Analytical Mean-field Theory Research Articles

Critical adsorption and charge reversal in polyelectrolyte solutions: Analytical mean-field theory.

An analytical linearized mean-field theory is presented to describe the adsorption behavior of polyelectrolytes near charged colloidal surfaces with additional short-ranged non-electrostatic interactions. The coupling between the polyelectrolyte segment density and electrostatic potential is explicitly accounted for in a self-consistent manner. This coupling gives rise to highly non-linear behavior, such as oscillations of the electrostatic potential. We derive analytical expressions for the critical surface charge density σc, after which adsorption takes place, and recover the well-known σc∼ns3/2 scaling regime, where ns is the salt concentration. In addition, the theory yields a new ns1 scaling regime if the surface is hard and a unified ns1 scaling regime if the surface also possesses some short-ranged attraction with the polyelectrolyte. Furthermore, we derive an analytical expression to describe the critical polyelectrolyte concentration φc to achieve complete charge reversal, which is found to scale as φc ∼ σ2/(f2c2), where c is related to the magnitude of short-ranged interactions and f is the average charge per monomer of the polyelectrolyte. It is observed that within our theory, complete charge reversal can only take place if the short-ranged interactions are sufficiently strong to completely compensate for the entropy loss of adsorption.

The Role of Receptor Uniformity in Multivalent Binding.

Multivalency is prevalent in various biological systems and applications due to the superselectivity that arises from the cooperativity of multivalent binding. Traditionally, it was thought that weaker individual binding would improve the selectivity in multivalent targeting. Here, using analytical mean field theory and Monte Carlo simulations, we discover that, for receptors that are highly uniformly distributed, the highest selectivity occurs at an intermediate binding energy and can be significantly greater than the weak binding limit. This is caused by an exponential relationship between the bound fraction and receptor concentration, which is influenced by both the strength and combinatorial entropy of binding. Our findings not only provide new guidelines for the rational design of biosensors using multivalent nanoparticles but also introduce a new perspective in understanding biological processes involving multivalency.

Modeling the Phase Transition in Hydrophobic Weak Polyelectrolyte Gels under Compression.

One of the emerging water desalination techniques relies on the compression of a polyelectrolyte gel. The pressures needed reach tens of bars, which are too high for many applications, damage the gel and prevent its reuse. Here, we study the process by means of coarse-grained simulations of hydrophobic weak polyelectrolyte gels and show that the necessary pressures can be lowered to only a few bars. We show that the dependence of applied pressure on the gel density contains a plateau indicating a phase separation. The phase separation was also confirmed by an analytical mean-field theory. The results of our study show that changes in the pH or salinity can induce the phase transition in the gel. We also found that ionization of the gel enhances its ion capacity, whereas increasing the gel hydrophobicity lowers the pressure required for gel compression. Therefore, combining both strategies enables the optimization of polyelectrolyte gel compression for water desalination purposes.

Colloidal particles interacting with a polymer brush: a self-consistent field theory.

The interaction of colloidal particles with a planar polymer brush immersed in a solvent of variable thermodynamic quality is studied by a numerical self-consistent field method combined with analytical mean-field theory. The effect of embedded particle on the distribution of polymer density in the brush is analyzed and the particle insertion free energy profiles are calculated for variable size and shape of the particles and sets of polymer-particle and polymer-solvent interaction parameters. In particular, both cases of repulsive and attractive interactions between particles and brush-forming chains are considered. It is demonstrated that for large particles the insertion free energy is dominated by repulsive (osmotic) contribution and is approximately proportional to the particle volume in accordance with earlier theoretical predictions [Halperin et al., Macromolecules, 2011, 44, 3622]. For the particles of smaller size or/and large shape asymmetry the adsorption or depletion of a polymer from the particle surface essentially contributes to the insertion free energy balance. As a result, depending on the set of polymer-solvent and polymer-particle interaction parameters and brush grafting density the insertion free energy profile may exhibit complex patterns, i.e., from a pure repulsive effective potential barrier to an attractive well. The results of our study allow for predicting equilibrium partitioning and controlling diffusive transport of (bio)nanocolloids across (bio)polymer brushes of arbitrary geometry including polymer-modified membranes or nanopores.

