Measuring virial coefficients

The virial coefficients Bk play a central role in the theory of dilute polymer solutions. But to measure them it is not necessary (or even desirable) to simulate a many-chain system rather, it suffices to simulate k independent polymer chains and then measure a suitable overlap observable. Consider, 

So we can run in parallel two independent simulations (using for example the pivot algorithm), and then every once in a while measure the observable 

Fortunately, there exist Monte Carlo algorithms which can produce an unbiased estimate of T uP uP ), with statistical errors comparable to or smaller than those already intrinsic in the observable r(o a ( ), in a mean CPU time of order N. So the idea is to perform a Monte Carlo within a Monte Carlo . At least two such algorithms are known the hit-or-miss algorithm, and the Karp-Luby algorithm.See Ref. 96, Section 5.3 for a preliminary discussion, and Ref. 39 for a fuller account. 

The hit-or-miss algorithm can easily be generalized to compute higher virial coefficients. I do not know whether the Karp-Luby algorithm can be generalized in this way. 

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The third virial coefficient C(7) depends upon tliree-body interactions, both additive and non-additive. The relationship is well understood [106. 107. 111]. If the pair potential is known precisely, then C(7) ought to serve as a good probe of the non-additive, tliree-body interaction energy. The importance of the non-additive contribution has been confimied by C(7) measurements. Unfortunately, large experimental uncertainties in C (7) have precluded unequivocal tests of details of the non-additive, tliree-body interaction. 

Equation (8.97) shows that the second virial coefficient is a measure of the excluded volume of the solute according to the model we have considered. From the assumption that solute molecules come into surface contact in defining the excluded volume, it is apparent that this concept is easier to apply to, say, compact protein molecules in which hydrogen bonding and disulfide bridges maintain the tertiary structure (see Sec. 1.4) than to random coils. We shall return to the latter presently, but for now let us consider the application of Eq. (8.97) to a globular protein. This is the objective of the following example. 

The parameter a which we introduced in Sec. 1.11 to measure the expansion which arises from solvent being imbibed into the coil domain can also be used to describe the second virial coefficient and excluded volume. We shall see in Sec. 9.7 that the difference 1/2 - x is proportional to. When the fully... 

Krigbaumf measured the second virial coefficient of polystyrene in cyclohexane at several different temperatures. The observed values of B as well as some pertinent volumes at those temperatures are listed below ... 

Here eK, gk are the force constants for the pure solute K, which can be determined from measurements of its second virial coefficient, and q, oq are similar, but as yet unknown, constants characteristic for the / -hydroquinone lattice. 

Equation 8 may be fitted to those results just described for which the vapor pressure of the pure solid is known. We show graphically the second virial coefficients derived from such fitting and those derived from conventional p-V-T measurements. 

Figure 15 shows the second virial coefficients derived independently by us and by Reuss and Beenakker66 from the measurements of solubility by Dokoupil, van Soest, and Swenker,18 and the coefficients at room temperature from the conventional measurements of Verschoyle,86 Michels and Wassenaar,49 and Michels and Boer-boom.47 These results are sufficient to give unambiguously the parameters of a 12-6 potential... 

Figure 17 shows two sets of virial coefficients—derived by us and by Reuss and Beenakker56 from the measurements of Dokoupil,... 

The virial coefficients at 190°K have been calculated from Ewald s results22 and may be combined with the measurements at room temperature of Michels and Boerboom,47 of Cottrell and his colleagues,11 12 and of Harper and Miller,32 to give the parameters for helium + carbon dioxide... 

The second virial coefficients at 155°K are, respectively, +15, —9, and —19 cm3/mole for these three systems.22 There are no measurements at room temperature. 

Intrinsic viscosity measurements revealed a conformational transition upon heating from 26 to 40 °C, while the UV absorbance of the solution was insensitive to the change. The entropy parameters for PA were also discussed in light of the Flory-Krigbaum correlation between the second virial coefficient and theta temper-... 

The same measurements also provide values of the second virial coefficient, which corresponds to the repulsive energy between micelles. The coefficient of the Na methyl a-sulfomyristate decreases from 7.30 x 10 3 to 3.05 x 10"4 ml/ g with increasing concentration of the electrolyte. The second virial coefficient of the calcium salt is small and changes to a negative value in 0.01 N Ca(N03)2. 

