J. Phys. Chem. 1984, 88, 6613-6617
6613
Polydispersity Correctlons on Excluded Volume Dependence in Flexible Polymers Jack F. Douglas and Karl F. Freed* The James Franck Institute and the Department of Chemistry, The University of Chicago, Chicago, Illinois 60637 (Received: June 1 , 1984; In Final Form: July 30, 1984)
Polydispersity corrections are determined for a wide variety of polymer properties that have previously been calculated from the two-parameter model in conjunction with the chain space renormalization group theory. Specific calculations are provided for the commonly encountered properties (S2),A2,P, P,and for linear flexible polymers using the Schulz-Zimm distribution when the molecular weight distribution is fairly narrow, Le., M,/M, I1.1. The solvent quality dependence of the polydispersity correction is explicitly evaluated, and the final predictions emerge in simple algebraic form for typical sharply peaked molecular weight distributions.
I. Introduction Accurate ~ o r k l to - ~test the two-parameter (TP) model4 of excluded volume in polymers has required the preparation of highly monodisperse samples in order to avoid the ambiguity of polydispersity corrections arising from the strong power law type dependences of physical properties on molecular weight. Many of these polymer properties are sensitive functions of the empirical TP theory dimensionless z parameter which is proportional to where M is the molecular weight. Therefore, the average of these properties over broad distributions in M complicates the comparison with theories which are least ambiguously made for idealized monodisperse distributions.2 The test of the recently developed renormalization group (RG) theory, which enables the extension of the two-parameter model to much larger values of the interaction variable z, has further necessitated the use of higher molecular weight polymers whose molecular weight distributions can no longer be reasonably considered monodisperse. It is important then to correct for any systematic trends introduced in the variation of properties with the tendency toward broader molecular weight distributions with increasing average molecular weight. The better experiments use “fairly monodisperse” samples, characterized by a polydispersity M,/M, I1.1, and the sharply peaked nature of these distributions allows for a “universal” description of polydispersity effects. For broader distributions this is no longer possible and the averaging should be performed numerically with the empirical molecular weight distribution. The methods developed here may also be applied in the general case, but our treatment of narrow distributions enables the determination of simple analytical formulas for the polydispersity corrections required to correct data obtained for fairly monodisperse samples over the full range of excluded volume interaction, z 2 0. Our main interest beyond obtaining these needed corrections is to investigate a discrepancy between the RG theory and experiments that emerges for certain properties in the limit of fairly large z. The properties of interest in these experiments are as follows: The penetration function4 is \k defined by \k=
@A2
(4n( S 2 ) ) 3 / 2 N A with A2 the second virial coefficient, N A Avogadro’s number, and ( S 2 )the square of the radius of gyration. The Flory function4 is given by (1) Norisuye, T.; Kawahara, K.; Teramoto, A.; Fujita, H. J. Chem. Phys. 1968, 49, 4330. (2) Matsumoto, T.; Nishioka, N.; Fujita, H. J . Polym. Sci., Polym. Phys. Ed. 1972, 10, 23. (3) Miyaki, Y.; Fujita, H. Macromolecules 1981, 14, 742. (4) Yamakawa, H. “Modern Theory of Polymer Solutions”; Harper and Row: New York, 197 1. (5) (a) Miyaki, Y.; Einaga, Y.; Hirosye, T.; Fujita, H. Macromolecules 1977,10, 1356. (b) Miyaki, Y.; Einaga, Y.; Fujita, H. Macromolecules 1978, 11, 1180. (c) Akita, S.; Einaga, Y.; Miyaki, Y.; Fujita, H. Macromolecules 1976, 9, 774.
