6983
J . Phys. Chem. 1989, 93, 6983-6991 was not very sensitive to any of these model variables. The exponent, a, was fixed a t 0.8; the fit for eq 5 was not sensitive to the value of a. The uncertainty in the viscosity was taken as the experimental uncertainty given in Table I, since the propagated uncertainty from the temperature error was shown to be negligible. When all parameters were left free, we obtained b3 = 0.085 f 0.009, b4 = -0.00040 f 0.00004, and b5 = 1.50 X lo4 5 X This function is drawn as the solid line in Figure 1. Clearly, the function and the data for sample 1 begin to deviate at high temperatures, which was reflected in nonrandom residuals and a reduced x2 of 35. A fit to eq 5 with only the linear and quadratic terms was less successful than the fit including the cubic term. W e note that the coefficients we found for eq 5 lead to a Huggins coefficient, k l , of -0.047; this coefficient is normally between +0.3 and If we fix k l at 0.3, thus fixing b4 at klb$, the cubic term becomes undetermined. The fit with just the first two terms, and thus only one free parameter, is shown as the dashed line in Figure 1; b3 = 0.0316 0.0006 and the reduced x2 is 71. While the fit is not as good as with three free parameters, the qualitative agreement is satisfying. We also attempted to fit eq 4, from the reptation model, to our data above T p We also tried putting the Arrhenius background viscosity into eq 4 as either a multiplicative factor or an additive
*
*
term. We were unable to get the nonlinear least-squares analysis to converge to any fit at all, perhaps because we have so little data in that temperature range.
Conclusions We believe that the extraordinary relaxations which we have observed in the viscosity of liquid sulfur are due to the reaction of the diradical polymeric sulfur with trace impurities. We have made new measurements near T p on very pure sulfur which we believe to be closer to the true values than are any reported previously. However, even in our measurements the relaxation of the viscosity was not entirely eliminated. It is possible that the remaining problem is due to the reaction of the sulfur with the glass cell. Better measurements will require a careful choice of the cell material. A viscosity measurement technique which is fast and in which the shear rate can be varied is desirable. Our measurements are consistent with current theories of polymer solutions and of equilibrium polymerization. Acknowledgment. This work was supported by the National Science Foundation under Grants CHE-8413404 and CHE8708426. We thank D. Thirumalai, G. Castellan, G. R. Miller, D. T. Jacobs, M. Cates, P. Pfeuty, W. Stockmayer, J. Wheeler, and R. Ziff for helpful discussions.
Critical Amplitude Scaling Laws for Polymer Solutions Isaac C . Sanchez Chemical Engineering Department and Center for Polymer Research, The University of Texas at Austin, Austin, Texas 78712 (Received: March 2, 1989)
Critical properties of polymer solutions have amplitudes (scale factors) that depend on the degree of polymerization, N . The amplitudes vanish or diverge with characteristic exponents on N . The traditional critical exponents (a,b, etc.) characterize the divergence or vanishing of critical properties with respect to e I( T - T,.)/Tcl. For a polymer solution a pair of critical indices is required (a, a ) , (b, @),etc., to characterize the behavior of the critical property. Scaling laws are derived for the N indices, and it is shown that a striking parallelism exists between the two sets of indices. For example, the scaling relationships, 3v = 2p + y = 1.1 + u, have a completely analogous form among the N indices: 3n = 26 + g = m n. Although the analogy is not complete, equations have been identified that transform one set of =sum rules" to the corresponding set of the other. In one approximation that is equivalent to assuming that the small correlation function index 7 equals 0, the N indices are shown to be related to the exponent r that characterizes the N dependence of the critical concentration (4c W ) .Heuristic arguments suggest that r is related to the Flory exponent that characterizes the size of the polymer chain at the critical point. The fundamental scaling variable for polymer solutions is identified as N"Iye rather than the classical N% This important identification allows for the experimental determination of the correlation length index n from existing coexistence curve data. Theoretical estimates of the N indices compare favorably with available experimental data with one exception. Data gaps and needed experiments are identified.
+
-
Introduction A growing body of experimental evidence1-15has convincingly shown that the critical point properties of a polymer solution under (1) Debye, P.; Coll, H.; Woermann, D. J . Chem. Phys. 1960, 32, 939. (2) Chu, B. Phys. Left. 1969, 28A, 654. (3) Kuwahara, N.; Fenby, D. V.;Tamsky, M.; Chu, B. J . Chem. Phys. 1971.55, 1140. (4) Kuwahara, N.; Kojima, J.; Kaneko, M. Phys. Reu. A 1975,12, 2606. (5) Hamano, K.; Kuwahara, N.; Nakata, M.; Kaneko, M. Phys. Left. 1977,63A, 121. (6) Hamano, K.; Kuwahara, N.; Kaneko, M. Phys. Reu. A 1979.20, 1135. (7) Hamano, K.; Kuwahara, N.; Kaneko, M. Phys. Reu. A 1980,21,1312. (8) Hamano, K.; Nomura, T.; Kuwahara, N. Phys. Reu. A 1982,26, 1153. (9) Shinozaki, K.; Hamada, T.; Nose, T. J. Chem. Phys. 1982,77,4734. (10) Kuwahara, N.; Kojima, J.; Kaneko, M.; Chu, B. J . Polym. Sci., Polym. Phys. Ed. 1973, 11, 2307. (11) Nakata, M.; Kuwahara, N.; Kaneko, M. J. Chem. Phys. 1975, 1 1 , 4278. (12) Nakata, M.; Dobashi, T.; Kuwahara, N.; Kaneko, M.; Chu, B. Phys. Reo. A 1978, 18, 2683. (13) Dobashi, T.; Nakata, M.; Kaneko, M.J. Chem. Phys. 1980,72,6685, 6692. (14) Shmmki, K.; van Tan, T.; Saito, Y.;Nose, T. Polymer 1982, 23,728. (15) Moldover, M. R.Phys. Rev. A 1985, 31, 1022.
