Grafted polymers with annealed excluded volume: A model for surfactant association in brushes

We present an analytical self-consistent-field (SCF) theory for a neutral polymer brush (a layer of long polymer chains end-grafted to a surface) with annealed excluded volume interactions between the monomer units. This model mimics the reversible adsorption of solute molecules or aggregates, such as small globular proteins or surfactant micelles, on the grafted chains. The equilibrium structural properties of the brush (the brush thickness, the monomer density profile, the distribution of the end segments of the grafted chains) as well as the overall adsorbed amount and the adsorbate density profile are analyzed as a function of the grafting density, the excluded volume parameters and the chemical potential (the concentration) of the adsorbate in the solution. We demonstrate that, when the grafting density is varied, the overall adsorbed amount always exhibits a maximum, whereas the root-mean-square brush thickness either increases monotonically or passes through a (local) minimum. At high grafting densities the chains are loaded by adsorbed aggregates preferentially in the distal region of the brush, whereas in the region proximal to the grafting surface depletion of aggregates occurs and the polymer brush retains an unperturbed structure. Depending on the relative strength of the excluded volume interactions between unloaded and loaded monomers both the degree of loading of the chains and the polymer density profile are either continuous or they exhibit a discontinuity as a function of the distance from the grafting surface. In the latter case intrinsic phase separation occurs in the brush: the dense phase consists of unloaded and weakly extended chains and occupies the region proximal to the surface, whereas a more dilute phase consisting of highly loaded and strongly extended chains forms the periphery of the brush. Received 26 November 1998 and Received in final form 2 April 1999     

Deformation of globular polysoaps: extension, confinement and extensional flow

A theoretical model of the extension and confinement of globular polysoaps predicts novel force laws. Polysoaps are polymers comprising of a flexible hydrophilic backbone incorporating, at intervals, amphiphilic monomers. The equilibrium configuration of long polysoaps, that form numerous spherical intrachain micelles, is a spherical globule of close packed micelles. The coupling of the deformation to the hierarchical self organization of the chain gives rise to a distinctive force law involving, for extension, two plateau regimes. When the chain is stretched by extensional flow the two regimes merge and the polysoap exhibits a single globule-stretch transition. Received 16 June 1998 and Received in final form 19 November 1998     

Effects of ionic strength and charge annealing in star-branched polyelectrolytes

We have developed a scaling theory that describes the conformations of weak star-branched polyelectrolytes in dilute solutions. The dependences of the overall star size on the number of branches and on the ionic strength of the solution (tuned by the addition of low molecular weight salt) are analyzed. The intrinsic structure of the polyelectrolyte stars in salt-free and salt-added solutions is discussed in terms of concentration and elastic blobs. In contrast to neutral stars, the swollen corona of the polyelectrolyte star consists of blobs which are not closely packed. We have shown that the size of star polyelectrolytes is less sensitive to the variation in the ionic strength than the size of linear polyelectrolytes. The effects of the ionization-recombination balance in the star polyelectrolyte were taken into account. For polyelectrolytes with small ionization constant, the size of the star depends nonmonotonically on the number of branches and on the ionic strength of the solution due to recombination of counterions with charged monomers. Received: 10 November 1997 / Revised: 16 February 1998 / Accepted: 1st April 1998     

Phase behavior of a suspension of hard spherocylinders plus ideal polymer chains

We study isotropic-isotropic and isotropic-nematic phase transitions of fluid mixtures containing hard spherocylinders (HSC) and added non-adsorbing ideal polymer chains using scaled particle theory (SPT). First, we investigate isotropic-nematic (I -N phase coexistence using SPT in the absence of polymer. We compare the results obtained using a Gaussian form of the orientational distribution function (ODF) to minimize the free energy versus minimizing numerically. We find that formal numerical minimization gives results that are much closer to computer simulation results. In order to describe mixtures of HSC plus ideal chains we studied the depletion of ideal chains around a HSC. We analyze the density profiles of ideal chains near a hard cylinder and find the depletion thickness δ is a function of the ratio of the polymer's radius of gyration R_g and the cylinder radius R_c. Our results are compared with a common approximation in which the depletion thickness is taken equal to the radius of gyration of the polymer chain. We incorporate the correct depletion thickness into SPT and find that for R _g/R _c < 1.56 using ideal chains gives phase transitions at smaller polymer concentrations, whereas for R _g/R _c > 1.56 , which is a common experimental situation, the phase transitions are found at larger polymer concentrations with respect to δ = R _g . The differences are significant, especially for R _g ≫ R _c , so we can conclude it is essential to take into account the properties of ideal polymer chains and the resulting depletion near a cylinder. Finally, we present phase diagrams for rod-polymer mixtures which could be realized under experimental conditions.     