0022-3654/84/2088-6613$01.50/0
[oIM
@= (6 ( S2))3/2
with [o] the intrinsic viscosity. The quantity P is defined4 by P=
f o,(6(S2))1/2
with f the polymer friction coefficient and 7, the solvent viscosity. and PIP, are dimensionless comThe three quantities \k, @/ao, binations of measurable polymer properties and are designed to focus on the z dependence of the more slowly varying “prefactor” [VI, andfand thereby provide highly stringent portions in A2, (p), tests of theory. The zero subscript on Qo and Po indicates the value of that property at the 8 point. The cause of our concern is illustrated by the peculiar hump in the penetration function data of Miyaki and Fujita3 that is displayed in Figure 1 along with our RG calculations6 which predict a monotone increase to the indicated limit P*for good solvents. Hence, one goal of the present work is to determine whether increasing polydispersity corrections with increasing M , (or z ) produce this hump as an artifact. If not, it indicates either an inadequacy of the second-order RG calculations and the necessity of higher order calculations6 or the inadequacy of the RG method or perhaps even the inadequacy of the two-parameter model. The hump is also frequently present2 in experimental plots An experimental alternative to the present theoretical of @’/ao. analysis involves measurements of \k for a fixed molecular weight sample over a substantial range of cys? = (S2)/(S2),. Another possible source of discrepancy in the penetration function data may arise from the customary assumption4 used in analyzing light-scattering data that the ratio r = A 3 / A 2 Mis equal to ‘ / 3 over the whole range of excluded volume interaction. The ratio is taken as 1 / 4 for analyzing osmotic pressure data. These ratios result from a semiempirical argument of Flory* for the good solvent limit, and there is no a priori justification for its use elsewhere. In fact, we have provided a calculation of the excluded volume dependence of F based on a combination of two-parameter theory calculation by Stockmayerg and by Yamakawa’O and on the extension of this calculation with the renormalization group theory.’ We find that the previous semiempirical estimates for r are in substantial error in the intermediate solvent regime; hence, the analysis of P in this intermediate region should be reinvestigated with the improved theory of r. We now turn to our specific goal of obtaining polydispersity corrections to determine their dependence on excluded volume. (6) (a) Douglas, J. F.; Freed, K. F. Macromolecules 1984,17, 1854. (b) Douglas, J. F.; Freed, K. F. Macromolecules, in press. (7) Douglas, J. F.; Freed, K. F. Macromolecules, in press. ( 8 ) Flory, P. J. “Principles of Polymer Chemistry”; Cornell University Press: Ithaca, NY, 1953. (9) Stockmayer, W. H. Makromol. Chem. 1960, 35, 54. (IO) Yamakawa, H. J . Chem. Phys. 1965, 42, 1764.
0 1984 American Chemical Society
6614
The Journal of Physical Chemistry, Vol. 88, No. 26, 1984
Douglas and Freed tribution. The most widely used is the Schulz-Zimm13 distribution
1
1
0.4
f(M)= ykMk-’e-YM/r(k) with the properties1’J2 k - 1 - - =k y = - -
Mn
3
5
7
9
II
Figure 1. Penetration function P data for highly ex ded coils. The The closed circles ( 0 )are from computer experiments by Gobush et other data are from Miyaki et aL5 for poly(o-8-hydroxybutyrate) in trifluoroethanol (TFE) at 25 O C (o,O), for the stereoisomer5PMPL (0) in TFE, and for polystyrene in benzene at 40 O C and 30 0Czo,2’ (A,?). The monotonically increasing solid line is the theoretical prediction obtained from (6.b), (8.c), and (16.b) where the variable 5 is eliminated between the equations and the concave down curve is a best experimental fit found by Miyaki et aL5 in the good solvent limit. The double-dotted line is obtained similarly by eliminating 3, between the corrected expressions for P and (upgiven in (1l.h) and (16.d) with k = 10. The polydispersity correction for polystyrene in benzene in the good solvent limit, for example, is discussed in ref 12 based upon the same data indicated in the figure where k is estimated to range between 10 and 20. The lowest k corresponds to the highest molecular weight sample.
