Search

We study the effect of polydispersity on the structure of polymer brushes by analytical theory, a numerical self-consistent field approach, and Monte Carlo simulations. The polydispersity is represented by the Schulz-Zimm chain-length distribution. We specifically focus on three different polydispersities representing sharp, moderate and extremely wide chain length distributions and derive explicit analytical expressions for the chain end distributions in these brushes. The results are in very good agreement with numerical data obtained with self-consistent field calculations and Monte Carlo simulations. With increasing polydispersity, the brush density profile changes from convex to concave, and for given average chain length $N_n$ and grafting density $\sigma$, the brush height $H$ is found to scale as $(H/H_{\mathrm{mono}}-1)\propto(N_w/N_n-1)^{1/2}$ over a wide range of polydispersity indices $N_w/N_n$ (here $H_{\mathrm{mono}}$ is the height of the corresponding monodisperse brush. Chain end fluctuations are found to be strongly suppressed already at very small polydispersity. Based on this observation, we introduce the concept of the brush as a near-critical system with two parameters (scaling variables), $(N_{n}\sigma^{2/3})^{-1}$ and $(N_w/N_n-1)^{1/2}$, controlling the distance from the critical point. This approach provides a good description of the simulation data. Finally we study the hydrodynamic penetration length $l_\mathrm{p}$ for brush-coated surfaces in flow. We find that it generally increases with polydispersity. The scaling behavior crosses over from $l_\mathrm{p}\sim N_\mathrm{n}^{1/2}\sigma^{-1/6}$ for monodisperse and weakly polydisperse brushes to $l_\mathrm{p}\sim N_{n}^{2/3}$ for strongly polydisperse brushes., Comment: 21pages, 9 figures. Accepted for publication in Macromolecues

We propose simple expressions II/IIo = 1 + and (omega/omega(ex))(3 alpha-1) and (delta(0)/delta)(2) = 1 + (omega/omega(ex))(2 alpha) for the osmotic pressure II and the depletion thickness 6 as a function of the polymer concentration omega. Here, IIo and delta 0 correspond to the dilute limit, and omega(ex) is an extrapolation concentration which is of the order of the overlap concentration omega(ov). The De Gennes exponent a describes the concentration dependence of the semidilute correlation length xi similar to omega(-alpha) it is related to the Flory exponent nu through alpha = nu/(3 nu - 1.). The quantity omega(ex) is experimentally accessible by extrapolating the semidilute limit towards II = IIo or delta = delta(o). These expressions are exact in mean field, where the ratio omega(ex)/omega(ov) (0.49 for II, 0.41 for delta) follows from established models. For excluded-volume chains they describe simulation data excellently: in this case omega(ex)/omega(ov) is 0.69 for II and again 0.41 for delta. We find also very good agreement with experimental data.

Upon compression between two pistons an end-tethered polymer chain undergoes an abrupt transition from a confined coil state to an inhomogeneous flower-like conformation that is partially escaped from the gap. In the thermodynamic limit the system demonstrates a first-order phase transition. A rigorous analytical theory of this phenomenon for a Gaussian chain is presented in two ensembles: a) the H-ensemble, in which the distance H between pistons plays the role of the control parameter, and b) the conjugate f-ensemble in which the external compression force f is the independent parameter. A loop region for (f(H)) with negative compressibility exists in the H-ensemble, while in the f-ensemble is strictly monotonic. The average lateral forces taken as functions of H (or (H), respectively) have distinctly different behavior in the two ensembles. This result is a clear counterexample of the main principles of statistical mechanics stating that all ensembles are equivalent in the thermodynamic limit. Another theorem states that the thermodynamic potential as a function of volume must be concave everywhere. We demonstrated that the exact free energy in the H-ensemble contradicts this statement. Inapplicability of these fundamental theorems to a macromolecule undergoing the escape transition is clearly related to the fact that phase coexistence in the present system is strictly impossible. This is a direct consequence of the tethering and the absence of global translational degrees of freedom of the polymer chain.

The analysis of the exact partition function of an end-fixed Gaussian chain compressed between two pistons reveals a loop-like curve of the average pressure as a function of the inter-piston volume. In the thermodynamic limit the system undergoes a first-order phase transition from a compressed coil to a partially escaped flower conformation. In the transition region the fixed volume ensemble is not equivalent to the fixed pressure ensemble, and for finite chains there appears a region with a negative compressibility. There is no conflict with the general notion that the pressure for a thermodynamic system should be a non-increasing function of the volume because the negative compressibility disappears in the thermodynamic limit. Our results are also expected to hold for self-avoiding chains and are experimentally accessible for real chains.

We derive a general equation for the depletion thickness delta next to a flat wall in a solution of nonadsorbing polymer, which is easily extended to spherical geometry. This equation has the simple form delta(-2) = delta(0)(-2) + xi(-2). Here, delta(0) is the value of delta in the limit of infinite dilution, which depends only on the chain length: delta(0) = 2R/rootpi, where R is the radius of gyration of the polymer. The parameter xi is a correlation length in solution, which depends on the polymer concentration phi(b) and the solvency chi, but not on the chain length. We use a mean-field form of xi = xi(chi,phi(b)) which provides a smooth crossover from good to 0 solvency conditions. We show that the depletion thickness is actually a generalized bulk solution correlation length which does depend on chain length. In all cases the profile for a flat wall is given by rho = phi/phi(b) = tanh(2)(z/delta). The extension to a sphere of radius a is also very simple: rho(s) = [z/a + tanh(z/delta)](2)/ [z/a + 1](2). These analytical results are compared to numerical self-consistent-field computations, whereby the segment-wall repulsion in the lattice model is chosen in accordance with the boundary condition P(0) = 0 in the continuum model. The agreement is nearly perfect for good solvents and large particles. For a Theta solvent our simple analytical model overestimates delta slightly; in this case the tanh(2) profile is not strictly valid, and we derive a corrected analytical form. For smaller particles, also a slight overestimation of the width of the depletion zone is found. However, in all cases the trends are predicted very well. Our simple equations allow, in principle, analytical expressions for the surface tension, for the distribution coefficient in size-exclusion chromatography, and for interaction potentials and phase diagrams of colloids with nonadsorbing polymer.

An end-tethered polymer chain squeezed between two pistons undergoes an abrupt transition from a confined coil state to an inhomogeneous flower-like conformation partially escaped from the gap. We present a rigorous analytical theory for the equilibrium and kinetic aspects of this phenomenon for a Gaussian chain. Applying the analogy with the problem of the adsorption of an ideal chain constrained by one of its ends, we obtain a closed analytical expression for the exact partition function. Various equilibrium thermodynamic characteristics (the fraction of imprisoned segments, the average compression, and lateral forces) are calculated as a function of the piston separation. The force versus separation curve is studied in two complementary statistical ensembles, the constant force and the constant confinement width ones. The differences in these force curves are significant in the transition region for large systems, but disappear for small systems. The effects of metastability are analyzed by introducing the Landau free energy as a function of the chain stretching, which serves as the order parameter. The phase diagram showing the binodal and two spinodal lines is presented. We obtain the barrier heights between the stable and metastable states in the free energy landscape. The mean first passage time, i.e., the lifetime of the metastable coil and flower states, is estimated on the basis of the Fokker-Planck formalism. Equilibrium analytical theory for a Gaussian chain is complemented by numerical calculations for a lattice freely jointed chain model.