Equilibrium values

It was determined, for example, that the surface tension of water relaxes to its equilibrium value with a relaxation time of 0.6 msec [104]. The oscillating jet method has been useful in studying the surface tension of surfactant solutions. Figure 11-21 illustrates the usual observation that at small times the jet appears to have the surface tension of pure water. The slowness in attaining the equilibrium value may partly be due to the times required for surfactant to diffuse to the surface and partly due to chemical rate processes at the interface. See Ref. 105 for similar studies with heptanoic acid and Ref. 106 for some anomalous effects. 

If the spreading is into a limited surface area, as in a laboratory experiment, the film front rather quickly reaches the boundaries of the trough. The film pressure at this stage is low, and the now essentially uniform film more slowly increases in v to the final equilibrium value. The rate of this second-stage process is mainly determined by the rate of release of material from the source, for example a crystal, and the surface concentration F [46]. Franses and co-workers [47] found that the rate of dissolution of hexadecanol particles sprinkled at the water surface controlled the increase in surface pressure here the slight solubility of hexadecanol in the bulk plays a role. 

One can, in fact, drive the piston in both directions from the equilibrium value / = (p = p ) and construct a... 

As t becomes large, this should approach the equilibrium value (fi), which for an ergodic system is... 

Let Ap, Au and AT denote the deviations of the mass density, p, the velocity field, u, and the temperature, T, fiom their fiill equilibrium values. The fluctuating, linearized Navier-Stokes equations are... 

The strong dependence of the PES on molecular orientation also leads to strong coupling between rotational states, and hence rotational excitation/de-excitation in the scattering. This has been observed experimentally for H2 scattering from Cu surfaces. Recent work has shown that for H2 the changes m rotational state occur almost exclusively when the molecular bond is extended, that is, longer than the gas-phase equilibrium value [ ]. 

Figure 3, Wavepacket dynamics of the photodissociation of NOCl, shown as snapshots of the density (wavepacket amplitude squared) at various times, The coordinates, in au, are described in Figure b, and the wavepacket is initially the ground-state vibronic wave function vertically excited onto the 5i state. Increasing corresponds to chlorine dissociation. The density has been integrated over the angular coordinate. The 5i PES is ploted for the geometry, 9 = 127, the ground-state equilibrium value,...

The matrix elements of these derivatives are to be evaluated with R equal to its equilibrium value Rq. However, to keep the notation simple, we shall still write R in place of Rq in later text unless ambiguity may occur. 

Many of the molecular modelling force fields in use today for molecular systems can be interpreted in terms of a relatively simple four-component picture of the intra- and inter-molecular forces within the system. Energetic penalties are associated with the deviation of bonds and angles away from their reference or equilibrium values, there is a function... 

The forces between bonded atoms are very strong and considerable energy is required to cause a bond to deviate significantly from its equilibrium value. This is reflected in the magnitude of the force constants for bond stretching some typical values from the MM2 force field are shown in Table 4.1, where it can be seen that those bonds one would... 

As a simple example of a normal mode calculation consider the linear triatomic system ir Figure 5.16. We shall just consider motion along the long axis of the molecule. The displace ments of the atoms from their equilibrium positions along this axis are denoted by It i assumed that the displacements are small compared with the equilibrium values Iq and th( system obeys Hooke s law with bond force constants k. The potential energy is given by ... 

For non-linear molecules, when treated as rigid (i.e., having fixed bond lengths, usually taken to be the equilibrium values or some vibrationally averaged values), the rotational Hamiltonian can be written in terms of rotation about three axes. If these axes (X,Y,Z) are located at the center of mass of the molecule but fixed in space such that they do not move with the molecule, then the rotational Hamiltonian can be expressed as ... 

You need to specify two parameters the equilibrium value of the internal coordinate and the force constant for the harmonic potential. The equilibrium restraint value depends on the reason you choose a restraint. If, for example, you would like a particular bond length to remain constant during a simulation, then the equilibrium restraint value would probably be the initial length of the bond. If you want to force an internal coordinate to anew value, the equilibrium internal coordinate is the new value. 

A larger value for the bending force constant K0 leads to a greater tendency for the angle to remain at its equilibrium value 0g. There may be cubic, quartic, etc. terms as with the corresponding bond stretch term in addition to the quadratic term shown here. 

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