Methods used previously to evaluate these corrections for the special limits of the good and 8 solvent limits are generalized to solvents of intermediate quality. The case of ( S 2 )is treated in section 11, A , in section 111, and other properties in section IV. The renormalization group results for intermediate solvents are used, but no knowledge of renormalization group theory is required in our analysis. 11. Polydispersity Corrections for Polymer Dimensions as a Function of Solvent Quality Polymer solutions are characterized by a weight distribution which is usually written in terms of the experimental moments
4S’(V) =
(11) Oberthiir, R. c. Makromol. Chem. 1978, 179, 2693. (12) Cotton, J. J . Phys. (Orsay, Fr.) 1980,41, L-231.
(3c)
r ( k + 2v + 2) r(k + 2) ( k + l)-zu
(4a)
k = [ ( M , - Mn)/Mn]-’ = [ ( M , - M w ) / M w ] - ’- 1 (4b) For k I 10 Stirling’s approximation with a leading asymptotic c ~ r r e c t i o n may ’ ~ be used to express (4a) in the more convenient form
k+2
qs2(v)=
(=)’”[
+
242v - 1) 2(k 2)
+
+ O((k + 2).)]
(4c)
The extension of the analysis of (3) and (4) to intermediatequality solvents uses the renormalization group theory predictions through first order in e = 4 - d, where d is the dimensionality of space; d = 3 in all our applications. As a simple example of the application of the t expansion perturbation theory, eq 4c is further simplified in the good solvent regime by using the RG expression of 2v - 1 to first order in t
1
The basic approach pursued here is to use the renormalization group predictions to define an effective molecular weight dependent exponent veff in the marginal solvent and crossover region that permits the extension of (2b) and expressions like it to those intermediate regions of excluded volume interaction. The evaluation of even the good solvent limiting case of (2b) requires the introduction of a particular molecular weight dis-
+ a)/F(k)y*
while k is conveniently written from (3b) as
+ O(tz),
{ + :[ &+
where (Sz), is the mean-square radius of gyration for a monodisperse sample of molecular weight Mw. Hence, theoretical expressions for (Sz) are to be compared with experimental (SZ), by using the measured M , and the conversion factor qs2. In good and 6 solvents ( S 2 ) , satisfies a simple power law (p), = CMwzUwith C and v constants, so that (2a) has the formI2
Mz
where r ( k ) is the r function. The distribution (3a) is fairly narrow for k 2 10 where its predictions should closely correspond to the experimental situation despite some departures from the precise shape of (3a). The limitations of this procedure are described in ref 12 where it is suggested that for sharp distributions the errors of measuring M,, and M , outweigh those of approximating the true (unknown) distribution function by the Schulz-Zimm distribution. The evaluation of (2b) with (3a) and (3c) follows simply in the 6 and good solvent limits as’,
2v - 1 = t/8
Light-scattering experiments provide the M , average of ( S 2 )and the M, average molecular weight. Given this experimental M,, , the it is useful to convert other measured data, such as ( S 2 ) =to M , form. Hence, the measured (Sz), is related to the theoretical through a polydispersity correctionl1*l2qsz quantity (p),
k + l
M w
(M’) = I’(k 1
(38)
qg(v) =
In
(-)
k+2 k + l
X
(s)] I
+ O [ ~ ~ , ( k + 2 ) - ~(4d) ]
a representation which is evidently restricted to large k (k 1 0 (10)) in order to be of reliable accuracy (see section IV). We now derive similar expressions for the intermediate excluded volume regime after discussing some matters of notation. The final form of our theoretical calculations is given in terms of a z variable6 which is to be treated in exactly the same way as the z variable of the T P theory where the molecular weight is taken as the Mnaverage. For the purposes of the calculations, on the other hand, it is convenient to use one of several6 variant representations of the RG theory which is due to Oono and Freed. l 5 To introduce the RG notation and to relate it to the more familiar z form, we note that to first order in the radius of gyration ( S 2 )expansion factor in the monodisperse limitI5 is
where N is a model parameter corresponding to the chain length, I is a Kuhn length, M is the molecular weight, and L / ~ Tis a temperature solvent dependent phenomenological parameter (13) Zimm, B. H. J . Chem. Phys. 1948,16, 1099. (14) Abramowitz, M.; Stegun, Z. “Handbook of Mathematical Functions”; Dover: New York, 1972; Section 6. (15) Oono, Y.;Freed, K. F. J . Phys. A: Math. Gen. 1982, 15, 1931.
