Rigid spherical particles models

A. q and o oq =const rigid spherical particles (Debye model) 83... 

Attempts to formulate a causal description of electron spin have not been completely successful. Two approaches were to model the motion on either a rigid sphere with the Pauli equation [102] as basis, or a point particle using Dirac s equation, which is pursued here no further. The methodology is nevertheless of interest and consistent with the spherical rotation model. The basic problem is to formulate a wave function in polar form E = RetS h as a spinor, by expressing each complex component in spinor form... 

An inevitable side effect of compliant elastomeric particles (or voids) dispersed in rigid polymer matrix is a reduction of the yield stress of the material. As a first approximation the Ishai-Cohen effective area model (Ishai and Cohen 1968), considering a unit cube with a spherical particle of radius R at its center, can be used for estimation of the reduction of the yield stress ... 

Now we employ this model to compute the membrane deformation due to contact with a spherical particle considering a predefined contact angle 6. Based on this deformation, we can compute the unknown parameters discussed in Section 5.5, and evaluate the equilibrium forces. In the following example, we consider a square membrane of dimensions 5L X 5L, in contact with a rigid sphere of radius = L fsee Figure 5.9). Ihe membrane is deformed under the distributed contact line load defined in Equation (5.32), applied along the contact line. Due to the symmetry of the system, it is enough to run the computations for one quarter of the system, after... 

Let us consider systems which consist of a mixture of spherical atoms and rigid rotators, i.e., linear N2 molecules and spherical Ar atoms. We denote the position (in D dimensions) and momentum of the (point) particles i with mass m (modeling an Ar atom) by r, and p, and the center-of-mass position and momentum of the linear molecule / with mass M and moment of inertia I (modeling the N2 molecule) by R/ and P/, the normalized director of the linear molecule by n/, and the angular momentum by L/. 

The simulations to investigate electro-osmosis were carried out using the molecular dynamics method of Murad and Powles [22] described earher. For nonionic polar fluids the solvent molecule was modeled as a rigid homo-nuclear diatomic with charges q and —q on the two active LJ sites. The solute molecules were modeled as spherical LJ particles [26], as were the molecules that constituted the single molecular layer membrane. The effect of uniform external fields with directions either perpendicular to the membrane or along the diagonal direction (i.e. Ex = Ey = E ) was monitored. The simulation system is shown in Fig. 2. The density profiles, mean squared displacement, and movement of the solvent molecules across the membrane were examined, with and without an external held, to establish whether electro-osmosis can take place in polar systems. The results clearly estab-hshed that electro-osmosis can indeed take place in such solutions. 

Fig. 3 A comparison of different coarse grain lipid models. The Shelley model " of DMPC, and Marrink and Essex models of DPPC are compared to their atomistic equivalents (for ease of comparison, hydrogen atoms of the atomistic models are not shown). Solid lines represent harmonic bonds connecting CG particles, and the CG particle types for the Shelley and Marrink models are labelled (the labels are the same as those used in the main text). The point charges (represented by + and —) and point dipoles (represented by arrows) are shown for the Essex model (the charges and dipoles are located at the centre of their associated CG particle). The Shelley and Marrink models use LJ particles (represented by spheres), while the Essex model uses a combination of LJ particles (spheres) and Gay-Berne particles (ellipsoids). Finally, the blob model proposed by Chao et al is also shown for comparison. This model represents groups of atoms as rigid non-spherical blobs that use interaction potentials based on multipole expansions.

We start with a model of polar molecules in which the effects of polarizability are neglected. More precisely, we assume that in the absence of external fields, the potential energy associated with N particles is a sum of pair potentials < >( /), each of which depends on the positions r, and tj and orientations S2, and itj of particles / and j. Thus the particles are regarded as rigid, with no internal coordinates, and we assume for simplicity that they are all identical. Extensions of the results of Section II to mixtures are for the most part straightforward, as discussed by Hoye and StelP and in references they cite. Pertinent references to the mean spherical approximation generalized to mixtures are also given at an appropriate point in this chapter. 

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