Theory of polyelectrolyte dendrigrafts

A mean-field approach is used to analyze equilibrium conformations of polyelectrolyte dendrigrafts comprising ionically charged dendrons attached by focal points to a flexible linear backbone. Power law dependences for local structural parameters, cross-sectional thickness and intergraft distance, are derived as a function of grafting density and degree of branching of the dendrons. The cases of quenched and pH-sensitive ionization of the dendrons are considered. The finite extensibility of the backbone is taken into account. It is demonstrated that an increase in the degree of branching of the dendrons leads to a decrease in the dendrigraft thickness compared with that of the polyelectrolyte molecular brush with the same degree of polymerization of the side chains, while intergraft distance either increases or stays close to counter length of fully extended backbone spacer. The analytical mean-field theory predictions are confirmed by results of numerical self-consistent field modelling.

Intramolecular micellization and nanopatterning in pH- and thermo-responsive molecular brushes.

Conformational transitions and nanoscale self-organization triggered in double pH- and thermo-responsive molecular brushes by varying environmental conditions are studied by means of analytical mean-field theory and numerical Scheutjens-Fleer self-consistent field modelling. Such molecular brushes are composed of multiple thermo-responsive side chains end-grafted onto the main chain (backbone) and are capable of acquiring ionic charges via reversible (de)protonation of the monomer units. Competition of long-range Coulomb repulsion with short-range solvophobic interactions leads to complex patterns in the intramolecular self-organization of molecular brushes. In particular, we observed formation of pearl necklace-like structures with multiple dense nanodomains formed by weakly ionized collapsed side chains and stabilized by a fraction protruding into the solution and strongly ionized ones. Such structures are thermodynamically stable in a certain parameter range and can be termed as intramolecular micelles. The stimuli-induced intramolecular nanopatterning occurs via a sequence of quasi-first order phase transitions corresponding to splitting/fusion of collapsed domains accompanied by jumps in the average degree of ionization and macromolecular dimensions. A re-entrant sequence of transitions is observed when the salt concentration is used as a control parameter. These theoretical predictions provide guidelines for design of smart unimolecular devices, for example multicompartment nanocarriers of active substances or nanosensors.

Tension-Induced Nematic Phase Separation in Bidisperse Homopolymer Melts.

We use an analytical mean-field theory and all-atom molecular dynamics (MD) simulations to predict that external tension, together with the nematic coupling interactions, can drive phase separation of long chains from short ones in bidisperse homopolymer melts. The nematic coupling parameter α for polyethylene (PE) oligomers under applied tension is extracted from the MD simulations and used in the mean-field free energy to predict the phase boundary for bidisperse melts in which the longer chains are stretched by uniaxial tension. The predicted phase diagram is validated by direct MD simulations. We also show that extensional flow, and possibly even shear flow, may lead to nematic phase separation in molten PE oligomers, because the flow can impose a stronger tension on the longer chains than the short ones.

How to fold back grafted chains in dipolar brushes

Recent fully atomistic molecular dynamics (MD) simulation of polymer nanocomposites based on polylactic acid (PLA) filled with cellulose nanocrystals (CNC) covalently modified by grafting of lactic acid oligomers (OLA) [A.D. Glova et al. Soft Matter 13 (2017) 6627] have revealed a non-trivial structure of the OLA brush. It was found that a segregation of grafted OLA chains into two groups, or populations, occurs due to the presence of partial charges so that a fraction of chains folds back to the surface and acquires a hairpin-like conformation. Motivated by these results, we study the structure of a brush made of polymers with dipole moments oriented along the main chain (dipolar chains of A type) by using the numerical self-consistent field method for lattice models. In full agreement with the results of MD simulation study, the segregation of brush chains in two populations is observed as the strength of dipole-dipole interactions increases: A part of the brush chains folds back to form hairpin-like conformation while the remaining chains are extended and directed towards the brush periphery. An increase of the strength of dipole-dipole interactions and/or grafting density leads to the increase in the fraction of backfolded chains. We develop an analytical mean-field theory for the brush in the “dry” state which demonstrates the thermodynamic advantage of backfolding for a fraction of the brush chains. We find that the key factor leading to the observed non-trivial brush structure is the grafting of the chains in the brush via the same end. In contrast, in the case of the brush with the equal amount of chains grafted via each of the two ends, the dipole-dipole interactions do not result in the backfolding. This prediction for the lattice model is confirmed by additional fully atomistic molecular dynamics simulations of the corresponding OLA brush grafted onto CNC surface and immersed in the PLA melt.