Polydispersity Corrections for Flexible Polymers
The Journal of Physical Chemistry, Vol. 88, No. 26, 1984 6615
corresponding to a thermal blob size.I6 Next we define a proportionality constant between N a n d the molecular weight, N = bM, and the molecular weight of thermal blob B = bL/27r. Also the 2 parameter, mentioned earlier, is written in the good solvent regime6 as ( d = 3)
z = (M/B)1/2/6.441
c
-(h(Mn) 8 + ;X(Mn)[I
- A(Mn)I[(M- Mn)/Mn] +
( Y S= ~ ~1.72Z2(2'1), i > 0.75 (6b) where 2v - 1 to second order" in e is 2v - 1 = €18 + ( 1 5 / 4 ) ( ~ / 8 ) ~ O(e3). At the end of the calculations all expressions are given in terms of this i variable for convenient comparison with experiment. For intermediate and weak excluded volume 0 5 i I 0.75, we introduce the RG crossover scaling variable 7 which ranges between 0 in the Gaussian chain limit and infinity in the good solvent limit. This 7 is expressed in terms of M / B and the quantity ii = u / u * by
+
- ii)-1
The nonnegative quantity X(M,)[ 1 - X(M,)] is always less than 1/4, while ( M - M,)/M,, is small for k 2 10 since k-' 1 ( M , M,)/M,,. Thus, the variation of veff with M is very slow over the range in which the k 1 10 distribution is nonnegligible. Furthermore, the second term on the right-hand side of (10) is formally of order e2, so it should not be retained without the second-order parts of v,ff itself. In addition, the variation of the prefactor portion of (8a) must be included in second order because ds/dM = O(c). The full second-order treatment is fairly cumbersome, but for narrow distributions we need only keep the first-order portion of (10). Retaining terms through order t in (10) converts (8a) to
with the first-order approximation e
2Yeff- 1 = -X(M,) 8
For 2 5 0.75 the first order in e relations6 -+
322/3,
(M/B)'vl*(l+V)
-+
(1
+ 322/3)1/4,
2
-+
0 (7c)
+ O(e2)
4s' =
with 2v(7) - 1 the "effective exponent" in the intermediate excluded volume region 2v(7) - 1 = e7/8(1 7) O(t2)E th/8 (8b)
Introducing an e expansion using (1 lb) and I'[k + 2 + (2veff- l ) ] = 1 + ih(Mn)J.(k I'(k + 21
+ +
(1lb)
making (1 l a ) of the general form leading to (2b), but with v replaced by vefP Hence, qS2 in the intermediate excluded volume region is given by (4a) with v replaced by veff. Using the relation I'(1 + z ) = zI'(z) in this expression leads to
are adequate to transform RG expressions to the i representation. In terms of these variables can be written in the equivalent forms6J6
+ 2) + O(e2) (114
or through use of (7c) as with $(x) X = (322/3)/(1
(k
+ 322/3)
(8c) While the form (8c) as an approximation to (sa) is strictly valid for 0 5 i 5 0.15, it is found to agree well with experiment over the broader range 0 I 2 I 0.75. Through a combination of (6b) and (8c) the whole excluded volume regime i? > 0 is described. To justify our interpretation of 2v(7) - 1 as an "effective exponent", we define an effective exponent, for instance, through as22
0:
p e f r l
which implies that
a In ffs22/a
[,",
In M = 2Veff- 1 + In M M-
(2Veff- i ) ] (8d)
In first order in e the calculations below show that 2veff- 1 equals 2v(7) - 1, but the effective exponent becomes more complicated in higher order. The molecular weight averaging is conveniently considered by using the 9 representation of (8a), (8b), and (7a) 2Veff- 1 = f[(M/B)'/2ii(l 8
- a)-'][1
(10)
(7a)
where u is a dimensionless measure of the excluded volume interaction and where17
7
*e.)
