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Temperature (K)

Temperature (K)

Figure 5. A PS/1-component solvent mixture with two hypercritical points. (a Left) 21.5% PS8300 in cis/trans//1/1-1,4-dimethylcyclohexane(DMCH) exhibiting only THYPL in this experimental range. (b Right) 7% PS575,000 in trans-1,4-DMCH showing both Thypl and PHYPL. Modified from ref. 7 and used with permission.

e. A reduced description of curvature in the (T,P) demixing plane. To our knowledge the examples above constitute the only weakly interacting polymer/solvent systems now known with two hypercritical points (homogeneous double critical points). To facilitate comparisons with other experiments or theory it is useful to employ fitting equations containing the minimum set of parameters. In the present case polynomial expansions are inconvenient because (P,T)CP loci in some regions are double valued. Higher order terms are required and the fits are no longer economical so far as number of parameters is concerned. We therefore elected rotation to a new coordinate system, observing that in the new system the demixing data set is symmetrically disposed about a single extremum. The transformation equations are

T = [T2+ P2]V cos{arctan(P/T) + a) and n = [T2 + P2]V sin{arctan(P/T) + a} (1)

where a is the angle of rotation and the new minimum is selected by a minimization routine. An logarithmic expansion centered at that minimum results in a scaling fit which is characterized by an economy of parametrization.

In Equation 2, A is a width factor, v an exponent, x and % are the transformed (T,P) coordinates, and xmin and %mta are coordinates of the new origin. Least squares parameters for the plots in Figures 4 and 5 are reported in Table 1. Figure 6a is a logarithmic representation of the PS/MCH/HE data. It is linear over~2 V orders of magnitude. Figure 6b compares that least squares fit, now transformed from the (tc,t) coordinate system back to (P,T), with experiment. The quality of fit to the PS/trans -1-4-DMCH data (not shown) is comparable. (For either set of solutions the use of a cubic polynomial in place of Equation 2 fails to represent the data set within experimental error although the number of parameters is the same.) The symmetry exhibited by the (T,P)cri loci after transformation to (n,i) space is a point of special interest. It will be important to determine whether the exponent which describes curvature along (tc,t)cr, v = 0.61±0.04, will carry over to other polymer/solvent systems, and to find whether the parameter a correlates with other thermodynamic properties of solution. It is interesting that the purely empirical scaling exponent describing the (tc,t)cr isopleth is numerically equal to the theoretically established scaling exponent which describes divergences in DLS and SANS correlation length and intensity on critical demixing.

Temperature (K) logrc red

Figure 6. Reduced representation of critical demixing isopleths of high curvature. The PS/MCH/HE system, (a) logarithmic representation in the (k,z) plane. (b) Same as (a) but after transformation back to (P,T) coordinates, see Equations (1) and (2), Figure 4, and text. Modified from ref.4 and used with permission.

Temperature (K) logrc red

Figure 6. Reduced representation of critical demixing isopleths of high curvature. The PS/MCH/HE system, (a) logarithmic representation in the (k,z) plane. (b) Same as (a) but after transformation back to (P,T) coordinates, see Equations (1) and (2), Figure 4, and text. Modified from ref.4 and used with permission.

f. The effect of H/D substitution. Curvature in (P,T,yD) space. We now turn to H/D substitution on solvent or polymer. It is now clear from data on PS/acetone(AC)56

Figure 7. (a) Solvent isotope effects on demixing of PS/propionitrile mixtures, CH3CH2CN/CH3CD2CN. (b) Demixing of PS/acetone mixtures at various MW's and isotope ratios, (CH3)2CO/(CD3)2CO. Notice the increased curvature as the hypercritical Mw is approached. Modified from refs. 6 and 18 and used with permission.

Figure 7. (a) Solvent isotope effects on demixing of PS/propionitrile mixtures, CH3CH2CN/CH3CD2CN. (b) Demixing of PS/acetone mixtures at various MW's and isotope ratios, (CH3)2CO/(CD3)2CO. Notice the increased curvature as the hypercritical Mw is approached. Modified from refs. 6 and 18 and used with permission.

PS/propionitrile(PPN),18 and PS/MCH 16'2%olutions that the demixing isopleths are very sensitive to solvent quality and such sensitivity extends to effects ofH/D substitution. Several examples follow.

