explicit Boltzmann distribution, providing a natural generalization of analytical results on pure nematic equilibria [P. Constantin, I. Kevrekidis, and E. S. Titi, Arch. Rat. Mech. Anal. 174, 365 (2004); P. Constantin, I. Kevrekidis, and E. S. Titi, Discrete and Continuous Dynamical Systems 11, 101 (2004); P. Constantin and J. Vukadinovic, Nonlinearity 18, 441 (2005); H. Liu, H. Zhang, and P. Zhang, Comm. Math. Sci. 3, 201 (2005); C. Luo, H. Zhang, and P. Zhang, Nonlinearity 18, 379 (2005); I. Fatkullin and V. Slastikov, Nonlinearity 18, 2565 (2005); H. Zhou, H. Wang, Q. Wang, and M. G. Forest, Nonlinearity 18, 2815 (2005)] and extensional flow-induced equilibria [Q. Wang, S. Sircar, and H. Zhou, Comm. Math. Sci. 4, 605 (2005)]. We predict large parameter regions of bi-stable equilibria; the lowest energy state always has principal axis aligned in the flow plane, while another minimum energy state often exists, with primary alignment transverse to the coplanar field.

Transitions to nematic states in homogeneous suspensions of high aspect ratio magnetic rods

Isotropic-nematic and nematic-nematic transitions from a homogeneous suspension of high aspect ratio magnetic rods are studied for both Maier-Saupe and Onsager excluded volume potentials. Asymptotic analysis in the vicinity of critical points yields insight into the stability and type of polarized nematic states emanating from nonpolarized equilibrium states. This, in conjunction with recently published global numerical results, yields a unified picture of the bifurcation diagram and provides a convenient base state to study effects of external orienting fields.

Simulations of Concentrated Suspensions of Semirigid Fibers: Effect of Bending on the Rotational Diffusivity

Brownian dynamics simulations of the behavior of semirigid polymers demonstrate that small deviations from rigid models affect the dynamics of rod suspensions and that the dynamics are sensitive to the hydrodynamic model. Simulations were performed with two models having different short-time diffusivities. The first was a slender-body dimer model, where two rigid slender-bodies are joined at their ends, and the second was a three-bead trimer model where the beads are connected by rigid rods. A potential was applied to hinder the bending of the rods. An aspect ratio (rod length over diameter) of 50 was used for all simulations, and an excluded volume potential was used to prevent the rods from crossing one another. Simulations of systems of stiff rods which are not quite rigid show a delayed onset of semidilute behavior in comparison to simulations of rigid rod systems, with different hydrodynamic models providing different power-law scalings of the rotational diffusivities within the semidilute concentrat...

A new proof on axisymmetric equilibria of a three-dimensional Smoluchowski equation

We consider equilibrium solutions of the Smoluchowski equation for rodlike nematic polymers with a Maier–Saupe excluded volume potential. The purpose of this paper is to present a new and simplified proof of classical well-known results: (1) all equilibria are axisymmetric and (2) modulo rotational symmetry, the number and type of axisymmetric equilibria are characterized with respect to the strength of the excluded volume potential. These results confirm the phase diagram of equilibria obtained previously by numerical simulations (Faraoni et al 1999 J. Rheol. 43 829–43, Forest et al 2004 Rheol. Acta 43 17–37, Larson and Ottinger 1991 Macromolecules 24 6270–82).

A united residue force-field for calcium-protein interactions.

United-residue potentials are derived for interactions of the calcium cation with polypeptide chains in energy-based prediction of protein structure with a united-residue (UNRES) force-field. Specific potentials were derived for the interaction of the calcium cation with the Asp, Glu, Asn, and Gln side chains and the peptide group. The analytical expressions for the interaction energies for each of these amino acids were obtained by averaging the electrostatic interaction energy, expressed by a multipole series over the dihedral angles not considered in the united-residue model, that is, the side-chain dihedral angles chi and the dihedral angles lambda for the rotation of peptide groups about the C(alpha)...C(alpha) virtual-bond axes. For the side-chains that do not interact favorably with calcium, simple excluded-volume potentials were introduced. The parameters of the potentials were obtained from ab initio quantum mechanical calculations of model systems at the Restricted Hartree-Fock (RHF) level with the 6-31G(d,p) basis set. The energy surfaces of pairs consisting of Ca(2+)-acetate, Ca(2+)-propionate, Ca(2+)-acetamide, Ca(2+)-propionamide, and Ca(2+)-N-methylacetamide systems (modeling the Ca(2+)-Asp(-), Ca(2+)-Glu(-), Ca(2+)-Asn, Ca(2+)-Gln, and Ca(2+)-peptide group interactions) at different distances and orientations were calculated. For each pair, the restricted free energy (RFE) surfaces were calculated by numerical integration over the degrees of freedom lost when switching from the all-atom model to the united-residue model. Finally, the analytical expressions for each pair were fitted to the RFE surfaces. This force-field was able to distinguish the EF-hand motif from all potential binding sites in the crystal structures of bovine alpha-lactalbumin, whiting parvalbumin, calbindin D9K, and apo-calbindin D9K.

