5704
Macromolecules 1994,27, 5704-5712
Second Virial Coefficient of Oligo- and Polystyrenes near the 0 Temperature. More on the Coil-to-Globule Transition Hiromi Yamakawa,' Fumiaki Abe, and Yoshiyuki Einaga Department of Polymer Chemistry, Kyoto University, Kyoto 606-01, J a p a n Received May 10, 1994; Revised Manuscript Received July 11, 1994'
ABSTRACT: The secondvirial coefficientA2 was determined for atactic oligo-and polystyrenesin cyclohexane below, at, and above 0 in the range of weight-average molecular weight M, from 3.70 X 102 to 4.00 x 104. It is found that A2 increases significantly with decreasing M , at any temperature T for M , < 5 X 103, while this dependence almost disappears for larger M , below 8, in agreement with the literature results. It is then shown that such molecular-weight dependence of A2 may be explained quantitatively by the Yamakawa theory that takes account of the effect of chain ends. An analysis gives values of the effective excess binarycluster integrals ($ and @Z associated with the chain end beads and also the binary-cluster integral @ between intermediate identical beads as functions of T , all of them except for @ above 0 being quadratic in T = 1BiT. With these values of @,the conventional and scaled excluded-volumeparameters z and i below 0 are of A2 without the effect of chain ends give a single composite recalculated. The results for the part AJHW) is plotted against z , being consistent with the two-parameter theory prediction. curve below 8 when A2(HW)Mw1/2 The deviation of the A&f,1/2vs z plot or the A2 vs 171plot from this prediction previously reported arises not only from the effect of chain ends but also from the previous assumption of the proportionality of p and hence z to T . The present and literature findings of the M , independence of A2 itself (except for small M,) below 0 are due to a cancellation of the M , dependence of A z ( by ~ ~that ) of the effect of chain ends. The composite curve above and also that of the gyration-radius expansion factor as vs i below 0 are found to be close to the respective first-order perturbation theory values. With these results, a supplementary discussion of the coil-to-globuletransition is also given.
Introduction Recently, Yamakawa' has investigated theoretically the second virial coefficient A2 and radius expansion factor as for flexible polymers below the 8 temperature within the new framework of a theory2based on the helical wormlike (HW) hai in.^^^ The main problem of A2 below 8 is to explain its molecular-weight M independence as observed for atactic polystyrene (a-PS),5q6which is inconsistent with the two-parameter theory p r e d i ~ t i o n .According ~ t o the new theory,2the chain stiffness has a significant effect on A2 or the interpenetration function \k appearing in it even for large M and, on the other hand, the effect of chain ends also becomes appreciable for relatively small M . Indeed, it has already been showng.10that the former effect explains well, although not quantitatively, the experimental result that \k is not a universal function of as, while the latter explains satisfactorily the strong M dependence of A2 observed for small M at 8 (at which A2 vanishes for large M ) and also in good solvents. These findings are also inconsistent with the two-parameter theory prediction. As for A2 below 8,from an analysis of the literature data5*''for a-PSby the use of the new theory, it has been suggested that the M independence of A2 may be explained by taking account of the effect of chain ends but not of chain stiffness.l However, an accurate determination of the effect of chain ends on A2 below 0 has been difficult because of the lack of data for A2 for small M. Thus, in the present work, we determine A2 for a-PS in cyclohexane below and also above 8 in the range of small M , including the oligomers. (The problem of A2 at 8 has already been resolved, as mentioned above.9) We evaluate first the contribution of the effect of chain ends to A2 below and above 6 and then that part of Az without this effect which vanishes at 8. The excludedvolume strength B or the binary-cluster integral 0is then
'Abstract published in AdoancP ACS Abstracts, September 1.
1994.
determined as a function of temperature from A2 in the oligomer region in which A2(HW)is independent of M.This enables us to calculate the conventional excluded-volume parameter z and the scaled excluded-volume parameter 2 without any assumption. Thus we obtain values of Az(Hw)W/2 and as as functions of z or Z to compare them with the theory. With these results, a supplementary discussion of the coil-to-globule transition is also given.