0022-3654/89/2093-6983$01.50/0
going liquid-liquid phase separation are in the same universality class as the three-dimensional Ising model.16 Critical exponents associated with the divergence or vanishing of various thermodynamic properties as the critical point is approached have been determined for polymer solutions; these include the y and u exponents associated with the divergence of the susceptibility ( x ) and the correlation length ([),'+ respectively, and the /3 and p exponents associated with the vanishing of the concentration '~ difference and interfacial tension ( o ) , ' ~ ,respectively. In all cases the values of these critical exponents (or indices) agree well with values those assigned to the 3-D Ising model. As is well-known, binary mixtures of small molecules, liquid-vapor, and ferromagnetic critical phenomena also fall into this same universality c1ass.l6 Since the early light scattering work of Debye' to determine correlation lengths in polymer solutions, it has been known that the amplitudes of these various properties depend on polymer molecular weight. The amplitudes vanish or diverge with N, the (16) Sengers, J. V.;Levelt-Sengers, J. M. H. Progress in Liquid Physics; Croxton, C. A., Ed.; Wiley: Chichester, 1978; Chapter 4.
0 1989 American Chemical Society
6984
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
number of monomer units in the polymer, with characteristic exponents. Thus, for polymer solutions, we have the general forms: heat capacity:
C
-
x-
conc diff (order param): A$ susceptibility: interfacial tension:
u
correlation length:
f
(la)
N-at-"
N-befi
(1b)
NgeY
(IC)
N-V'
(Id) (le)
where e is defined by
and Tc and T a r e the critical temperature and temperature, respectively. The coefficients of the t factors are defined as the amplitudes that are N dependent for polymers; it is believed that only two of the amplitudes are independent (the concept of two-scale-factor u n i v e r ~ a l i t y ~ ~ JWe ~ ) .have also included in the above list the heat capacity, although to my knowledge this property has not been measured for polymer solutions (there are two other important exponents, 6 and q, and the corresponding N induces that will be defined later). As a mnemonic aid, the exponents on N are defined as the lower case English alphabet equivalents of the corresponding Greek indices; thus, we have the corresponding pairs (a, a ) ,(b, p), etc. Additionally, all exponents are defined to be intrinsically positive, although g = 0 and may be slightly negative. Now it is well-known that the critical indices on e are related and only two are independent. For example, two well-known relationships arela 3v = 2p
+y =p+
v
(3)
Sanchez Scaling Laws Preliminaries. Much of what will be developed here depends on an important paper by Shinozaki and Nose.22 These authors (S&N) were the first to point out that the critical N indices were related to one another and derived relationships between them. However, for reasons that will be elaborated upon below, they did not arrive at eq 4, although it is implicit in their results. One contributing factor was the way they chose to define the susceptibility. So we begin this section with a digression on definitions. Chemists and physicists sometimes use the terms chemical potential, osmotic pressure, and osmotic compressibility rather loosely and inconsistently. Usually no harm is done, but in the present case it can become a problem as illustrated below. The inverse of the response function or susceptibility (x)is defined as a second derivative of an appropriate thermodynamic potential (f) with respect to the order parameter (4):23 (7)
For a ferromagnetic spin system, 4 is the magnetization and H is the applied magnetic field, whereas in the present case, 4 represents the polymer or solvent concentration (4 is usually the volume fraction of polymer) and we have yet to identify the appropriate thermodynamic field variable that is conjugate to 4; x measures the ease with which the order parameter is changed in response to the conjugate field. In a two-component mixture with n , molecules of type 1 (solvent) and n2 molecules of type 2 (polymer), the Gibbs potential (C) is related to the associated and p2 by chemical potentials llL1 dC = kI dnl + p2 dn2 (8) or the Gibbs free energy density (g) by
What will be shown in this paper is that an analogous relationship exists ("sum rules") for the N indices: 3n = 26 + g = m
+n
(4)
Although the analogy is not complete, for example, p + v = 2 # 2 - a , a set of transformation equations have been discovered that connect the e and N sum rules. The indices can be expressed as functions of the correlation length index n and the index h that corresponds to the familiar q index associated with the decay of the correlation function. When q = 0 the N indices become functions of n only and the relationships take on a particularly simple form. The scaling arguments used to obtain the above results (the homogeneity or scaling are the same as those that can be used to obtain the various relationships among the e indices. It will be argued that the fundamental exponent is the one associated with the correlation length (n) and the appropriate scaling variable for polymer solutions is N"IYe rather than the classical N112e.This is an important identification because it allows n to be determined from existing coexistence curve data. There is another exponent ( r ) that governs the dependence of the critical polymer concentration (&) on N:
- a,but m + n
4c
-
N-'
(5)
The classical value of r is but its experimental value is 0.38 f 0.01. It will be argued that the correlation length critical index n is related to r by 2n z r
(9)
where u1 and v2 are the respective molar volumes and q51 and 42 are the respective volume fractions (& + 42 = 1). Dropping the subscript on 42,we have
where p is often defined by physicists as the exchange chemical p ~ t e n t i a l or ~ ~more s ~ ~ambiguously as the chemical potential (this 11 is not to be confused with the p critical index). Thus, the susceptibility is dr$/dp (see eq 7) and p is the field variable conjugate to 4:
In a polymer solution the difference in chemical potential between pure solvent ( p o l )and solvent in solution (pI)is related to the osmotic pressure (T) that develops in a system that is permeable to solvent molecules and impermeable to polymer molecules: TU1
= Pol - 111
(12)
With the help of a thermodynamic Gibbs-Duhem relationship, it follows that
(6)
(17) Stauffer, D.; Ferer, M.; Wortis, M. Phys. Rev. Lett. 1972,29, 345.18. (18) ,Rowlinson, J. S.;Widom, B. Molecular Theory Capillarity; Oxford University Press: Oxford, 1982; Chapter 9. (19) Widom, B. J . Chem. Phys. 1%5,43, 3898. (20) Vicentini-Missoni, M.; Levelt-Sengers, J. M. H.; Green, M. S. Phys. Rev. Lett. 1967, 18, 1 1 13. (21) Fisher, M. E. Rep. Prog. Phys. 1967, 30, 615.
