Excluded Volume Constraints Research Articles

Generalized geometric criteria for the absence of effective many-body interactions in the Asakura–Oosawa model

We investigate variants of the Asakura–Oosawa (AO) model for colloid-polymer mixtures, represented by hard classical particles interacting via their excluded volume. The interaction between the polymers is neglected but the colloid-polymer and colloid-colloid interactions are present and can be condensed into an effective depletion interaction among the colloids alone. The original AO model involves hard spherical particles in three spatial dimensions with colloidal radii R and the so-called depletion radius δ of the polymers, such that the minimum possible center-to-center distance between polymers and colloids allowed by the excluded-volume constraints is R + δ. It is common knowledge among physicists that there are only pairwise effective depletion interactions between the colloids if the geometric condition δ/R<2/3−1 is fulfilled. In this case, triplet and higher-order many body interactions are vanishing and the equilibrium statistics of the binary mixture can exactly be mapped onto that of an effective one-component system with the effective depletion pair-potential. Here we rigorously prove that the criterion δ/R<2/3−1 is both sufficient and necessary to guarantee the absence of triplet and higher-order many body interactions among the colloids. For an external hard wall confining the system, we also include a criterion which guarantees that the system can be exactly mapped onto one with effective external one-body interactions. Our general formulation also accounts for polydisperse mixtures and anisotropic shapes of colloids in any spatial dimension. In those cases where the resulting condition is only sufficient, we further demonstrate how to specify improved bounds.

High-dimensional optimization under nonconvex excluded volume constraints.

We consider high-dimensional random optimization problems where the dynamical variables are subjected to nonconvex excluded volume constraints. We focus on the case in which the cost function is a simple quadratic cost and the excluded volume constraints are modeled by a perceptron constraint satisfaction problem. We show that depending on the density of constraints, one can have different situations. If the number of constraints is small, one typically has a phase where the ground state of the cost function is unique and sits on the boundary of the island of configurations allowed by the constraints. In this case, there is a hypostatic number of marginally satisfied constraints. If the number of constraints is increased one enters a glassy phase where the cost function has many local minima sitting again on the boundary of the regions of allowed configurations. At the phase transition point, the total number of marginally satisfied constraints becomes equal to the number of degrees of freedom in the problem and therefore we say that these minima are isostatic. We conjecture that by increasing further the constraints the system stays isostatic up to the point where the volume of available phase space shrinks to zero. We derive our results using the replica method and we also analyze a dynamical algorithm, the Karush-Kuhn-Tucker algorithm, through dynamical mean-field theory and we show how to recover the results of the replica approach in the replica symmetric phase.

Surfing on minima of isostatic landscapes: avalanches and unjamming transition

Recently, we showed that optimization problems, both in infinite as well as in finite dimensions, for continuous variables and soft excluded volume constraints, can display entire isostatic phases where local minima of the cost function are marginally stable configurations endowed with non-linear excitations [, ]. In this work we describe an athermal adiabatic algorithm to explore with continuity the corresponding rough high-dimensional landscape. We concentrate on a prototype problem of this kind, the spherical perceptron optimization problem with linear cost function (hinge loss). This algorithm allows to ‘surf’ between isostatic marginally stable configurations and to investigate some properties of such landscape. In particular we focus on the statistics of avalanches occurring when local minima are destabilized. We show that when perturbing such minima, the system undergoes plastic rearrangements whose size is power law distributed and we characterize the corresponding critical exponent. Finally we investigate the critical properties of the unjamming transition, showing that the linear interaction potential gives rise to logarithmic behavior in the scaling of energy and pressure as a function of the distance from the unjamming point. For some quantities, the logarithmic corrections can be gauged out. This is the case of the number of soft constraints that are violated as a function of the distance from jamming which follows a non-trivial power law behavior.

Role of Grafting Mechanism on the Polymer Coverage and Self-Assembly of Hairy Nanoparticles.

