Stretching equations

Note the similarity between eqn [8] for the electrostatic blob and eqn [7] for the onset of the chain stretching.) Equation [8] leads to the following relations between the number of monomers in a blob gY, its size dY, and the fraction of charged... 

Equations 11.23 through 11.26 are the counterparts to Eqs. 11.14 through 11.17 of the MED theory. In Eq. 11.23, the CCR term is just Ir S, similar to the CCR term K S - XlA) in the MED theory, but without the transient chain retraction rate A/ /I. (In Eq. 11.23, an absolute value must be taken of the CCR term at S to keep its value positive, while in the MED theory, this term is kept positive through the stretch equation 11.16.) The expression Eq. 11.23 for the orientational relaxation time contains not only the reptation time and the rate of convective constraint release k S, but also the stretch time t. This guarantees that even for velocity gradients greater than 1 /Tj, the rate of orientational relaxation remains bounded by 1 /Tj. This effectively switches off the CCR effect for fast flows, and so functions in much the same way as the switch function/(A) in the MED theory. Hence, no explicit switch function is present in Eq. 11.23. 

Finally, Eq. 11.2 5 is a stretch equation, similar to Eq. 11.16 in the MED theory, except that there is no CCR term in Eq. 11.25. lanniruberto and Marrucci argued that even when constraints on the chain are released rapidly, the chain as a whole cannot respond on a time scale faster than the retraction time of the chain, which is already included in Eq. 11.25. Note that the nonlinearity ofthe spring constant used in Eq. 11.25isgivenby A ((/l) = - l)KA - A)... 

Equation 11.45 is the equation for the orientation tensor S. This equation is similar to that for the Doi-Edwards or DEMG theory see Eq. 11.8. Equation 11.46 is the stretch equation. It is also similar to its counterpart in the DEMG theory see Eq. 11.9. The main difference in the theory for the pom-pom is that the stretch X is limited to be equal to or less than q, the number of arms on each end of the backbone. If the stretch attains the value q, the arms start to be pulled into the backbone tube. The length of arm pulled into the backbone tube is defined by S, which is measured in units of numbers of entanglements. Thus, 1 - c = the fraction of each arm that has been pulled into the backbone tube. The evolution equation... 

Hammen equation A correlation between the structure and reactivity in the side chain derivatives of aromatic compounds. Its derivation follows from many comparisons between rate constants for various reactions and the equilibrium constants for other reactions, or other functions of molecules which can be measured (e g. the i.r. carbonyl group stretching frequency). For example the dissociation constants of a series of para substituted (O2N —, MeO —, Cl —, etc.) benzoic acids correlate with the rate constant k for the alkaline hydrolysis of para substituted benzyl chlorides. If log Kq is plotted against log k, the data fall on a straight line. Similar results are obtained for meta substituted derivatives but not for orthosubstituted derivatives. 

There are a number of relatively simple experiments with soap films that illustrate beautifully some of the implications of the Young-Laplace equation. Two of these have already been mentioned. Neglecting gravitational effects, a film stretched across a frame as in Fig. II-1 will be planar because the pressure is the same as both sides of the film. The experiment depicted in Fig. II-2 illustrates the relation between the pressure inside a spherical soap bubble and its radius of curvature by attaching a manometer, AP could be measured directly. 

The equations of electrocapillarity become complicated in the case of the solid metal-electrolyte interface. The problem is that the work spent in a differential stretching of the interface is not equal to that in forming an infinitesimal amount of new surface, if the surface is under elastic strain. Couchman and co-workers [142, 143] and Mobliner and Beck [144] have, among others, discussed the thermodynamics of the situation, including some of the problems of terminology. 

Figure Al.2.7. Trajectory of two coupled stretches, obtained by integrating Hamilton s equations for motion on a PES for the two modes. The system has stable anhamionic synmretric and antisyimnetric stretch modes, like those illustrated in figrne Al.2.6. In this trajectory, semiclassically there is one quantum of energy in each mode, so the trajectory corresponds to a combination state with quantum numbers nj = [1, 1]. The woven pattern shows that the trajectory is regular rather than chaotic, corresponding to motion in phase space on an invariant torus.

The above expressions are empirical approaches, with m and D. as parameters, for including an anliamionic correction in the RRKM rate constant. The utility of these equations is that they provide an analytic fomi for the anliamionic correction. Clearly, other analytic fomis are possible and may be more appropriate. For example, classical sums of states for Fl-C-C, F1-C=C, and F1-C=C hydrocarbon fragments with Morse stretching and bend-stretch coupling anhamionicity [M ] are fit accurately by the exponential... 