(22) Shinozaki, K.; Nose, T. Polym. J . 1981, 23, 1 1 19. (23) Fisher, M. E. Lecture Notes in Physics: Critical Phenomena;Hahne, F. J. W., Ed.; Springer-Verlag: Berlin, 1983. (24) de Gennes, P.-G. Scaling Concepts in PoIymer Physics; Cornell University Press: Ithaca, NY, 1979; p 105. (25) If mole fraction is chosen as the concentration variable, as is customary for nonplymer mixtures, then fi = fi2 - w , .
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6985
Critical Amplitude Scaling Laws for Polymer Solutions
-
x = d(A$)/dp
where Ag is the free energy of mixing: Ag = 4l(Pl - rO1)/ul + 4J2(P2 - P 0 2 ) / U 2
(14)
N-zf+Ttt~r”l(x)
(21)
Setting x = 0 and comparing eq 21 with I C yields
-
Now from eq 13
g = y t - 2f
(22)
As can be seen from eq 16, as x 0, the free energy itself becomes (an obvious generalization to D dimensions is possible here): Note the scale factor of 4. If d r / d + is defined as x - l , as it is occasionally,this factor of 4J must be considered because it becomes molecular weight dependent near the critical point (4 c$~ N r ) .Further confusion arises when the symbol r is used for the exchange chemical potential. Still other authors define x-l as 4J d r / d d , which really equals 4J2x-’.Since all of these definitions of the inverse susceptibility (also called the inverse osmotic compressibility) are proportional to d2g/dq!J2,they are acceptable, but care must be taken to account for the concentration-dependent scale factors which become molecular weight dependent. For whatever reasons known only to them, S & N define the inverse susceptibility as 4Jc times the usual definition of x-l (see eq 15) and an extra term of r (see eq 5 ) appears in their critical index equations that tends to obscure their results (their g r is equal to g in this paper). However, even after removing this extra term and setting their s parameter to zero (the polymer/ solvent limit), differences still remain in the S& N results and those obtained here. My main contribution will be to show that a set of simple relationships exists between the indices and that they are intimately related to 4Jc. We begin by restating and modifying some of S&N’s scaling arguments in the present notation. Theoretical Development. At the critical point the free energy is nonanalytic; the scaling hypothesis states that the singular part of the free energy (7,) can be expressed as26
- -
-
r,(O,c)
- (N-‘C)~”T~(X)
(16)
where x = N-fp/(N-‘e)3V-8
(17)
p is a dimensionless exchange potential, p
pul/kTc,andfand t are arbitrary exponents on N (anticipating the results, we have defined f and t so that they are positive). It is assumed from the beginning, by analogy with the c indices, that only two independent N exponents (fand t ) are required. Note that the free energy has been expressed as a function of p rather than 4. By a Legendre transformation we can convert the Gibbs potential to a new potential r = g - 6~so that
ar/aP
= -4
(18)
This is a convenient potential for investigating the order parameter A4J 4 - &. Actually this new potential is the solvent chemical potential ( ~ l / v l )as can be verified through eq 10 and 14; thus, rsis the singular part of the osmotic pressure as defined by eq 12. As the critical point is approached, p 0 (Le., x 0) on the coexistence curve.27 An important property of r l ( 0 )and its first two derivatives, ~ ’ ~ ( and 0 ) ~ ” ~ ( 0 )is, that they exist; this is true Thus, from because of the assumed homogeneity property of rs.26 eq 16 we have
-
A+
-
ar,/ap
-
-
N-V+~~V~’~(X)
(19)
Setting x = 0 and comparing eq 19 to l b and equating exponents yields b=f+pt For the susceptibility, we have (see eq 11)
(20)
(26) Stanley, E. Introduction to Phase Transitions and Critical Phenomena; Oxford University Press: Oxford, 1971; Chapters 1 1 and 12. (27) For the classical Landau model p = 0 everywhere on the coexistence curve because the free energy is symmetric around the critical concentration. In general fi # 0 due to asymmetry which in principle vanishes sufficiently close to the critical point. See ref 40 and references therein on the relationship between choice of order parameter and asymmetry.