It is now well-accepted that controlling the spatial dispersion of nanoparticles (NPs), which can be achieved by grafting them with polymers of different chain lengths and grafting densities, is central to optimizing the thermomechanical properties of the resulting polymer nanocomposites. In general, there are two methods for creating such polymer-grafted NPs: "grafting to" and "grafting from". The conventional wisdom is that the "grafting from" mechanism, where monomer-sized initiator/functional groups are attached to the surface followed by growing the chains, allows for higher polymer grafting densities and hence a more uniform polymer coverage of the NP surface. Here, we perform calculations and instead show that the "grafting to" strategy surprisingly leads to a more uniform polymer coverage of the NP surface at a given grafting density since the brush is formed while respecting the excluded volume constraints of the previously grafted chains. This conclusion is especially clear in the limit of low-to-moderate grafting density. Thus, at a given grafting density, the "grafting to" mechanism leads to an enhanced miscibility of the NPs in the matrix (which has the same chemistry as the grafts) and lower propensity to create self-assembled structures. Another important factor is that the dispersity in the number of grafted chains on the NPs is also smaller in the case of "grafting to" systems, thus leading to better defined materials. These two conclusions imply that the "grafting to" mechanism may provide better control over the NP dispersion state and hence the thermomechanical properties of polymer nanocomposites.

Lattice cluster theory for dense, thin polymer films.

While the application of the lattice cluster theory (LCT) to study the miscibility of polymer blends has greatly expanded our understanding of the monomer scale molecular details influencing miscibility, the corresponding theory for inhomogeneous systems has not yet emerged because of considerable technical difficulties and much greater complexity. Here, we present a general formulation enabling the extension of the LCT to describe the thermodynamic properties of dense, thin polymer films using a high dimension, high temperature expansion. Whereas the leading order of the LCT for bulk polymer systems is essentially simple Flory-Huggins theory, the highly non-trivial leading order inhomogeneous LCT (ILCT) for a film with L layers already involves the numerical solution of 3(L - 1) coupled, highly nonlinear equations for the various density profiles in the film. The new theory incorporates the essential "transport" constraints of Helfand and focuses on the strict imposition of excluded volume constraints, appropriate to dense polymer systems, rather than the maintenance of chain connectivity as appropriate for lower densities and as implemented in self-consistent theories of polymer adsorption at interfaces. The ILCT is illustrated by presenting examples of the computed profiles of the density, the parallel and perpendicular bonds, and the chain ends for free standing and supported films as a function of average film density, chain length, temperature, interaction with support, and chain stiffness. The results generally agree with expected general trends.

Microstructure and Rheology of Rigid Rod Suspensions

Nonspherical particles dispersed in a fluid have a tendency to align because of particle–fluid interactions. At high particle concentrations and in the absence of fluid dynamic couples, spontaneous self-alignment can occur due to excluded volume constraints on rotary Brownian motion. This phenomenon prevents a return-to-isotropy from an anisotropic state. In this communication, the low-order moments of the rotary Smoluchowski (S) equation for a rigid rod suspension are used to explore the effect of the Péclet number on the shear viscosity of a suspension. A closure model for the fourth-order orientation tetradic uses an algebraic fully symmetric quadratic (FSQ) mapping of the second-order orientation moment into the fourth-order orientation moment. The algebraic mapping preserves the 6-fold symmetry and the 6-fold contraction properties of the exact fourth-order orientation moment. The theory predicts that, if the orientation director is in the tumbling regime, shear thickening may occur and, if the orientation director is in the wagging regime, shear thinning may occur.

Theory of Anisotropic Diffusion of Entangled and Unentangled Polymers in Rod-Sphere Mixtures.

We present a microscopic self-consistent theory for the long-time diffusion of infinitely thin rods in a hard sphere matrix based on the simultaneous dynamical treatment of topological uncrossability and finite excluded volume constraints. Distinctive regimes of coupled anisotropic longitudinal and transverse diffusion are predicted, and steric blocking of the latter leads to a tube-like localization transition largely controlled by the ratio of the sphere diameter to rod length and tube diameter. For entangled polymers, in a limited regime of strongly retarded dynamics a "doubly renormalized" reptation law is predicted where the confinement tube is compressed and longitudinal motion is partially blocked. At high sphere volume fractions, strong suppression of rod motion results in dynamic localization in the unentangled regime. The present advance provides a theoretical foundation to treat differential mobility effects and flexible chain dynamics in diverse polymer-particle mixtures.

Self-avoiding wormlike chain model for double-stranded-DNA loop formation.