Multiple conformations in polyampholytes

We study the configurational statistics of a ring polyampholyte chain made of N randomly charged monomers with elementary charge .To a large extent, the overall structure of a polyampholyte is controlled by a total sum Q of all charges. When the total charge is smaller than , the polyampholyte has a compact globular structure. At charges larger than , the configuration has the form of a ring of small globules (beads) connected by strings. Between Q₁ and Q₂ we find a remarkable diversity of meta-stable configurations having the shapes of irregular clusters of small globules connected by the strings. We estimate the number of these configurations and the energy barriers between them. Between Q₁ and Q₂, the minimum energy configurations are completely controlled by randomness in the charge distribution along the chain. There are hysteresis effects in the shapes of the clusters. As the total charge increases, the linearly extended configurations become dynamically more preferable. When the charge decreases, the circular shapes are preferred. We remark on the probable connection with the multiple phase transitions observed in polyampholyte gels. Received: 10 July 1997 / Accepted: 13 October 1997     

Excluded-volume polymer-induced depletion interaction between parallel flat plates

The interaction between two parallel plates due to non-adsorbing polymer chains with excluded volume is calculated using the adsorption method. The adsorption is calculated from the profile of the polymer segment concentration between the plates, which is obtained from the product function of the concentration profile near a single wall, involving the correlation length. The renormalization group theory provides expressions for the osmotic pressure and consequently for the osmotic compressibility, chemical potential and correlation length of a polymer solution. Both the local polymer concentration profiles as well as the minimum of the interaction potential between the plates agree with recently published self-avoiding random walk computer simulations. Received 9 August 2001     

Secondary super-structures in random copolymers

Secondary domain superstructures in correlated random block copolymers are considered theoretically using the concept of the second order parameter related to fluctuations of the local mean block length. It is shown that the size of secondary domains, , is much larger than the primary domain size, L: , while , where is a small parameter defining the composition asymmetry. Different secondary morphologies are characterized. It is also shown that separation of the system in two macroscopic phases with different primary morphologies predicted earlier using the free energy expansion up to ( is the usual order parameter related to local composition) is an artifact of this widely accepted theoretical model. Received 15 July 1998 and Received in final form 18 January 1999     

Curved polymer and polyelectrolyte brushes beyond the Daoud-Cotton model

We revise the classical Daoud-Cotton (DC) model to describe conformations of polymer and polyelectrolyte chains end-grafted to convex spherical and cylindrical surfaces. In the framework of the DC model, local stretching of chains in the brush does not depend on the degree of polymerization of grafted chains, and the polymer density profile follows a single-exponent power law. This model, however, does not correspond to a minimum in free energy of the curved brush. The nonlocal (NL) approximation exploited in the present paper implies the minimization of the overall free energy of the brush and predicts that the polymer density profile does not follow a single-exponent power law. In the limit of large surface curvature the NL approximation provides the same scaling laws for brush thickness and free energy as the local DC model. Numerical prefactors are however different. Extra extension of chains in the brush interior region leads to larger equilibrium brush thickness and lower free energy per chain. A significant difference between outcomes of the two models is found for brushes formed by ionic polymers, particularly for weakly dissociating (p H-sensitive) polyelectrolytes at low solution salinity.     

