Motion, molecular

The application of relaxation time measurements to study segmental motion (in polymers) as well as diffusional chain motion is very well documented but is still a subject of study, particularly using the frequency dependence of relaxation times to test the detailed predictions of models (McBriety and Packer 1993). The anisotropy of reorientation can also be studied conveniently, and recent interest in motion of molecules on surfaces (e.g. water on porous silica) has been investigated with great sueeess (Gladden 1993). Since the dipolar interaction is usually both intermolecular and intramolecular, the relaxation of spin- /2 nuclei (e.g. H) in the same molecule as a quadrupolar nucleus (e.g. H) can permit a complete study of reorientation and translation at a microscopic level (Schmidt-Rohr and Spiess 1994). 

Finally, it should be noted that, although the kinetic theories can explain much observed experimental data, other theories are also under consideration, including some that postulate that crystallisation may take place via metastable phases, i.e. types of unit cell that occur only in the early stages of crystallisation and are subsequently converted into the observed final type of unit cell. These theories may have implications for the xmderstanding of chain folding. 

The description of the structure of polymers given so far in this chapter has concentrated on the spatial arrangements of the molecules. Just as important for the physical properties of the materials are the various motions that the molecules can undergo in later chapters their elfects on the mechanical and dielectric properties are examined in detail. This section provides an overview of the types of motion to be found, with evidence taken mainly from NMR experiments. 

The range of frequencies of the motions observable in solid polymers is extremely wide, from about 10 to lO Hz. The reason for placing the word frequencies between quote marks is that, although the higher frequencies, in the range 10 -10 Hz, correspond to the true vibrational modes of the polymer chains that are studied by means of infrared and Raman spectroscopy, many of the lower frequencies are the frequencies of jumps between different relatively discrete states. 

NMR experiments are capable of probing motions in the range from about 10 to about 10 Hz and the methods employed fall into three principal groups two-dimensional exchange measurements, line-shape measurements and relaxation-rate measurements. The three types of measurement are applicable in the approximate frequency ranges of motion 

10 to 3 X 10, 10 to 3 X 10 and 10 to 10 Hz, respectively. In section 5.7.4 an example of each of these methods is discussed and in section 5.7.5 a brief description of some of the general findings from NMR spectroscopy with relevance to motion in polymers is given. Before these descriptions are given it will be useful to understand the relationship between the relaxation phenomena observed in NMR, described in section 2.7, and those observed in mechanical and electrical measurements. This is described in the next section, which is followed by a description of a simple model for the temperature dependence of the relaxation time observed in mechanical or electrical measurements for a particularly simple type of relaxation. 

Let us next consider how the blob chains move in the semiconcentrated solution. The chain cannot move freely because it is entangled with other chains in the neighborhood. Such a constraint is called a topological constraint, since the force originates in the entanglements and has a more topological nature than a geometrical one. 

When there is a fluctuation in the concentration, the polymer under study tries to move to fill the vacancy in the low-concentration region, but it is impossible for the whole chain to move simultaneously due to the topological constraints. Instead, a blob plays a role of the moving unit. It can diffuse into the neighborhood to restore the concentration back to the average value. This movement can be seen as a diffusion of a rigid sphere of radius f in the solvent, so that the diffusion constant Dc is estimated to be 

Cooperative diffusion can be seen in a different way using the scaling idea. Let us assume that it takes a scaling form 

Since there are n/g blobs in a chain, the total length of the tube is Lx = nf /g. The friction coefficient of the viscous resistance working on a blob is fb = ( 7tr]o, the total friction is given by Hence, the diffusion coefficient A of the reptation is 

The time required for the original tube to disappear by the motion of the blob is the time for the chain to reptate the distance Lx, and hence it is given by xx = Lx/Dx. Sustituting the relation = a ( x / ) and = (r into this equation, we And 

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This section will concentrate on the motions of atoms within molecules— internal molecular motions —as comprehended by the revolutionary quantum ideas of the 20th century. Necessarily, limitations of space prevent many topics from being treated in the detail they deserve. Some of these are treated in more detail in... 

