Profile hyperbolic-tangent

The simulated free surface of liquid water is relatively stable for several nanoseconds [68-72] because of the strong hydrogen bonds formed by liquid water. The density decrease near the interface is smooth it is possible to describe it by a hyperbolic tangent function [70]. The width of the interface, measured by the distance between the positions where the density equals 90% and 10% of the bulk density, is about 5 A at room temperature [70,71]. The left side of Fig. 3 shows a typical density profile of the free interface for the TIP4P water model [73]. 

Figure 16. Water density profile along the z direction in the membrane/vapor interface (system 11) at X = 4.4, 6.4, 9.6 and 12.8. Hyperbolic tangents have been fitted to determine the interface thickness.

Equation [2.5.251 describes p [z) and on the basis of the present or other models this relation can be made explicit. Differential equations of this type give rise to a hyperbolic tangent-type of P [z] relation. Various elaborations of the present situation, which we shall not discuss here, also give rise to such behaviour for the profile. In our case, the result can be represented as... 

Figure 2.6. Hyperbolic tangent density profile. The zero point of z is placed in the centre of the profile. At that point p (z] = +P )-...

The translational order parameter permits an estimate of the width of the interfaces. The 10-90 width is defined to be the length over which a specific interfacial order parameter changes from 10% to 90% of the bulk solid value. We have estimated the 10-90 widths of the interfaces using a fit by a simple hyperbolic tangent function, used frequently in earlier studies [17]. In the case of the mass-density profile, the translational order parameter may be extracted from a fitting procedure,... 

Recently, Binder et al. [118] considered the Ising lattice of a binary atomic (N=l) mixture confined in a very thin film by antisymmetric surfaces each attracting a different component. It was shown that the segregation of each blend component to opposite surfaces may create antisymmetric (with respect to the center of the film z=D/2) profiles ( >(z) even for temperatures above critical point T>TC, where flat profiles are expected when external interfaces are neglected. Such antisymmetric profiles would not be distinguished in experiments (with limited depth resolution) from coexisting profiles described by a hyperbolic tangent. 

Radial density profiles perpendicular to the fiber axis can be fitted to a hyperbolic tangent function. Equation (1). For fibers with diameters in the range 5-8 nm, the correlation lengths, are about 0.6 nm, which is close to the value obtained with the models of the free-standing thin films. The end beads are enriched in the surface region, as was also the case with the free-standing thin films. The anisotropy of the chord vectors, as assessed by the order parameter, S, is also similar to the result obtained with the free-standing thin films. 

The right-hand side of this equation approximates the curvature-driven motion for the case when the function (f> approximately possesses a hyperbolic tangent profile near the interface in the form... 

For cases when one does not desire the interface to have any curvature-driven motion, the term on the right-hand side is still maintained in order to keep the sharp, hyperbolic tangent profile near the interface however, a so-called counter term is subtracted from the right-hand side in order to cancel the main curvamre-driven flow. The resulting equation then becomes [4]... 

Figure 2.17. The hyperbolic tangent surface density profile.

Mean-field theories of the surface tension of polymer solutions have been developed using the Cahn square gradient approach for interfacial properties of solutions and mixtures both for attractive and for repulsive air/liquid interfaces (Cahn and Hilliard 1958), in a way analogous to the treatment of surface segregation in polymer blends given in section 5.1. For situations in which a surface excess was formed, the volume fraction profile was a hyperbolic cotangent, whereas repulsive profiles were described by hyperbolic tangent functions. Values of the surface tension of semi-dilute solutions of polyst)n ene in toluene (a depletion layer) and polydimethyl siloxane in toluene (an attractive interface, a surface excess formed) were well described by this theory. 

Optimal control of a batch distillation column consists in the determination of the suitable reflux policy with respect to a particular objective function (e.g. profit) and set of constraints. In the purpose of the present work, the optimisation problem is defined with an operating time objective function and purity constraints set on the recovery ratio (90%) and on the propylene glycol final purity (80% molar). Different basis fimctions have been adopted for the control vector parameterisation of the problem piecewise constant and linear, hyperbolic tangent function. Optimal reflux profiles are determined with the final conditions of the previous optimal reactions as initial conditions. The optimal profiles of the resultant distillations are presented on figure 2. 

In a series of papers [7,106,107], we have combined our EoS model with the density gradient approximation of inhomogeneous systans [99-105]. In Refs. [7,106,107], we have addressed in three alternative ways the problon of consistency and equivalence of the various methods of calculating the interfacial tension. In the first case [106], we have simulated the number density profile across the interface with the classical hyperbolic tangent expression [92] (Equation 2.138). In the second case [7], this profile was obtained from the free-energy minimization condition [103,105]. 

This is the historic hyperbolic tangent profile, implicit in van der Waals and explicit in Landau and Lifshitz and in Cahn and Hilliard, " of which wc have already had an intimation in (5.103) and which we shall encounter again in Chapter 9. By the obvious symmetry of (8.50) and (8.51), the profile of the density p in the afi or Py interfooe far from the three-phase line is also given by (8.54), with p or p replaced by p, and with z then the distance from the mid-plane of the aB or ary interface. 

Theoretical considerations for block copolymers lead to a hyperbolic tangent shape for the sigmoidal density profile so that in this case the correcting factor is given by... 

---