Continuous-spin models

Nickel, B.G., Meiron, D.I., and Baker, G.A. (1977). Compilation o/2pt and 4pt graphs for continuous spin models. University of Guelph Report, Canada. 

An even coarser description is attempted in Ginzburg-Landau-type models. These continuum models describe the system configuration in temis of one or several, continuous order parameter fields. These fields are thought to describe the spatial variation of the composition. Similar to spin models, the amphiphilic properties are incorporated into the Flamiltonian by construction. The Flamiltonians are motivated by fiindamental synnnetry and stability criteria and offer a unified view on the general features of self-assembly. The universal, generic behaviour—tlie possible morphologies and effects of fluctuations, for instance—rather than the description of a specific material is the subject of these models. 

Figure 1.40. A schematic of a SRS model with a continuous spin state [166].

Fig. 8 Molecular model of PFPE Zdol (A) SRS model with a discrete spin state (B) bead-spring model, where gray spheres represent the backbone and black spheres represent the polar end-groups and (C) SRS model with a continuous spin state (small end group sphere with the large backbone sphere). (View this art in color at www.dekker. com.)...

To better examine the disjoining pressure driving force behind the nanoscale spreading process, it is more advantageous to adopt a continuous spin state SRS model for an off-lattice scenario. The end-groups in this case are... 

Sarma s model deals with continuous spins of fixed length it is defined by the probability law... 

Lawrence [14, 15] studied the effect of interactions with the environment, using a unitary 3D = 0 model of a particle confined by a delfa-shell potential at r = fl. Once outside the barrier, the particle interacts with N continuous spins. The interaction Hamiltonian is FI = rj r — a) Scontinuous spectrum, Si /u. ) = i diXi), with /r, e [—1,1]. This... 

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

To summarize, the properties of triplet and singlet diradicals are closely related to the effectiveness of through-bond and through-space interactions, which are governed by the orbital phase continuity/discontinuity properties. In the next two sections, we will utilize this simple model to predict the spin preference and intramolecular reactivity for a broad range of diradicals. 

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