The second virial coefficient B in Eq. 17 refers to the static case. In the ultracentrifuge the measured value can show a speed dependence [39], an effect which can be minimized by using low speeds and short solution columns. If present it will not affect the value of after extrapolation to zero concentration. 

Theta temperature (Flory temperature or ideal temperature) is the temperature at which, for a given polymer-solvent pair, the polymer exists in its unperturbed dimensions. The theta temperature, , can be determined by colligative property measurements, by determining the second virial coefficient. At theta temperature the second virial coefficient becomes zero. More rapid methods use turbidity and cloud point temperature measurements. In this method, the linearity of the reciprocal cloud point temperature (l/Tcp) against the logarithm of the polymer volume fraction (( )) is observed. Extrapolation to log ( ) = 0 gives the reciprocal theta temperature (Guner and Kara 1998). 

Theta temperature is one of the most important thermodynamic parameters of polymer solutions. At theta temperature, the long-range interactions vanish, segmental interactions become more effective and the polymer chains assume their unperturbed dimensions. It can be determined by light scattering and osmotic pressure measurements. These techniques are based on the fact that the second virial coefficient, A2, becomes zero at the theta conditions. 

Virial coefficients (GC) 5 Viscosity detector (SEC) 452 Viscous fingering (SEC) 447 Visualization techniques (TLC) (see derivatization (TLC)] Void volume, column (LC) 371 measurement 372 Volman trap (GC) 211 Volume of a theoretical plate 49... 

Saunders, A.E. and Korgel, B.A. (2004) Second virial coefficient measurements of dilute gold nanocrystal dispersions using small-angle x-ray scattering. Journal of Physical Chemistry B, 108 (43), 16732-16738. 

Special care has to be taken if the polymer is only soluble in a solvent mixture or if a certain property, e.g., a definite value of the second virial coefficient, needs to be adjusted by adding another solvent. In this case the analysis is complicated due to the different refractive indices of the solvent components [32]. In case of a binary solvent mixture we find, that formally Equation (42) is still valid. The refractive index increment needs to be replaced by an increment accounting for a complex formation of the polymer and the solvent mixture, when one of the solvents adsorbs preferentially on the polymer. Instead of measuring the true molar mass Mw the apparent molar mass Mapp is measured. How large the difference is depends on the difference between the refractive index increments ([dn/dc) — (dn/dc)A>0. (dn/dc)fl is the increment determined in the mixed solvents in osmotic equilibrium, while (dn/dc)A0 is determined for infinite dilution of the polymer in solvent A. For clarity we omitted the fixed parameters such as temperature, T, and pressure, p. 

The theoretical foundations of these rules are, however, rather weak the first one is supposed to result from a formula derived by London for dispersion forces between unlike molecules, the validity of which is actually restricted to distances much larger than r the second one would only be true for molecules acting as rigid spheres. Many authors tried to check the validity of the combination rules by measuring the second virial coefficients of mixtures. It seems that within the experimental accuracy (unfortunately not very high) both rules are roughly verified.24... 

The virial coefficient A2, A3, etc. is a measure of the resultant interaction between the Polymer chains,... 

For accuracy in light-scattering measurement the proper choice of solvent is necessary. The difference in refractive index between polymer and solvent should be as large as possible. Moreover, the solvent should itself have relatively low scattering and the polymer-solvent system must not have too high a second virial coefficient as the extrapolation to zero polymer concentration becomes less certain for high A2. Mixed solvent should be avoided unless both components have the same refractive index. 

Archibald (1947) showed that measurement of c and dc/dr at the cell boundaries permit Molecular weight determination at any stage in the equilibrium process. However, when measurements are made early enough, before the molecular species have time to redistribute in the cell, the weight-average Molecular weight and second virial coefficient can be evaluated. In practice, measurements can be made... 

Of the preponderance of small ions, the colligative properties of polyelectrolytes in ionising solvents measure counterion activities rather than Molecular weight. In the presence of added salt, however, correct Molecular weights of polyelectrolytes can be measured by membrane osmometry, since the small ions can move across the membrane. The second virial coefficient differs from that previously defined, since it is determined by both ionic and non-ionic polymer-solvent interactions. 

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