Polymer coupling and theory of on-random crosslinking: an analytical solution

Gel formation is an important feature in free-radical polymer coupling. Due to the different possible combination reactivities of each polymer backbone radical, polymer chains are crosslinked in a non-random manner. Equations of the moments have been derived to predict the pregel molecular weight development and the crosslink density at gel point. This work provides an analytical solution for the differential equations. The model agrees with the Flory-Stockmayer gelation theory under the condition of random crosslinking. The magnitude of deviations from the classical theory for non-random crosslinking depends on the product of the radical termination reactivity ratios (r₁r₂), the ratio of the rate constants of backbone radical generation (k), the ratio of the weight-average chain lengths of primary polymers (y), and the polymer weight fractions (w₂).     

Stochastic dynamics simulation of grafted polymer brushes under shear deformation

We present results of computer simulations of polymer brushes (layers of polymer chains attached at one end onto an impermeable planar surface) under shear deformation at constant shear rate. As the first stage of calculations the behavior of a single brush was studied. The monomer density profile, the distribution of the chain ends, the positions and orientations of different monomers along the chain were calculated. Dimensions of the polymer chains as functions of the shear rate were obtained for different grafting densities. An increase in the brush thickness over the grafting plane with an increase in the shear rate as predicted by the theory of Barrat was observed. However, the magnitude of the effect appears to be small. We explain this by finite extensibility of the grafted chains.     

Polymer Polydispersity Effect on Depletion Interaction between Colloidal Particles

The effect of polymer polydispersity on the polymer‐induced interaction between colloidal particles due to non‐adsorbing ideal chains is investigated. An analytical theory is developed for the polymer‐segment density between two plates and in the space surrounding two spheres by extending a recently proposed superposition approximation to include polymer polydispersity. Monte Carlo computer simulations were made to test the validity of the analytical theory. The polymer densities predicted by the superposition approximation are in reasonable agreement with simulation results for the polydisperse case. The simulations show that depletion leads to a size fractionation of the polymers. It is shown that size polydispersity has a small effect on the interaction between two parallel plates but a more significant effect on the interaction between two spheres. The range of the potential increases and the contact potential drops with increasing polydispersity.

Polymer‐segment density as a function of y for three values of x, as indicated, in the space surrounding two colloidal spheres with radius R = R_g0 and h = 0.48R_g0. Symbols are the MC results: polydisperse polymer (○; z = 1) and monodisperse polymer (•) samples. Curves are the predictions of the product‐function approximation for monodisperse polymer (solid lines) and polydisperse polymer (z = 1, dashed lines).     

Swelling and shrinking of polymer chains in homopolymer blends

The radius of gyration, R, of polymer chains in homopolymer blends is studied in the framework of a self-consistent one-loop approximation. We show that the polymer chains can shrink or swell in comparison to the Gaussian chain. Swelling of the polymer chains in a region far away from the critical point is caused by the steric repulsive forces that were included in the model as the constraint of incompressibility. The chains shrink progressively, as we approach the critical region passing through the Gaussian limit, $ R_0 = \sqrt{\frac{N}{6}}l $, far away from the critical point (N − degree of polymerization, l − length of monomer). The correction responsible for the swelling and the shrinking is small when the concentrations of components ϕ are comparable ($ N = 1000,\bar\phi = 0.5,\frac{{R_0^2 - R^2}}{{R_0^2}} = \pm 0.02\%$). This effect, although small, leads to a local demixing, a sharp shrinking of chains in both components accompanied by a strong change of the global inter-monomer distance, which should be observable experimentally. When the local demixing occurs there is a sudden increase in the scattering intensity (of the order of 30% for N = 1000, and ϕ _A = 0.2). The increase of the degree of polymerization of the same type of chains leads to an increase of the swelling-shrinking effects. In addition, the critical concentration of the shorter chains component is smaller in comparison to the value obtained in the Flory-Huggins theory. The self-consistent determination of the radius of gyration and the upper wave-vector cutoff make our model free from any divergences. In the limit of N → ∞ this theory reduces to the random phase approximation (RPA) of de Gennes.     