The miderstanding of molecular motions is necessarily based on quaiitum mechanics, the theory of microscopic physical behaviour worked out in the first quarter of the 20th century. This is because molecules are microscopic systems in which it is impossible—or at least very dangerous —to ignore the dual wave-particle nature of matter first recognized in quaiitum theory by Einstein (in the case of classical waves) and de Broglie (in the case of classical particles). 

Classically, the nuclei vibrate in die potential V(R), much like two steel balls coimected by a spring which is stretched or compressed and then allowed to vibrate freely. This vibration along the nuclear coordinated is our first example of internal molecular motion. Most of the rest of this section is concerned with different aspects of molecular vibrations in increasingly complicated sittiations. 

Even with these complications due to anliannonicity, tlie vibrating diatomic molecule is a relatively simple mechanical system. In polyatomics, the problem is fiindamentally more complicated with the presence of more than two atoms. The anliannonicity leads to many extremely interestmg effects in tlie internal molecular motion, including the possibility of chaotic dynamics. 

The view of this author is that knowledge of the internal molecular motions, perhaps as outlined in this chapter, is likely to be important in achieving successfiil control, in approaches that make use of coherent light sources and quantum mechanical coherence. However, at this point, opinions on these issues may not be much more than speculation. 

Knowledge of internal molecular motions became a serious quest with Boyle and Newton, at the very dawn of modem natural science. Flowever, real progress only became possible with the advent of quantum theory in the 20th century. The study of internal molecular motion for most of the century was concerned primarily with molecules near their equilibrium configuration on the PES. This gave an enonnous amount of inunensely valuable infonuation, especially on the stmctural properties of molecules. 

This is a comprehensive survey of algebraic methods for internal molecular motions. 

The axis labels (p,q,r) are chosen In order not to confuse this axis system with other systems, such as the molecule fixed axes (x,y,z) discussed below, used to describe molecular motion. 

This section begins with a brief description of the basic light-molecule interaction. As already indicated, coherent light pulses excite coherent superpositions of molecular eigenstates, known as wavepackets , and we will give a description of their motion, their coherence properties, and their interplay with the light. Then we will turn to linear and nonlinear spectroscopy, and, finally, to a brief account of coherent control of molecular motion. 

Conservation laws at a microscopic level of molecular interactions play an important role. In particular, energy as a conserved variable plays a central role in statistical mechanics. Another important concept for equilibrium systems is the law of detailed balance. Molecular motion can be viewed as a sequence of collisions, each of which is akin to a reaction. Most often it is the momentum, energy and angrilar momentum of each of the constituents that is changed during a collision if the molecular structure is altered, one has a chemical reaction. The law of detailed balance implies that, in equilibrium, the number of each reaction in the forward direction is the same as that in the reverse direction i.e. each microscopic reaction is in equilibrium. This is a consequence of the time reversal syimnetry of mechanics. 

In equilibrium statistical mechanics, one is concerned with the thennodynamic and other macroscopic properties of matter. The aim is to derive these properties from the laws of molecular dynamics and thus create a link between microscopic molecular motion and thennodynamic behaviour. A typical macroscopic system is composed of a large number A of molecules occupying a volume V which is large compared to that occupied by a molecule ... 

Marquardt R and Quack M 1989 Molecular motion under the Influence of a coherent Infrared-laser field infrared Phys. 29 485-501... 

Both infrared and Raman spectroscopy provide infonnation on the vibrational motion of molecules. The teclmiques employed differ, but the underlying molecular motion is the same. A qualitative description of IR and Raman spectroscopies is first presented. Then a slightly more rigorous development will be described. For both IR and Raman spectroscopy, the fiindamental interaction is between a dipole moment and an electromagnetic field. Ultimately, the two... 

