Hard-disk model

Figure 11 Illustration of the nonadditive hard-disk model for lipid mixtures.

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Note that large density fluctuations are suppressed by construction in a random lattice model. In order to include them, one could simply simulate a mixture of hard disks with internal conformational degrees of freedom. Very simple models of this kind, where the conformational degrees of freedom affect only the size or the shape of the disks, have been studied by Fraser et al. [206]. They are found to exhibit a broad spectrum of possible phase transitions. 

TFL is an important sub-discipline of nano tribology. TFL in an ultra-thin clearance exists extensively in micro/nano components, integrated circuit (IC), micro-electromechanical system (MEMS), computer hard disks, etc. The impressive developments of these techniques present a challenge to develop a theory of TFL with an ordered structure at nano scale. In TFL modeling, two factors to be addressed are the microstructure of the fluids and the surface effects due to the very small clearance between two solid walls in relative motion [40]. 

FIG. 4 ir—A isotherms measured for DSPC at water-1,2-DCE (O) and water-air ( ) interfaces from Ref. 41 and simulated with a real gas model [40] ideal gas with A = 0 and Ug = 0 (thin solid line), hard disks gas with A = 40 and ug = 0 (thick solid line), vdW gas with = 40 and Ug/kT = 3 (thin dashed line), and vdW gas with = 40 A and UgjkT = 7 (thick dashed line). The inset shows part of the thick dashed line. (Reproduced from Ref 40 with permission from Elsevier Science.)... 

When applied to spatially extended dynamical systems, the PoUicott-Ruelle resonances give the dispersion relations of the hydrodynamic and kinetic modes of relaxation toward the equilibrium state. This can be illustrated in models of deterministic diffusion such as the multibaker map, the hard-disk Lorentz gas, or the Yukawa-potential Lorentz gas [1, 23]. These systems are spatially periodic. Their time evolution Frobenius-Perron operator... 

In contrast to the discussion above with amorphous barriers, it is possible to use first-principles electron-structure calculations to describe TMR with crystalline tunnel barriers. In the Julliere model the TMR is dependent only on the polarization of the electrodes, and not on the properties of the barrier. In contrast, theoretical work by Butler and coworkers showed that the transmission probability for the tunneling electrons depends on the symmetry of the barrier, which has a dramatic influence on the calculated TMR values [20]. In the case of Fe(100)/Mg0(100)/Fe (100) the majority of electrons in the Fe are spin-up. They are derived from a band of delta-symmetry. In 2004 these theoretical predictions were experimentally confirmed by Parkin et al. and Yusha et al. [21, 22]. Remarkably, by 2005 TMR read heads were introduced into commercial hard disk drives. 

The second contribution spans an even larger range of length and times scales. Two benchmark examples illustrate the design approach polymer electrolyte fuel cells and hard disk drive (HDD) systems. In the current HDDs, the read/write head flies about 6.5 nm above the surface via the air bearing design. Multi-scale modeling tools include quantum mechanical (i.e., density functional theory (DFT)), atomistic (i.e., Monte Carlo (MC) and molecular dynamics (MD)), mesoscopic (i.e., dissipative particle dynamics (DPD) and lattice Boltzmann method (LBM)), and macroscopic (i.e., LBM, computational fluid mechanics, and system optimization) levels. 

Widom [9] realized the importance of this problem for statistical mechanics and showed that the centers of the particles of a hard disk gas, in an equilibrium position, are not uniformly random distributed. The available area for a nevt particle power series in particle density 6 = Nnr2/A, where N is the number of adsorbed panicles, r their radius and A the total area of the surface. The coefficients of the series terms are identical up to the second power of 9 for the equilibrium and the RSA models. The differences in the higher powers coefficients lead for RSA to jamming for Op = 0.76, 0.547 and 0.38 for the ID (segments on a line), 2D (disks on a surface) and 3D (spheres on a volume), respectively, while for the equilibrium configurations the close-packing occurs at 9 = 1, 0.91 and 0.74, respectively. 