Compression of Polymer Brushes Under a Lateral Force

Summary: The influence of a lateral force (or lateral shear) acting on chains in a polymer brush is investigated theoretically. Brushes consisting of chains with temperature dependent anisotropic interactions between monomers (main‐chain mesogenic groups) are considered. It is shown that a lateral force applied to polymer brush induces its compression. In contrast to a conventional brush, the compression of brush, capable of forming a liquid crystalline (LC) state, can be caused by comparatively small shear forces. Moreover, such shear forces can induce a phase transition of a brush into the tilted LC state with a several‐fold decrease in brush thickness. These results allow us to predict a possibility to observe a decrease in brush thickness in a real experiment with reasonably values of shear rate.

Model of a chain in a polymer brush under an influence of lateral force p.     

Viscoelasticity of a stretched polymer chain with ends exposed to a constant force

The viscoelasticity of a stretched polymer chain with its end particles exposed to oppositely acting forces is studied via collisional molecular dynamics and analytically. A simple model according to which a polymer molecule is a chain of particles linked through freely jointed elastic bonds is adopted. The analytical theory is in good agreement with the results of the computer simulation of time correlation functions in the range of large-scale motions of a polymer molecule. It is found that the decay of correlation functions K _αβμν of fluctuations of the microscopic stress tensor of a chain, K _zzzz, K _zαzα , = K _zαzα , (α = x, y; z is the axis along which the forces act), is slowed down, and their value increases relative to the respective correlation functions of a chain with fixed ends. The greater the force, the higher this difference. The correlation functions that are transverse with respect to the z axis do not differ from those for chains with fixed ends. The results show that, in the calculation of time correlation functions of strongly stretched polymer chains, different statistical ensembles are not equivalent; this must be taken into account in the dynamic theories of heavily deformed polymers.     

Amphiphilic polymer brush in a mixture of incompatible liquids

An analogue of the Alexander‐DeGennes box model is used for theoretical investigation of polymer brushes in a mixture of two solvents. The basic solvent A and the admixture B are assumed to be highly incompatible (Flory‐Huggins parameter χ_AB = 3.5). Thermodynamics of a polymer in the solvents A and B are described by parameters χ_B < χ_A ≤ 1/2. The equilibrium behavior of a brush is investigated in dependence on solvent composition, grafting density, polymer‐solvents and solvent‐solvent interactions. The possibility of a phase transition related with a strong preferential solvation of a brush by a minor solvent component with higher affinity to polymer is shown and examined. Microphase segregation inside a brush is also demonstrated despite overestimating of the brush homogeneity given by the box model. A further simplification of the model permits to obtain scaling formulas and to investigate main regularities in the brush behavior. This offers a clear physical picture of the phase segregation inside a brush in correlation with the phase state of a bulk solvent.     

Nonlinear dynamics of melted polymer layers

A theory for non-linear rheology of molten polymer layers between solid surfaces in the Rouse regime is discussed. It is shown that the effect of finite extensibility of polymer chains leads to the characteristic 1/3 power law for the shear stress vs. shear velocity in the regime of high velocities. It is also shown that bridging polymer fragments connecting the two surfaces play an important role for the rheology if the effective monomer friction in the immediate vicinity of the surfaces is much higher than far from the surfaces. In particular we predict that shear stress is decreasing with shear velocity u in a limited range between u₁ and u, _min. This effect results in a possibility of stick-slip periodical dynamics of the layer under a constant imposed velocity.     

Self‐Propagating Association of Zwitterionic Polymers Initiated by Ionene Polymers

A self‐propagating association of zwitterionic polymers is observed when a small amount of x,y‐ionene bromide (x = 3 or 6; y = 3, 4, 6, 10 or 12) polymer is added to aqueous solutions of zwitterionic polymer, poly[3‐dimethyl(methacryloyloxyethyl)ammoniumpropanesulfonate] (PDMAPS), to give large amount of PDMAPS precipitate. The self‐propagating association initiated by ionene polymers is explained in terms of the electrostatic interaction between the ionene polymers and the zwitterionic polymers whereupon the geometry of the charges on the polymer chains plays an important role.     

