Correlation volume

The large size of now is responsible for mean-field theory being reliable for large N Invoking the Ginzburg criterion one says mean field theory is self-consistent if the order parameter fluctuation in a correlation volume is much smaller than the order parameter itself. 

Wun and Prins (32) have used a Gaussian form for the correlation function, and then defined a non-random index in terms of a correlation volume rather than in terms of correlation length. 

Debye plots, vs. q, also provide a convenient measurement of a (slope/intercept)Thus, the correlation volume for random two-phase systems is given by... 

K depends on V through N, whereas V depends on K through Scorr The correlation volume ( ycorr = S3COrr) is derived self-consistently. One obtains... 

The above conclusion was verified by checking whether the and a2 values were consistent with the composition of the rubber in the plastic matrix (2). The volume of one scattering particle (t>) is 4/3 rai3. If there are n such scatterers in the correlation volume (V) given by 4/3 ir and if c is the percentage of the rubber filler, then... 

Kirkwood-Buff integrals in aqueous alcohol systems aggregation, correlation volume and local composition. 

Kirkwood—Buff Integrals in Aqueous Alcohol Systems Aggregation, Correlation Volume, and Local Composition... 

The Kirkwood—Buff theory of solution was used to investigate the formation of clusters in aqueous alcohol solutions. The correlation volume (volume in which the composition differs from the bulk one) was calculated for the systems 1-propanol—water and fert-butyl alcohol—water and compared with the sizes of clusters determined by various physical techniques. The calculations indicated that two types of clusters, alcohol- and water-rich clusters, are present in the solutions. Their sizes, which depend on composition in a similar way, exhibit maxima in the water-rich region. The calculated values are in a satisfactory agreement with experiment. The composition inside the clusters (the local composition) was calculated as a function of the correlation volume for dilute aqueous methanol, ethanol, propanols, and terf-butyl alcohol solutions. The results were compared with the local compositions provided by the Wilson and NRTL equations. 

Similar equations can be written for the local composition near a central molecule j. To calculate the correlation volumes with eqs 16 aud 17, it is necessary to express x, x, G , G (, as a function of composition. The values of Gij, G j for the systems iuvestigated were calculated in our previous paper." Forxa andx the Wilson and NRTL equations can be used. Although not always satisfactory, the NRTL expression represents the local composition better than the Wilson equation. The NRTL provides the following expressions for the local compositions ... 

The correlation volumes were calculated using eqs 16 and 17 for the aqueous systems 1-PrOH (T= 303.15 K) and t-BuOH T = 323.15 K). These systems were chosen because reliable data about their clustering are available (Table 1). The values of the KBIs were taken from our previous paper," and the... 

The clustering in aqueous solutions of alcohols was examined by combining the Kirkwood—Buff theory of solution with the Wilson and the NRTL equations. The correlation volumes were calculated for the aqueous systems of 1-PrOH and f-BuOH. Two type of clusters, alcohol- and water-rich, were found with similar dependencies of size on composition. Satisfactory agreement was found between the calculated cluster sizes and those provided by the SAXS, SANS, and LS experiments. 

An analytical expression was derived for the local composition in the dilute region, which was used for the dilute aqueous solutions of MeOH, EtOH, 1-PrOH, 2-PrOH, and f-BuOH. The results were compared with those obtained with the Wilson and NRTL equations and on this basis the correlation volume in the dilute region evaluated. It was found that small clusters (such as dimers and trimers) can be present in the dilute region of alcohols. 

Correlation Volume. We consider a molecule of species i and its surrounding correlation volume in which the composition and/or the structure differ from the bulk one. The total number of molecules i and j in this volume is given by the expressions... 

Inserting eqs 16—17 into eqs 18—19, one obtains at infinite dilution, by assuming that the derivative of the correlation volume with respect to the molar fraction is negligible... 

The calculated correlation volumes and energetic parameters for the alcohol—water and hydrocarbon—water systems are hsted in Tables 4 and 5. The calculated volumes are compared with the sizes of clusters in several alcohol/water systems determined by small-angle X-ray scattering or light scattering " at low concentrations (Table 6). Table 6 shows that there is reasonable agreement between them and the calculated correlation volumes at infinite dilution. 

One can see from Tables 4 and 5 that the correlation volume at infinite dilution increases for both normal hydrocarbons and normal alcohols with the number of carbon atoms. A comparison between the two shows that they are several times larger for hydrocarbons than for the corresponding alcohols, but that the difference between them decreases as the number of carbon atoms increases (Figure 1). 

TABLE 1 Data Used for the Calculation of the Correlation Volumes of Alcohols at Infinite Dilution in Alcohol/Water Solutions... 

TABLE 4 Correlation Volumes and Intermolecular Interaction Energy Parameters in AlcoholAVater Systems at Infinite Dilution"... 

The correlation volume was not accounted in the Butler s scheme. Butler s scheme for dissolution in water accoimted only for the formation of a cavity, introduction of the solute molecule in that cavity, and its interactions with the nearest-neighbor water molecules. He assumed, however, that the water molecules are distributed around a solute molecule as randomly as in its absence. One more step should be added, namely, the formation of a hydrophobic layer of volume around the cavity, in which the water molecules are reorganized and are no longer randomly distributed (Figure 2). While this layer is similar to that suggested by Frank and Evans in their iceberg ... 

TABLE 6 Comparison between the Radius of the Correlation Volume at Infinite Dilution of Alcohol in ... 

The estimation of the thickness of the water layer affected by the presence of a solute molecule was made for two geometries (1) the cavity containing the solute and the correlation volume have the shape of a sphere, (2) both have the shape of a cylinder. The results of the calculations are listed in Table 7, which shows that the water layer is formed of several molecular shells (between 4 and 8, Table 7). The correlation volumes for cyclic hydrocarbon are much lower than for aliphatic hydrocarbons, but, among the cychc hydrocarbons, the... 

Figure 1. Dependence of correlation volume at infinite dilution on the number of hydrocarbon atoms n. Hydrocarbon in water (A) and alcohol in water ( ).

According to the local composition (LC) concept, the composition of solution in the vicinity of any molecule differs from the overall (bulk) composition. For binary mixtures composed of components 1 and 2 with mole fractions x and X2, respectively, four LCs can be considered the local mole fractions of components 1 and 2 around a central molecule 1 (jCn and JC21) and the local mole fractions of components 1 and 2 around a central molecule 2 (JC12 and JC22). In terms of the total number of molecules i and j in the correlation volume, the local mole fractions are given by... 

Equations 19 and 20 coupled with expressions for the local compositions and for the activity coefficients at infinite dilution allowed the evaluation of the correlation volume and the unlike interaction energy parameter at infinite dilution. ... 

The correlation volume and the energy of interaction between two unlike molecules in the systems HFB—B were calculated as for the binary aqueous solutions of alcohols and hydrocarbons, using eqs 19—20 and expressions for the local compositions and activity coefficients at infinite dilution. It should be however emphasized that the calculation procedure is not very accurate when the activity coefficients at infinite dilution are close to unity. For the system HFB (1)—B (2) at 40 °C, =... 

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