Langevin networks

We consider instead only an eight-chain model proposed by Arruda and Boyce (1993), in which eight identical chains that are connected at the center of an initial cube radiate out to the eight corners (Fig. 6.6). As with the three-chain model of Wang and Guth (1952), the eight-chain model considers deformation in the principal-axis system of the cube on the basis of the argument that for any other 

The eight-chain-network model is centered aroimd following the entropy of any one of its identical eight chains as the assembly deforms. Thus, the consideration starts with eqs. (6.43) and (6.44) or (6.45), in which rc now represents the end-to-end distance of the chain. In the unstretched network the initial length of the chain is ro = /nl in the cube with initial edge length oq, giving 

In any one of the modes of deformation of the network the cube edge lengths are subject to extension ratios Ai,l2, andl3 in thex, y, andz directions. The length of the generic chain rc becomes 

Insertion of this current chain length into eq. (6.43) gives the work of deformation 

Using the defining expressions of the true stresses acting on the cube faces, 

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We note the structural similarity between eqs. (6.39) and (6.54) for clear reasons. The main difference between them resides in the term of eq. (6.55) that introduces the unique response form of Langevin-network statistics. 

Comparison of the Langevin-network model with experiments... 

Equations (28) and (29) are derived from the statistical theory based on the Gaussian statistics which describes the network behaviour if the network is not deformed beyond the limit of the applicability of the Gaussian approximation33). For long chains, this limit is close to 30 % of the maximum chain extension. For values of r, which are comparable with rmax, the force-strain dependence is usually expressed using the inverse Langevin function 33,34)... 

For a network of Gaussian chains having the same number n of links, uniaxially stretched by an amount L/Lo = A., the assumptions of affine displacement of jimction points and initial Gaussiein distribution of end-to-end vectors allows one to calculate the optical anisotropy of the network by integrating Eq.lO over the distribution of end-to-end vectors in the stretched state. By taking Treloar s expansion [11] for the inverse Langevin function, the orientation distribution function for the network can be put into the form of a power series of the number of Unks per chain ... 

Examples of learning mechanisms are neural networks and ant algorithms. Heuristics remains a reliable approach for the solution of practical instances of hard combinatorial problem such as the vehicle routing problem (Langevin and Riopel 2005). 

In the above, we have considered only static properties at the single-chain level. The dynamics of individual chains exhibit rich behavior that can have important consequences even at the level of bulk solutions and networks. The principal dynamic modes come from the transverse motion. Thus, we consider the transverse equation of motion of the chain that can be fotmd from Hbend above, together with the hydrodynamic drag of the filaments through the solvent. This is done via a Langevin equation describing the net force per unit length on the chain at position x,... 

The transition of bulk network from Gaussian to Langevin statistics accompanied with transfer of the part of a load to inner hard polymer layer at the filler surface (in the range of Mooney-Rivlin curve minimum). 

Fig. 4.7 Tension-elongation curve of a cross-linked rubber. Experimental data (circles), affine network theory by Gaussian chain (broken line), affine network theory (4.107) by Langevin chain (solid line).

The assumption of Gaussian chains in the affine network theory can be removed by using nonlinear chains, such as the RF model (Langevin chain), stiff chain model (KP chain), etc. These models show enhanced stress in the high-stretching region. The effect of nonlinear stretching will be detailed in Section 4.6. 

The elastic free energy of an affine network made up of such Langevin chains is written as... 

Comparison of the Monte-Carlo p r) with those predicted on the basis of the Gaussian distribution function (Equation (3.18) above) and the Langevin function (Equation above) show clear differences. In particular, the molecular structure-based Monte-Carlo p r) reflects clearly the limited extensibility of chains in the true network. 

We hence start with a cubic network formed by beads which have an identical friction constant, f, and which are connected to each other by means of elastic Hookean springs with elasticity constant K, see Fig. 3. The network is embedded into an effective viscous medium and is a regular structure in the sense of connectivity only. Every site of the cubic network is denoted by a three-dimensional index 12 = (a,/3,y). The Langevin equation of motion, Eq. 2, can be rewritten here as ... 

Inserting Eq. 73 into the Langevin equations of motions, Eq. 72, immediately leads to the eigenvalues of the simple cubic network [24,30,62,63] ... 

The dynamics of a topologically square network can be treated along the lines used for the cubic network above. The Langevin equations of motion for a square network are similar to those of the cubic network ... 

In order to obtain the intrachain phase shift as well as the transformation from the Cartesian coordinates to the normal coordinates, seven constants, Ai, A2, A3, Bo, Bi, B2, and Bs, have to be determined, see Eqs. 84 and 85. For these purposes one can use the Langevin equations of motion for the network junctions (cross-links), Eq. 80. Formally we also add the following six conditions at the junction points ... 

It turns out that under these conditions the modes of the network system can be determined analytically. In this case the Langevin equations for the network junctions, Eq. 80, may be represented as a superposition of three equations of motion for chain beads, Eq. 79. Considering Eq. 80 and the boundary conditions for the cross-links points, Eqs. 87 to 92, jointly, leads to the following sets of allowed values for the intrachain phase shift (which, in... 

To summarize, the relaxation times (or eigenvalues) of a rather complex system such as a 3-D topologically-regular network end-Unked from Rouse chains were determined analytically. In fact, one can do even better it is possible to construct all of the eigenfunctions of the network analytically (which amounts to the transformation from Cartesian coordinates to normal coordinates). Briefly, to construct the normal mode transformation, see Eqs. 84 and 85, one has to combine the Langevin equations of motion of a network jimction, Eq. 80, and the boundary conditions in the network junctions, Eqs. 87 to 92. After some algebra one finds [25,66] ... 

Below we consider a (topologically) cubic network (following [31,74] closely). An elementary cubic cell of the network is denoted by 52 = (a, y) (here a, and y range from 1 to N) and it contains s beads, which we number by the index e 1,..., s. The whole network consists then of Nm = sN beads, denoted by ( , 52) = (j, a, p, y). All of the beads (which have identical friction constants f) are connected to their neighbors by means of elastic springs all of which have the same elasticity constant K. The Langevin equation of mo-... 

The matrices B(k) = Bji(k) include all relevant information concerning the intra-cell topology and the way in which the ceUs of the network are connected to each other. All in all there are k-values and therefore different B(k) matrices. Using the B(k) matrices, the Langevin equations, Eq. 2, are reduced to [31,73,74] ... 

It should be emphasized that the Langevin equations in the form given by Eqs. 2 and 146 are not simple to solve because one needs to average over both the stochastic Brownian forces fi(t) and the random part of the locaUza-tion parameter 8qo(R) (the crosslink density SM R)). Such calculations were performed in [137,138] with the use of perturbation theory, taking Sqo(R) as being a small parameter. As an illustration, here we present the final expression for the mean square displacement of a network bead [ 137,138] ... 

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