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Rubbers are highly elastic solids formed by the random permanent linkages between long chain polymers. Their elasticity is largely entropic in origin, so that the molecular theory is largely a matter of counting the configurations available to the polymers and how this number changes under deformation. To d o this one needs to know the way in which the crosslinks are formed, which need not be related to any equilibrium process, and to know the restriction placed on the chains by one another's presence, i.e. their entanglements.Recent theoretical developments suggest that the problem is by no means as intractable as it might at first sight appear, and simple models of tube confinement seem to give very good agreement with experiment.

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Reinforced rubber allows the production of passenger car tires with improved rolling resistance and wet grip. This book provides in-depth coverage of the physics behind elastomer reinforcement, with a particular focus on the modification of polymer properties using active fillers such as carbon black and silica. The authors build a firm theoretical base through a detailed discussion of the physics of polymer chains and matrices before moving on to describe reinforcing fillers and their applications in the improvement of the mechanical properties of high-performance rubber materials. Reinforcement is explored on all relevant length scales, from molecular to macroscopic, using a variety of methods ranging from statistical physics and computer simulations to experimental techniques. Presenting numerous technological applications of reinforcement in rubber such as tire tread compounds, this book is ideal for academic researchers and professionals working in polymer science.

A rigorous molecular statistical model of filled polymer networks with quenched structural disorder coming from the conventional chemical cross-links between polymers and from an ensemble of rigid and highly dispersed multifunctional filler domains is presented. Specific surface and structure of the filler and its effect of "hydrodynamic" disturbance of intrinsic strain distribution are explicitly taken into account. The conformational constraints (entanglements, packing effects) of polymer chains in the mobile rubber phase are described by a mean-field-like tube model. The calculation of the elastic free energy follows a straightforward generalization of the non-Gibbsian statistical mechanics using the replica technique. Within the model proposed, the network is characterized by four typical length scales: the Kuhn's statistical segment length l, of the polymers; the root-mean-square end-to-end distance of the mobile network chains Rc\ the square root Z>pr of the average area available to a couple site between mobile polymer phase and filler surface; and the lateral tube dimension d0, which is equal to the mean spacing between two successive entanglements in the mobile rubber phase. Application of the theory to stress-strain experiments of unfilled and carbon black filled vulcanízales (styrene-butadiene copolymer rubbers, butadiene rubbers) yields the characteristic length scales Rc, bpR, and d". The typical relations 6?r = l, and l,

The translocation dynamics of a polymer chain through a nanopore in the absence of an external driving force is analyzed by means of scaling arguments, fractional calculus, and computer simulations. The problem at hand is mapped on a one-dimensional anomalous diffusion process in terms of the reaction coordinate s ͑i.e., the translocated number of segments at time t͒ and shown to be governed by a universal exponent ␣ =2/͑2 +2−␥ 1 ͒, where is the Flory exponent and ␥ 1 is the surface exponent. Remarkably, it turns out that the value of ␣ is nearly the same in two and three dimensions. The process is described by a fractional diffusion equation which is solved exactly in the interval 0 Ͻ s Ͻ N with appropriate boundary and initial conditions. The solution gives the probability distribution of translocation times as well as the variation with time of the statistical moments ͗s͑t͒͘ and ͗s 2 ͑t͒͘ − ͗s͑t͒͘ 2 , which provide a full description of the diffusion process. The comparison of the analytic results with data derived from extensive Monte Carlo simulations reveals very good agreement and proves that the diffusion dynamics of unbiased translocation through a nanopore is anomalous in its nature. The dynamics of polymer translocation through a pore has recently received a lot of attention and appears highly relevant in both chemical and biological processes ͓1͔. The theoretical cosideration is usually based on the assumption ͓2-4͔ that the problem can be mapped onto a onedimensional diffusion process. The so-called translocation coordinate ͑i.e., reaction coordinate s͒ is considered as the only relevant dynamic variable. The whole polymer chain of length N is assumed to be in equilibrium with a corresponding free energy F͑s͒ of an entropic nature. The onedimensional ͑1D͒ dynamics of the translocation coordinate then follows the conventional Brownian motion, and the onedimensional Smoluchowski equation ͓5͔ can be used with the free energy F͑s͒ playing the role of an external potential. In the absence of external driving force ͑unbiased translocation͒, the corresponding average first-passage time follows the law ͑N͒ ϰ a 2 N 2 / D, where a is the length of a polymer Kuhn segment and D stands for the proper diffusion coefficient. The question of the choice of the proper diffusion coefficient D, and the nature of the diffusion process, is controversial. Some authors ͓2,3͔ adopt D ϰ N −1 , as for Rouse diffusion, which yields ϰ N 3 as for polymer reptation ͓8͔, albeit the short pore constraint is less severe than that for a tube of length N. In Ref. ͓4͔ it is assumed that D is not the diffusion coefficient of the whole chain but rather that of the monomer just passing through the pore. The unbiased translocation time is then predicted to vary as ϰ N 2 . The latter assumption has been questioned ͓6,7͔ too. Indeed, on the one hand, the mean translocation time scales ͓4͔ as ϳ N 2 , but on the other hand the characteristic Rouse time ͑i.e., the time it takes for a free polymer to diffuse a distance of the order of its gyrati...

-We study the translocation dynamics of a polymer chain threaded through a nanopore by an external force. By means of diverse methods (scaling arguments, fractional calculus and Monte Carlo simulation) we show that the relevant dynamic variable, the translocated number of segments s(t), displays an anomalous diffusive behavior even in the presence of an external force. The anomalous dynamics of the translocation process is governed by the same universal exponent α = 2/(2ν + 2 − γ1), where ν is the Flory exponent and γ1 the surface exponent, which was established recently for the case of non-driven polymer chain threading through a nanopore. A closed analytic expression for the probability distribution function W (s, t), which follows from the relevant fractional Fokker-Planck equation, is derived in terms of the polymer chain length N and the applied drag force f . It is found that the average translocation time scales as τ ∝ f −1 N 2 α −1 . Also the corresponding time-dependent statistical moments, s(t) ∝ t α and s(t) 2 ∝ t 2α reveal unambiguously the anomalous nature of the translocation dynamics and permit direct measurement of α in experiments. These findings are tested and found to be in perfect agreement with extensive Monte Carlo (MC) simulations.