-
(N-ncy)3
(23)
Equating exponents yields
n = ut The interfacial tension (a) is also related to the free energy by u
-
-
&,(O,t)
kTc/12
-
(N-ne”)2
-
(25)
Equating exponents yields m = 2n
The heat capacity is given by
+
r,(P,t)/kT,
-
( N - ‘ C ) ~ ~ kTc/E3
C--
d2r,(0,t) de2
N-3~1~3v-2
-
-
~ - 3 n ~ a wae-a
(27)
or a = 3n
(28) Eliminating f from eq 20 and 22 and then using eq 24 to eliminate t yields 26 + g = (2p y ) n / u (29)
+
which by virtue of eq 3 can also be written as 3n = 26 g
+
(30)
As has already mentioned in the Introduction, eq 30 and 26 are the analogues of eq 3. There is another critical exponent pair (d, 6) that is related to the behavior of the exchange chemical potential along the critical isotherm (c = 0):
-
p
N~~A+(*
(31)
The classical values for d and 6 are 1 / 2 and 3, respectively. The 4 in the 4J - 4c in eq 31 is not a concentration on the coexistence line, but one that is in the one-phase region of the phase diagram along the critical isotherm. However, as 4 4Jc, we expect that A@ will behave as in eq 1b. We can relate both d and 6 to the other indices by computing the susceptibility from eq 31 and comparing that with eq IC. Thus
-
but as 4
-
dp x-1 = d4J
-
NdlA+lb’
N
N-geY
(32)
4JC eq 1b holds or t
-
Nbl@IA4(‘/@
(33)
which when substituted into eq 32 and equating exponents yields the Widom relation’*J9 s=1+y/p
(34)
and d = y b / P - g = ( 6 - 1)b-g (35) Finally, there is one other exponent pair (h, q ) that must be considered. This pair is related to the decay of the correlation function, C(R,e),and its Fourier transform, the structure factor or wave vector dependent susceptibility, x(q). At zero wave vector, the thermodynamic limit, x(0) = x, the susceptibility. At large m, values of q, corresponding to R/E 0 or equivalently, q l C(R,e) falls off as K(l+q) and the behavior of x ( q ) becomes
-
x(q)
N-r‘+h 4-2+1
-
(36) where we have anticipated the form of the N dependence of x(q). N
Sanchez
6986 The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 TABLE I: Scaling Laws and Sum Rules for the N Indices general v = 0 approx a = 3n a = 3r/2 b = (3n - h)/2 b = 3r/4 g=h g=O d = 3 4 6 - 1)/4 d = [3n(6 - 1) - h(6 + 1)]/2 m = 2n m=r
TABLE II: Corresponding Scaling Laws and Sum Rules for the e Indices index
nonclassical
a=2-3v P = (u/2)(1 + 7) Y = 4 2 - 7) 6 = ( 5 - 7 ) I U + 7) /.I= 2v
0.1 1 0.326 1.24 4.80 1.26 0.035 0.630
e
Sum Rules 3n = 2b + g = m + n = a
1 P
d+(gorh)=(6-l)b
-
x ( q ) = N-HP/’G(tq) The scaling function G has the following properties:
IW’+~ constant
G(tq) =
(37b)
‘11
1 3
’I1 0 ‘11
Corresponding “Sum Rules” = + v = 2 - (Y 3v = 28 Y = (6 - 1)8
+
According to the scaling hypothesis, x ( q ) can be expressed in the scaled formz8 x(4) t’f’G(t4) (37a) Since the proportionality constant may be N dependent, we write
classical 0 or
where K is the coefficient of the square gradient term and scales the free energy cost associated with concentration fluctuations. Expressing V+(R) in terms of its Fourier components it can be shown that x ( q ) is given byz8
q=0
tq
>> 1
and the correlation length t by [=
The latter implies that u = of [ q we have
Kl/2t-l/z
and
K
~
/
(49) N“. ~ At large values
-
(43)
which implies that 7 = 0 and that h = 0. The result h = 0 obtains because K - I N-2n= N7*Iv,which is to be compared with eq 36. The n index depends on the N dependence of K . Debye3I ~ ~ ~ convincingly for K suggested K N a n d de G e n n e argued N1I2. The de Gennes result is the commonly accepted one and using it yields n = Thus, the classical value of H = 2n = Also note that if 9 = 0, the y = 2v and eq 37b becomes
(44)
lim x ( q ) = N-H[2(tq)-z = N-H/qz = W m h / q 2 €e-
and thus g=h
Comparison of eq 43 with 22 indicates that H = 2f From eq 20 and eq 39, 41, and 44, we also have b = (3n - h ) / 2
(45) This completes our calculation of the N indices. A summary is given in Table I. As will be seen in the next section, h = 0 and H = 2n in the classical theory and h = 0 also when 9 = 0, which suggests that h is very small.
Theoretical Estimates of the N Indices Classical Treatment. A Landau free energy expansion around the critical point for a polymer solution takes the formzz~30 f / k T c = AtA4’ + BN’fZAd4 (46) where A and E are unimportant positive constants independent of N and e is defined as in eq 2, but with its sign. From this free energy it is easy to determine the pairs (a, a),(b, p), (g, y), and (d, 6).zz The results are summarized in Tables I1 and 111. There are two caveats worth mentioning. The first is that the predicted discontinuity in the heat capacity is finite (a = 0). The second is that the value of g = 0 is consistent with the usual definition of susceptibility given in eq 7. To obtain the exponent pairs ( m p ) , (n, u), and (h, 7) associated with the correlation function, a gradient energy term is added to eq 46 and the fourth-order term is dropped to obtain a LandauGinsburg free energy functional: F / k T c = $[eA&(R)
+ K ~ V ~ ( R ) d3R I’]
(47)
(28) Ma, S.-K. Modern Theory of Critical Phenomena; Benjamin: Reading, MA, 1976; Chapters 3 and 4. (29) Fisher, M. E. J. Marh. Phys. 1964, 5,944. (30) Sanchez, I. C. Macromolecules 1984, 17, 967.
