Macromolecules 1990,23, 2050-2057
2050
Polymer-Polymer Interactions in Dilute Solution P. M. Cotts* IBM Research Division, Almaden Research Center, 650 Harry Road, Sun Jose, California 95120-6099
J. C. Selser Department of Physics, University of Nevada, Las Vegas, Nevada 89154. Received July 22, 1989; Revised Manuscript Received October 16, 1989 ABSTRACT: Coil-coil interactions in dilute solution for poly(a-methylstyrene) in sub-8, 8, marginal, and good solvents were studied using static and dynamic light scattering. The behavior observed across the solvent quality range is well described by a semiempirical fit to the interchain parameter k,' versus the reduced radius X SIR, using a recent theoretical treatment developed by Akcasu and co-workers.
I. Introduction Light scattering is an elegant and noninvasive method of investigating polymer-polymer interactions in dilute solution. This paper presents the results of a light-scattering study of PaMS-PaMS (poly(a-methylstyrene)) interactions in good, marginal, 0, and sub-0 solvents. When time-averaged or static light scattering is used, the polymer weight-averaged molecular weight, Mw, and the solution second virial coefficient, A,, can be determined, while dynamic light scattering (specifically photon correlation spectroscopy, PCS) provides the hydrodynamic radius, RH, along with the concentration coefficient, k," 1-3
D ( c ) = Do(1 + kDCc+ ...)
(1)
Do = k T / 6 * ~ & , (2) where mutual diffusion coefficients D(c) are determined from measured autocorrelation functions. As the fluctuations in scattering intensity arise from the random formation and relaxation of concentration gradients in the solution, the concentration dependence contains both thermodynamic and transport or hydrodynamic factors. The Gibbs-Duhem equation permits expression of the concentration dependence of D ( c ) as
or
kT D ( c ) = -(1f(c)
+
D c ) ( l + ~ A @ c ...)
(3b)
where the osmotic compressibility (a.rr/ac),which may be obtained from time-averaged light scattering, has been expanded in a virial series including the second virial coefficient, A,, and molecular weight, M , explicitly. The thermodynamic and hydrodynamic parameters incorporated in k , are then related according to
kDC= 2A2M - k,C - D
(4) expressed in concentration units. The polymer partial specific volume in solution is D, while k," is the linear term in the concentration expansion for the friction coefficient f(c) = fo(l
+ k,Cc + ...)
(5)
0024-9297/90/ 2223- 2050$02.50/O
with c the solution concentration. Theoretical calculations use the volume fraction, 4, where
4 = CNAVH/M with the hydrodynamic volume, V,, given by
(6)
VH = ( 4 / 3 ) * R ~ ~
(7) Recast in volume fraction units and ignoring the negligible contribution of D to k,' for large polymers, eq 4 reads
kD' = 8x3- 12,' (8) with X E SIR, and S defined as an effective hardsphere radius determined by the second virial coefficient, A:
s = (3hf'A2/16aNA)113
(9)
and N A Avogadro's number. The thermodynamic excluded-volume term in eq 8, 8X3,may be measured by intensity light scattering or colligative methods and is positive in thermodynamically good solvents; decreasing to zero at To and becoming negative below To. The hydrodynamic term, k,', has been the focus of several earlier theoriesa4-' While these transport theories differ in their quantitative predictions of k,+', they all predict that k,+'remains positive even at To so that k,+' is expected to be negative at To,as has been observed experimentally. Accurate measurements of k," are more accessible than measurements of 12,' from transport properties such as sedimentation or tracer diffusion. An especially compact and useful way of employing kD+', and RH in the study of polymer-polymer interactions in dilute solution has recently been developed by Akcasu and Benmouna' and by Akcasu and Han,g.lo and this method is used to develop the principal conclusions of the present study. The Akcasu theory differs from the earlier transport theories for k,+'in that both the thermodynamic and hydrodynamic contributions to kDVare evaluated from the static structure factor obtained from light scattering. In this method, experimentally determined values of kD+'are plotted against experimentaly determined values of X over a range of polymer molecular weights and for several solution solvent qualities, which may be varied by varying solvent or temperature. Note that both the temperature and the polymer molecular weight dependences of kD+'are incorporated in X.lo
s,
0 1990 American Chemical Society
Macromolecules, Vol. 23, No. 7, 1990
Polymer-Polymer Interactions in Dilute Solution
Table I PaMS Molecular Weight Results sample no.
