Thermal Motion Effects

In the forgoing we have assumed that the atoms are in fixed positions within the crystal lattice. However, in reality the atoms are not fixed but are in at least an oscillation about their equilibrium positions that depends on temperature. An X-ray or neutron diffraction experiment records a time average of the possible instantaneous atom positions this smears the scattering density about the 

In terms of the Debye Waller factor, B, which is used in some crystallographic computing codes. Equation (21) is  

Notably, the anisotropic thermal displacement factors form the elements of a 3x3 symmetric matrix. The physically meaningful form of this matrix when it is positive-definite is that of an ellipsoidal probability surface centered at the equilibrium atom position. An alternative form for Equation (22) frequently used in crystallography  

---

Application to hexacyanobenzene indicates an improved fit to the 120 K experimental data (Druck and Kotuglu 1984). But interpretation of the results is not straightforward, because such a model does not deconvolute charge density and thermal motions effects, and is not well suited for comparison with theory and derivation of electrostatic properties. 

Johnson CK (1969) Addition of higher cumulants to the crystallographic structure-factor equation a generalized treatment for thermal-motion effects. Acta Crystallogr A 25 187-194... 

While the results of the thermal motion effect are somewhat exaggerated in the top two ellipsoids in Figure 2.4, this thermal motion leads to errors, often in the range of 0.005-0.010 A, with the experimental bond length always being found to be too short. When one reduces the temperature, the thermal ellipsoids become smaller, as shown with the lower set of ellipsoids. Even so, the two atoms are still following curved paths, and the bond length obtained will be too short, but the effect will obviously be much reduced at lower temperatures. 

A molecular dynamics simulation samples the phase space of a molecule (defined by the position of the atoms and their velocities) by integrating Newton s equations of motion. Because MD accounts for thermal motion, the molecules simulated may possess enough thermal energy to overcome potential barriers, which makes the technique suitable in principle for conformational analysis of especially large molecules. In the case of small molecules, other techniques such as systematic, random. Genetic Algorithm-based, or Monte Carlo searches may be better suited for effectively sampling conformational space. 

Effect of Temperature and pH. The temperature dependence of enzymes often follows the rule that a 10°C increase in temperature doubles the activity. However, this is only tme as long as the enzyme is not deactivated by the thermal denaturation characteristic for enzymes and other proteins. The three-dimensional stmcture of an enzyme molecule, which is vital for the activity of the molecule, is governed by many forces and interactions such as hydrogen bonding, hydrophobic interactions, and van der Waals forces. At low temperatures the molecule is constrained by these forces as the temperature increases, the thermal motion of the various regions of the enzyme increases until finally the molecule is no longer able to maintain its stmcture or its activity. Most enzymes have temperature optima between 40 and 60°C. However, thermostable enzymes exist with optima near 100°C. 

The outer layer (beyond the compact layer), referred to as the diffuse layer (or Gouy layer), is a three-dimensional region of scattered ions, which extends from the OHP into the bulk solution. Such an ionic distribution reflects the counterbalance between ordering forces of the electrical field and the disorder caused by a random thermal motion. Based on the equilibrium between these two opposing effects, the concentration of ionic species at a given distance from the surface, C(x), decays exponentially with the ratio between the electro static energy (zF) and the thermal energy (R 7). in accordance with the Boltzmann equation ... 

Solvated ions have a complicated structure. The solvent molecules nearest to the ion form the primary, or nearest, solvation sheath (Fig. 7.2). Owing to the small distances, ion-dipole interaction in this sheath is strong and the sheath is stable. It is unaffected by thermal motion of the ion or solvent molecules, and when an ion moves it carries along its entire primary shell. In the secondary, or farther shells, interactions are weaker one notices an orientation of the solvent molecules under the effect of the ion. The disturbance among the solvent molecules caused by the ions becomes weaker with increasing distance and with increasing temperature. 

Hydrophobic colloidal particles move readily in the liqnid phase under the effect of thermal motion of the solvent molelcnles (in this case the motion is called Brownian) or under the effect of an external electric field. The surfaces of such particles as a rule are charged (for the same reasons for which the snrfaces of larger metal and insnlator particles in contact with a solution are charged). As a result, an EDL is formed and a certain valne of the zeta potential developed. 

Studies of the effect of permeant s size on the translational diffusion in membranes suggest that a free-volume model is appropriate for the description of diffusion processes in the bilayers [93]. The dynamic motion of the chains of the membrane lipids and proteins may result in the formation of transient pockets of free volume or cavities into which a permeant molecule can enter. Diffusion occurs when a permeant jumps from a donor to an acceptor cavity. Results from recent molecular dynamics simulations suggest that the free volume transport mechanism is more likely to be operative in the core of the bilayer [84]. In the more ordered region of the bilayer, a kink shift diffusion mechanism is more likely to occur [84,94]. Kinks may be pictured as dynamic structural defects representing small, mobile free volumes in the hydrocarbon phase of the membrane, i.e., conformational kink g tg ) isomers of the hydrocarbon chains resulting from thermal motion [52] (Fig. 8). Small molecules can enter the small free volumes of the kinks and migrate across the membrane together with the kinks. 

When an electric field is applied, jumps of the ions in the direction of the field are somewhat preferred over those in other directions. This leads to migration. It should be noted that the absolute effect of the field on the ionic motion is small but constant. For example, an external field of 1 V m-1 in water leads to ionic motion with a velocity of the order of 50 nm s 1, while the instantaneous velocity of ions as a result of thermal motion is of the order of 100 ms-1. 

The conductivity of membranes that do not contain dissolved ionophores or lipophilic ions is often affected by cracking and impurities. The value for a completely compact membrane under reproducible conditions excluding these effects varies from 10-8 to 10 10 Q 1 cm-2. The conductivity of these simple unmodified membranes is probably statistical in nature (as a result of thermal motion), due to stochastically formed pores filled with water for an instant and thus accessible for the electrolytes in the solution with which the membrane is in contact. Various active (natural or synthetic) substances... 

The terms involving the subscript j represents the contribution of atom j to the computed structure factor, where nj is the occupancy, fj is the atomic scattering factor, and Ris the coordinate of atom i. In Eq. (13-4) the thermal effects are treated as anisotropic harmonic vibrational motion and U =< U U. > is the mean-square atomic displacement tensor when the thermal motion is treated as isotropic, Eq. (13-4) reduces to ... 

Another molecule for which there seems to be an appreciable conformational difference between the free molecule, as determined by calculation, and the molecule in the crystal phase, but where the effect is somewhat masked by the large thermal motion in the latter, is 2-bromo-l, 1-di-p-tolylethylene (14) (45). 

The application of an electric field E to a conducting material results in an average velocity v of free charge carriers parallel to the field superimposed on their random thermal motion. The motion of charge carriers is retarded by scattering events, for example with acoustic phonons or ionized impurities. From the mean time t between such events, the effective mass m of the relevant charge carrier and the elementary charge e, the velocity v can be calculated ... 

---