-
(51)
which is identical in form with eq 50 and implies that h = 0 when 7 = 0 even though n # 1/4. Near the critical point the interfacial tension varies as33.34 0
N
N-’/2~1/2e-3/2
-
N
I\rl/4€-3/2
(52)
which yields m = l / q with K N112. The classical indices are summarized in Table 111. Note that x(0) = 6It2= e-l and is independent of K and yields g = 0 and y = 1 as previously obtained from the Landau free energy, eq 46. Also notice that the classical x(q), eq 48, has the same functional form as the nonclassical x ( q ) given by eq 37. Concentration fluctuations near the critical point, which the above classical treatment ignores, are responsible for the classical values yielding the incorrect values of the e indices and presumably the N indices as well. However, within the classical analysis, fluctuations can be crudely taken into account in a Gaussian approximation.28 Only the specific heat is affected by the fluctuations and it becomes C
N
~-3/2e-1/2 N N-3/4e-I/’
(53)
so that with fluctuations (a, a) = (3/4, as compared to (‘/z, 0) without fluctuations. The fluctuations turn the finite discontinuity into a infinite one, but the effect of fluctuations is overestimated in the Gaussian approximation. Sum rules are relationships among the N indices that hold for all values of the index h. As can be easily verified, not all (31) (32) (33) (34)
Debye, P. J. Chem. Phys. 1959, 31, 680. de Gennes, P.-G. Phys. Leu. 1968, 26A, 313. Sanchez, I. C. Annu. Rev. Mater. Sei. 1983, 13, 387. Widom, B. J . Srar. Phys. 1988, 52, 1343.
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6987
Critical Amplitude Scaling Laws for Polymer Solutions the sum rules in Table I are satisfied by the classical N indices. This is not surprising because the classical e indices do not satisfy all of the relationships for the e indices either. In the next section a set of N indices ill be developed that satisfy all of the sum rules. Scaling Arguments a la de Gennes. de G e n n e ~suggested ~~.~~ a scaling argument for determining the index b, ( 1 - p ) / 2 , and the index n, ( 1 - v ) / 2 . More recently Widom" used the de Gennes type argument to estimate m, ( 1 - p / 2 ) . Following Widom's recipe, we have completed the calculation of the N indices. The scaling procedure begins with the classical expression, say the susceptibility, which is rewritten in the form34 x = Me-' = N1/2(N'/2€)-' (54) Now replace the classical value of y = 1 with its nonclassical value, i.e. x = N1/2(N1/2e)Y = N'/2Y/2eY (55) which yields g = ( 1 - y ) / 2 . The variable N1I2eis the natural scale variable in the classical theory, and it is rescaled by the proper nonclassical exponent. Interestingly, the same result for the a index, a = ( 1 a)/2, is obtained for either pair of (a, CY) values listed in Table 11. The value of the d index is obtained by replacing A4 in eq 31 with Ar"e8 and proceeding as before to rescale N112e and then returning to the A+ representation to determine d . The h index is obtained by rescaling the classical N1/Z(N1/2q)-2 to the nonclassical N1/2(N'/2q)r/v and comparing the resulting exponent with y n / v - k = ( y / v ) ( l - v ) / 2 - h. The results are summarized in Table 111. Rather remarkably, all of the sum rules in Table I1 are satisfied by the de Gennes estimates. For example, 3n - 2b g yields 3v = 28 y , a = 3n yields 3v = 2 - a;Le., the N index sum rules yield the corresponding sum rules for the c indices shown in Table 11. Also notice that although the de Gennes estimates of the N indices all reduce to their classical values when the classical values of the c indices are substituted, the classical N indices do not satisfy all of the sum rules. LGW Model. Recently S t e p a n ~ wcalculated ~~ the b and n indices using a Landau-Ginsburg-Wilson model. The Hamiltonian for this model is obtained by adding the fourth-order term in eq 46 to the Landau-Ginsburg functional in eq 47, i.e. (all numerical factors are ignored)
+
+
+
TABLE 111: de Gennes Approximation for the N Indices
N index
de Gennes approx
a
(1 a ) / 2 = 0.555 (1 - /3)/2 = 0.337 (1 - y)/2 = -0.12 (6 - 2)/2 = 1.40 (1 - y ) / 2 = -0.12 (2 - p)/2 = 0.37 (1 - u)/2 = 0.185
classical
+
b g
d h m
n
'11 or
'14
'14
0 '12
0
'14 '14
TABLE I V Comparison of Experimental and Theoretical Values for the N Indices
N classical de Gennes index exptl values values approx a or 0.555 b 0.2840 '14 0.337 g 0.08: 0.03' 0 -0.12 d '11 1.40 h 0.03c 0 -0.12 m 0.39d '14 0.370 n 0.28,1*9 '14 0.185 0.197e
scaling laws LGW q = o model" generalb approx 0.51 0.585 0.57 0.27 0.285 -0.03 0.025 0 1.06 1.04 1.09 -0.03 0.025 0 0.34 0.38 0.17 0.195 0.19
'The indices a, d, h, and m were calculated by using the sum rules in Table I. Numerical estimates are based on experimental r = 0.38. bExperimental values of b and m were used to estimate the remaining indices by using the scaling laws of Table I. CAn average of three estimates; see eq 73 and subsequent discussion. "Determined in this study from interfacial tension data from ref 14. CDeterminedin this study from A@ data from ref 13 and the data analysis in ref 40. /Determined in this study from the experimental value of h and the scaling law g = h.
-
assumed that K N112as in the classical analysis, then the N indices b, g , and n that are obtained are identical with those obtained by the de Gennes argument (see Table 111). It can also easily be shown that the sum rule 3n = 26 g is satisfied for any assumed dependence of K on N . Actually Stepanow used the +I; N' which yields mean-field result that K
+
- -
(63)
+
H = eA42 + ~ ( 0 4 ) N'/2A44 ~
(64)
which can be recast into the standard LGW form that is amenable to analysis:
H , = H N - 3 / 2 ~=6 mo@
+ (V,M)2 + M4
(56)
where M =
(57)
&,3/2N-'/4
€K3wI
(58)
x = RNl/2~-2
(59)
C;
-
w I / ZK 2-3u t -v
(60)
A4
-
NlI4-8K38-3/2,8
mQ =
Stepanow then obtained and
(61) The susceptibility was not calculated, but it can also be obtained:
Notice that unlike the classical Gaussian approximation, the LGW model calculation yields results for A 4 and x that depend on K . However, if the e indices assume their classical values, A 4 and x become independent of K and E K ' / ~ as before. If it is
-
(35) Reference 24, pp 121 and 213. (36) Stepanow, S . J . Phys. 1987, 48, 2037.
n = (1 - v ) / 2 - ( 2 - 3v)(Y2 - r )
(65)
Notice that the leading term is the de Gennes approximation plus a correction term. Using the experimental value of r = 0.38 yields b = 0.27, g = -0.03, and n = 0.17, which should be compared with the values in Table 111. See also Table IV. It will be seen in the next section that existing experimental data are in good agreement with the mean-field interpretation that t2 K $;I.