The estimation of the jamming coverage for the RSA of monodisperse disks is not an important issue, because its value is already accurately known from Monte Carlo simulations [12], However, it is of interest to develop a procedure that can predict the available area and the jamming coverage for a mixture of disks, for which much Less information is available. Even at equilibrium, for which reasonable accurate equations of state for binary mixtures of hard disks are known for low densities [ 19,20], the available area vanishes only for the unphysical total coverage 9 = 9 +0p = 1 (where the subscripts S and L stand for small and large disk radii, respectively), hence there is no jamming . Exact analytical expressions are known only for the first three virial coefficients of a binary mixture of disks [21], The fourth and fifth coefficients were computed numerically for some diameter ratios and molar fractions for an equilibrium gas [22], However, there are no such calculations for the RSA model. 

The only approximate analytical solution for the RSA of a binary mixture of hard disks was proposed by Talbot and Schaaf [27], Their theory is exact in the limit of vanishing small disks radius rs — 0, but fails when the ratio y = r Jrs of the two kinds of disk radii is less than 3.3 its accuracy for intermediate values is not known. Later, Talbot et al. [28] observed that an approximate expression for the available area derived from the equilibrium Scaled Panicle Theory (SPT) [19] provided a reasonable approximation for the available area for a non-equilibrium RSA model, up to the vicinity of the jamming coverage. While this expression can be used to calculate accurately the initial kinetics of adsorption, it invariably predicts that the abundant particles will be adsorbed on the surface until 6=1, because the Scaled Particle Theory cannot predict jamming. 

H. Meirovitch and H. A. Scheraga,/. Chem. Phys., 84, 6369 (1986). Computer Simulation of the Entropy of Continuum Chain Models The Two-Dimensional Freely Jointed Chain of Hard Disks. 

In a somewhat similar paper, diffusion through a 2D porous solid modeled by a regular array of hard disks was evaluated [65] using non-equilibrium molecular dynamics. It was found that Pick s law is not obeyed in this system unless one takes different diffusion constants for different regions in the flow system. Other non-equilibrium molecular dynamics simulations of diffusion for gases within a membrane have been presented [66]. The membrane was modeled as a randomly... 

The reason why a rigid-wall model is appealing can be seen from Fig. la In order to initiate sliding through the application of a lateral force F, the upper solid has to be lifted up a maximum slope 0. This slope is independent of the normal load L if we assume simple hard-disk interactions that is, as soon as the two solids overlap, the interaction energy increases from zero to infinity. The maximum lateral force Fj between the two solids would then simply be... 

Further results of the overlap model [61] are as follows (iv) The prefactor that determines the strength of the exponential repulsion has no effect on Fj at fixed normal load L, (v) the lateral force scales linearly with L for any fixed lateral displacement between slider and substrate, (vi) allowing for moderate elastic interactions within the bulk does not necessarily increase Fj, because the roughness may decrease as the surfaces become more compliant, and (vii) the prefactor of F for nonidentical commensurate surfaces decreases exponentially with the length of the common period 5 . This last result had already been found by Lee and Rice [62] for a yet different model system. We note that the derivation of properties (iv) and (v) relied strongly on the assumption of exponential repulsion or hard disk interactions. Therefore one must expect charged objects to behave differently concerning these two points. 

Israelachvili considered the possibility of Langmuir monolayers of any sort of amphiphiles forming surface micelles in 1994 [18]. In his model, there is a critical micellar area (CMA or Ac), below which few micelles form and the concentration of the system is nearly equal to the concentration of discrete molecules, but above which, the concentration of micelles increases while the concentration of discrete molecules is constant. Below Ac, the total average area per molecule, A, will be the same as the area per molecule of the discrete molecules, defined A. If Ao is defined as the hard-disk excluded area of a molecule in a micelle, and N is the number of molecules in a micelle, then the n — A isotherm for a system forming surface micelles can be written as... 

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