We compute the effects of excluded volume on the probability for double-stranded DNA to form a loop. We utilize a Monte Carlo algorithm for generation of large ensembles of self-avoiding wormlike chains, which are used to compute the J factor for varying length scales. In the entropic regime, we confirm the scaling-theory prediction of a power-law drop off of -1.92, which is significantly stronger than the -1.5 power law predicted by the non-self-avoiding wormlike chain model. In the elastic regime, we find that the angle-independent end-to-end chain distribution is highly anisotropic. This anisotropy, combined with the excluded volume constraints, leads to an increase in the J factor of the self-avoiding wormlike chain by about half an order of magnitude relative to its non-self-avoiding counterpart. This increase could partially explain the anomalous results of recent cyclization experiments, in which short dsDNA molecules were found to have an increased propensity to form a loop.

Theory of Entanglements and Tube Confinement in Rod-Sphere Nanocomposites.

We formulate a microscopic theory for the polymer transverse confinement length and associated dynamic potential for a mixture of infinitely thin rods and hard spheres based solely on topological entanglements and excluded volume constraints. For fixed spheres, the needle effective tube diameter decreases with particle loading, and is largely controlled by a single dimensionless parameter involving all three key length-scales in the problem. A crossover from polymer entanglement to nanoparticle-controlled tube localization with increased loading is predicted. A preliminary extension to chain melts exhibits reasonable agreement with a recent simulation, and experimentally testable predictions are made. This work establishes a first-principles theoretical foundation to investigate a variety of dynamical problems in entangled polymer nanocomposites.

Monte Carlo simulations of a polymer chain conformation. The effectiveness of local moves algorithms and estimation of entropy

A linear chain on a simple cubic lattice was simulated by the Metropolis Monte Carlo method using a combination of local and non-local chain modifications. Kink-jump, crankshaft, reptation and end-segment moves were used for local changes of the chain conformation, while for non-local chain rearrangements the "cut-and-paste" algorithm was employed. The statistics of local micromodifications was examined. An approximate method for estimating the conformational entropy of a polymer chain, based on the efficiency of the kink-jump motion respecting chain continuity and excluded volume constraints, was proposed. The method was tested by calculating the conformational entropy of the undisturbed chain, the chain under tension and in different solvent conditions (athermal, theta and poor) and also of the chain confined in a slit. The results of these test calculations are qualitatively consistent with expectations. Moreover, the obtained values of the conformational entropy of self avoiding chain with ends fixed over different separations, agree very well with the available literature data. FigureVisualization of the neighborhood of two local chain microconformations containing a bead (indicated by the arrow) which a) can and b) cannot be moved by the kink-jump. In red there are marked the possible trajectories of chain fragment which can block the adjacent site

Dynamics simulations for engineering macromolecular interactions

The predictable engineering of well-behaved transcriptional circuits is a central goal of synthetic biology. The artificial attachment of promoters to transcription factor genes usually results in noisy or chaotic behaviors, and such systems are unlikely to be useful in practical applications. Natural transcriptional regulation relies extensively on protein-protein interactions to insure tightly controlled behavior, but such tight control has been elusive in engineered systems. To help engineer protein-protein interactions, we have developed a molecular dynamics simulation framework that simplifies features of proteins moving by constrained Brownian motion, with the goal of performing long simulations. The behavior of a simulated protein system is determined by summation of forces that include a Brownian force, a drag force, excluded volume constraints, relative position constraints, and binding constraints that relate to experimentally determined on-rates and off-rates for chosen protein elements in a system. Proteins are abstracted as spheres. Binding surfaces are defined radially within a protein. Peptide linkers are abstracted as small protein-like spheres with rigid connections. To address whether our framework could generate useful predictions, we simulated the behavior of an engineered fusion protein consisting of two 20,000 Da proteins attached by flexible glycine/serine-type linkers. The two protein elements remained closely associated, as if constrained by a random walk in three dimensions of the peptide linker, as opposed to showing a distribution of distances expected if movement were dominated by Brownian motion of the protein domains only. We also simulated the behavior of fluorescent proteins tethered by a linker of varying length, compared the predicted Förster resonance energy transfer with previous experimental observations, and obtained a good correspondence. Finally, we simulated the binding behavior of a fusion of two ligands that could simultaneously bind to distinct cell-surface receptors, and explored the landscape of linker lengths and stiffnesses that could enhance receptor binding of one ligand when the other ligand has already bound to its receptor, thus, addressing potential mechanisms for improving targeted signal transduction proteins. These specific results have implications for the design of targeted fusion proteins and artificial transcription factors involving fusion of natural domains. More broadly, the simulation framework described here could be extended to include more detailed system features such as non-spherical protein shapes and electrostatics, without requiring detailed, computationally expensive specifications. This framework should be useful in predicting behavior of engineered protein systems including binding and dissociation reactions.