--
Comparison with Experiment General Comments. Experimental values of three exponents (r, b, and m ) have been determined with a good deal of confidence in their accuracy. The molecular weight dependence of 4c for polystyrene in cyclohexane from three different studies were analyzed39 with the conclusion that r = 0.38 f 0.01 (see eq 5). Critical concentration data for polystyrene in another solvent, methylcyclohexane, are also a ~ a i l a b l e 'and ~ * ~are shown in Figure 1 . A least squares analysis again yields r = 0.38. The N index b associated with the order parameter has also been determined by a comprehensive analysism of high precision coexistence curve data for polystyrene in methylcyclohexane." 4 0327 = N'0.102e0.327. Us'1% It was determined that A4/4c (1vo.313 r = 0.38 yields a value of b = 0.28 (see eq lb).
-
'
(37) Reference 24, pp 115 and 214. (38) Sanchez, I. C. Encyclopedia of Physical Science and Technology; Academic Press: New York, 1987; Vol. XI. (39) Perzynski, R.; Delsanti, M.; Adam, M. J. Phys. 1987, 48, 115. (40) Sanchez, I. C. J. Appl. Phys. 1985, 58, 2871.
6988
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
Sanchez 50 0
-0.7
40
M-17,000 M-37.000
0
A
M-110,000
A
M-233.000
0
A
0
-0.9
0
13 0
A
0
30
-1.1
A
0 0
A
A
OD
-1.3
20
m 0%
*
A
A A
A
a -1.5
3.8
4.8
5.8
10
Log Molecular Weight
Figure 1. Log-log plot of the critical volume fraction (4,) versus molecular weight ( M ) for polystyrene in methylcy~lohexane.'~Tabulated data from ref 40 used to construct figure. Equation of least-squares line is log bC= 0.823 - 0.379 log M.
The N index m has been obtained from interfacial tension data of polystyrene/methylcyclohexane solution^'^ shown in Figure 2. These data were fitted according to eq I C with p fixed at its accepted nonclassical value of 1.26, Le., u = ufl-me1.26. The least-squares fit yields m = 0.39 f 0.01 and uo = 53.9 dynfcm (the actual best fit value of I.L from these data is 1.32). Reference 14 initially concluded that m = 0.44 f 0.03 but later acknowledged that their estimate was biased because they had included some data in their analysis that was out of the critical r e g i ~ n . ~ The correlation length index n has been determined,'^^ but its value of n = 0.28 is in significant disagreement with the theoretical estimates of de Genned5 and S t e p a n ~ wand ~ ~in disagreement with its estimated value based on the scaling laws in conjunction with the reliable experimental values of b and m (see Table IV). The theoretical and scaling law predictions all point to a value of n = 0.19 As will be discussed below, there is another experimental determination of n (0.197) that agrees well with theory. The index g associated with the susceptibility was determined in the same light scattering experiments in which the index n was determined.9 A value of g = +0.08 was obtained (my interpretation). As previously emphasized, S & N adopted a definition of susceptibility that differs from the commonly accepted one and that definition is also used in ref 9. Their susceptibility index is larger by a term of r from the usual one, and they obtained a value of 0.48 for their index using r = 0.40; thus, the conclusion that g = +0.08. Theoretical estimates of g calculated from the de Gennes scaling procedure, eq 5 5 , or from the LGW model, eq 62, indicate that g is slightly negative (see Table IV). The scaling laws in conjunction with the experimental values of b and m predict that g = +0.025. All of these comparisons are summarized in Table IV. The experimental determination of g, or equivalently h, may prove to be as difficult as the direct determination of the corresponding 9 index because g and h are probably of the order of in magnitude. Experimental values of the heat capacity ( a ) and the exchange chemical potential index (d)have not been experimentally determined. The remaining index h can be estimated from existing coexistence curve measurements as explained below. Table IV summarizes the theoretical and experimental estimates of the N indices. With the exception of the light scattering and to a lesser degree g, determined experimental value for there is good agreement between measured values and theoretical estimates. The general scaling law estimates used the experimental b and m values to calculate the other indices. Using the n = r / 2 approximation (h and 71 = 0) and the experimental value of r, the remaining indices can be calculated from the scaling laws. Note n,'99
0 Epsilon x 100
Figure 2. Interfacial tension data (a)plotted against e
( T - T,)/T, for
polystyrene in methylcyclohexane for the indicated molecular weights. Data from ref 14. that although in one case experimental values of b and m were used while in the other only the experimental value of r was used, good agreement is obtained between the estimated values of a, d , and n. This is taken as good evidence that the experimental values of b, m, and r are highly reliable. The theoretical estimates of b and m are also in excellent agreement with experiment. W i d ~ has n ~ suggested ~ that the de Gennes approximations for m and n may be highly accurate. Note that it is these m and n estimates that agree best with the scaling law estimates. By contrast, the de Gennes estimate for b is 20% higher than the experimental value. The LGW model estimates are in good agreement with the scaling law estimates and with experimental values (except for n determined from light scattering measurements). However, only b, n, and g values were directly calculated from the model (which satisfy 3n = 26 g ) and the rest were obtained via the sum rules. There is no reason to believe that the LGW model does not satisfy the sum rules especially since the de Gennes relations (which satisfy all of the sum rules) appear to be a special case of the LGW model. Support for n rf2. As discussed in the previous section, classical theory suggests that the coefficient of the square gradient term, K , varies as 4;'. The implication is that the N dependence of the correlation length, E, is completely determined by K (see eq 50):
+
E
&-1/2
MI2
(66) or n = r / 2 = 0.19, which agrees well with the theoretical estimates shown in Table 1V. In this approximation the remaining indices are all related to the exponent r (see Table I). For example, m = 2n = r, which implies N
N
-
N
&-'u eu (67) Thus, &-Iu should be a function of e only. In Figure 3 the interfacial tension data of Figure 2 are replotted in this manner, and as can be seen, the data nearly collapse into a single master curve as suggested by eq 67. We can test the approximation of n = r / 2 in still another way. If the h index is small and nearly zero, then b = 3 n / 2 by eq 45 or b = 3rf4 = 0.285 in excellent agreement with the experimental value of 0.28 (see Table IV).
