User:Davidhanson471/sandbox/Rubber Elasticity

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Integrated Rubber Network Models To study the mechanical properties of rubber requires not only chain force-extension models but also a method to account for the geometric effects of the network, specifically, how the chain end-to-end distance changes with a macroscopic tensile strain (the ratio of the increase in length to the original length). Historically, elasticity theories began with the ansatz that a volume element of a rubber network could be represented by a single crosslink node as a connection point for a few chains. Early versions consisted of a crosslink node with 3 or more equal chains having orthogonal end-to-end vectors, oriented symetrically with the strain axis. To relate the network chain extension to the macroscopic strain, it was assumed that the crosslink node coordinates undergo an affine transformation with respect to the applied strain. With these assumptions, formulas could be derived for the macroscopic stress vs. strain. A new theory of rubber elasticity, 'The Molecular Kink Paradigm' has recently been introduced that associates elastic chain forces with molecule-specific physical mechanisms (entropic and enthalpic) that occur as a network chain is put in tension. The theory also includes an explicit polymer network model that captures the complex morphology of a rubber network, including chain rupture and network failure.

The Molecular Kink Paradigm for Rubber Elasticity The Molecular Kink Paradigm proceeds from the intuitive notion that the chains that make up a natural rubber (polyisoprene) network are constrained by surrounding chains to remain within a ‘tube’, and that elastic forces produced in a chain, as a result of some applied strain, are propagated along the chain contour within this tube. Over experimental time scales, only short sections of the chain, comprised of a few backbone units, are free to occupy all allowed rotational conformations given by an equilibrium Boltzmann distribution. Changes in the entropy of a chain are then associated with the thermal motion of short regions that can move more or less freely within the tube. These non-straight regions evoke the concept of ‘kinks’ that in fact manifest the random-walk nature of the chain. As a network is subjected to strain, some kinks are forced into more extended conformations, causing a decrease in entropy that produces an elastic force along the chain. There are three distinct molecular mechanisms that produce these forces, two of which arise from changes in entropy that we shall refer to as low chain extension regime, Ia and moderate chain extension regime, Ib. The third mechanism occurs at high chain extension as, it is extended beyond its initial equilibrium contour length by the distortion of the chemical bonds along its backbone. In this case, the restoring force is enthalpic and we shall refer to it as regime II. The three force mechanisms are found to roughly correspond to the three regions observed in tensile stress vs. strain experiments, shown in Fig. 1. The initial morphology of the network, immediately after chemical crosslinking, is governed by two random processes: (1) the probability for a crosslink to occur at any isoprene unit and, (2) the random walk nature of the chain conformation. The end-to-end distance distribution for a fixed chain length, i.e. fixed number of isoprene units, is described by a Markov walk. It is the joint probability distribution of the network chain lengths and the end-to-end distances between their crosslink nodes that characterizes the network morphology. Because both the molecular physics mechanisms that produce the elastic forces and the complex morphology of the network must be treated simultaneously, simple analytic elasticity models are not possible; an explicit 3-dimensional numerical model is required to simulate the effects of strain on a representative volume element of a network.

Low chain extension regime, Ia At very low strain, the molecular mechanism for elasticity arises from the distortion, or stretching, of kinks along the chain contour. Physically, the applied strain causes the kinks to be forced beyond from their thermal equilibrium end-to-end distances. A force constant for this regime can be estimated by sampling Molecular Dynamics (MD) trajectories of short chains. From these MD trajectories, the probability distributions versus end-to-end distance for short kinks, comprised of 2-4 isoprene units can be obtained. Since the distribution (which turns out to be approximately Gaussian) is directly related to the number of states at each distance, we may associate it with an entropy change of the kink. By numerically differentiating the probability distribution, the change in entropy, and hence free energy, with respect to the kink end-to-end distance can be found.

Moderate chain extension regime, Ib The physical process that gives rise to the elastic force in the moderate chain extension regime, is the gradual straightening of the chain. At full chain extension, (i.e., the onset of regime II), the applied tension forces all of the isoprene units along the chain backbone to lie along piece-wise straight lines by. Numerous experiments strongly suggest that the molecular mechanism responsible for the elastic force must be associated with a change in chain entropy. How does a chain become straight? From Molecular Dynamics simulations of free natural rubber molecules at temperatures near 300 K, we can study the conformations of the chain backbone. We find that departures of the chain backbone from linearity occur over contour lengths of just a few isoprene units (a kink). Although an isoprene unit (Fig. 2) is free to rotate about each of its single C-C bonds, there are typically 3 favored rotational conformations, separated by ~120 degrees, that correspond to energy minima. An isoprene unit has three single C-C bonds and 18 allowed rotational conformations, each one with a unique end-to-end distance and energy. States with shorter end-to-end distances tend to have a higher energy. We designate these as ‘compact’ states and those having greater end-to-end distances as ‘extended’. As a network chain is gradually extended toward linear (but still confined by a surrounding tube), kinks must be straightened. Six of the 18 isoprene rotational conformations are extended states and, as the chain is straightened, more and more of the isoprene units are forced to spend more time these states. It is the decrease in entropy associated with reducing the number of rotational states allowed for each isoprene unit that gives rise to the elastic force in this regime. A force constant for chain extension can be estimated from the change in free energy associated with the entropy change that occurs as the occupancy of some rotational states is decreased.