Impact of Macromolecular Architecture on Bending Rigidity of Dendronized Surfaces

Nanomechanical properties of natural and artificial nanomembranes can be strongly affected by anchored or tethered macromolecules. The intermolecular interactions in polymeric layers give rise to so-called induced bending rigidity which complements the bare rigidity of the membrane. Using analytical mean-field theory, we explore how macromolecular architecture of tethered polymers affects the bending rigidities of the polymer-decorated membranes. The developed theory enables us to consider explicitly various polymer architectures including regular dendrons, Ψ-shaped, star- and comblike macromolecules as well as macrocycles. Numerical self-consistent field computations for selected (regular dendritic) topology complement the analytical theory and support its predictions. We consider both cases of (i) quenched symmetric distribution of tethered molecules on both sides of the membrane and (ii) annealing distribution in which the tethered polymers can relocate from the concave to the convex side of the membra...

Stochastic Resonance in Protein Folding Dynamics.

Although protein folding reactions are usually studied under static external conditions, it is likely that proteins fold in a locally fluctuating cellular environment in vivo. To mimic such behavior in in vitro experiments, the local temperature of the solvent can be modulated either harmonically or using correlated noise. In this study, coarse-grained molecular simulations are used to investigate these possibilities, and it is found that both periodic and correlated random fluctuations of the environment can indeed accelerate folding kinetics if the characteristic frequencies of the applied fluctuations are commensurate with the internal timescale of the folding reaction; this is consistent with the phenomenon of stochastic resonance observed in many other condensed-matter processes. To test this theoretical prediction, the folding dynamics of phosphoglycerate kinase under harmonic temperature fluctuations are experimentally probed using Förster resonance energy transfer fluorescence measurements. To analyze these experiments, a combination of theoretical approaches is developed, including stochastic simulations of folding kinetics and an analytical mean-field kinetic theory. The experimental observations are consistent with the theoretical predictions of stochastic resonance in phosphoglycerate kinase folding. When combined with an alternative experiment on the protein VlsE using a power spectrum analysis, elaborated in Dave et al., ChemPhysChem 2016, 10.1002/cphc.201501041, the overall data overwhelmingly point to the experimental confirmation of stochastic resonance in protein folding dynamics.

Persistence length of dendronized polymers: the self-consistent field theory.

We present numerical results for the thermodynamic rigidity and induced persistence length of dendronized polymers with systematically varied topology of their grafts obtained by the Scheutjens-Fleer self-consistent field method. The results were compared to predictions of an analytical mean-field theory. The two approaches have marked different predictions. In particular, the analytical theory predicts that the induced persistence length and the effective segment aspect ratio of dendronized polymers are increasing functions of the degree of branching of their side chains, whereas numerical calculations provide evidence of the opposite dependences. This discrepancy is argued to be due to the ability of side chains to repartition from the compressed to the dilated regions of a curved bottle brush, which is accounted for by the numerical, but not by the analytical method. The difference is most crucial in the light of the expected ability of dendronized polymers to have a liquid crystalline ordering in semi-dilute solutions.

Analytical mean-field scaling theory of radio-frequency heating in a Paul trap

While the microscopic origins of radio-frequency (rf) heating of simultaneously stored, charged particles in a Paul trap are not yet understood in detail, a universal heating curve [J. D. Tarnas, Y. S. Nam, and R. Bl\"umel, Phys. Rev. A 88, 041401 (2013)] was recently discovered that collapses scaled rf heating data onto a single universal curve. Based on a simple analytical mean-field theory, we derive an analytical expression for the universal heating curve, which is in excellent agreement with numerical data. We find that for spherical clouds the universal curve depends only on a single scaling parameter, $\ensuremath{\lambda}={[q(N\ensuremath{-}1)]}^{2/3}/T$, where $N$ is the number of trapped particles, $q$ is the Paul-trap control parameter, and $T$ is the temperature.

