We use scaling theory to derive the time dependence of the mean-square displacement 〈Δr2〉 of a spherical probe particle of size d experiencing thermal motion in polymer solutions and melts. Particles with size smaller than solution correlation length ξ undergo ordinary diffusion (〈Δr2 (t)〉 ~ t) with diffusion coefficient similar to that in pure solvent. The motion of particles of intermediate size (ξ < d < a), where a is the tube diameter for entangled polymer liquids, is sub-diffusive (〈Δr2 (t)〉 ~ t1/2) at short time scales since their motion is affected by sub-sections of polymer chains. At long time scales the motion of these particles is diffusive and their diffusion coefficient is determined by the effective viscosity of a polymer liquid with chains of size comparable to the particle diameter d. The motion of particles larger than the tube diameter a at time scales shorter than the relaxation time τe of an entanglement strand is similar to the motion of particles of intermediate size. At longer time scales (t > τe) large particles (d > a) are trapped by entanglement mesh and to move further they have to wait for the surrounding polymer chains to relax at the reptation time scale τrep. At longer times t > τrep, the motion of such large particles (d > a) is diffusive with diffusion coefficient determined by the bulk viscosity of the entangled polymer liquids. Our predictions are in agreement with the results of experiments and computer simulations.

We develop and solve a new molecular model for nonlinear elasticity of entangled polymer networks. This model combines and generalizes several succeseful ideas introduced over the years in the field of the rubber elasticity. The topological constraints imposed by the neighboring network chains on a given network are represented by the confining potential that changes upon network deformation. This topological potential restricts fluctuations of the network chain to the nonaffinely deformed confining tube. Network chains are allowed to fluctuate and redistribute their length along the contour of their confining tubes. The dependence of the stress σ on the elongation coefficient λ for the uniaxially deformed network is usially represented in the form of the Mooney stress, f*(1/λ) = σ/(λ − 1/λ2). We find a simple expression for the Mooney stress, f*(1/λ) = G c + G e/(0.74λ + 0.61λ-1/2 − 0.35), where G c and G e are phantom and entangled network moduli. This allows one to analyze the experimental data in the form of the universal plot and to obtain the two moduli G c and G e related to the densities of cross-links and entanglements of the individual networks. The predictions of our new model are in good agreement with experimental data for uniaxially deformed polybutadiene, poly(dimethylsiloxane), and natural rubber networks, as well as with recent computer simulations.

We propose a hopping mechanism for diffusion of large nonsticky nanoparticles subjected to topological constraints in both unentangled and entangled polymer solids (networks and gels) and entangled polymer liquids (melts and solutions). Probe particles with size larger than the mesh size ax of unentangled polymer networks or tube diameter ae of entangled polymer liquids are trapped by the network or entanglement cells. At long time scales, however, these particles can diffuse by overcoming free energy barrier between neighboring confinement cells. The terminal particle diffusion coefficient dominated by this hopping diffusion is appreciable for particles with size moderately larger than the network mesh size ax or tube diameter ae. Much larger particles in polymer solids will be permanently trapped by local network cells, whereas they can still move in polymer liquids by waiting for entanglement cells to rearrange on the relaxation time scales of these liquids. Hopping diffusion in entangled polymer liquids and networks has a weaker dependence on particle size than that in unentangled networks as entanglements can slide along chains under polymer deformation. The proposed novel hopping model enables understanding the motion of large nanoparticles in polymeric nanocomposites and the transport of nano drug carriers in complex biological gels such as mucus.