Mw f 6% 6 300 24 600 71 000 109 300 293 000 584 000 726 000 1010 000 2 900 000
0.5
2 l
6
4
I
I
/
l
2051
8
1
1
MwlM,., (SEC) 1.35 1.17 1.09 1.09 1.08 1.14 1.16 1.14 1.18
Theoretical treatments of solution binary polymer interactions may then be tested b y formulating ',k as a function of X and comparing the observed dependence of ',k on X with that predicted b y eq 8.
0.31
I I 0.0 A-AA-A-A-A 0.0 0.5
I
1/74
I
1 .o C
W
1.5
2.0
)
Figure 1. Kc Re,,. versus concentration for P a M S samples 5-9 in margin solvent (cyclohexane at 50 "C). By sample number: 0,5 ; 0,6; 0 , 7; A, 8; A, 9.
d
I
20
40
60
80
I
I
I
I
I
I
10
20
30
40
100
11. Experimental Section Nine samples of poly(a-methylstyrene) (PaMS) with molecular weights in the range 6 X lo3-3 X lo6 were used in this study (see Table I). These samples were prepared by anionic polymerization a t the Pressure Chemical Co.; relative amounts of the different tactic forms as determined by 'H or 13C NMR were 46 f 6% isotactic plus heterotactic and 53 f 6% syndiotactic." The fraction of isotactic triads was very small, 5 f 2%. These values are very similar to those reported by Kat0 and co-workers" for anioncally polymerized samples of PaMS. The steric hindrance toward isotactic addition is evident although the fraction of syndiotactic is similar to the smallest values reported for a variety of ~yntheses.'~The fraction of syndiotactic triads was somewhat larger a t lower molecular weights as was also reported by Kat0 and co-workers." Stock solutions for light scattering were prepared gravimetrically a t room temperature in spectrophotometric-grade toluene and a t 50 "C in spectrophotometric-grade cyclohexane. Samples of reduced concentration were prepared by serial dilution of the stock solution and filtered through 0.5-fim Fluoropore filters directly into cells that had been cleaned by refluxing in an isopropyl alcohol drier. Light-scattering measurements were then made a t 25, 34.5, and 50 "C in cyclohexane (sub-8, 0, and marginal solv e n t ~ ) ' ~ *and ' ~ ~a' t~ 25 "C in toluene (good solvent; see ref 16 and below). During measurement, samples were maintained within 0.05 OC of the reported temperatures by thermostatic control. Static light-scattering measurements were made either in a Chromatix KMX-6 low-angle laser light-scattering photometer or in a Brookhaven BI-2030 light-scattering photometer. PCS measurements were made either in the Brookhaven photometer using its 136-channel digital correlator a t IBM or a t UNLV in a homebuilt system that has been described in some detail earlier.I7 The UNLV system was modified by the substitution of a 136channel Brookhaven correlator for the earlier Langley-Ford or Saicor correlators. All autocorrelation functions measured were "unclipped". Mw and A , values for P a M S were determined from the concentration dgpendence of the absolute intensity of scattered light, while Mw/MN values were determined by size-exclusion chromatography (SEC) in tetrahydrofuran. Values of M, and A , were obtained by extrapolation of meajurements a t 4-6 concentrations to infinite dilution, obtaining Mw and A , from the intercept and slope of a square-root plot
where the subscript 0 indicates extrapolation to zero scattering angle and R, at 4" scattering angle is taken as equivalent to R,. The constant K = 4 ~ n ~ ( d n / d c ) ' / X ~ NExperimental ~. data in Cyclohexane a t 50 "C are shown in Figures 1 and 2. The values of Mw reported in Table I are the averages of the light-scattering results in cyclohexane and toluene and the SEC results. The refractive index increments (dnldc) measured were 0.194 and 0.124 mL/g for cyclohexane and toluene, respectively. These values were obtained a t 30 "C and A = 632.8 nm using a Chro-
0
0
50
cig/L)
Figure 2. Kc/R,,,. versus concentration for P a M S samples 1-4 in marginal solvent (cyclohexane at 50 "C). By sample number: +, 1; V, 2; ,. 3; v, 4.