Figure 7a shows demixing isopleths for PS/PPN solutions as a function of D substitution on PPN in the methylene position. As yD,™ increases 0.00 to 0.75 (yD,cH2= solvent fraction methylene deuteration, and Zd = polymer fraction deuterated), Phyp l shifts markedly and nonlinearly from its origin at (383K, -1.0MPa, yD =0.00) to (402K, 3.8MPa, yD =0.75). (Although not shown on this figure, it is this yD,cH2=0 solution which was studied under tension to establish continuity of state for demixing at P<0 (Figure 3)). In Figure 7b we turn attention from curvature of the demixing isopleths at selected values of yD, and show curvature in the (T,yD) v cr plane at several Mwfor PS/AC mixtures. The figures are analogous to those described in the (T,P) plane, above, and mean-field or scaling descriptions of isotope effects on the demixing loci follow directly.6

DYNAMIC LIGHT SCATTERLNG (DLS). AND SMALL ANGLE NEUTRON SCATTERING (SANS) NEAR CRITICAL DEMIXING ISOPLETHS.

The development above gives examples of thermodynamic information which can be extracted from studies of the T, P, Mw, yD and Zd dependences of critical demixing. More recently we turned attention from the purely thermodynamic description of LL transitions to studies of the mechanism of LL precipitation employing dynamic light scattering (DLS) and small angle neutron scattering (SANS). It is interesting to determine scattering intensities and correlation lengths near LL transitions, especially in regions of high curvature, because such measurements permit exploration of the proper thermodynamic path to employ in multidimensional scaling descriptions of the approach to criticality.23 The example shown in Figure 8 reports DLS and SANS data for PS30k in MCHh„ (1 1.5 wt%, 12.1 segment%) and MCHd14 (1 1.4 wt%, 13.3 segment%). Although referring to slightly different concentrations the (T,y) curves are flat near the maximum and both solutions are near critical. The data nets are shown in Figure 8a. We chose deuterated solvent to permit direct comparison with SANS data. Deuteration is widely employed in SANS to set the contrast. Figures 8b and 8c show

Lsssw CO.C so»® SsatfSs-an cSs-gram

Lsssw CO.C so»® SsatfSs-an cSs-gram

FSJSNl

P/MPa P/Mpa

Figure 8. (a) DLS and SANS investigations in the (T,P) vcr plane for PS30k/MCHh and PS30k/MCHd solutions. Open and shaded symbols = DLS, Closed and centered symbols = SANS. (b,c) DLS and SANS correlation radii and scattering intensity at several isotherms for the data shown in part (a). Notice the divergences observed along several isotherms as pressure is either raised or lowered toward the critical line. Modified from ref. 16 and used with permission.

P/MPa P/Mpa

Figure 8. (a) DLS and SANS investigations in the (T,P) vcr plane for PS30k/MCHh and PS30k/MCHd solutions. Open and shaded symbols = DLS, Closed and centered symbols = SANS. (b,c) DLS and SANS correlation radii and scattering intensity at several isotherms for the data shown in part (a). Notice the divergences observed along several isotherms as pressure is either raised or lowered toward the critical line. Modified from ref. 16 and used with permission.

DLS and SANS correlation lengths, and scattering intensities, I, for PS30k/MCHd and PS30k/MCHh. Divergences in \ and I are found as the critical line is approached by increasing or decreasing P, or by decreasing T. Similar data is available at other Mw's in MCH and for a variety of other PS/solvent systems.16'22

A scaling representation of DLS and SANS correlation radii. In a review of reentrant LL phase transitions, Narayanan and Kumar12 comment, "...the most compelling issue concerning experimental investigations of reentrant phase transitions (RPT's) is how to recover universal exponents for the RPT. In other words which field variable should be used in place of ?", and elsewhere, "In other words, it is difficult to obtain the correct thermodynamic path in the multidimensional field space." In common with others they demonstrate that a doubling of critical exponents follows from the geometrical model of RPT's for data sufficiently close to the critical line (provided that line be quadratic in the vicinity of the hypercritical turning point). Near Phypl or Phypu it is a properly defined reduced temperature which should demonstrate the doubling, near Thypl or Thypu it is a properly reduced pressure. They remind us that the properly defined scaling variable can nowhere be tangent to the critical curve.

The present data extend over a wide range and in no sense carefully sample the immediate vicinity of the hypercritical points. Further, over the range of these experiments, the critical line is markedly asymmetric for an expansion about Phypl We require a different approach, and have developed an empirical two dimensional scaling formalism to describe scattering data in the (TP plane. Mindful of the admonition that at no point should the relevant field variable be tangent to the critical line we have elected to develop scaling in terms of a variable, Red, defined as the minimum distance from the experimental point of interest to the LL critical line in an appropriately reduced space.