A scaling model of self-similar conformations and dynamics of nonconcatenated entangled ring polymers is developed. Topological constraints force these ring polymers into compact conformations with fractal dimension df = 3 that we call fractal loopy globules (FLGs). This result is based on the conjecture that the overlap parameter of subsections of rings on all length scales is the same and equal to the Kavassalis–Noolandi number OKN ≈ 10–20. The dynamics of entangled rings is self-similar and proceeds as loops of increasing sizes are rearranged progressively at their respective diffusion times. The topological constraints associated with smaller rearranged loops affect the dynamics of larger loops through increasing the effective friction coefficient but have no influence on the entanglement tubes confining larger loops. As a result, the tube diameter defined as the average spacing between relevant topological constraints increases with time t, leading to “tube dilation”. Analysis of the primitive paths in molecular dynamics simulations suggests a complete tube dilation with the tube diameter on the order of the time-dependent characteristic loop size. A characteristic loop at time t is defined as a ring section that has diffused a distance equal to its size during time t. We derive dynamic scaling exponents in terms of fractal dimensions of an entangled ring and the underlying primitive path and a parameter characterizing the extent of tube dilation. The results reproduce the predictions of different dynamic models of a single nonconcatenated entangled ring. We demonstrate that traditional generalization of single-ring models to multi-ring dynamics is not self-consistent and develop a FLG model with self-consistent multi-ring dynamics and complete tube dilation. This selfconsistent FLG model predicts that the longest relaxation time of nonconcatenated entangled ring polymers scales with their degree of polymerization N as τrelax ~ N7/3, while the diffusion coefficient of these rings scales as D3d ~ N−5/3. For the entangled solutions and melts of rings, we predict power law stress relaxation function G(t) ~ t−3/7 at t < τrelax without a rubbery plateau and the corresponding viscosity scaling with the degree of polymerization N as η ~ N4/3. These theoretical predictions are in good agreement with recent computer simulations and are consistent with experiments of melts of nonconcatenated entangled rings.

We demonstrate that the origin of the nonlinear elasticity of polymer networks rests in their nonaffine deformations. We introduce the affine length R aff, which separates the solid-like elastic deformations on larger scales from liquid-like nonaffine deformations on smaller scales. This affine length grows with elongation λ as R aff ∼ λ3/2 and decreases upon compression as R aff ∼ λ1/2. The behavior of networks on scales up to R aff is that of stretched or compressed individual chains (we call them affine strands). The affine strands are stretched in the elongation direction and confined and folded in the effective tubes in the compression direction. The fluctuations of affine strands determine the diameters of the confining tubes a, which change nonaffinely with the network deformation a ∼ λ1/2. Our model gives a unified picture of deformations of both phantom and entangled networks and leads to a stress−strain relation that is in excellent agreement with experiments.

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Steric repulsion between grafted side chains inhibits interpenetration of bottlebrushes, transforming them into flexible filaments.

Molecular bottle-brushes are highly branched macromolecules with side chains densely grafted to a long polymer backbone. The brush-like architecture allows focusing of the side-chain tension to the backbone and its amplification from the picoNewton to nanoNewton range. The backbone tension depends on the overall molecular conformation and the surrounding environment. Here we study the relation between the tension and conformation of the molecular brushes in solutions, melts, and on substrates. In solutions, we find that the backbone tension in dense brushes with side chains attached to every backbone monomer is on the order of f 0 N 3/8 in athermal solvents, f 0 N 1/3 in θ-solvents, and f 0 in poor solvents and melts, where N is the degree of polymerization of side chains, f 0 ≃ k B T/b is the maximum tension in side chains, b is the Kuhn length, k B is Boltzmann constant, and T is absolute temperature. Depending on the side chain length and solvent quality, molecular brushes in solutions develop tension on the order of 10-100 picoNewtons, which is sufficient to break hydrogen bonds. Significant amplification of tension occurs upon adsorption of brushes onto a substrate. On a strongly attractive substrate, maximum tension in the brush backbone is ~ f 0 N, reaching values on the order of several nanoNewtons which exceed the strength of a typical covalent bond. At low grafting density and high spreading parameter the cross-sectional profile of adsorbed molecular brush is approximately rectangular with thicknes , where A is the Hamaker constant and S is the spreading parameter. At a very high spreading parameter (S > A), the brush thickness saturates at monolayer ~ b. At a low spreading parameter, the cross-sectional profile of adsorbed molecular brush has triangular tent-like shape. In the cross-over between these two opposite cases, covering a wide range of parameter space, the adsorbed molecular brush consists of two layers. Side chains in the lower layer gain surface energy due to the direct interaction with the substrate, while the second layer spreads on the top of the first layer. Scaling theory predicts that this second layer has a triangular cross-section with width R ~ N 3/5 and height h ~ N 2/5 . Using self-consistent field theory we calculate the cap profile y (x) = h (1 − x 2 /R 2 ) 2 , where x is the transverse distance from the backbone. The predicted cap shape is in excellent agreement with both computer simulation and experiment.