(64
which brings (sa) into the more familiar looking form
7 = (M/Byii(l
the narrowness of the molecular weight distribution, eq 9 2v(7) - 1, about the maximum at M,, of the Schulz-Zimm distribution 2Veff- 1 =
+ (M/B)'/%(l
- a)- ]
-
I -I =
ih(W (9)
with the M dependence explicit. Because of the assumption of (16) Douglas, J. F.; Freed, K. F.; Macromolecules 1983, 16, 1800. (17) Kholodenko, A.; Freed, K. F. J . Chem. Phys. 1983, 78, 7390.
d In I'(x)/dx and
+ l)-(2uefrL)= 1 - ?h(M,) 8
In (k
+ 1) + O(c2) ( l l e )
gives the form
I
J.(k
1)
.
+ 2) - In (k + 1) + + O [ ~ ~ , ( k + 2 ) -(~l l]f ) k+2
-
-
When k -, the Schulz-Zimm distribution becomes a mono1 in this limit as it must. The latter is disperse one and qsz obtained by using the large k limit14 of 1 $(k) In k - - 0 ( k 2 ) , k 2 O(10) (llg) 2k
-
+
which may be introduced into (1 If) to produce the simpler form
O [ ~ ~ , ( k + 2 ) -(1 ~ ]lh)
-
This is to be compared with (4d) which is identical in the good solvent limit X(M,,) 1. The z representation follows from (7c) through the replacement
h(M,) = (322,,/3)[1
+ (32~,,/3)]-'
(lli)
where Z, = (Mn/M,,,)1/2iw for d = 3. Broader distributions require the calculation of ( 1 I f ) to second order in e as well as a more careful consideration of the particular molecular weight distri-
6616 The Journal of Physical Chemistry, Vol. 88, No. 26, 1984
bution. An approximate method of treating this situation is described in section IV.
Douglas and Freed
4k '){ 1 + i [ l n k - $(k + 2)]X(Mn) 3 I'(k + 2 j 3 ) I'(k + 4/3)
qA2 = (k
111. Polydispersity Corrections for the Second Virial Coefficient and Penetration Function as a Function of Solvent Quality The second virial coefficient polydispersity correction is defined analogous to (2a) as
+
r(k
4
+ 1) r(k+ 1) 4/3)
1
+ ;[In
k - (2/3)$(k
+
\
- (1/3)+(k + 2 / 3 ) 1 ~ ~ n +) }o(E2) (15a)
where 14e has been used. Using ( l l d ) and (6.1.47) of ref 14, we further reduce (15a) to where A,, is the value for a monodisperse sample of molecular weight M , and the superscript (R) denotes light scattering. In the good solvent limit the second virial coefficient behaves asymptotically as A2 Mdw2,and Cassasa (see ref 4) has obtained an approximate expression for qA2 as k2-du I'(k dv) r ( k ) 3I' k qA2(u) = 4[I'(k 1)12
-
[
+
+
?dv) 3
+
+
(
I'(k
+ idv)]
-;[In
k + 2/3)III) + 4/3)2/3(k
((k
(12b)
for good solvents. This result is extended into the intermediate excluded volume region by using the second-order expression for A2 for linear flexible polymers as
6k
+4
+'* +
- -
by expansion in powers of k-l. Again as k a, qA2 1 for a monodisperse distribution. However, for all k 1 10 the expression (15b) remains close to unity in good and poor solvents in conformity with the conclusions noted previously by Yamakawa4 and Cotton.12 Now we turn to the determination of the polydispersity correction for the penetration function defined in d space by \k = 2M2A,(d/3)d/2/N~(4..)d/2(SZ)d/2 (16a)
