Phase boundary concentration

Chen s analysis is more accurate than the procedure in which the Onsager trial function is used with a(N) given by Eq. (18), but it is very involved to carry through. On the other hand, the Onsager trial function procedure is simple enough for practical purposes. As shown in Appendix A, it predicts the isotropic-nematic phase boundary concentrations that can be favorably compared with those by Chen s procedure. 

The Gaussian trial function for f(a) used by Odijk [6] is mathematically simpler than Onsager s and allows p to be expressed by the leading term of Eq. (22) and ct(N) to be derived analytically. However, it becomes less accurate as the orientation gets weaker. As shown in Appendix A, its use leads to the isotropic-nematic phase boundary concentrations largely different from those by Chen s method and hence is not always relevant for quantitative discussion. 

Now we compare the isotropic-liquid crystal phase boundary concentrations for various polymer solution systems with the scaled particle theory for the wormlike spherocylinder. If the equilibrium orientational distribution function f(a) in the coexisting liquid crystal phase is approximated by the Onsager trial... 

When all lengths associated with polymers are measured in units of the Kuhn statistical segment length 2q, the thermodynamic functions AF, II, and g, given by Eqs. (19)-(21), contain two molecular parameters N = L/2q and d s d/2q and two state variables c = (2q)3 c and a. Thus, numerical solution to Eqs. (23) and (31) provides ci, cA, and a as functions of N and d. The results for the phase boundary concentrations have been found to be represented to a good approximation by the following empirical expressions ... 

Equation (32) has been compared with phase boundary concentration data in the following way. For each solution, N of the polymer sample is estimated from Mw or the viscosity-average molecular weight Mv along with the molecular parameters ML and q listed in Table 1, and d is calculated with d from II or 0II/0c data. For systems which lack these data, the values of d from the (partial) specific volume vsp may be substituted. Table 2 lists the resulting values of d from II, 0II/0c, or vsp for various systems. The phase boundary volume fractions vc v ( = vc v v = I and A) are calculated from experimental phase boundary weight fractions (or mass concentrations) with d, Mw (or Mv), and Ml. Finally, with these numerical results, [vc v/dav(d)] — AV(N, d) is computed... 

Table S. Numerical parameters appearing in Eqs. (32)—(3 5) for the phase boundary concentrations C and cA...

Fig. 7. Comparison of experimental phase boundary concentrations between the isotropic and biphasic regions for various liquid-crystalline polymer solutions with the scaled particle theory for wormlike hard spherocylinders. ( ) schizophyllan water [65] (A) poly y-benzyl L-glutamate) (PBLG)-dimethylformamide (DMF) [66-69] (A) PBLG-m-cresoI [70] ( ) PBLG-dioxane [71] (O) PBLG-methylene chloride [71] (o) po y(n-hexyl isocyanate) (PHICH°Iuene at 10,25,30,40 °C [64] (O) PHIC-dichloromethane (DCM) at 20 °C [64] (5) a po y(yne)-platinum polymer (PYPt)-tuchIoroethane (TCE) [33] ( ) (hydroxypropyl)-cellulose (HPC)-water [34] ( ) HPC-dimethylacetamide (DMAc) [34] (N) (acetoxypropyl) cellulose (APC)-dibutylphthalate (DBP) [35] ( ) cellulose triacetate (CTA)-trifluoroacetic acid [72]...

Fig. 8. Comparison of experimental phase boundary concentrations between the biphasic and liquid crystal regions for various liquid crystalline polymer solutions with the scaled particle theory for hard wormlike spherocylinders. The symbols are the same as those in Fig. 7...

Figure 11 shows the phase boundary concentration data for aqueous Na salt xanthan [78], fd-virus [24], and TMV [23] with added salt. In all these systems, Ci and cA are very low at low added salt concentration Cs or ionic strength I, and increase with Cs or I. Since such low phase boundary concentrations are not usually observed for neutral liquid-crystalline polymer solutions, it is apparent that polyion electrostatic interactions play an important role in the phase equilibria of these systems. 

In order to calculate the phase boundary concentrations for stiff polyelectrolyte solutions, we express the total intermolecular interaction u for the polyion as the sum of the hard-core interaction u0 and the electrostatic interaction wel, and assume Eq. (1) for u0 and the following for wd ... 

Rill and Strzelecka [85-88] studied isotropic-cholesteric phase equilibria in aqueous DNA, with the phase boundary concentration results which are compared with those from the perturbation theory in Table 7. Their experimental... 

Table 7. Comparison of experimental phase boundary concentrations for aqueous DNA (N — 0.49) with the perturbation theory...