Exciton condensation in strongly correlated electron bilayers

We studied the possibility of exciton condensation in Mott insulating bilayers. In these strongly correlated systems, an exciton is the bound state of a double occupied and empty site. In the strong coupling limit, the exciton acts as a hard-core boson. Its physics is captured by the exciton $t$-$J$ model, containing an effective $XXZ$ model describing the exciton dynamics only. Using numerical simulations and analytical mean-field theory, we constructed the ground-state phase diagram. Three homogeneous phases can be distinguished: the antiferromagnet, the exciton checkerboard crystal, and the exciton superfluid. For most model parameters, however, we predict macroscopic phase separation between these phases. The exciton superfluid exists only for large exciton hopping energy. Additionally, we studied the collective modes and susceptibilities of the three phases. In the superfluid phase, we find the striking feature that the bandwidth of the spin-triplet excitations, potentially detectable by resonant inelastic x-ray scattering (RIXS), is proportional to the superfluid density. The superfluid phase mode is visible in the charge susceptibility, measurable by RIXS or electron energy loss spectroscopy (EELS).

Conformations of Amphiphilic Polyelectrolyte Stars with Diblock Copolymer Arms

We consider conformations and intra-molecular conformational transitions in amphiphilic starlike polymers formed by diblock copolymer arms with inner hydrophobic and outer polyelectrolyte blocks. A combination of an analytical mean-field theory with the assumption-free numerical self-consistent field (SCF) modeling approach is applied. It is demonstrated that unimolecular micelles with collapsed hydrophobic cores and swollen polyelectrolyte coronae are formed in dilute aqueous solutions at high ionic strength or/and low degree of ionization of the outer hydrophilic block. An intra-molecular conformational transition related to the unfolding of the hydrophobic core of the unimolecular micelles can be triggered by a decrease in the ionic strength of the solution or/and increase in the degree of ionization of the coronal blocks. In the stars with large number of diblock copolymer arms the transition between conformations with collapsed or stretched core-forming blocks occurs continuously by progressive unfolding of the core domain. By contrast, in the stars with relatively small number of arms the continuous unfolding of the core is interrupted by an abrupt unravelling transition. A detailed SCF analysis indicates that under both unfolding scenario the arms of the star are extended fairly equally, i.e., no intra-molecular disproportionation occurs.

Confinement-deconfinement transition and spin correlations in a generalized Kitaev model

We present a spin model, namely, the Kitaev model augmented by a loop term and perturbed by an Ising Hamiltonian, and show that it exhibits both confinement-deconfinement transitions from spin liquid to antiferromagnetic/spin-chain/ferromagnetic phases and topological quantum phase transitions between gapped and gapless spin-liquid phases. We develop a fermionic resonating-valence-bonds (RVB) mean-field theory to chart out the phase diagram of the model and estimate the stability of its spin-liquid phases, which might be relevant for attempts to realize the model in optical lattices and other spin systems. We present an analytical mean-field theory to study the confinement-deconfinement transition for large coefficient of the loop term and show that this transition is first order within such mean-field analysis in this limit. We also conjecture that in some other regimes, the confinement-deconfinement transitions in the model, predicted to be first order within the mean-field theory, may become second order via a defect condensation mechanism. Finally, we present a general classification of the perturbations to the Kitaev model on the basis of their effect on it's spin correlation functions and derive a necessary and sufficient condition, within the regime of validity of perturbation theory, for the spin correlators to exhibit a long-ranged power-law behavior in the presence of such perturbations. Our results reproduce those of Tikhonov et al. [Phys. Rev. Lett. 106, 067203 (2011)] as a special case.

