Ppam Paul van der Schoot

Continuum percolation of carbon nanotubes in polymeric and colloidal media

We apply continuum connectedness percolation theory to realistic carbon nanotube systems and predict how bending flexibility, length polydispersity, and attractive interactions between them influence the percolation threshold, demonstrating that it can be used as a predictive tool for designing nanotube-based composite materials. We argue that the host matrix in which the nanotubes are dispersed controls this threshold through the interactions it induces between them during processing and through the degree of connectedness that must be set by the tunneling distance of electrons, at least in the context of conductivity percolation. This provides routes to manipulate the percolation threshold and the level of conductivity in the final product. We find that the percolation threshold of carbon nanotubes is very sensitive to the degree of connectedness, to the presence of small quantities of longer rods, and to very weak attractive interactions between them. Bending flexibility or tortuosity, on the other hand, has only a fairly weak impact on the percolation threshold.

Controlling electrical percolation in multicomponent carbon nanotube dispersions

Carbon nanotube reinforced polymeric composites can have favourable electrical properties, which make them useful for applications such as flat-panel displays and photovoltaic devices. However, using aqueous dispersions to fabricate composites with specific physical properties requires that the processing of the nanotube dispersion be understood and controlled while in the liquid phase. Here, using a combination of experiment and theory, we study the electrical percolation of carbon nanotubes introduced into a polymer matrix, and show that the percolation threshold can be substantially lowered by adding small quantities of a conductive polymer latex. Mixing colloidal particles of different sizes and shapes (in this case, spherical latex particles and rod-like nanotubes) introduces competing length scales that can strongly influence the formation of the system-spanning networks that are needed to produce electrically conductive composites. Interplay between the different species in the dispersions leads to synergetic or antagonistic percolation, depending on the ease of charge transport between the various conductive components.

Connectivity percolation of polydisperse anisotropic nanofillers

We present a generalized connectedness percolation theory reduced to a compact form for a large class of anisotropic particle mixtures with variable degrees of connectivity. Even though allowing for an infinite number of components, we derive a compact yet exact expression for the mean cluster size of connected particles. We apply our theory to rodlike particles taken as a model for carbon nanotubes and find that the percolation threshold is sensitive to polydispersity in length, diameter, and the level of connectivity, which may explain large variations in the experimental values for the electrical percolation threshold in carbon-nanotube composites. The calculated connectedness percolation threshold depends only on a few moments of the full distribution function. If the distribution function factorizes, then the percolation threshold is raised by the presence of thicker rods, whereas it is lowered by any length polydispersity relative to the one with the same average length and diameter. We show that for a given average length, a length distribution that is strongly skewed to shorter lengths produces the lowest threshold relative to the equivalent monodisperse one. However, if the lengths and diameters of the particles are linearly correlated, polydispersity raises the percolation threshold and more so for a more skewed distribution toward smaller lengths. The effect of connectivity polydispersity is studied by considering nonadditive mixtures of conductive and insulating particles, and we present tentative predictions for the percolation threshold of graphene sheets modeled as perfectly rigid, disklike particles.

Lowering the percolation threshold of single-walled carbon nanotubes using polystyrene/poly(3,4-ethylenedioxythiophene): poly(styrene sulfonate) blends

Single-walled carbon nanotubes (SWCNTs) are introduced into a polymer matrix via a latex-based route resulting in a conductive composite. The percolation threshold for a polystyrene (PS)-based composite prepared with SWCNTs dispersed in water using a conventional surfactant like sodium dodecyl sulfate (SDS) is approximately 0.4 wt%. In this study, SDS is substituted by a conductive polymer latex, poly(3,4-ethylenedioxythiophene):poly(styrene sulfonate) also known as PEDOT:PSS. This latex can effectively stabilize individual SWCNTs in water and composites prepared with these dispersions show a lower percolation threshold value of 0.2 wt%. The percolation of PEDOT:PSS in PS in a binary polymer blend without SWCNTs is also investigated, and found to occur at a remarkably low loading of 2.2 wt% of the conductive latex. The morphology of the final polymer-filled blend is further investigated and the findings provide an explanation as to why PEDOT:PSS lowers the percolation threshold of the SWCNTs, and in fact has such a low threshold itself without the presence of the nanotubes.

Size Regulation of ss-RNA Viruses

While a monodisperse size distribution is common within one kind of spherical virus, the size of viral shells varies from one type of virus to another. In this article, we investigate the physical mechanisms underlying the size selection among spherical viruses. In particular, we study the effect of genome length and genome and protein concentrations on the size of spherical viral capsids in the absence of spontaneous curvature and bending energy. We find that the coat proteins could well adjust the size of the shell to the size of their genome, which in turn depends on the number of charges on it. Furthermore, we find that different stoichiometric mixtures of proteins and genome can produce virus particles of various sizes, consistent with in vitro experiments.

Kinetic theory of virus capsid assembly

A phenomenological theory is presented for the kinetics of the in vitro assembly and disassembly of icosahedral virus capsids in solutions of coat proteins. The focus is on conditions where nucleation-type processes can be ignored. We find that the kinetics of assembly is strongly concentration dependent and that the late-stage relaxation time varies as the inverse of the square of the concentration. These findings are corroborated by experimental observations on a number of viruses. Further, our theory shows that hysteresis observed in some experiments could be a direct effect of the kinetics of a high-order mass action law, not necessarily the result of a free energy barrier between assembled and disassembled states.