The elasticity of nematic networks

Nematic rubbers are composed of crosslinked polymer chains with stiff rods either incorporated into their backbones or pendant as side chains. When nematic effects axe strong, such rubbers exhibit discontinuous stress-strain relationships and spontaneous shape changes. We model such a rubber using Gaussian elasticity theory, including the nematic interaction via a mean field. Results are presented for the cases of uniaxial extension and compression. Under uniaxial extension the rubber can undergo a first order phase transition to a uniaxial nematic phase. Under uniaxial compression first or second order transitions are possible to genuinely biaxial nematic states with biaxial strains. When nematic effects are very small (i.e. T >> T_c where T_c is the nematic-isotropic phase transition temperature of the rubber) we postulate that the model is a good approximation to a conventional, non-nematic elastomer, and fit our model to data from an isoprene rubber.     

Angle‐resolved XPS of fluorinated and semi‐fluorinated side‐chain polymers

Angle‐resolved XPS data (elemental quantification and high‐energy‐resolution C 1s) are presented for ten polymers with side‐chains of the form ? OCO(CF₂)_yF, ? COO(CH₂)₂OCO(CF₂)_yF (y = 1, 2, 3) and ? COO(CH₂)_x(CF₂)_yF (x = 1, y = 1, 2, 3; x = 2, y = 8). Particular attention was paid to charge compensation and speed of data acquisition, with co‐addition from multiple fresh samples to give spectra with good energy resolution and good signal‐to‐noise ratio free from the effects of x‐ray‐induced degradation. Water contact angles for the polymers are also reported. The XPS data demonstrate preferential surface segregation of fluorine‐containing groups for all but the shortest side‐chain polymer, where the ? OCOCF₃ side‐chain either does not surface segregate or is too short for surface segregation to be detectable by angle‐resolved XPS. In the other polymers studied the relative positions of functional groups in the side‐chains correlate with the angle‐resolved behaviour of the corresponding C 1s components. This shows that the surface side‐chains are oriented towards the polymer surface. For the ? COO(CH₂)₂OCO(CF₂)_yF (y = 1) side‐chain, the angle‐resolved C 1s data suggest reduced ordering and linearity compared with y = 2 and 3. For any particular series of polymers, e.g. ? COO(CH₂)_x(CF₂)_yF, the water contact angles increase with y, consistent with burying of the hydrophilic ester groups as y increases. For any particular value of y the sequence of water contact angles is ? COO(CH₂)_x(CF₂)_yF > ? OCO(CF₂)_yF ～ ? COO(CH₂)₂OCO(CF₂)_yF, suggesting greater ordering and density of fluorocarbon species at the surface of the ? COO(CH₂)_x(CF₂)_yF side‐chain polymers compared with the other polymers studied. For the ? COO(CH₂)₂(CF₂)₈F polymer a water contact angle of 124° is measured, which is greater than that of poly(tetrafluoroethene). The ? COO(CH₂)₂OCO(CF₂)F polymer is unusual in that it shows a particularly low water contact angle (83° ), suggesting that the probe fluid is able to sense both ester groups, consistent with the reduced ordering of the side‐chain detected by angle‐resolved XPS. Copyright © 2004 John Wiley & Sons, Ltd.     

When tethered chains meet free ones; the stability of polymer wetting films on polymer brushes

We present a combined experimental and theoretical self-consistent field (SCF) investigation of the wetting behavior of a polystyrene melt (composed of chains with degree of polymerization P) on top of a polystyrene brush (composed of chains with length N) grafted onto a silica surface. The control variables are the grafting density σ of the brush chains and the length of mobile chains P. Experiments show in agreement with the theory that there is a window of complete wetting. Both at very low and at high grafting densities the system remains partial wet. At large degree of polymerization P, there is a difference between the experimental and theoretical results. Theory predicts partial wetting only, whereas the window of complete wetting persists in the experiments even when P >> N. This difference is attributed to the double-well structure of the disjoining pressure as revealed by the SCF theory. With this type of disjoining pressure it is conceivable that a metastable zero contact angle remains present for very long times.     