Dielectrophoresis of nanocolloids: A molecular dynamics study

Dielectrophoresis (DEP), the motion of polarizable particles in non-uniform electric fields, has become an important tool for the transport, separation, and characterization of microparticles in biomedical and nanoelectronics research. In this article we present, to our knowledge, the first molecular dynamics simulations of DEP of nanometer-sized colloidal particles. We introduce a simplified model for a polarizable nanoparticle, consisting of a large charged macroion and oppositely charged microions, in an explicit solvent. The model is then used to study DEP motion of the particle at different combinations of temperature and electric field strength. In accord with linear response theory, the particle drift velocities are shown to be proportional to the DEP force. Analysis of the colloid DEP mobility shows a clear time dependence, demonstrating the variation of friction under non-equilibrium. The time dependence of the mobility further results in an apparent weak variation of the DEP displacements with temperature.     

Charge relaxation in polyampholytes of various statistics

We discuss theoretically the relaxation of charge fluctuations in polyampholyte solutions. It has been shown previously by some of us (J. Wittmer et al. Europhys. Lett. 24, 263 (1993)) that the charge distribution along the polyampholyte backbone has a dramatic influence on the polarization energy and hence on the solubility. Here it is demonstrated that a similar effect exists for the charge relaxation. The charge relaxation mechanism qualitatively depends on the statistics: for alternating polyampholytes the relaxation is mainly due to local dipole inversion and is not primarily driven by electrostatic interactions, whereas for random polyampholytes it is driven by electrostatic interactions. Intermediate statistics (with short-ranged (exponential) correlations) appear as a combination of these two limiting cases: short-wavelength modes are insensitive to the loss of correlations along the backbone, whereas long-wavelength modes correspond to a random statistics with renormalized charges. The relaxation of the dielectric constant is also calculated. Received: 20 December 2002 / Accepted: 13 March 2003 / Published online: 24 April 2003 RID="a" ID="a"e-mail: johner@ics.u-strasbg.fr     

Langevin dynamics of the glass forming polymer melt: Fluctuations around the random phase approximation

In this paper the Martin-Siggia-Rose (MSR) functional integral representation is used for the study of the Langevin dynamics of a polymer melt in terms of collective variables: mass density and response field density. The resulting generating functional (GF) takes into account fluctuations around the random phase approximation (RPA) up to an arbitrary order. The set of equations for the correlation and response functions is derived. It is generally shown that for cases whenever the fluctuation-dissipation theorem (FDT) holds we arrive at equations similar to those derived by Mori-Zwanzig. The case when FDT in the glassy phase is violated is also qualitatively considered and it is shown that this results in a smearing out of the ideal glass transition. The memory kernel is specified for the ideal glass transition as a sum of all “water-melon” diagrams. For the Gaussian chain model the explicit expression for the memory kernel was obtained and discussed in a qualitative link to the mode-coupling equation. Received: 9 January 1998 / Revised: 24 April 1998 / Accepted: 2 July 1998     

Softening of the interactions between surfactant bilayers in a lamellar phase due to the presence of a polymer

The compressibility modulus of a lamellar phase containing a neutral polymer guest molecule was measured directly using a surface force apparatus. The system studied consisted of sodium dodecyl sulphate (SDS), pentanol, water and polyethylene glycol (PEG) . The lamellar phase was induced from a micellar phase in situ via a confinement induced isotropic to lamellar phase transition. This avoided problems resulting from the viscosity and turbidity normally characteristic of these lamellar phase samples. Increasing the amount of PEG resulted in a marked decrease in the layer compressibility modulus indicating a decrease in the repulsive forces between the lamellae. The origin of such a phenomenon is discussed in terms of different mechanisms including depletion interactions, bridging interactions and modification of the electrostatic interaction between the lamellae by the polymer. Received 2 February 1998     

