Molecular displacements

We tieat this case first, since it is simpler than the trigonal case. The molecular displacements are denoted by x and y (with suitable choice of their origins and of scaling). Then, without loss of generality we can denote the positions of the ci pairs in Cartesian coordinates by... 

Newtonian behavior the rate of shear is small compared to the rate constant for the flow process. When molecular displacements occur very much faster than the rate of shear (7 < kj ), the molecules show maximum efficiency in dissipating the applied forces. When the molecules cannot move fast enough to keep pace with the external forces, they couple with and dissipate those forces to a lesser extent. Thus there is a decrease in viscosity from its upper, Newtonian limit with increasing 7/kj. The rate constant for the flow process is therefore seen to define a standard against which the rate of shear is to be judged large or small. In the next section we shall consider a molecular model in terms of which this rate constant can be analyzed. 

In a liquid that is in thermodynamic equilibrium and which contains only one chemical species, the particles are in translational motion due to thermal agitation. The term for this motion, which can be characterized as a random walk of the particles, is self-diffusion. It can be quantified by observing the molecular displacements of the single particles. The self-diffusion coefficient is introduced by the Einstein relationship... 

Figure 3.1.1, bottom left, illustrates a situation where PFG NMR may provide immediate evidence about the existence and intensity of additional transport resistances on the surface of the individual crystallites, the so-called surface barriers [60, 61]. This option is based on the sensitivity of PFG NMR towards molecular displacements. Molecules traveling over distances exceeding the typical crystallite sizes have to leave the individual crystallites (and are captured by some other crystallite(s) on their further trajectory). This fraction of molecules contributes to the broad part of the propagator. Plotting the relative intensity of the broad part of the propagator as a function of t we thus obtain the relative number y (t) of molecules, which have left their (starting) crystallites at time t. The function y(t) is... 

Owing to its ability to monitor the probability distribution of molecular displacements over microscopic scales from hundreds of nanometers up to several millimeters, PFG NMR is a most versatile technique for probing the internal structure of complex materials. As this probing is based on an analysis of the effect of the structural properties on molecular propagation, the properties of the material studied are those which are mainly of relevance for the transport processes inherent to their technical application. 

The shear work done for one atomic (molecular) displacement, b is the applied force times the displacement, or xb3. This work must equal the promotion energy 2Eg. Therefore, letting b3 equal the molecular volume, Vm, the required shear stress is approximately 2Eg/Vm. The parameter [Eg/Vm] is called the bond modulus. It has the dimensions of stress (energy per unit volume). The numerator is a measure of the resistance of a crystal to kink movement, while the denominator is proportional to the work done by the applied stress when a kink moves one unit distance. Overall, the bond modulus is a measure of the shear strengths of covalent bonds. 

Crystal phase transitions are a possible target with present day computational means, when the transition is a smooth one and does not involve melting of the mother phase and subsequent recrystallization into the daughter phase. For crystalline OL-norleucine, an MD simulation has provided a detailed picture of the mechanism of a solid-solid second-order transition between two polymorphic crystal forms, showing concerted molecular displacements involving entire bilayers [61]. 

In addition, an Ionic or a molecular displacement associated with this equilibrium would explain the observed memory phenomena and the fact that all the devices show only two stable resistive states. 

When pulsed magnetic field gradients are applied to study diffusive processes, the MR technique is often referred to as pulsed field gradient or pulsed gradient spin echo (PGSE) MR. Application of PGSE MR techniques to quantify molecular diffusion was pioneered by Stejskal and Tanner 17,18), and the techniques typically probe molecular displacements of 10 -10 m over time scales of the order 10 M s. 

The flowing units for molecular displacements are larger, than the well known sizes of segments and branches in macromolecules. 

Independent molecular displacements lead simultaneously or consecutively to a system of coupled differential equations characterized by longer relaxation times or by several different relaxation processes. 

The next n flowing units (segments or molecules) will also change their elastic potential due to one single molecular displacement so that the total energy transfer is given by A ei = n A.i ei,. The shift r0 is related to n flowing units and with the law of Hooke we can write... 

According to the assumptions about structure with the superposing of thermal vibrations and the elastic energy A caused by external forces we obtain the well known Eyring differential equations4 for the difference z between the molecular displacements in the direction of the external shearing stress and that of the opposite direction. 

Let us refer to the alternative assumption, that the energy hyperplane will remain unchanged by molecular displacements, which means implying that there must be adjacent positions preserving the structure. We speak of reversible flow processes (meaning relaxation processes). 

The laws of Fick are concerned with the concentration and the change of concentration with respect to space and time. In this paper we are interested in the number of molecular displacements. If we calculate the change of concentration we must use the laws of Fide. The diffusion constant in all cases is given by... 

Fluctuation Rate and Molecular Displacements in the Direction of External Stress... 

In contrast to this the expectation time rex for any molecular displacement contributing to the observed deformation y is ... 

We subject a sample of solid polymer material to a sudden deformation process y0, with an elastical stress o0 = Gy0 (G shear modules). The original strain gives rise to an increase of the molecular valency angles and the intermolecular distances. From these molecular deformations an elastic molecular potential A0 arises which in turn causes molecular displacements. These prevail in the direction of the original strain, decreasing the elastic potential A in that neighborhood. We can calculate... 

The relaxation time Tstress depends on B and differs from 7strain. In first approximation we can calculate a reduction of the stress due to reversible molecular displacements. According to Hooke we have A a = Go y where Ay is given by z/z0. Thus we find... 

In many cases molecular displacements depend on other transition processes occurring previously in the neighborhood. Let us consider Fig. 27. The double kink may move from 1 to l Then segment 2 has the possibility to go from 2 to 2. Flowing units 4,5 and 6 may undergo similar processes. At last the molecule may reach the dashed structure. In every case the final position will be reached after some successive place exchange processes each determining the next step. We therefore get a system of simultaneous differential equations. 

The main effect for coupled molecular displacements is that the relaxation processes are prolonged. 

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