Modeling cytoskeletal flow over adhesion sites: competition between stochastic bonddynamics and intracellular relaxation

In migrating cells, retrograde flow of the actin cytoskeleton is related to traction atadhesion sites located at the base of the lamellipodium. The coupling between the movingcytoskeleton and the stationary adhesions is mediated by the continuous association anddissociation of molecular bonds. We introduce a simple model for the competition betweenthe stochastic dynamics of elastic bonds at the moving interface and relaxationwithin the moving actin cytoskeleton represented by an internal viscous frictioncoefficient. Using exact stochastic simulations and an analytical mean field theory, weshow that the stochastic bond dynamics lead to biphasic friction laws as observedexperimentally. At low internal dissipation, stochastic bond dynamics lead to aregime of irregular stick-and-slip motion. High internal dissipation effectivelysuppresses cooperative effects among bonds and hence stabilizes the adhesion.

Robust transport by multiple motors with nonlinear force–velocity relations and stochastic load sharing

Transport by processive molecular motors plays an important role in many cell biological phenomena. In many cases, motors work together to transport cargos in the cell, so it is important to understand the mechanics of the multiple motors. Based on earlier modeling efforts, here we study effects of nonlinear force–velocity relations and stochastic load sharing on multiple motor transport. We find that when two or three motors transport the cargo, then the nonlinear and stochastic effects compensate so that the mechanical properties of the transport are robust. Similarly, the transport is insensitive to compliance of the cargo-motor links. Furthermore, the rate of movement against moderate loads is not improved by increasing the small number of motors. When the motor number is greater than 4, correlations between the motors become negligible, and the earlier analytical mean-field theory of the multiple motor transport holds. We predict that the effective diffusion of the cargo driven by the multiple motors under load increases by an order of magnitude compared to that for the single motor. Finally, our simulations predict that the stochastic effects are responsible for a significant dispersion of velocities generated by the ‘tug-of-war’ of the multiple opposing motors.

Contact process in disordered and periodic binary two-dimensional lattices

The critical behavior of the contact process (CP) in disordered and periodic binary two-dimensional (2D) lattices is investigated numerically by means of Monte Carlo simulations as well as via an analytical approximation and standard mean field theory. Phase-separation lines calculated numerically are found to agree well with analytical predictions around the homogeneous point. For the disordered case, values of static scaling exponents obtained via quasistationary simulations are found to change with disorder strength. In particular, the finite-size scaling exponent of the density of infected sites approaches a value consistent with the existence of an infinite-randomness fixed point as conjectured before for the 2D disordered CP. At the same time, both dynamical and static scaling exponents are found to coincide with the values established for the homogeneous case thus confirming that the contact process in a heterogeneous environment belongs to the directed percolation universality class.

Counterion Localization in Solutions of Starlike Polyelectrolytes and Colloidal Polyelectrolyte Brushes: A Self-Consistent Field Theory

A quantitative analysis of the distribution of counterions in salt-free solutions of colloidal polyelectrolyte brushes and starlike polyelectrolytes is performed on the level of the Poisson-Boltzmann approximation. Exact numerical solutions are obtained for starlike polyelectrolyte molecules composed of f = 20, . . ., 50 arms with a fixed fractional charge alpha per segment by applying the self-consistent field method of Scheutjens and Fleer (SF-SCF). The Wigner-Seitz cell dimension defines the concentration of polyelectrolyte stars in the system. The numerical results are compared to predictions of an analytical mean field theory and related to experimental observations on the osmotic pressure in solutions of starlike polyelectrolytes and colloidal polyelectrolyte brushes.

Noise-induced phase transitions in field-dependent relaxational dynamics: The Gaussian ansatz

We present an analytic mean-field theory for relaxational dynamics in spatially extended systems that undergo purely noise-induced phase transitions to ordered states. The theory augments the usual mean-field approach with a Gaussian ansatz that yields quantitatively accurate results for strong coupling. We obtain analytic results not only for steady-state mean fields and distribution widths, but also for the dynamical approach to a steady state or to collective oscillatory behaviors in multifield systems. Because the theory yields dynamical information, it can also predict the initial-condition-dependent final state (disordered state, steady or oscillatory ordered state) in multistable arrays.