Swelling behavior and viscoelasticity of ultrathin grafted hyaluronic acid films

In this paper we study the effect of monovalent and divalent ions on the swelling behavior and viscoelastic parameters of ultrathin layers of the natural polyelectrolyte hyaluronic acid covalently coupled to glass substrates. A colloidal probe technique is applied for this purpose based on latex beads, hovering over the polymer cushion. By analyzing the vertical Brownian motion of these beads with reflection interference contrast microscopy (RICM) we determined the equilibrium layer thickness (with 3 nm vertical resolution), the interfacial interaction potential, and the characteristic mesh size limiting the hydrodynamic flow within the polyelectrolyte film as a function of the ionic strength. The experimental results are interpreted in terms of three different theoretical models: the polyelectrolyte brush approximation of Pincus [#!ref1!#], a modified polyelectrolyte brush approximation in the high salt concentration limit of Ross and Pincus [#!ref2!#] and the simple scaling approximation for neutral adsorbed polymers of de Gennes [#!ref3!#]. Within experimental error all of these different models fit our experimental data and yield comparable results for the equilibrium layer thickness. Moreover we determine a thickness dependent, effective surface coverage from both brush models. The hydrodynamic properties of the films are interpreted in terms of the Brinkmann model of elastic porous media by assuming an effective mesh size, which depends linearly on the Debye screening length. The salt induced condensation of the polyelectrolyte films can be described microscopically in terms of a progressive contraction of the mesh size with increasing salt concentration. Received 10 September 1998 and Received in final form 30 November 1998     

Globules of annealed amphiphilic copolymers: Surface structure and interactions

A mean-field theory of globules of random amphiphilic copolymers in selective solvents is developed for the case of an annealed copolymer sequence: each unit can be in one of two states, H (insoluble) or P (soluble or less insoluble). The study is focussed on the regime when H and P units tend to form long blocks, and when P units dominate in the dilute phase, but are rare in the globule core. A first-order coil-to-globule transition is predicted at some T = T _cg. The globule core density at the transition point increases as the affinity of P units to the solvent, ˜, is increased. Two collapse transitions, coil → “loose” globule and “loose” globule → “dense” globule, are predicted if ˜ is high enough and P units are marginally soluble or weakly insoluble. H and P concentration profiles near the globule surface are obtained and analyzed in detail. It is shown that the surface excess of P units rises as ˜ is increased. The surface tension decreases in parallel. Considering the interaction between close enough surfaces of two globules, we show that they always attract each other at a complete equilibrium. It is pointed out, however, that such equilibrium may be difficult to reach, so that partially equilibrium structures (defined by the condition that a chain forming one globule does not penetrate into the core of the other globule) are relevant. It is shown that at such partial equilibrium the interaction is repulsive, so the globules may be stabilized from aggregation. The strongest repulsion is predicted at the coil-to-globule transition point T _cg: the repulsion force decreases with the distance between the surfaces according to a power law. In the general case (apart from T _cg) the force vs. distance decay becomes exponential; the decay length ξ diverges as T → T _cg. The developed theory explains certain anomalous properties observed for globules of amphiphilic homopolymers.     

Charge distribution on annealed polyelectrolytes

We investigate the equilibrium charge distribution along a single annealed polyelectrolyte chain under different conditions. The coupling between the conformation of the chain and the local charge distribution is described for various solvent qualities and salt concentration. In salt free solution, we find a slight charge depletion in the central part of the chain: the charges accumulate at the ends. The effect is less important if salt is added to the solution since the charge inhomogeneity is localized close to the chain ends over a distance of order of the Debye length. In the case of poor solvent conditions we find a different charge per monomer in the beads and in the strings in the framework of the necklace model. This inhomogeneity leads to a charge instability and a first order transition between spherical globules and elongated chains. Received 19 March 1999 and Received in final form 2 August 1999     

Nematic ordering in anisotropic elastomers: Effect of frozen anisotropy

Nematic ordering in anisotropic non-Gaussian elastomers is considered theoretically using mean field approximation. We focus on the effect of anisotropy during network cross-linking on the system elasticity and, in particular, on the so-called soft deformation mode. As the main result, we calculate the dependence of the elastomer free energy on the angle between the axis of “frozen” anisotropy and the nematic director. The dependence of the isotropic-nematic transition point on the orientational field acting on the monomers during the cross-linking process is also calculated. Received: 5 November 1997 / Revised and Accepted: 29 June 1998     

Conformation of poly(L-lysine)-graft-poly(ethylene glycol) molecular brushes in aqueous solution studied by small-angle neutron scattering