Micromechanics of temperature sensitive microgels: dip in the Poisson ratio near the LCST

Microgels of poly-N-isopropylacrylamide (pNIPAM) exhibit a remarkable sensitivity to environmental conditions, most strikingly a pronounced deswelling that occurs close to the lower critical solution temperature (LCST) of the polymer at ≈32 °C. This transition has been widely studied and exploited in a range of applications. Along with changes in size, significant changes are also expected for the mechanical response of the particles. However, the full elastic properties of these particles as a function of temperature, T, have not yet been assessed at the single-particle level. Here we present measurements of the elastic properties of pNIPAM particles as a function of both temperature and cross-linking density using capillary micromechanics, a technique based on the pressure-dependent deformation of particles trapped in a tapered glass capillary. The shear elastic modulus G increased monotonously upon increasing temperature. In contrast, but in qualitative agreement with previous experiments on macroscopic pNIPAM hydrogels, we found that the compressive elastic modulus K of our microgels exhibits a dip close to the LCST. Remarkably, this dip is less sharp and deep than that observed in macroscopic hydrogels. The Poisson ratio of the particles also exhibits a pronounced dip close to the LCST, reaching unusually low minimum values of σ ≈ 0.15. To rationalize this behavior, we compared our experimental data to Flory–Rehner theory; the theory is able to qualitatively predict the general mechanical behavior observed, thus indicating that the observed dip in the Poisson ratio can be accounted for by simple thermodynamic arguments.

Percolation in suspensions of polydisperse hard rods: Quasi universality and finite-size effects

We present a study of connectivity percolation in suspensions of hard spherocylinders by means of Monte Carlo simulation and connectedness percolation theory. We focus attention on polydispersity in the length, the diameter, and the connectedness criterion, and we invoke bimodal, Gaussian, and Weibull distributions for these. The main finding from our simulations is that the percolation threshold shows quasi universal behaviour, i.e., to a good approximation, it depends only on certain cumulants of the full size and connectivity distribution. Our connectedness percolation theory hinges on a Lee-Parsons type of closure recently put forward that improves upon the often-used second virial approximation [T. Schilling, M. Miller, and P. van der Schoot, e-print arXiv:1505.07660 (2015)]. The theory predicts exact universality. Theory and simulation agree quantitatively for aspect ratios in excess of 20, if we include the connectivity range in our definition of the aspect ratio of the particles. We further discuss the mechanism of cluster growth that, remarkably, differs between systems that are polydisperse in length and in width, and exhibits non-universal aspects.

Theory of supramolecular co-polymerization in a two-component system

As a first step to understand the role of molecular or chemical polydispersity in self-assembly, we put forward a coarse-grained model that describes the spontaneous formation of quasi-linear polymers in solutions containing two self-assembling species. Our theoretical framework is based on a two-component self-assembled Ising model in which the chemical bidispersity, i.e., the presence of two distinct chemical entities, is parameterized in terms of the strengths of the binding free energies that depend on the monomer species involved in the pairing interaction. Depending upon the relative values of the binding free energies involved, different morphologies of assemblies that include both components are formed, exhibiting random, blocky or alternating ordering of the two components in the assemblies. Analyzing the model for the case of blocky ordering, which is of most practical interest, we find that the transition from conditions of minimal assembly to those characterized by strong polymerization can be described by a critical concentration that depends on the concentration ratio of the two species. Interestingly, the distribution of monomers in the assemblies is different from that in the original distribution, i.e., the ratio of the concentrations of the two components put into the system. The monomers with a smaller binding free energy are more abundant in short assemblies and monomers with a larger binding affinity are more abundant in longer assemblies. Under certain conditions the two components congregate into separate supramolecular polymeric species and in that sense phase separate. We find strong deviations from the expected growth law for supramolecular polymers even for modest amounts of a second component, provided it is chemically sufficiently distinct from the main one.As a first step to understanding the role of molecular or chemical polydispersity in self-assembly, we put forward a coarse-grained model that describes the spontaneous formation of quasi-linear polymers in solutions containing two self-assembling species. Our theoretical framework is based on a two-component self-assembled Ising model in which the bidispersity is parameterized in terms of the strengths of the binding free energies that depend on the monomer species involved in the pairing interaction. Depending upon the relative values of the binding free energies involved, different morphologies of assemblies that include both components are formed, exhibiting paramagnetic, ferromagneticor anti ferromagnetic-like order, i.e., random, blocky or alternating ordering of the two components in the assemblies. Analyzing the model for the case of ferromagnetic ordering, which is of most practical interest, we find that the transition from conditions of minimal assembly to those characterized by strong polymerization can be described by a critical concentration that depends on the concentration ratio of the two species. Interestingly, the distribution of monomers in the assemblies is different from that in the original distribution, i.e., the ratio of the concentrations of the two components put into the system. The monomers with a smaller binding free energy are more abundant in short assemblies and monomers with a larger binding affinity are more abundant in longer assemblies. Under certain conditions the two components congregate into separate supramolecular polymeric species and in that sense phase separate. We find strong deviations from the expected growth law for supramolecular polymers even for modest amounts of a second component, provided it is chemically sufficiently distinct from the main one.

Amplification of chirality in helical supramolecular polymers beyond the long-chain limit

The optical activity of helical homopolymers devoid of chiral centers increases drastically when a small amount of homochiral monomers is incorporated into them. We study this so-called sergeants-and-soldiers effect of chirality amplification in solutions of helical supramolecular polymers with a theoretical model that bears a strong resemblance to a one-dimensional, two-component Ising model. In the limit of very long self-assembled helical polymers, the strength of the sergeants-and-soldiers effect depends strongly on the free energy of a helix reversal and less so on the concentration of aggregating material. Outside the long-chain limit, we find the reverse--that is, a strong concentration dependence and a weak dependence on the helix-reversal energy. Our treatment gives an excellent agreement with recently published circular-dichroism measurements on mixed aggregates of discotic molecules in the solvents water and n-butanol, at two different overall concentrations.