We again find an excluded volume dependent correction for polydispersity by defining an effective exponent by the relation A2(M,u,B) a MdvcAM-2 (14) The effective exponent in (14) is found in a similar way as for as22
du(7)
-
d In ~ d ~ ( dlnM
- 2 = -: 2 +4
d In fa2/d In M =
4 - 2d
( 14b)
+ O(t2)
I(1 - X) 2
4
( 14c)
+ O(t2)
(:);(4
t2, (14a-14d)
combine
+ O(cZ)= dveff- 2
(14e)
+
'[ 4
[r(k
I'(k
+ 4/3)
+ 1)12
1
r[k
(16 ~ )
-+
-
where use of (1 li) produces the convenient representation in terms of z,. In second order in E perturbation expressions such as (16d) we make the replacement X X2 = 6.4412/(1 6.4412) (16e) as discussed in ref 7. An idea of the magnitude of the polydispersity effect is obtained by choosing M,/Mn = 1.1 or k = 10. Then the asymptotic limit or X(M,) 1 of (16d) is 7 \ k * k = l o = 0.233 (160 which is about 13% lower than the monodisperse Q*. A curve of (16d) for k = 10 is included in Figure 1 along with the data of Miyaki and Fujita which has a k ranging between about 10 and 20.5,'2 Polydispersity evidently accounts for the lowering of the asymptotic value of Q* seen in the data, but the hump at smaller values of aSzis still evident and is even more exaggerated after correction for polydispersity. This effect is very likely due for the intermediate to the improper choice of A3/A22M=
-
-+
+ 2/3)
3)d/2/N~(47r)d12 (s2) wd12q,pd/2
Substitution of (1 lb) and (15b), use of d = 4 - E, expansion to order c2, and extraction of qu to order e yield a lengthy expression for qe in terms of k, X(M,,) of (1 li), and E 1 for d = 3. Since qA2departs little from unity for k 2 10, we employ the approximation qA2 1 to simplify matters. After some algebra, we find for d = 3 that
(1 4 4
for polymers of molecular weight M . Expanding around the peak of the Schulz-Zimm distribution again, as in (lo), yields the effective exponent as a function of the number average molecular weight. Upon substitution of dveff for du in (12b) and rearranging terms by using r(1 x ) = xI'(x) several times yields
I'(k
+ In 2 + 7/6)$(1 + q)-2 + 21q/4(1 + 1))+ O(t3) 1
\k = 2M;q~,A2,(d/
1nfA2
+-d l n M
Because d[du(v) - 2 ]/d In M i s of order to give d In A2/d In M = -:A
€
=-8 1+q
,,k,\
( 16b) for monodisperse distributions. The measured Q, however, involves the ratio of AiR) and ( s,),. Introducing polydispersity corrections for each gives
which upon use of (1 3a) becomes d In A, -dlnM To order e we have
and evaluated to order e2 by the RG chain space formulation as6
+ 2(dueff- 2)/3 + 4/31 I'(k + 4/3)
Equation 15 can be expanded into the convenient form
-+
+
The Journal of Physical Chemistry, Vol. 88, No. 26, 1984 6617
Polydispersity Corrections for Flexible Polymers excluded volume regime and/or to less polydispersity in the lower molecular weight samples. These points will require separate experimental and theoretical studies.
