Chain-length distribution in a model of equilibrium polymerization

Microstructural Capture of Living Ultrathin Polymer Brush Evolution via Kinetic Simulation Studies

Performing dynamic off-lattice multicanonical Monte Carlo simulations, we study the statics, dynamics, and scission-recombination kinetics of a self-assembled in situ-polymerized polydisperse living polymer brush (LPB), designed by surface-initiated living polymerization. The living brush is initially grown from a two-dimensional substrate by end-monomer polymerization-depolymerization reactions through seeding of initiator arrays on the grafting plane which come in contact with a solution of nonbonded monomers under good solvent conditions. The polydispersity is shown to significantly deviate from the Flory-Schulz type for low temperatures because of pronounced diffusion limitation effects on the rate of the equilibration reaction. The self-avoiding chains take up fairly compact structures of typical size R_g(N) ∼ N^ν in rigorously two-dimensional (d = 2) melt, with ν being the inverse fractal dimension (ν = 1/d). The Kratky description of the intramolecular structure factor F(q), in keeping with the concept of generalized Porod scattering from compact particles with fractal contour, discloses a robust nonmonotonic fashion with q^dF(q) ∼ (qR_g)^-3/4 in the intermediate-q regime. It is found that the kinetics of LPB growth, given by the variation of the mean chain length, follows a power law ⟨N(t)⟩ ∝ t^1/3 with elapsed time after the onset of polymerization, whereby the instantaneous molecular weight distribution (MWD) of the chains c(N) retains its functional form. The variation of ⟨N(t)⟩ during quenches of the LPB to different temperatures T can be described by a single master curve in units of dimensionless time t/τ_∞, where τ_∞ is the typical (final temperature T^∞-dependent) relaxation time which is found to scale as τ_∞ ∝ ⟨N(t = ∞)⟩⁵ with the ultimate average length of the chains. The equilibrium monomer density profile ϕ(z) of the LPB varies as ϕ(z) ∝ ϕ^-α with the concentration of segments ϕ in the system and the probability distribution c(N) of chain lengths N in the brush layer scales as c(N) ∝ N^-τ. The computed exponents α ≈ 0.64 and τ ≈ 1.70 are in good agreement with those predicted within the context of the Diffusion-Limited Aggregation theory, α = 2/3 and τ = 7/4.

Entropy of polydisperse chains: solution on the Husimi lattice

We consider the entropy of polydisperse chains placed on a lattice. In particular, we study a model for equilibrium polymerization, where the polydispersity is determined by two activities, for internal and endpoint monomers of a chain. We solve the problem exactly on a Husimi lattice built with squares and with arbitrary coordination number, obtaining an expression for the entropy as a function of the density of monomers and mean molecular weight of the chains. We compare this entropy with the one for the monodisperse case, and find that the excess of entropy due to polydispersity is identical to the one obtained for the one-dimensional case. Finally, we obtain a distribution of molecular weights with a rather complex behavior, but which becomes exponential for very large mean molecular weight of the chains, as required by scaling properties, which should apply in this limit.

Entropy of polydisperse chains: solution on the Bethe lattice

We consider the entropy of polydisperse chains placed on a lattice. In particular, we study a model for equilibrium polymerization, where the polydispersivity is determined by two activities, for internal and endpoint monomers of a chain. We solve the problem exactly on a Bethe lattice with arbitrary coordination number, obtaining an expression for the entropy as a function of the density of monomers and mean molecular weight of the chains. We compare this entropy with the one for the monodisperse case and find that the excess of entropy due to polydispersivity is identical to the one obtained for the one-dimensional case. Finally, we obtain an exponential distribution of molecular weights.

Impact of steric interactions on the helical transition in assemblies of discotic molecules

We investigate theoretically the effect of excluded-volume interactions on the helical configurational transition of supramolecular assemblies in solutions of chiral disklike molecules. To this end, we set up a second-virial theory within the context of the helical self-assembly of rodlike objects. We find that interaggregate interactions shift both the helical-transition point and the sharpness of the transition. For realistic values of the model parameters, the helical-transition temperature shifts by several degrees, and the more so the higher the concentration of assembling material. The mean aggregation number is also affected by the interactions, albeit only by a modest amount, unless the solution becomes very concentrated.

The ultrasensitivity of living polymers

Synthetic and biological living polymers are self-assembling chains whose chain length distributions (CLDs) are dynamic. We show these dynamics are ultrasensitive: Even a small perturbation (e.g., temperature jump) nonlinearly distorts the CLD, eliminating or massively augmenting short chains. The origin is fast relaxation of mass variables (mean chain length, monomer concentration) which perturbs CLD shape variables before these can relax via slow chain growth rate fluctuations. Viscosity relaxation predictions agree with experiments on the best-studied synthetic system, alpha-methylstyrene.

Reversible polymerizations and aggregations

The aggregation of monomers into polymers, whether by covalent or noncovalent interactions, is often reversible and frequently occurs with the entropy and enthalpy of the aggregation sharing the same sign. In such a case, the aggregation goes forward or reverses, depending on such variables as temperature and composition, rather like a phase transition. We explore the physical chemistry of three such systems: an organic monomer (alpha-methylstyrene), an inorganic monomer (sulfur), and a biopolymer (actin). We compare the available theories and experiments and list issues still open.