Monte Carlo studies of tethered chains

Monte Carlo computer simulations of end-tethered chains grafted onto a hard wall have been performed. The chains were modeled as self-avoiding chains on a cubic lattice at athermal solvent conditions. The simulations spanned a wide range of chain lengths, N (100–1000, i.e., up to molecular weights of a few hundred thousands), and anchoring densities, σ (2 × 10⁻⁴ to 0.4), to properly chart the relevant parameter space. It is shown that the reduced surface coverage σ* = σπR is the most appropriate variable that quantitatively determines the mushroom, overlapping mushroom and brush regimes, where R_g is the radius of gyration of a free chain in solution. The simulation data are analyzed to determine the conformational characteristics and shape of the anchored chains and to compare them with the predictions of the analytical self consistent field theory. The strong stretching limit of the theoretical predictions is obtained only for σ* > 8. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47:2449–2461, 2009     

Postgel properties in the statistical crosslinking of heterochains. I. Systems with N types of chains

The heterochain crosslinking model describes nonrandom crosslinking of polymer chains and is an extension of the classical Flory/Stockmayer gelation theory. We consider the postgelation relationship for the system consisting of N types of polymer chains, in which the probability that a crosslink point on an i‐type chain is connected to a j‐type chain is explicitly given by p_ij. The analytical solutions for the weight fraction of the sol, the number‐average and weight‐average molecular weights within the sol fraction, and the crosslinking density within the sol and gel fractions are derived for the systems, with each type of chain conforming to the Schulz–Zimm distribution. Illustrative calculations are shown for the systems consisting of two and three types of chains, and the obtained results agree with those from the Monte Carlo method. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2333–2341, 2000     

Simple model for grafted polymer brushes

The first theories of grafted polymer brushes assumed a step profile for the monomer density. Later, the real density profile was obtained from Monte Carlo or molecular dynamics simulations and calculated numerically using a self-consistent field theory. The analytical approximations of the solutions of the self-consistent field equations provided a parabolic dependence of the self-consistent field, which in turn led to a parabolic distribution for the monomer density in neutral brushes. As shown by numerical simulations, this model is not accurate for dense polymer brushes, with highly stretched polymers. In addition, the scaling laws obtained from the analytical approximations of the self-consistent field theory are identical to those derived from the earlier step-profile-approximation and predict a vanishing thickness of the brush at low graft densities, and a thickness exceeding the length of the polymer chains at high graft densities. Here a simple model is suggested to calculate the monomer density and the interaction between surfaces with grafted polymer brushes, based on an approximate calculation of the partition function of the polymer chains. The present model can be employed for both good and poor solvents, is compatible with a parabolic-like profile at moderate graft densities, and leads to an almost steplike density for highly stretched brushes. While the thickness of the brush depends strongly on solvent quality, it is a continuous function in the vicinity of the temperature. In good and moderately poor solvents, the interactions between surfaces with grafted polymer brushes are always repulsive, whereas in poor solvents the interactions are repulsive at small separations and become attractive at intermediate separation distances, in agreement with experiment. At large separations, a very weak repulsion is predicted.     

Nanomechanical properties of polymer brushes by colloidal AFM probes

Nanomechanical properties of end grafted polymer layers were studied by AFM based, colloidal probe compression measurements. Zwitterionic poly(sulfobetaine methacrylate) (PSBMA) brush was grafted from planar Si surface and poly(methyl methacrylate) (PMAA) brush was grown on colloidal probe by surface initiated atom transfer radical polymerization. PMAA brush was further modified with adhesion promoting arginyl-glycyl-aspartic acid (RGD) peptide sequences. Force–distance curves were obtained for systems where the polymer brushes were probed on unmodified surfaces or face to each other. For each systems the grafting density of the polymer brush was determined applying a ‘box’ like polymer brush model based on the theory by de Gennes. ‘Average’ grafting density was calculated in cases when two polymer brushes face each other: RGD functionalized PMAA or PMAA against PSBMA. For our systems the values for the grafting density was between 0.04 and 0.11 nm^?2. Furthermore the measured approach force–distance curves were fitted according to the Hertz model and the apparent Young’s modulus was determined for all measurements being in a range of around 250 kPa at physiological conditions.     