Analysis of Shape Optimization for Magnetic Microswimmers

We analyze an infinite dimensional, geometrically constrained shape optimization problem for magnetically driven microswimmers (locomotors) in three-dimensional (3-D) Stokes flow and give a well-posed descent scheme for computing optimal shapes. The problem is inspired by recent experimental work in this area. We show the existence of a minimizer of the optimization problem using analytical tools for elastic rods that respect the excluded volume constraint. We derive a variational gradient descent method for computing optimal locomotor shapes using the tools of shape differential calculus. The descent direction is obtained by solving a saddle-point system, which we prove is well-posed. We also introduce a finite element approximation of the gradient descent method and prove its stability. We present numerical results illustrating our method and the effect that finite aspect ratio and external cargo can have on the optimal shape. The 3-D Stokes equations are solved by a boundary integral method.

Drift effect and “negative” mass transport in an inhomogeneous medium: Limiting case of a two-component lattice gas

The mass transport in an inhomogeneous medium is modeled as the limiting case of a two-component lattice gas with excluded volume constraint and one of the components fixed. In the long-wavelength approximation, the density relaxation of mobile particles is governed by diffusion and interaction with a medium inhomogeneity represented by the static component distribution. It is shown that the density relaxation can be locally accompanied by density distribution compression, i.e., the local mass transport directed from low-to high-density regions. The origin of such a "negative" mass transport is shown to be associated with the presence of a stationary drift flow defined by the medium inhomogeneity. In the quasi-one-dimensional case, the compression dynamics manifests itself in the hoppinglike motion of packet front position of diffusing substance due to staged passing through inhomogeneity barriers, and it leads to fragmentation of the packet and retardation of its spreading. The root-mean-square displacement reflects only the averaged packet front dynamics and becomes inappropriate as the transport characteristic in this regime. In the stationary case, the mass transport throughout the whole system may be directed from the boundary with lower concentration towards the boundary with higher concentration. Implications of the excluded volume constraint and particle distinguishability for these effects are discussed.

Localization phenomena in models of ion-conducting glass formers

The mass transport in soft-sphere mixtures of small and big particles as well as in the disordered Lorentz gas (LG) model is studied using molecular dynamics (MD) computer simulations. The soft-sphere mixture shows anomalous small-particle diffusion signifying a localization transition separate from the big-particle glass transition. Switching off small-particle excluded volume constraints slows down the small-particle dynamics, as indicated by incoherent intermediate scattering functions. A comparison of logarithmic time derivatives of the mean-squared displacements reveals qualitative similarities between the localization transition in the soft-sphere mixture and its counterpart in the LG. Nevertheless, qualitative differences emphasize the need for further research elucidating the connection between both models.

Double Transition Scenario for Anomalous Diffusion in Glass-Forming Mixtures

We study by molecular dynamics computer simulation a binary soft-sphere mixture that shows a pronounced difference in the species' long-time dynamics. Anomalous, power-law-like diffusion of small particles arises that can be understood as a precursor of a double-transition scenario, combining a glass transition and a separate small-particle localization transition. Switching off small-particle excluded-volume constraints slows down, rather than enhances, small-particle transport.

Coarse-Grained Structural Modeling of Molecular Motors Using Multibody Dynamics

Experimental and computational approaches are needed to uncover the mechanisms by which molecular motors convert chemical energy into mechanical work. In this article, we describe methods and software to generate structurally realistic models of molecular motor conformations compatible with experimental data from different sources. Coarse-grained models of molecular structures are constructed by combining groups of atoms into a system of rigid bodies connected by joints. Contacts between rigid bodies enforce excluded volume constraints, and spring potentials model system elasticity. This simplified representation allows the conformations of complex molecular motors to be simulated interactively, providing a tool for hypothesis building and quantitative comparisons between models and experiments. In an example calculation, we have used the software to construct atomically detailed models of the myosin V molecular motor bound to its actin track. The software is available at www.simtk.org.

