Constraining potential

F.A. Bornemann and Ch. Schiitte. Homogenization of Hamiltonian systems with a strong constraining potential. Physica D, 102 57-77, 1997. 

The advantage of such an optimization scheme is that the SCF iterations do not converge to a non-stationary energy on the unconstrained potential energy surface that may only represent an energy minimum on the constrained potential energy surface, but to a true energy minimum, where the final local spin values—within a certain threshold—may differ from the ideal ones. 

Ubration in a Constraining Potential. In most crystals and many liquids the mean field of force due to the neighbours of a given molecide tends to hold it in a fixed orientation for longer or shorter times. Superposed on this time-smoothed force field is the rapid fluctuation due to the actual molecular movements. Even in a liquid, it may be worthwhile to approximate the force on a molecule as a random force of simple character superposed on a steady orienting field. 

In our work on argon clusters we have adopted a modification of the Lee, Barker, and Abraham S reflecting potential. We assume the clusters to form an ideal polyatomic gas and confine them by a continuous constraining potential of the form... 

In Eq. (3.2) m is the mass of particle i at coordinate r,-, M is the total mass, and Rc is the constraining radius. We have chosen a continuous constraining potential because it is well suited to the Fourier path integral Monte Carlo calculations to be described in the next section. 

In practice T2 is taken to be the temperature of interest and is chosen to be sufficiently high that the clusters behave as an ideal gas of noninteracting particles confined by the constraining potential. Under such high-temperature conditions the system is classical and the partition function takes the form... 

At high temp>eratures only the constraining potential contributes and Eq. (3.19) becomes... 

In both the affine and phantom network models, chains are only aware that they are strands of a network because their ends are constrained by crosslinks. Strand ends are either fixed in space, as in the affine network model, or allowed to fluctuate by a certain amplitude around some fixed position in space, as in the phantom network model. Monomers other than chain ends do not feel any constraining potential in these simple network models. 

The constraining potential represented by virtual chains must be set up so that the fluctuations of junction points are restricted, but the virtual chains must not store any stress. If the number of monomers in each virtual chain is independent of network deformation, these virtual chains would act as real chains and would store elastic energy when the network is deformed. A principal assumption of the constrained-junction model is that the constraining potential acting on junction points changes with network deformation. In the virtual chain representation of this con-... 

The constrained-junction model relies on an additional parameter that determines the strength of the constraining potential, and can be thought of as the ratio of the number of monomers in real network strands and in wirtual chains NjnQ. If this ratio is small, the virtual-chain is relatively long... 

One of the main assumptions of the Edwards tube model is that the number n of monomers in the virtual chains (the strength of the constraining potential) is independent of network deformation. This assumption implies that the amplitude of monomer fluctuations (the tube diameter a) does not change upon network deformation [Eq. (7.61)]. As mentioned in Section 7.3.1. this assumption of deformation-independent... 

Fig 6.5 Representation of the 2D type constraining potential energy left, the potential energy as a function of the polar angle d centre, a representation of the potential energy function as a cut in the Acz-plane right, the three-dimensional representation of the potential in polar coordinates. 

Shading by trcc/shrub canopies reduces soil temperatures relative to those in grassland (Archer, 1995b), thus constraining potential mineralization (Q q effect). 

Among the cited models, the tube model with harmonic constraints seems to be the most natural. Furthermore, it allows an approximate self-consistent calculation of the strength and the deformation dependence of the constraining potential. The following model is used in Refs. 

The strength of the constraints is determined in this model by the prefactors of [R (s) - ft (s)] It is assumed that for the undeformed isotropic system the constraining potential is independent of the direction of the constrained chain. Consequently, the constraining potential has to be diagonal in the main axis system of the deformation tensor in the case of external deformation. 

The simplest case is that of high crosslink density or small coil interpenetration (Np 1). In this case, the restrictions on the configurations of network chains caused by the crosslinks dominate, and the constraints acting on the constraining chains may be omitted in the course of the calculation of the constraining potential. With the assumption of an affine displacement of the crosslink positions with the deformation of the sample, Eq. (12) was obtained with... 

A number of theoretical models use a single-chain approach to simulate topological constraints in real polymer networks. The basic idea is that one starts from the statistical mechanics of a single network chain which is subjected to a spatial domain of constraints. The constraining potential is introduced in a heuristic manner and cannot be calculated within the frame of the chosen model self-consistently. Hence, the strength of the topological interaction must be characterized by best-fit parameters of the model. 

In both cases an expression for the free energy results which is identical with tq. (37) derived by the replica method From the congruent results obtained by different theoretical methods for physically equivalent models, it can be concluded that Eq. (37) is representative for network models with harmonic constraining potentials. Equation (37) will be used extensively in all further discussions concerning the properties of the tube model of polymer networks. 

In summary, the common feature of all constrained chain models is that they impose only limited constraints on chain fluctuations. [101] The constrained-junction fluctuation model restricts fluctuations of junctions and of the center of mass of network chains. The diffused constraint model restricts fluctuations of a single randomly chosen monomer for each network strand. Consequently, all these models can only represent the crossover between the phantom and afflne limits. [101] The phantom limit corresponds to a weak constraining case, while the affine limit corresponds to a very strong constraining potential. 

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