In concluding this section, we should touch upon phase boundary concentration data for poly(p-benzamide) dimethylacetamide + 4% LiCl [89], poly(p-phenylene terephthalamide) (PPTA Kevlar)-sulfuric acid [90], and (hydroxy-propyl)cellulose-dichloroacetic acid solutions [91]. Although not included in Figs. 7 and 8, they show appreciable downward deviations from the prediction by the scaled particle theory for the wormlike hard spherocylinder. Arpin and Strazielle [30] found a negative concentration dependence of the reduced viscosity for PPTA in dilute Solution of sulfuric acid, as often reported on polyelectrolyte systems. Therefore, the deviation of the Ci data for PPTA in sulfuric acid from the scaled particle theory may be attributed to the electrostatic interaction. For the other two systems too, the low C] values may be due to the protonation of the polymer, because the solvents of these systems are very polar. 

Fig. 12a-c. Polymer concentration dependence of the orientational order parameters S for three liquid-crystalline polymer systems a PBLG-DMF [92,93] b PHIC-toluene [94] c PYPt-TCE [33], Marks experimental data solid curves, theoretical values calculated from the scaled particle theory. The left end of each curve gives the phase boundary concentration cA... 

The equilibrium value of a in the nematic phase can be determined by minimizing AF. With Eq. (19) for AF from the scaled particle theory, S has been computed as a function of c, and the results are shown by the curves in Fig. 12. Here, the molecular parameters Lc and N were estimated from the viscosity average molecular weight Mv along with ML and q listed in Table 1, and d was chosen to be 1.40 nm (PBLG), 1.15 nm (PHIC), and 1.08 nm (PYPt), as in the comparison of the experimental phase boundary concentrations with the scaled particle theory (cf. Table 2). 

Abe et al. pointed out that the experimental S of PBLG [93] and PYPt [33] solutions at the phase boundary concentration cA largely departed from the prediction of Khokhlov and Semenov s second virial approximation theory [7,44]. Similar deviations of the scaled particle theory from experiment are seen in Fig. 12a,c, where the left ends of the theoretical curves and the experimental data points at the lowest c correspond to cA. Sato et al. [17] showed theoretically that ternary solutions containing two polymer species with different... 

Fig. 22. Plot of In [(H,1 2 — 1)/LjC ] vs L c constructed from the q0 data of aqueous xanthan presented in Fig. 20a. The vertical segments in the upper panel indicate the phase boundary concentration ct between the isotropic and biphasic regions...

This disagreement in D 1 may be attributed to the choice of the strength U (= 2L2dc ) of the molecular field Vscf (a). In fact, the U value of 10.67 was used to calculate the theoretical curves in Fig. 28b, which is the value at the theoretical phase boundary concentration cA, while U at the experimental cA is 24.5. Larson [154] showed that the peak position of the ctn1 curve for U = 12.0 appears at a T larger than that for U = 10.57, but that the peak heights for these... 

Chen [47] calculated the isotropic-nematic phase boundary concentrations using not any trial function but f(a) determined by an numerical-iteration procedure. This method is more rigorous than the one resorting to a trial... 

Fig. Al. Comparison of the reduced phase boundary concentrations (ch cA) obtained by Chen s procedure [47] and the Onsager trial function procedures for f (a)...

Since Chen used the second virial approximation in his phase boundary calculation, it is adequate to compare his results with the phase boundary concentrations calculated from Onsager s expression of S (cf. Sect. 2.5) and cj(N) of Eq. (18) together with the OTF. Chen expressed the phase boundary concentrations in terms of reduced quantities Cj = L2dc j and cA = L2dc A, which depend only on N. As shown in Fig. Al, the relative differences in both Q and cA between the two procedures are less than 13% over the whole range of N. This confirms the relevance of the OTF for semiflexible polymer systems. 

Odijk [6] calculated ct(N) by using a Gaussian trial function for f(a). This trial function, as well as the second virial approximation for use in calculating the phase boundary concentrations, however, leads to c, and cA which differ more than 70% at N l from Chen s calculations. In addition, cj from the Gaussian trial function appears beyond the critical concentration at which the isotropic phase becomes unstable for large N [9]. Therefore, the Gaussian trial function for f(a) is inadequate. 

Rod-like 6, adopting an almost 73 helical conformation, belongs to a unique set of stiff polymers, exhibiting both TchLC and lyotropic liquid crystallinity. Indeed, experiments demonstrated that solutions of 6 became cholesteric at high concentrations [99]. The isotropic-biphasic phase boundary concentration increases as the molecular weight is increased. This increase has been described theoretically using the molecular parameters determined from dilute solution data. 

FIGURE 22.4 Loss of membrane components from a nanopipette-type ISE (tip 0 100 nm half opening angle 5°, height of the IS membrane 100 pm, for other conditions, see main text) (a) Time until the phaseboundary concentration of the lipophilic component drops below 90% of the initial concentration, (b) Change in the normalized phase-boundary concentration of o-NPOE. 

FIGURE 22.8 (a) Cross-section averaged phase-boundary concentrations of and Cl in the nanopore... 

Fig. 9.19 Calculated response functions and associated membrane phase boundary concentration changes of primary ion. The detection limit is attained at a mole fraction of 50 % when half of the primary ions have been displaced by interfering ones of the same charge...

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