Mean-square moment

Most usually, polymer molecules will not be in a single, fixed conformation, and the experimentally observable quantity, the mean-square dipole moment, is an average over many different conformations. At any given instant the 

The mean-square moment for an assembly of such molecules will be defined theoretically by 

Intramolecular forces must dominate the correlation, and this notion provides the basis for a theoretical discussion of g,--values. Intermolecular forces may be taken into account as a secondary perturbation. An important case concerns the arrangement of segmental dipoles, aligned perpendicularly with respect to the chain contour, along a polymeric molecule which has a backbone of carbon atoms. Part of the dipolar correlation is fixed, of course, by the tetrahedral nature of the carbon valence, but part depends on the possible rotation about the C-C bonds of the chain. For completely free rotation it may be shown that gT = 11/12. 

Only rarely is this situation approached in real molecules, and in the re-paraffins, for example, the trans-isomer is about 3 kJ mol 1 lower in energy than the gauche-isomtr, so that at room temperature the /rares-isomer is appreciably favoured, and the gr-value for the CH2 dipoles is reduced. 

In the next member of the series, polyethylene oxide), the dipoles are further apart. An intermediate M1 jn value with negligible temperature dependence is observed, suggesting that the trans- and gawc/ze-isomers have approximately equal energies. At room temperature e f=a 4.5.  

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There are two basic problems associated with the MC or MD calculation of the dielectric constant. First, the relationship between e and the mean square moment obtained in the computer calculation will depend on exactly how the dipolar interactions are handled. For example, the Kirkwood formula (3.7a) only holds if A" (/ ) is that of an infinite sample, and hence does not apply in most computer situations. To find the correct relationship for a given simulation method is not a trivial problem, but for several commonly applied procedures the appropriate formulas are now known. The second, and perhaps more fundamental question, concerns whether the dielectric constant given by a particular simulation is really the true infinite-system value, or whether it is it seriously influenced by the approximate methods used in the calculation. In the absence of exact results, this question is obviously difficult to answer fully, but a detailed and, we hope, useful examination of the problem appears in Section III.D.2. 

The computer simulations employ periodic boundary conditions as well as a spherical cutoff, hence do not exactly correspond to the system just described. Nevertheless, the situations are very similar, and we would not expect the periodicity to influence the formal results. It is clear that for the infinite system with a truncated potential is the mean square moment of the entire sample, or... 

To obtain appropriate expressions relating the dielectric constant to the mean square moment, De Leeuw et al. consider the periodic sphere formed by replicating the central cell to be embedded in an infinite continuum of dielectric constant e. This is reminiscent of techniques first applied by Kirkwood, and the inJ inite limits must be taken properly. One then finds that the effective pair interaction for this system, (e 12), can be written as... 

Fig. 314. Cu(en)3S04 single crystal. T perature dependence of mean molecular susceptibility K shown by full line and that of the effective mean square moment [73M45].

Specific pairwise dipole-dipole interactions can be accounted for by introducing the Kirkwood correlation factor gi, such that the mean square dipole moment is replaced by an effective mean square moment defined by ... 

Evaluating the mean square moment of a sphere (Eq. (31)), and using the definition of the correlation factor above gives the anisotropic version of the Kirkwood-Froh-lich equation ... 

In the collapse phase the monomer density p = N/R is constant (for large N). Thus, the only confonnation dependent tenn in (C2.5.A1) comes from the random two-body tenn. Because this tenn is a linear combination of Gaussian variables we expect that its distribution is also Gaussian and, hence, can be specified by the two moments. Let us calculate the correlation i,) / between the energies and E2 of two confonnations rj ]and ry jof the chain in the collapsed state. The mean square of E is... 

The conformational characteristics of PVF are the subject of several studies (53,65). The rotational isomeric state (RIS) model has been used to calculate mean square end-to-end distance, dipole moments, and conformational entropies. C-nmr chemical shifts are in agreement with these predictions (66). The stiffness parameter (5) has been calculated (67) using the relationship between chain stiffness and cross-sectional area (68). In comparison to polyethylene, PVF has greater chain stiffness which decreases melting entropy, ie, (AS ) = 8.58 J/(molK) [2.05 cal/(molK)] versus... 