IV. Other Properties Excluded volume dependent polydispersity corrections are obtained for other properties in a similar fashion. For instance, consider an equilibrium property Q or a preaveraged, nondraining hydrodynamic one that scales as a polymer radius to thepth power. We show elsewhere6that the expansion factor Q/Qo = ad is given by the renormalization group theory as a d = (M/B)p[2Y(q)-1]/2[1 aeq(l + 7)-l] + 0(c2) (17)
+
where aQ = O(E) and v(7) can be given to 0(c2). The same arguments as in the case of ag2in section I1 lead to the conclusion that 2veM- 1 for any of these properties is given by (1 1b) to O(E). Hence, good solvent polydispersity corrections for k 1 10 can be extended to intermediate regimes of excluded volume interaction through the replacement of v by veff in the same fashion as that leading from (4a) to (1 IC). For example, Flory et a1.I8 calculate the polydispersity correction factors q* and qp
= q*[71Mw/(6(s2),)3/2
(18a)
P = q~v1/(6(S~),)~/~
(19a)
for nondraining polymers in good and 0 solvents where v] = f/Qs. Their results
[
+ 1)3/2
q p = -(k r(k) k r[k qP =
r(k
+ 2 +2u - 1) r ( k + 2)
1
3/2
/r[k
+ 1/2 +
3(2v - 1)/2] (18b)
+ 1 / 2 - (2v I'(k
X
+ 1/2) r(k
+ 2 + 2~ - 1) I'(k
+ 2)
are readily extended to the intermediate excluded volume limit by replacement of 2 v - 1 by 2 vcff - 1 = eX(Mn)/8. Furthermore, in the large k limit eq 6.1.47 of ref 14 gives
so (18b) and (19b) are converted to the intermediate excluded volume and large k forms k+ 1
9e = ( 4 l l 2 [ 1 +
ln---2 k + 4 2k+ 1
1
+
1 2k+4
-I[-4 1 128k2
+ 2k+1
3 1 + -eX(M,) 16
+ 0(2,k-3)
x (18c)
(18) Hunt, M. L.; Newman, S.; Scheraga, H. A.; Flory, P. J. J . Phys. Chem. 1956, 60, 1278. (19) Gobush, W . ;Sole, K.; Stockmayer, W. H. J. Chem. Phys. 1974, 60, 12. (20) Slagowski, E. Ph.D. Thesis, University of Akron, 1972. (21) Fakuda, M.; Fukutomi, M.; Kato, Y.; Hashimoto, T. J. Polym. Sci., Polym. Phys. Ed. 1974, 12, 871.
qp =
( k k) 1+/ 2 1 [
1-
2k+4 In--2k+1
-I[
-L + i28k2 1 8k
1 2k+4
1
+ %(Mn) 16
J-)] +
+ 2k+
1
x
0(2,k-3) ( m )
While the first-order good solvent v differs considerably from the more accurate second-order one, the expressions (18b) and (19b) differ little when these two values of v are used provided k is large. Hence, eq 18c and 19c can be used over the whole range of excluded volume interactions when k L 10 and the molecular weight distribution is fairly sharp. For distributions having a polydispersity index k < 10 and a known distribution type, a simple attempt at improving upon the theoretical expressions by "resumming" (1 If), (18c), and (19c) and similar expressions involves avoiding perturbative expansion in E and k-' and replacement of quantitities by their higher order analogues. Thus, as,, for example, equation (1 IC) with (9) can be written in the form
+
where 2v - 1 = e/8 0(c2). As a simple ad hoc approximation we either replace the first-order expression for 2v - 1 by its second-order expression (see (6.b)) and do not E expand or use the phenomenological exponent 2v - 1. The same procedure can be applied for any other chosen distribution type, leading to convenient analytical approximations for less monodisperse systems. Obtaining the rigorous second-order expressions in an analytic form is a much more difficult matter.
V. Discussion We derive excluded volume dependent polydispersity correction factors for a wide variety of polymer properties in the limit of narrow molecular weight distributions where the polydispersity corrections become universal functions. Previous theory has been limited to only the 8 and good solvent limits, while our generalization covers the intermediate excluded volume range. Effective exponents are derived in this intermediate regime from our previous renormalization group calculations, and we show that the polydispersity correction factors emerge as the same functions appropriate to the 0 and good solvent limits, but with the replacement of molecular weight dependent effective exponents for the limiting 8 and good solvent ones. This analysis is part of a more general study of the source of secondary structure exhibited by experimental data for the penetration function 9 and reduced Flory factor @/@, for very high molecular weight polymers. Our theory qualitatively explains the data, but a more quantitative comparison is best provided by using a single polymer sample with a narrow molecular weight distribution over a wide range of ap2. This type of experiment would provide better comparison with theory than that given by applying separate polydispersity corrections to data gathered from several narrow distribution polymer samples. Acknowledgment. This research is supported, in part, by NSF Grant DMR83- 18560 (polymers program).