Universal consequences of the presence of excluded volume interactions in dilute polymer solutions undergoing shear flow.

The role of solvent quality in determining the universal material properties of dilute polymer solutions undergoing steady simple shear flow is examined. A bead-spring chain representation of the polymer molecule is used, and the influence of solvent molecules on polymer conformations is modelled by a narrow Gaussian excluded volume potential that acts pairwise between the beads of the chain. Brownian dynamics simulations data, acquired for chains of finite length, and extrapolated to the limit of infinite chain length, are shown to be model independent. This feature of the narrow Gaussian potential, which leads to results identical to a delta-function repulsive potential, enables the prediction of both universal crossover scaling functions and asymptotic behavior in the excluded volume limit. Universal viscometric functions, obtained by this procedure, are found to exhibit increased shear thinning with increasing solvent quality. In the excluded volume limit, they are found to obey power law scaling with the characteristic shear rate beta, in close agreement with previously obtained renormalization group results. The presence of excluded volume interactions is also shown to lead to a weakening of the alignment of the polymer chain with the flow direction.

Equilibrium Swelling and Universal Ratios in Dilute Polymer Solutions: Exact Brownian Dynamics Simulations for a Delta Function Excluded Volume Potential

A narrow Gaussian excluded volume potential, which tends to a delta-function repulsive potential in the limit of a width parameter d* going to zero, has been used to examine the universal consequences of excluded volume interactions on the equilibrium and linear viscoelastic properties of dilute polymer solutions. Brownian dynamics simulations data, acquired for chains of finite length, has been extrapolated to the limit of infinite chain length to obtain model independent predictions. The success of the method in predicting well known aspects of static solution properties suggests that it can be used as a systematic means by which the influence of solvent quality on both equilibrium and non-equilibrium properties can be studied.

Hydrodynamic interactions in long chain polymers: Application of the Chebyshev polynomial approximation in stochastic simulations

We have simulated Brownian bead-spring chains of up to 125 units with fluctuating hydrodynamic and excluded volume interactions using the Chebyshev polynomial approximation proposed by Fixman [Macromolecules 19, 1204 (1986)] for the square root of the diffusion tensor. We have developed a fast method to continuously determine the validity of the eigenvalue range used in the polynomial approximation, and demonstrated how this range may be quickly updated when necessary. We have also developed a weak first order semiimplicit time integration scheme which offers increased stability in the presence of steep excluded volume potentials. The full algorithm scales roughly as O(N2.25) and offers substantial computational savings over the standard Cholesky decomposition. The above algorithm was used to obtain scaling exponents for various static and zero shear rate dynamical properties, which are found to be consistent with theoretical and/or experimental predictions.

Viscoelastic properties of dilute polymer solutions: The effect of varying the concentration

Dynamic viscoelastic behavior was investigated for solutions of polystyrene in tricresyl phosphate, a good solvent, at concentrations, c, less than the coil-overlapping concentration, c*. At the infinite dilution limit, the behavior was in accord with the theory of Doi and Edwards involving the excluded volume potential and hydrodynamic interaction (HDI). Thus, the viscoelastic functions were completely derived from the intrinsic viscosity–molecular weight relation. At finite c, the complex modulus was represented by the sum of two terms. One was a Rouse–Zimm (RZ) term conveniently represented by the Zimm theory with an arbitrarily chosen value of the HDI parameter. The other was a term with a single relaxation time, longer than the longest RZ relaxation time, and with a high-frequency modulus proportional to the square of c [the long-time (LT) term]. The behavior of the RZ term indicated the stronger screening of HDI with increasing c. Using the experimental c dependence of the longest RZ relaxation time to get the relevant parameter, we compared the RZ viscoelastic function with the Muthukumar–Freed theory. The agreement was good at low concentrations, c < c*. The contribution of the LT term, which was not included in the theory, was quite significant at low frequencies; about 60% of the Huggins coefficient was attributable to this term. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 211–217, 2001

Generating moment equations in the Doi model of liquid-crystalline polymers.

We present a self-consistent method for deriving moment equations for kinetic models of polymer dynamics. The Doi model [J. Polym. Sci., Polym. Phys. Ed. 19, 229 (1981)] of liquid-crystalline polymers with the Onsager excluded-volume potential is considered as an example. To lowest order, this method amounts to a simple effective potential different from the Maier-Saupe form. Analytical results are presented which indicate that this effective potential provides a better approximation to the Onsager potential than the Maier-Saupe potential. Corrections to the effective potential are obtained.