0
I
I
I
I
I
1
1
2
3
4
5
6
c(g/L)
Figure 3. D ( c ) versus c for P a M S samples 4-9 in marginal solvent (cyclohexane a t 50 "C). Symbols denote sample numbers as in Figures 1 and 2. matix KMX-16. This dn/dc in toluene is significantly larger than that we reported in an earlier study.16 We believe the earlier value to be in error, as ha_s also been suggested recently by Lindner et al." The A , and Mw values of samples 1-7 measured in toluene were taken from the earlier study and have been corrected to reflect the change in dn f dc. R, and k," values were determined using cumulant analyses of scattered light autocorrelation functions." In the cumulant expansion, deviations of the correlation function from a single exponential are approximated by a series expansion In Ig(')(q,t)l= -Fit + p2t2/2+ ...
(11)
in powers of the delay time t. The scattering vector q = (47rn/ A,) sin (0/2) with A, the wavelength of light, 632.8 nm in a vacuum, and n the refractive index of the solvent. (12) where g(,)(t) is the normalized measured intensity time correlation function ( C ( t ) / B ) and , b is an optical factor related to the number of coherence areas viewed by the detector. The base line B was determined by an average of 4 channels delayed to 1024 times the last correlator channel. Measured and calcylated base lines agreed within 0.02%. The first cumulant rl
Macromolecules, Vol. 23, No. 7, 1990
2052 Cotts and Selser 2,"
20hl
4:
,
I
1
"0
I
1
1
,
YO
O:1
-m . -. ,
/
I
5
0
15
10
1
/
20
25
c i g LI
0 01
Figure 4. D ( c ) versus c for PaMS samples 1-3 in marginal solvent (cyclohexane a t 50 "C). Symbols denote sample numbers as in Figure 2.
-zE
3
01
10
100
1000
i o 5 a,,
Figure 7. A as a function of Mw ror P a M S samples 1-9 in good solvent ?toluene, 25 "C) (0). Also included are measurements made by Kat0 et a1.l' (A). Equation 16 i n the text presents the power law fit for PaMS samples with Mw > lo5
I-
:- t
' I
00
3
2
1
5
4
clg, Ll
Figure 5. D ( c ) versus c for PaMS samples 4-9 at T (cyclohexane at 35 "C). Note that the slopes are the same. Aymbols denote sample numbers as in Figures 1 and 2. 161
I
20 I
I
40 I
I -
0 01
01
10
10 0
100 o
60
Figure 8. A , as a function of Mw for P a M S samples 1-9 in marginal solvent (cyclohexane a t 50 "C) (0). Also included are measurements made by Kat0 et al.14 (0).Equation 15 i_n the text presents the power law fit for P a M S samples with M, > 105.
4 0
.-.-.--.-
loo0 I
10
1
20
1
30
clg/L)
Figure 6. D ( c ) versus c for P a M S samples 1-3 at To (cyclohexane at 35 "C), as determined by PCS. Note that the slopes are the same (two scales are used). Symbols denote sample numbers as in Figure 2.
has been shown by Koppel to lead to the z-average diffusion coefficient in the limit of infinite dilution"
r, = D,q2
(13)
The normalized second cumulant, w2/rl2, was 0.07 or less for these samples reflecting their narrow molecular weight distribution (see Table I). All but the lowest M sample had polydispersity indices M,/M,, of