Experimental Section Materials. Most of the a-PS samples used in this work are the same as those used in the previous studies of the meansquare optical anisotropy ( r2),12 the intrinsic viscosities [&,13 and [7],14J5 the mean-square radii of gyration (S2),$6and (S*),17 the scattering function P,(k),18the translational diffusion coefLe., the fractions separated by ficient D,19 and A2 (or preparative gel permeation chromatography (GPC)or fractional precipitation from the standard samples supplied by Tosoh Co., Ltd. In this work, however, some additional samples were prepared similarly by separation from the Tosoh standard samples. All the sampleshave a fixed stereochemicalcomposition (the fraction of racemicdiadsf, = 0.59) independent of molecular weight, possessing an n-butyl group at one end of the chain (the initiating end) and a hydrogen atom at the other (the terminating end). The values of the weight-average molecular weight M,, the weight-averagedegree of polymerization n,, and the ratio of M , to the number-average molecular weight M. are listed in Table 1. Samples OS8a, A2500a-2, Fla-2, and F2-2 are the additional ones, and their A44 were determined from light scattering (LS) measurements in cyclohexane at 34.5 O C (8).As seen from the values of M,/M,, all the samples are sufficiently narrow in molecular weight distribution, and, in particular, samples OS3, OS4, and OS5 are completely monodisperse. The solvent cyclohexane was purified according to a standard procedure prior to use. Light Scattering. LS measurements were carried out to determine A2(and alsoM,) for all the a-PS samplesin cyclohexane at various temperatures ranging from 15.0 to 50.0 OC. A Fica 50 light-scattering photometer was used for all the measurements with vertically polarized incident light of wavelength 436 nm. For a calibration of the apparatus,the intensity of light scattered from pure benzene was measured at 25.0 "C at a scattering angle
0024-9297/9412227-~704$04.5010 0 1994 American Chemical Society
A2 of Oligo- and Polystyrenes near 8 5705
Macromolecules, Vol. 27, No. 20, 1994 Table 1. Values of M,, x, and M,/M. for Atactic Oligoand Polystyrenes sample Mw xw MwIMn 3.70 X lo2 3 1.00 OS3a 4 1.00 OS4 4.74 x 102 5.78 X lo2 5 1.00 OS5 5.98 1.00 OS6 6.80 X lo2 8.29 1.01 OS8a 9.20 X lo2 11.3 1.03 A1OOO-ab 1.23x 103 26.7 1.03 A2500a-2 2.83 x 103 51.2 1.03 A5000-3' 5.38 x 103 Fla-2 9.98 x 103 95.4 1.03 194 1.02 F2-2 2.02 x 104 F4b 4.00 x 104 384 1.02 4 M d of OS3 through OS6 had been determined from GPC.12J3 b M,s of A1000-aandF4 had been determined from LS in cyclohexane at 34.5 "C.l53 M , of A5000-3 had been determined from LS in methyl ethyl ketone at 25.0 0C.12
0.0
'
0
I
0.2
I
0.4 L
of 90°, where the Rayleigh ratio RvU(90") of pure benzene was taken as 46.5 X 10-6 cm-l. The depolarization ratio pu of pure benzene at 25.0 "C was determined to be 0.41 h 0.01 by the method of Rubingh and YuS2O The conventionalmethod was used for solutionsof the samples with M , > 103, while the new procedure previously21presented was applied to those of the oligomer samples with M, C lo3 as beforegJOsince then the concentration dependencesof the density scattering Rd and the optical constant K have significant effects on the determination of A2 (and also of M,). To determine A2 by the latter procedure, we measured the reduced total intensity Ruv* of the unpolarized scattered light for vertically polarized incident light, the depolarization ratio pu, the ratio KT/KT,Oof the isothermal compressibility of a given solution to that of the solvent,and the refractive index increment ( a f i / d c ) ~at, ~constant (absolute)temperature Tand pressurep for the oligomer solutions and also the first two quantities for the solvent. The values of the refractive index fi at finite concentrations c, which were required to calculate K , were calculated with the values of (afi/ d c ) ~for , ~each oligomer sample,as describedin the Results section. Measurements of Ru,* were carried out at scattering angles 0 ranging from 45 to 135", and the mean of the values obtained at different 0 was adopted as its value, since it must be independent of 0 for oligomers. The values of pu were obtained by the same method as employed in the calibration of the apparatus. All the LS data obtained were analyzed by using the Berry square-root plot22and also the Bawn plot.23924 The most concentrated solutions of each samplewere prepared by continuous stirring at ca. 50 "C for 1-4 days. They were optically purified by filtration through a Teflon membrane of pore size 0.45 or 0.10 km. The solutions of lower concentrations were obtained by successive dilution. The polymer mass concentrations c were calculated from the weight fractions with the densities of the solutions. The densities of the solvents and solutions were measured with a pycnometer of the LipkinDavison type. Isothermal Compressibility. Isothermal compressibility measurements were carried out to determine KT/KT,Ofor oligomer sample OS4 in cyclohexane at 15.0 and 50.0 "C. The apparatus and the method of measurements are the same as those described in the previous paper.g This ratio was determined as a function of c and p . The latter was varied from 1 to ca. 50 atm. In this range of p , it was independent of p within experimental error, so that we adopted the mean of the values obtained at various pressures as its value at 1 atm. Refractive Index Increment. The refractive index increment (afi/&)~,,,was determined as a function of c and Tfor each oligomer sample for M, C 3 X lo3at temperatures ranging from 15.0to50.0 "C by theuse of aShimadzudifferentialrefractometer.