Small-angle neutron scattering (SANS) has been employed for the analysis of conformations of poly(L-lysine)-graft-poly(ethylene glycol) (PLL-g -PEG) molecular bottle brushes in aqueous solutions. The degree of polymerisation of the PEG chains was systematically varied in order to unravel dependence of the conformational properties of the bottle brushes on the molecular weight of the grafted chains. The grafting density was kept constant and high enough to ensure strong overlap of the PEG chains. The scattering spectra were fitted on the basis of the model of an effective worm-like chain with the account of average radial distribution and local fluctuations of the PEG density in the bottle brush. The results of the fits indicate that molecular brushes retain weakly bent configuration on the length scale of the order of (or larger than) the brush thickness. This finding is in agreement with earlier simulation and recent theoretical results.     

Shape transformations of protein-like copolymer globules

Shapes of globules formed by amphiphilic multi-block-copolymers in a selective solvent are considered theoretically. We focus on copolymers consisting mostly of insoluble H-units forming large core surrounded by a shell of soluble P-blocks. It is shown that the globule becomes non-spherical when the effective shell tension is low enough. The resultant shape depends on the shell bending energy: it is prolate if this energy is larger than the elastic energy of the core, and oblate in the opposite case. The central result is the prediction of the formation of a surface pattern of fingers accompanying or even preempting the shape transition mentioned above. We elucidate and discuss the following finger morphologies: 1) nearly spherical knob; 2) a necklace of spherical beads extending away from the surface; 3) mostly cylindrical fingers; 4) large thorn-like fingers. The first 3 morphologies develop at equilibrium as the shell area increases (or, equivalently, the shell tension decreases). Considering the relevant kinetical aspects we show that formation of fingers is a nucleation and growth process, and that the energy of their equilibrium nucleation is likely to be high. Therefore, the finger formation may be delayed, and may actually occur in the regime where the plain spherical surface is metastable. It is the last morphology (thorn-like fingers) that characterizes the metastable regimes when the finger formation is controlled by a high activation energy. The universal features of the above predictions inviting experimental tests are discussed.     

Variational theory for a single polyelectrolyte chain

Variational methods are applied to a single polyelectrolyte chain. The polymer is modeled as a Gaussian chain with screened electrostatic repulsion between all monomers. As a variational Hamiltonian, the most general Gaussian kernel, including the possibility of a classical or mean polymer path, is employed. The resulting self-consistent equations are systematically solved both for large and small monomer-monomer separations along the chain. In the absence of screening, the polymer is stretched on average. It is described by a straight classical path with Gaussian fluctuations around it. If the electrostatic repulsion is screened, the polymer is isotropically swollen for large separations, and for small separations the polymer correlation function is calculated as an analytic expansion in terms of the monomer-monomer separation along the chain. The electrostatic persistence length and the electrostatic blobsize are inferred from the crossover between distinct scaling ranges. We perform a global analysis of the scaling behavior as a function of the screening length and electrostatic interaction strength , where is the Bjerrum length and A is the distance of charges along the polymer chain. We find three different scaling regimes. i) A Gaussian-persistent regime with Gaussian behavior at small, persistent behavior at intermediate, and isotropically swollen behavior at large length scales. This regime occurs for weakly charged polymers and only for intermediate values of the screening length. The electrostatic persistence length is defined as the crossover length between the persistent and the asymptotically swollen behavior and is given by and thus disagrees with previous (restricted) variational treatments which predict a linear dependence on the screening length .ii) A Gaussian regime with Gaussian behavior at small and isotropically swollen behavior at large length scales. This regime occurs for weakly charged polymers and/or strong screening, and the electrostatic repulsion between monomers only leads to subfluent corrections to Gaussian scaling at small separations. The concept of a persistence length is without meaning in this regime. iii) A persistent regime , where the chain resembles a stretched rod on intermediate and small scales. Here the persistence length is given by the original Odijk prediction, , if the overstretching of the chain is avoided. We also investigate the effects of a finite polymer length and of an additional excluded-volume interaction, which modify the resultant scaling behavior. Applications to experiments and computer simulations are discussed. Received 24 December 1997