Time series plots give a useful overview of the processes studied. However, in order to compare different simulations to one another or to compare the simulation to experimental results it is necessary to calculate average values and measure fluctuations. The most common average is the root-mean-square (rms) average, which is given by the second moment of the distribution. 

In impact limit (k — 0) this difference disappears and t = tj must be the same as in Eq. (1.22). Therefore the relative mean-square change in the moment caused by a collision is... 

In table I we present the molar Kerr constants and mean square dipole moments of three fluorinated polymers, poly (trifluoroethylene) (PFjE), polylvinylidene fluoride) (PVF2) and poly(fluoromethylene) (PFM), dissolved in p-dioxane. The results show the sensitivity of mK to the degree and type of fluorination varying over an order of magnitude and also changing sign. Calculations of mK and for comparison are in progress (5). 

The linear response of a medium to a weak applied electric field is characterized by the dielectric constant <0. From this experiment we deduce the molar polarization mP which is related to the mean square dipole moment by... 

Tables 1, 2, and 3 present a set of five alcohols. In Table 1, it should be noted that while MM3(96) calculates the magnitude of the dipole moment to be essentially the same for the entire set of molecules, MM3(2000) is superior in reproducing the experimental dipole moments. This is demonstrated by comparing the root mean squared deviation of 0.0878 Debye in MM3 to the 0.0524 Debye deviation in MM3(2000). (All of the experimental values except where notes are stark effect measurements determined from microwave spectra.)...

It is known that ab initio methods are not accurate in reproducing or predicting molecular dipole moments. For example, a typical basis set minimization with no additional keywords was carried out, and the results show that the computed magnitude of the dipole moment is not particularly accurate when compared with experimental values. For alcohols, MP2 has a root mean squared deviation of 0.146 Debye, while HF had a deviation of 0.0734 Debye when measured against the experimental values. 

The differences between ab initio and molecular mechanics generated dipole moments were discussed. The MM3(2000) force field is better able to reproduce experimental dipole moments for a set of forty-four molecules with a root mean squared deviation (rmsd) of 0.145 Debye compared with Hartree-Fock (rmsd 0.236 Debye), M0ller-Plesset 2 (rmsd 0.263 Debye) or MM3(96) force field (rmsd 0.164 Debye). The orientation of the dipole moment shows that all methods give comparable angle measurements with only small differences for the most part. This consistency within methods is important information and encouraging since the direction of the dipole moment cannot be measured experimentally. 

From a well-known result of calculus, the definite integral on the right-hand side is s/n so M is just equal to the quantity of diffusing substance. The present solution is therefore applicable to the case where M grams (or moles) per unit surface is deposited on the plane x=x at t=0. In terms of concentration, the initial distribution is an impulse function (point source) centered at x=x which evolves with time towards a gaussian distribution with standard deviation JlQit (Figure 8. 13). Since the standard deviation is the square-root of the second moment, it is often stated that the mean squared distance traveled by the diffusion species is 22t. 

According to the Kirkwood theory of polar dielectrics, simple relations (23) between molecular dipole moment vectors and the mean-square total dipole moment of water clusters can be used to compute the static dielectric constant of water. As the normalized mean-square total dipole moment increases towards unity, theory predicts decreases in the static dielectric constant. Since MD results indicate that the mean-square total dipole moment of interfacial water is greater than that for bulk water (48), the static dielectric... 

If electron clouds of cations and anions are deformable in an electric field and thus have polarizabilities of a+ and a, the value of the dipole moment induced in the cations by the presence of anions (p +) and induced in the anions by the presence of cations (p+ ) are, respectively, proportional to a+/P and a /r. The electric field acting on the cations is proportional to -a+]l L + /r, where 71I+ is the mean square dipole moment of the cations. 

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