Monte Carlo Simulations of Self-Assembling Copolymer Brushes

We studied a simplified model of a polymer brush formed by linear chains, which were restricted to vertices of a simple cubic lattice. The macromolecules consisted of a sequence of two different kinds of united atoms arranged in a specific sequence. The chains were grafted to an impenetrable surface, i.e. they were terminally attached to the surface with one end. The model system was studied at different solvent quality from good to poor solvent. The properties of this model system were determined by means of Monte Carlo simulation using a Metropolis-like sampling algorithm based on local changes of chain's conformations. The size and the structure of the brush were determined.     

Stimuli‐responsive diblock copolymer brushes via combination of “click chemistry” and living radical polymerization

pH‐ and temperature‐responsive poly(N‐isopropylacrylamide‐block?4‐vinylbenzoic acid) (poly(NIPAAm‐b‐VBA)) diblock copolymer brushes on silicon wafers have been successfully prepared by combining click reaction, single‐electron transfer‐living radical polymerization (SET‐LRP), and reversible addition‐fragmentation chain‐transfer (RAFT) polymerization. Azide‐terminated poly(NIPAAm) brushes were obtained by SET‐LRP followed by reaction with sodium azide. A click reaction was utilized to exchange the azide end group of a poly(NIPAAm) brushes to form a surface‐immobilized macro‐RAFT agent, which was successfully chain extended via RAFT polymerization to produce poly(NIPAAm‐b‐VBA) brushes. The addition of sacrificial initiator and/or chain‐transfer agent permitted the formation of well‐defined diblock copolymer brushes and free polymer chains in solution. The free polymer chains were isolated and used to estimate the molecular weights and polydispersity index of chains attached to the surface. Ellipsometry, contact angle measurements, grazing angle‐Fourier transform infrared spectroscopy, and X‐ray photoelectron spectroscopy were used to characterize the immobilization of initiator on the silicon wafer, poly(NIPAAm) brush formation via SET‐LRP, click reaction, and poly(NIPAAm‐b‐VBA) brush formation via RAFT polymerization. The poly(NIPAAm‐b‐VBA) brushes demonstrate stimuli‐responsive behavior with respect to pH and temperature. The swollen brush thickness of poly(NIPAAm‐b‐VBA) brush increases with increasing pH, and decreases with increasing temperature. These results can provide guidance for the design of smart materials based on copolymer brushes. © 2013 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2013, 51, 2677–2685     

Grafted polymers under strong shear: Scaling and non-equilibrium Monte Carlo studies

End-grafted polymer chains exposed to strong shear solvent flow in the x-direction are investigated by a non-equilibrium Monte Carlo method using the bond-fluctuating model. The solvent flow is modelled by an enhanced jump rate of monomers in the flow direction. Under strong enough shear flow, the shear force is non-linear with the blob size ζ depending on the shear force and not on the monomer volume fraction ϕ. For the case of a grafted single chain, our data on the end-to-end distance of the polymer for a wide range of shear rate γ agree well with the proposed scaling form and compare reasonably well with the Langevin force law in the extreme strong shear case. Also the scaling for the blob size is derived with ζ = aN^vK(γN^2+v) for some scaling function K, and verified by our simulation data. For the case of a polymer brush, we derive the crossover force scale F^* separating the weak and strong shear regime with F^*a/(kT) = ϕ^v/(3v−1) where v is the usual self-avoiding walk exponent. For the case of a polymer brush under strong shear, the velocity profile, v(z), inside and above the brush is calculated analytically by treating the flow as in a porous medium and solving the Brinkman equation. The solution of the velocity profile is then combined with non-equilibrium Monte Carlo simulation data which allow a self-consistent determination of the chain end-to-end length and the incline angle of the chain. Also we derive the scaling form for the positions x(n) of the nth monomer and verify it by our simulation data. We further obtain an analytical expression for x(n) in terms of the effective viscosity η of the model, given by F = ηav(z). The effective viscosity is also found from our simulation to be ηa²/(kT) ≃ 0.02 ± 0.005.