In eq. 3 (texp,pexp) and (Up»)™ are the coordinates of the experimental point of interest and that point on the critical line at minimum distance, respectively, and t# and p# are reduction parameters.

Further discussion is appropriate. Should the field variable lie parallel to t, an appropriate scaling equation for the correlation radius might be ^ = At [ | (t»p -tcr)/(tcr+t#)|]v = At xv where x is a reduced temperature and v a scaling exponent. By including the constant, t#, difficulties are avoided, most obviously for the case when tcr~0. Along the temperature coordinate that issue is often avoided by choosing t(oC) and setting t#=273.2, i.e. assuming that it is T/K which properly scales temperature. In the pressure domain we write, similarly (should the field variable lie parallel to p), i^A^ + ^n^A^, (4)

i.e. after choosing a coordinate set which is orthogonalized by the transformation, the correlation radius is assumed to scale logarithmically. The reduction parameters t# and p# are chosen commensurately in order to insure that unit steps along tred= (texp- tcr)min/(texp+ t#) and pred= (p«p - pcr)min/(pexp + p") are thermodynamically equivalent. This approach avoids the canonization of any point along the critical (T,P) locus as a unique scaling reference (for

Figure 9. Logarithmic scaling representation of DLS and SANS correlation radii for demixing of PS/MCHh and PS/MCHd solutions of several MW's including the data shown in Figure 8. Solid circles = SANS PS30k/MCHd, open circles = DLS PS30k/MCHd, shaded circles = DLS PS30k/MCHh, shaded squares = DLS PS90k/MCHh, shaded triangles = DLS PS400k/MCHh,Rled is defined by Equation (3) and the least squares parameters of fit reported in Table 1. Modified from ref. 22 and used with permission.

Figure 9. Logarithmic scaling representation of DLS and SANS correlation radii for demixing of PS/MCHh and PS/MCHd solutions of several MW's including the data shown in Figure 8. Solid circles = SANS PS30k/MCHd, open circles = DLS PS30k/MCHd, shaded circles = DLS PS30k/MCHh, shaded squares = DLS PS90k/MCHh, shaded triangles = DLS PS400k/MCHh,Rled is defined by Equation (3) and the least squares parameters of fit reported in Table 1. Modified from ref. 22 and used with permission.

example a hypercritical temperature or pressure). As a consequence, in (E^ Rred) space one avoids the doubling ofcritical exponents implied by such selection (see refs. 12, 22 and 23 for further discussion).

The scaling fits. R,ed was calculated from the LL equilibrium lines shown in Figure 8, setting t#=273.2°C as is customary, and selecting p#(MCH)=55MPa by smoothing. The calculation is not sensitive to the choice of p# and a universal value of p# can be employed for all MCH solutions. The scaling fits to DLS and SANS, which include data at all Mw's studied is shown in Figure 9. Least squares regression lines are reported in the caption. The scaling exponents for both DLS and SANS are within experimental error of the theoretical value, v= 0.63 forthree PS/solvent systems (PS/MCH, Figure 9, and PS/CH and PS/AC, not shown).22 Dispersions of the least squares fits lie in the range 0.1 to 0.2 logio(R,ed) units, and for each solvent system the DLS and SANS plots, while parallel, are offset by factors which vary from 3 (in AC) to 20 (MCH). It was this observation which led to to the development of viscosity correctionstothe (^dls)° data. The corrected DLS are now in better agreement with SANS, (i.e. within a factor of~3), and in each case this holds over nearly hundred fold changes in Rred It is remarkable that data in both H and D substituted solvents, and which extend over more than a decade in MW and more than two decades in R„d, lie on common scaling lines even though the ratio of wave lengths ofthe probe radiations is more than 103 (X/X SANS = 1332). We conclude the two techniques are measuring identical or at closely related structural correlations.

Table 1. Least squares parameters for scaling fits of demixing (T,P) isopleths for PS/MCH/HE and PS/trans-1,4-DMCH solutions (see Equations (1) and (2)) or scaling fits of DLS and SANS correlation radii in PS/MCHh and PS/MCHd solutions (see Equations (3) and (4).

Solution

PS/MCh/HE

PS30k in MCHh and MCHd

Property

10-3(polymer Mw)

Cr. Demixing Isopleth

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