Many of the fiindamental physical and chemical processes at surfaces and interfaces occur on extremely fast time scales. For example, atomic and molecular motions take place on time scales as short as 100 fs, while surface electronic states may have lifetimes as short as 10 fs. With the dramatic recent advances in laser tecluiology, however, such time scales have become increasingly accessible. Surface nonlinear optics provides an attractive approach to capture such events directly in the time domain. Some examples of application of the method include probing the dynamics of melting on the time scale of phonon vibrations [82], photoisomerization of molecules [88], molecular dynamics of adsorbates [89, 90], interfacial solvent dynamics [91], transient band-flattening in semiconductors [92] and laser-induced desorption [93]. A review article discussing such time-resolved studies in metals can be found in... 

The principal dilTerence from liquid-state NMR is that the interactions which are averaged by molecular motion on the NMR timescale in liquids lead, because of their anisotropic nature, to much wider lines in solids. Extra infonnation is, in principle, available but is often masked by the lower resolution. Thus, many of the teclmiques developed for liquid-state NMR are not currently feasible in the solid state. Furthemiore, the increased linewidth and the methods used to achieve high resolution put more demands on the spectrometer. Nevertheless, the field of solid-state NMR is advancing rapidly, with a steady stream of new experiments forthcoming. 

We begm tliis section by looking at the Solomon equations, which are the simplest fomuilation of the essential aspects of relaxation as studied by NMR spectroscopy of today. A more general Redfield theory is introduced in the next section, followed by the discussion of the coimections between the relaxation and molecular motions and of physical mechanisms behind the nuclear relaxation. 

ELDOR has been employed to study a number of systems such as inorganic compounds, organic compounds, biologically important compounds and glasses. The potential of ELDOR for studying slow molecular motions has been recognized by Freed and coworkers [29, 30]. 

Quack M 1995 Molecular infrared spectra and molecular motion J. Mol. Struct. 347 245-66... 

In its most fiindamental fonn, quantum molecular dynamics is associated with solving the Sclirodinger equation for molecular motion, whether using a single electronic surface (as in the Bom-Oppenlieimer approximation— section B3.4.2 or with the inclusion of multiple electronic states, which is important when discussing non-adiabatic effects, in which tire electronic state is changed [15,16, YL, 18 and 19]. 

The preceding sections were concerned with the description of molecular motion. An ambitious goal is to proceed further and influence molecular motion. This lofty goal has been at the centrepiece of quantum dynamics in the past decade and is still under intense investigation [182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193 and 194]. Here we will only describe some general concepts and schemes. 

Neuhauser D and Rabitz H 1993 Paradigms and algorithms for controlling molecular motion Acc. Chem. Res. 26 496... 

Gusev A A and Suter U W 1995 Relationship between helium transport and molecular motions in a glassy polycarbonate Macromolecules 28 2582- 4... 

Apart from the sheer complexity of the static stmctures of biomolecules, they are also rather labile. On the one hand this means that especial consideration must be given to the fact (for example in electron microscopy) that samples have to be dried, possibly stained, and then measured in high vacuum, which may introduce artifacts into the observed images [5]. On the other, apart from the vexing question of whether a protein in a crystal has the same stmcture as one freely diffusing in solution, the static stmcture resulting from an x-ray diffraction experiment gives few clues to the molecular motions on which operation of an enzyme depends [6]. 

As a consequence of this observation, the essential dynamics of the molecular process could as well be modelled by probabilities describing mean durations of stay within different conformations of the system. This idea is not new, cf. [10]. Even the phrase essential dynamics has already been coined in [2] it has been chosen for the reformulation of molecular motion in terms of its almost invariant degrees of freedom. But unlike the former approaches, which aim in the same direction, we herein advocate a different line of method we suggest to directly attack the computation of the conformations and their stability time spans, which means some global approach clearly differing from any kind of statistical analysis based on long term trajectories. 

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