High chain extension regime, II When a rubber sample is stretched sufficiently far, we know from experience that it breaks more or less cleanly in a plane perpendicular to the strain axis. It follows that covalent bonds on network chains must undergo a bond rupture as a consequence of the imposed strain. Some network chains can rupture before the entire sample completely fails but, as more and more chains break, too few network chains remain intact to support the imposed tensile stress, causing the sample to abruptly fail. The intrinsic molecular mechanisms that give rise to the strong elastic chain force in this region are bond distortions, e.g., bond angle increases, bond stretches and dihedral angle rotations. These forces are spring-like and are not associated with entropy changes. The tensile force along a chain required to cause bond rupture has been calculated from Quantum Chemistry simulations and it is approximately 7 nN, about a factor of a thousand greater than the entropic chain forces at low strain. The angles between adjacent backbone C-C bonds in an isoprene unit vary between about 115-120 degrees and the forces associated with maintaining these angles are quite large, so within each unit, the chain backbone always follows a zigzag path. The same Quantum Chemical simulations also predict that a natural rubber chain can be stretched by about 40% beyond its sensibly straight state 6 (before bond rupture) and also provide a force extension curve that can be used in a numerical network model. The steep upturn in the elastic stress, observed at moderate to high strains Fig. 2, is due network chains being extended beyond their sensibly straight state.

Network morphology The initial morphology of the network is dictated by two random processes: the probability for a crosslink to occur at any isoprene unit and the Markov random walk nature of a chain conformation. The end-to-end distance distribution for a fixed chain length is generated by a Markov sequence. This probability density correlates the chain length, n in units of the Kuhn length b, a statistical de-correlation length along the backbone, to the end-to-end distance, r.

                     P=4π∏r2(1⁄πnb2)exp(-1⁄3r2/2nb2)  Eq. 1     

The probability that any isoprene unit becomes part of a crosslink node is proportional to the ratio of the concentrations of the crosslinker molecules (e.g., dicumyl-peroxide) to the isoprene units.

 			 equation 2 

The factor of two comes about because two isoprene units (one from each chain) participate in the crosslink. The probability for finding a chain containing N isoprene units is proportional to:

     equation 3     

Note that the number of statistically independent backbone segments is not the same as the number of isoprene units. For natural rubber networks, the Kuhn length contains about 2.2 isoprene units so N~ 2.2 n. It is the product of Eq. 1 and Eq. 3 (the joint probability distribution) that relates the network chain lengths and the end-to-end distance between their crosslink nodes.

     equation 4 

The complex morphology of a natural rubber network can be seen in Fig. 3, which shows the probability density vs. end-to-end distance, in units of mean node spacing, for an ‘average’ chain. For the common experimental crosslink density of 4x10¹⁹ cm^-3, an average chain contains about 116 isoprene units (52 Kuhn lengths) and has a contour length of about 50 nm. Fig. 3 shows that a significant fraction of chains span several node spacings, i.e., the chain ends overlap other network chains. As the network is strained, paths composed of these more extended chains emerge that span the entire sample, and it is these paths that carry most of the stress at high strains.

Numerical network simulation model

 To calculate the elastic response of a rubber sample, the three chain force models (regimes Ia, Ib and II) and the network morpholgy must be combined in a micro-mechanical network model.   Using the joint probability distribution in Eq. 4 and the force extension models, it is possible to devise numerical algorithms to both construct a faithful representative volume element of a network and to simulate the resulting mechanical stress as it is subjected to strain. An iterative relaxation algorithm is used to maintain approximate force equilibrium at each network node as strain is imposed.  When the force constant obtained for kinks having 2 or 3 isoprene units is used in numerical simulations, the predicted stress is found to be consistent with experiments.  Such a calculation is shown in Fig. 1 by the solid blue line.  The network simulations shown in Fig. 1 also predict a steep upturn in the stress as network chains are forced into extension regime II, and ultimate material failure due to bond rupture.

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