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6989
Critical Amplitude Scaling Laws for Polymer Solutions
r
/
\
.
A
0
A
M-17,500 M-37,000
A
M-110.000
A
M-233.000
where the experimental value of r = 0.38 and the accepted nonclassical values of y and 0 (see Table 11) have been used. The h index can also be calculated from eq 45 using the experimental values of b = 0.28 and n = 0.197 which yields h = 0.031 or using the experimental n = m / 2 = 0.195 to obtain h = 0.025. This illustrates the self-consistency of the experimental values of r = 0.38, m = 0.39, b = 0.28, and n = 0.197. The value of h = 0.03 quoted in Table IV is the average of these three estimates of h.
A
d
I
A&
ABo* go
I
:
AnA
Epsilon x 100
Figure 3. Plot of #;’u versus c = ( T - Tc)/Tc for the interfacial tension data in Figure 2. Scaling relation, eq 67, suggests this result.
More support for n = r / 2 comes from a different analysis. As mentioned previously, it was found empirically that A4/& scaled asm A,#,/& (~.313f0.M)4t)0.327 as opposed to the classical result: A4/&
-
(N’/’t)‘/’
(69)
This raises the question: how is the empirical exponent 0.3 13 related to the N a n d e indices? Inspection of the correlation length [t Nz“(Na)-”]and the functional form of A$ given in eq 19 implies that t = n / v = 0.313, Le.,
-
Ad/&
-
Wf-’B‘(Nt)@
(70)
However, this scaling form must yield r - f - 2Pt = 0:
Using the c index relations in Table I1 yields
n=
-2r + -14( h - nq)
(72)
or in the q = 0 approximation, n = r / 2 . If n = r / 2 , then n / v = r / 2 v = 0.30 for r = 0.38 in good agreement with the experimental 0.31. Also note that n / v = in the classical limit as expected ( n = ‘I4and v = I/‘). This calculation also suggests that 2n = r is correct in the q = 0 approximation only. Determination of the n and h Indices. The identification of the empirical exponent 0.313 in eq 68 with t = n / v yields a new method for experimentally determining the exponent n from A$ data. The result is n = 0.313(0.630) = 0.197 f 0.002. To within experimental error this value of n equals the value obtained independently from the experimental value of m ( m = 2n) and is also in good agreement with the approximate value of n = 0.19 obtained independently from the exponent r. It is also interesting to note that if the classical value of n, = is rescaled by the intuitively appealing vo/v, where yo = ‘ / 2 is the classical value of v, then n = no(uo/v) = 1/8v = 0.198. We can use the experimental result, eq 68, along with this identification of n / v = 0.3 13 to calculate h from eq 7 1. Solving for h yields
Discussion and Conclusions Relation to Previous Work. Amplitudes of the critical properties of polymer solutions depend on the polymer’s degree of polymerization, N . The amplitudes vanish or diverge with characteristic exponents on N (herein called the N indices) in analogy with the usual critical exponents familiar in critical phenomena (herein called the t indices). Building on an important paper by Shinozaki and Nose,22we have derived a new set of scaling laws and associated “sum rules” for the N indices which are tabulated in Table I. These relations differ from those of S & N in two ways: The first is largely cosmetic. The S&N relations contain the critical exponent r, which has no analogue among the t indices, that characterizes the dependence of the critical concentration & on N (see eq s), whereas the relations in Table I do not. This extra term arises because of the unconventional manner that S&N define the susceptibility. Using the conventional definition yields scaling laws that are explicitly independent of r. However, there is a strong implicit dependence of the indices on r which was not recognized by S & N (see below). The second and more important difference is that we have identified a new N index h that corresponds to the t index q. As is well-known, q is the smallest of the t indices. If q is set equal to zero, the t indices can all be expressed in terms of the v index associated with the correlation length, l . The associated error involved in this approximation averages about 3% for the three indices (8, y, and 6) that depend explicitly on q (see Table 11). The q = 0 approximation corresponds to the h = 0 approximation for the N indices, and in analogy with the t indices, all of the N indices can be expressed as functions of the n index which characterizes the N dependence of correlation length amplitude (see Table I). Only three of the N indices depend explicitly on h ( b , g, and d ) , and we expect the associated error in this approximation to be comparable to that of the t indices, Le., small. S & N did not identify the h exponent, but rather a related exponent, herein defined as H,which is related to the more fundamental h by eq 41. This identification greatly simplifies the results and suggests the q = 0 approximation as discussed above. Experiments. The most serious disagreement between experimental and predicted values of the N indices happens to be the most important one, the n index associated with the correlation length. The experimental value of 0.28 obtained from light scattering mea~urements’*~ is significantly larger than the predicted values (0.17-0.195). However, the experimental value of n obtained from A4 data (0.197) is in excellent agreement with theory. Nevertheless the existing discrepancy is cause for concern. Future experiments will hopefully resolve the issue. In regard to future experiments, there are some identifiable needs. We have already discussed the need for more experimental measurements of n. Scattering experiments will yield not only the n index but in principle three other indices as well ( d , g, and h). However, the scaling analysis of this paper indicates that g and h are so small in magnitude (-0.03) that their direct determination offers a real experimental challenge which may be more frustrating than rewarding. On the other hand, the d index is predicted to be largest of all the N indices (- 1.O) and should be easily detectable. The corresponding 6 index has already been measured for polymer solutions4 by light scattering and an extension of that experimental protocol to a range of polymer molecular weights would yield the d index as well. An accurate determination of the d index, which has never been determined,