Figure A3.13.9. Probability density of a microcanonical distribution of the CH cliromophore in CHF within the multiplet with cliromophore quantum nmnber V= 6 (A. g = V+ 1 = 7). Representations in configuration space of stretching and bending (Q coordinates (see text following (equation (A3.13.62)1 and figure A3.13.10). Left-hand side typical member of the microcanonical ensemble of the multiplet with V= 6...

For example, the SHAKE algorithm [17] freezes out particular motions, such as bond stretching, using holonomic constraints. One of the differences between SHAKE and the present approach is that in SHAKE we have to know in advance the identity of the fast modes. No such restriction is imposed in the present investigation. Another related algorithm is the Backward Euler approach [18], in which a Langevin equation is solved and the slow modes are constantly cooled down. However, the Backward Euler scheme employs an initial value solver of the differential equation and therefore the increase in step size is limited. 

Molecu lar mechari ical force fields use the equation s of classical mech an ics to describe th e poteri tial energy surfaces and physical properties of m olecii Ies. A molecu le is described as a collection of atom slhal in teracl with each other by sim pic an alytical fiiriclions. I h is description is called a force field. One component of a force field is th e eri ergy arisiri g from com pression and stretch in g a bond. 

In sid e ih e poiii I of in flection of equation (3 1 ) eq nation (32) is identical to MM2 with the cubic stretch tenn turned on. At very long bond distances, it is identical to MM2 with the cubic stretch term turned off. 

Th c fun ction al form for bon d stretch in g in HlOa, as in CHARMM, is quadratic only and is identical to that shown in equation (1 1) on page I 75. Th e bond stretch in g force con stan ts are in units of... 

The stretching of the two bonds adjoining an angle could be modelled using an equation o the following form (as in MM2, MM3 and MM4) ... 

There are forces other than bond stretching forces acting within a typical polyatomic molecule. They include bending forces and interatomic repulsions. Each force adds a dimension to the space. Although the concept of a surface in a many-dimensional space is rather abstract, its application is simple. Each dimension has a potential energy equation that can be solved easily and rapidly by computer. The sum of potential energies from all sources within the molecule is the potential energy of the molecule relative to some arbitrary reference point. A... 

This Schrodinger equation forms the basis for our thinking about bond stretching and angle bending vibrations as well as collective phonon motions in solids... 

Bond stretching is most often described by a harmonic oscillator equation. It is sometimes described by a Morse potential. In rare cases, bond stretching will be described by a Leonard-Jones or quartic potential. Cubic equations have been used for describing bond stretching, but suffer from becoming completely repulsive once the bond has been stretched past a certain point. 

When the friction coefficient is set to zero, HyperChem performs regular molecular dynamics, and one should use a time step that is appropriate for a molecular dynamics run. With larger values of the friction coefficient, larger time steps can be used. This is because the solution to the Langevin equation in effect separates the motions of the atoms into two time scales the short-time (fast) motions, like bond stretches, which are approximated, and longtime (slow) motions, such as torsional motions, which are accurately evaluated. As one increases the friction coefficient, the short-time motions become more approximate, and thus it is less important to have a small timestep. 

The functional form for bond stretching in AMBER is quadratic only and is identical to that shown in equation (11) on page 175. The bond stretching force constants are in units of kcal/mol per A and are in the file pointed to by the QuadraticStretch entry for the parameter set in the Registry or the chem.ini file, usually called =>istr.txt(dbf). 

Equation (3.16) shows that the force required to stretch a sample can be broken into two contributions one that measures how the enthalpy of the sample changes with elongation and one which measures the same effect on entropy. The pressure of a system also reflects two parallel contributions, except that the coefficients are associated with volume changes. It will help to pursue the analogy with a gas a bit further. The internal energy of an ideal gas is independent of volume The molecules are noninteracting so it makes no difference how far apart they are. Therefore, for an ideal gas (3U/3V)j = 0 and the thermodynamic equation of state becomes... 

We might be tempted to equate the forces given by Eqs. (9.61) and (3.38) and solve for a from the resulting expression. However, Eq. (3.38) is not suitable for the present problem, since it was derived for a cross-linked polymer stretched in one direction with no volume change. We are concerned with a single, un-cross-linked molecule whose volume changes in a spherically symmetrical way. The precursor to Eq. (3.36) in a more general derivation than that presented in Chap. 3 is... 

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