Some Applications of Fractional Calculus to Polymer Science. In: Prigogine I, Rice SA, editors. Advances in Chemical Physics. Hoboken: John Wiley & Sons, Inc.; 1997. pp. 121-91. [DOI: 10.1002/9780470141618.ch3]

Structure and Dynamics of a Graphene Melt

We explore the structural and dynamic properties of bulk materials composed of graphene nanosheets using coarse-grained molecular dynamics simulations. Remarkably, our results show clear evidence that bulk graphene materials exhibit a fluid-like behavior similar to linear polymer melts at elevated temperatures and that these materials transform into a glassy-like "foam" state at temperatures below the glass-transition temperature ( T_g) of these materials. Distinct from an isolated graphene sheet, which exhibits a relatively flat shape with fluctuations, we find that graphene sheets in a melt state structurally adopt more "crumpled" configurations and correspondingly smaller sizes, as normally found for ordinary polymers in the melt. Upon approaching the glass transition, these two-dimensional polymeric materials exhibit a dramatic slowing down of their dynamics that is likewise similar to ordinary linear polymer glass-forming liquids. Bulk graphene materials in their glassy foam state have an exceptionally large free-volume and high thermal stability due to their high T_g (≈ 1600 K) as compared to conventional polymer materials. Our findings show that graphene melts have interesting lubricating and "plastic" flow properties at elevated temperatures, and suggest that graphene foams are highly promising as high surface filtration materials and fire suppression additives for improving the thermal conductivities and mechanical reinforcement of polymer materials.

End-anchored polymers in good solvents from the single chain limit to high anchoring densities

An increasing number of applications utilize grafted polymer layers to alter the interfacial properties of solid substrates, motivating refinement in our theoretical understanding of such layers. To assess existing theoretical models of them, we have investigated end-anchored polymer layers over a wide range of grafting densities, σ, ranging from a single chain to high anchoring density limits, chain lengths ranging over two orders of magnitude, for very good and marginally good solvent conditions. We compare Monte Carlo and molecular dynamics simulations, numerical self-consistent field calculations, and experimental measurements of the average layer thickness, h, with renormalization group theory, the Alexander-de Gennes mushroom theory, and the classical brush theory. Our simulations clearly indicate that appreciable inter-chain interactions exist at all simulated areal anchoring densities so that there is no mushroom regime in which the layer thickness is independent of σ. Moreover, we find that there is no high coverage regime in which h follows the predicted scaling, h ∼ Nσ^1/3, for classical polymer brushes either. Given that no completely adequate analytic theory seems to exist that spans wide ranges of N and σ, we applied scaling arguments for h as a function of a suitably defined reduced anchoring density, defined in terms of the solution radius of gyration of the polymer chains and N. We find that such a scaling approach enables a smooth, unified description of h in very good solvents over the full range of anchoring density and chain lengths, although this type of data reduction does not apply to marginal solvent quality conditions.

FRACTIONAL INTEGRATION TOOLBOX

The problems formulated in the fractional calculus framework often require numerical fractional integration/differentiation of large data sets. Several existing fractional control toolboxes are capable of performing fractional calculus operations, however, none of them can efficiently perform numerical integration on multiple large data sequences. We developed a Fractional Integration Toolbox (FIT), which efficiently performs fractional numerical integration/differentiation of the Riemann-Liouville type on large data sequences. The toolbox allows parallelization and is designed to be deployed on both CPU and GPU platforms.

Properties of knotted ring polymers. II. Transport properties

We have calculated the hydrodynamic radius R(h) and intrinsic viscosity [eta] of both lattice self-avoiding rings and lattice theta-state rings that are confined to specific knot states by our path-integration technique. We observe that naive scaling arguments based on the equilibrium polymer size fail for both the hydrodynamic radius and the intrinsic viscosity, at least over accessible chain lengths. (However, we do conjecture that scaling laws will nevertheless prevail at sufficiently large N.) This failure is attributed to a "double" cross-over. One cross-over effect is the transition from delocalized to localized knotting: in short chains, the knot is distributed throughout the chain, while in long chains it becomes localized in only a portion of the chain. This transition occurs slowly with increasing N. The other cross-over, superimposed upon the first, is the so-called "draining" effect, in which transport properties maintain dependence on local structure out to very large N. The hydrodynamic mobility of knotted rings of the same length and backbone structure is correlated with the average crossing number X of the knots. The same correlation between mobility and knot complexity X has been observed for the gel-electrophoretic mobility of cyclic DNA molecules.

Influence of variable hydrodynamic interaction strength on the transport properties of coiled polymers

We have computed the hydrodynamic radius Rh and intrinsic viscosity [eta] of a large number of random coil polymer models by path integration. We examine the effects of chain length, solvent quality, monomer size and shape, and chain stiffness on the approach of these transport properties to the limit of large molecular mass. For many of the models, we have also calculated the ensemble-averaged solvent velocity field in the vicinity of the coil as it moves with constant drift velocity under the action of an external force. Naive scaling arguments predict alpha=nu and beta=3(nu-1) , where alpha , beta , and nu are the exponents controlling the chain length behavior of Rh , [eta] , and Rg , respectively. We present evidence for a "draining crossover" that quantifies the slow convergence of the transport properties to their asymptotic scaling behavior. Indeed, the convergence is so slow that effective alpha and beta exponents rarely agree with the naive predictions at typical molecular masses. For the same chain models, Rg converges rapidly to its asymptotic behavior, indicating that the effect is not due to a crossover from theta to swollen behavior, as often stated. Solvent quality, monomer size, and chain stiffness all influence the draining crossover. Our results call into question the common practice of extracting metrical data, e.g., characteristic ratios, directly from polymer solution transport properties.

Crowding induced self-assembly and enthalpy-entropy compensation

We develop a general virial expansion to describe the influence of molecular additives on the equilibrium self-assembly of proteins or other supermolecularly assembling species M in solution. When specialized to high molar mass polymer additives, the cross-virial coefficient between the polymer and M, which dominates this effect, is found to vanish at a particular temperature T_{Theta} corresponding to an enthalpy-entropy compensation condition. Specifically, the increased stability of the assembled form of M, due to the modification of the entropy of the assembly by repulsive polymer-protein interactions, is progressively compensated by attractive interactions that alter the enthalpy of assembly.

Improved path integration method for estimating the intrinsic viscosity of arbitrarily shaped particles

In previous work, we have established that the intrinsic viscosity [eta] of an object is nearly proportional to the average electrical polarizability tensor alphae = tr(alphae)/3 of a conducting object having the same shape, or equivalently, to the intrinsic conductivity [sigma]=alphae/V , which characterizes the conductivity of a dilute mixture of randomly oriented conducting objects (V being the volume of the object). This hydrodynamic-electrostatic analogy is useful because alphae can be determined accurately and efficiently by numerical path integration for objects of arbitrary shape. Here, we show that the uncertainty in [eta] can be reduced to a relatively small value (< 1.5% relative uncertainty) by utilizing additional information from the full tensor alphae, rather than just its average. Specifically, we determine the exact constant of proportionality between [eta] and [sigma] for triaxial ellipsoids as a function of the ratios of the eigenvalues of alphae and apply this relation to particles of general shape. In addition to an improved estimation of [eta] , the ratios of the components of alphae provide useful measures of particle anisotropy. We also present an improved method for applying the technique to flexible particles, which requires performing a conformational ensemble average. Conformational averages of alphae generate systematic errors that can be avoided by performing the conformational average at an earlier stage in the computation.