Prediction of Solute Partition Coefficients between Polyolefins and Alcohols Using a Generalized Flory−Huggins Approach

The partition coefficients of n-alkanes, n-alcohols, volatiles, and typical antioxidants between a low density polyethylene and several alcohols (methanol and ethanol) were predicted without fitting variables using an off-lattice Flory−Huggins approach. The main advantage of the proposed excluded volume constraint method was to sample at atomistic scale pairwise contact energies and the residual entropic contribution due to possible different conformers and reorientations. The positional entropy was considered both in the polymer and in the liquid as in the original Flory−Huggins theory. Possible biases due to the formulation and conformational sampling were analyzed for molecules different in size, in shape, and in stiffness. The predictions were close to the experimental uncertainty on the partition coefficients between the amorphous part of the polymer and the tested liquids. The present work confirmed that plastic additives, which are by design highly compatible with the polymer, had also a significan...

Interactions between bottle-brush polyelectrolyte layers: Effects of ionic strength and oppositely charged surfactant

Interactions between cationic bottle-brush polyelectrolyte layers adsorbed on mica across salt and oppositely charged surfactant solutions were investigated with the interferometric surface force apparatus, and the results were compared with what is known for similarly charged linear polyelectrolytes. Ellipsometric measurements demonstrated that the bottle-brush polyelectrolytes, which contain 45 units long poly(ethylene oxide) side chains, are more readily desorbed than linear equivalents when the ionic strength of the solution is increased. It is argued that this is due to the steric repulsion between the poly(ethylene oxide) side chains that reduces the surface affinity. The preadsorbed bottle-brush polyelectrolyte layers were also exposed to sodium dodecyl sulfate (SDS) solutions. It was found that the presence of SDS affected the force profiles less than observed for similarly charged linear polyelectrolytes. This observation was attributed to excluded volume constraints imposed by the poly(ethylene oxide) side chains that reduces the accessibility of the charged polyelectrolyte segments and counteracts formation of large aggregates within the layer.

Exfoliation of stacked sheets: Effects of temperature, platelet size, and quality of solvent by a Monte Carlo simulation

AbstractCan a layer of stacked sheets (coarse grained description of clay platelets) exfoliate in a given solvent? Computer simulations are performed to address this question with a stacked layer of four sheets. A sheet is described by a set of nodes (N) tethered together by flexible (fluctuating) bonds in a planar structure with linear scale Ls, N = Ls2 on a cubic lattice of size L3. The initial distance between the sheets is small and kept constant for different sets of stacked sheets. Quality of the effective solvent is controlled by a short range interaction (ε) between sheets, that is, their nodes and the empty lattice sites (ε = −2, −1, 1, 2). Nodes execute stochastic movement with the Metropolis algorithm subject to bond fluctuation and excluded volume constraints. Simulations are also performed at different temperatures. Visual analysis shows that the exfoliation depends on the size of the sheet and temperature. We find that it is easier to exfoliate smaller sheets and that the exfoliation is enhanced by raising the temperature. The exfoliation becomes very slow for large sheets at low temperatures because of longer relaxation times. Exfoliation is enhanced by increasing the attractiveness (i.e., the quality) of the solvent interaction with the platelets. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 3580–3589, 2006

Optimum folding pathways of proteins: Their determination and properties

We develop a dynamic optimization technique for determining optimum folding pathways of proteins starting from different initial configurations. A coarse-grained Go model is used. Forces acting on each bead are (i) the friction force, (ii) forces from bond length constraints, (iii) excluded volume constraints, and (iv) attractive forces between residue pairs that are in contact in the native state. An objective function is defined as the total attractive energy between nonbonded residues, which are neighbors in the native state. The objective function is minimized over all feasible paths, satisfying bond length and excluded volume constraints. The optimization problem is nonconvex and contains a large number of constraints. An augmented Lagrangian method with a penalty barrier function was used to solve the problem. The method is applied to a 36-residue protein, chicken villin headpiece. Sequences of events during folding of the protein are determined for various pathways and analyzed. The relative time scales are compared and scaled according to experimentally measured events. Formation times of the helices, turn, and the loop agree with experimental data. We obtain the overall folding time of the protein in the range of 600 ns-1.2 micros that is smaller than the experimental result of 4-5 micros, showing that the optimal folding times that we obtain may be possible lower bounds. Time dependent variables during folding and energies associated with short- and long-range interactions between secondary structures are analyzed in modal space using Karhunen-Loeve expansion.