Results Light Scattering from Oligostyrene Solutions. In this subsection we give the results for KT/KT,O and (aA/ ~ c ) Tand , ~ the LS data for the oligomer samples with M , 5 5.38 x 103. Figure 1shows plots of K T / K T , O against c for
I
I
I
0.6
0.8
1 .o
I
(gicrn')
Figure 1. Plots of KT/KT,O against c for a-PS in cyclohexane: (e) OS4 at 15.0 "C; ( 8 )OS4 at 50.0 "C; ( 0 )OS2 (M= 266) at 34.5 0C;9(0) OS3 at 34.5 0C;9 (pip down) x , = 2-5 in the bulk at 15.0 (q),34.5 (q),and 50.0 "C (9);25 (pip up) M,,= 5.1 X lo4 and 8.2 X lO4in the bulk at 15.0 (d),34.5 (b),and 50.0 "C (6)26 (see text). Table 2. Values of k1 in Equation 2 for Atactic Oligostyrene in Cyclohexane sample kl, cm3k sample kl, cm3k OS3 0.146 OS8a 0.1715 OS4 0.154 A1000-a 0.1765 OS5 0.1575 A2500a-2 0.179 OS6 0.1635 sample OS4 in cyclohexane at 15.0 (top-half-filled circles) and 50.0 "C (bottom-half-filled circles) along with the literature data obtained by Allen et al.25(circles with pip down) for a mixture of styrene oligomers including the dimer through the pentamer in the bulk and by Hocker et a1.26 (circles with pip up) for high-molecular-weight samples with M , = 5.1 X lo4and 8.2 X lo4in the bulk. For these literature data, the values of KT/KT,Ohave been calculated by the use of the values of KT,Ogiven by Holder and WhalleyZ7for pure cyclohexane at the respective temperatures and the densities have been taken as the values of c. The figure also includes the results obtained previouslyg for OS2 ( M = 266) (circles with horizontal bar) and OS3 (circles with vertical bar) in cyclohexane at 34.5 "C and the literature data by the above authors25.26 (unfilled circles) for their respective samples at 34.5 "C. As seen from Figure 1, the plot of KT/KT,Oagainst c at each temperature follows a single straight line irrespective of the difference in molecular weight and may then be represented by the equation KT/KT= ,~1
+ kc
(1)
where the values of k determined from the straight lines indicated are -0.554, -0.591, and -0.630 cm3/g at 15.0,34.5, and 50.0 "C, respectively. It is also seen that for c < 0.3 g/cm3, where LS measurements for the oligomer samples were carried out, the variation of KT/KT,Owith temperature is very small and almost within experimental uncertainty. Thus we have evaluated KT/KT,Oat temperatures ranging from 15.0 to 50.0 "C except at 34.5 "C by interpolation using eq 1 with the values of k above. The results for ( a f i / & ) ~at , ~436 nm for each oligomer sample in cyclohexane are independent of c for c < 0.15 g/cm3 and may be expressed as a function of T by the equation (ait/a~), = k,
+ k , ( -~ e)
(2)
Macromolecules, Vol. 27, No. 20, 1994
5706 Yamakawa et al.
-0
0.2
04
06
I 0 I 0 1 h
-
0 (see text).
Temperature Dependence of Binary-Cluster Integrals. With the values of B, al, and a2 obtained in the last subsection, we have calculated P, 81, and PZ a t the respective temperatures from eqs 6 and 16 with eq 17 by taking the repeat unit of the chain as a single bead (Mo = 104). For the calculation of 0,we have used the values of the HW model parameters determined p r e v i o ~ s l y ~ ~ J ~ from (l?) and (S2)0,i.e., X-~KO= 3.0, X-%O= 6.0, h-l= 20.6 A, and M L = 35.8 A-l. The values of /3 obtained at various temperatures above and below 8are represented by the unfilled circles inFigure 10, where T is defined by
3 h
T=l-e/T
It also includes the values determined by Miyaki and Fujita30 (filled circles) from as. (We note that their values of 0 plotted in this figure are somewhat lower than the original values because of the difference between our and their coil-limiting values of (S2)0/xw.17)For T I 0 (T I e),the present values of 0from Az are in good agreement with theirs from as and are found to be proportional to 7. However, it is important to see that for 7 < 0, the plot deviates significantly downward from a linear extension of the straight line fitted to the data points for 7 I 0 (dashed line), the deviation being larger for lower T. With these results, for later use, we have constructed an empirical equation for 6 (in A3) as a function of 7 as follows:
0 = 657 = 657 - 6107'
3
(19)
for
T
IO
for
T