6990 The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
would be a valuable and welcomed addition to the reliably known b and m indices and the critical concentration exponent r. That leaves the a index, the one associated with heat capacity as an unknown. It is predicted to have a reasonable magnitude (-0.6) and should offer a reasonable expectation of being measurable by high precision calorimetry. The important n index can also be determined directly by osmotic pressure measurements, at least in principle if not in practice, since a - ac k T C / t 3 N-3ne3u. As will be discussed below, the N indices may be directly related to the size of the polymer chain at the critical point and the Flory exponent. Small-angle neutron scattering measurements using protonated polymer "doped" with a small amount of deuterated polymer of comparable molecular weight could address this issue. The measurements will probably not be as difficult as obtaining or synthesizing the requisite deuterated polymers over a range of molecular weights. A major conclusion of this study is that the experimental values of r, b, and m, as well as the value of n determinedfrom A4 data (0.197), appear highly accurate with uncertainties of fO.O1 at most. This conclusion is based on the self-consistency of using pairs of these values and the scaling laws to predict the remaining N indices. The best example of this self-consistency comes in the estimation of the index h. Using r and n, b and n, or b and m yields small positive values of h between 0.025 and 0.036. Theory. A classical analysis of the correlation length indicates that the N dependence of [ is determined by the dependence of the critical concentration q5c on N (see eq 66) so that n = r / 2 . Independent of the classical analysis, it is shown that the 7 = 0 approximation requires n = r / 2 . The argument is based on the empirical observation@that A ~ / C $ ~(1\10.313e)0.327 and identifying the empirical exponent 0.313 as n / v (see eq 70-72). Classically A4/& scales as (N1/2e)t/2, which suggests that N"/ve is the appropriate nonclassical scaling variable for polymer solutions. This identification in conjunction with the 7) = 0 approximation demands that n = r / 2 . Note that n / v equals 0.30 when n = r / 2 = 0.19 (the experimental value of r = 0.38)in excellent agreement when n and u take with the experimental 0.31. Also n / v = on their classical values of 1/4 and 1/2, respectively. Other reliable experimental data ( b and m values) are also in accord with n = r / 2 (see Table IV). A graphic demonstration is provided by the a'; interfacial tension data shown in Figure 2. If r = 4 2 , then 4 should be a function of e only. To a good approximation this result obtains as illustrated in Figure 3 . Our conclusion is that, to a excellent approximation, n = r / 2 and all of the N indices, with the possible exception of h and g, are related to r. Sum rules are relationships among the N indices that hold for any value of the index h. Similarly, sum rules among the t indices hold for any value of the index 7. The sum rules given in Tables I and I1 are by no means exhaustive, but are instead a kind of minimal basis set from which other sum rules can be derived. For example, from the basis set we also have 3n + 2d + g = 26b, which corresponds to 3v + y = 26@(this correspondence is explained below). It is unknown how many nontrivial sum rules exist, but it is doubtful that any more of them are exact analogues. de Gennes has proposed a procedure that yields nonclassical values of the N indices. Heretofore, the b, m, and n indices had been calculated and we have completed the calculation of the remaining four indices ( a , g, d , and h), which are all tabulated in Table 111. These theoretical approximations, unlike the classical indices, satisfy all of the sum rules in Table I. The de Gennes relations are remarkable because they transform a N index sum rule into the correct sum rule for the e indices and vice versa. For example, 3n = a yields 3v = 2 - a and inverting the de Gennes relations for a and n ( a = 2a - 1 and v = 1 - 2n) and forming 3v = 2 - a yields 3n = a. The de Gennes relations are the "transformationequations" between the e and N sum rules. Their significance is that although the de Gennes relations are approximations with unknown errors, the errors cancel exactly when forming the sums in Tables I or / I . A Landau-Ginsburg-Wilson model has been used by Stepato calculate the n and b indices and used by the author to calculate the g index (see eq 63-65). The theoretically calculated
-
-
-
Sanchez N indices depend explicitly on the exponent r in 4c as well the 6 indices. It can be shown that the sum rule 3n = 26 g is satisfied for arbitrary values of r. If r is set equal to its classical value of the de Gennes approximations are recovered. Satisfied that the LGW model must obey the remaining sum rules,the remaining four indices (a, d, h, and m) were calculated by using the sum rules. Numerical estimates for the LGW model based on the experimental r = 0.38 are tabulated in Table IV. Overall, the LGW estimates appear comparable to those of de Gennes which do not depend on r. The parallelisms between the N a n d e indices, heretofore unrecognized, is striking (see Tables I and 11). The five dependent t indices can be expressed in terms of the two independent and fundamental indices v and 7. Similarly, the corresponding five N indices can be expressed in terms of n and h. If 7 = 0, then h = 0 and there is only one independent t index (v) and only one independent N index (n), and both are associated with the correlation length. Only three of the five dependent e indices (& y, and 6) depend on 7, and the corresponding three dependent N indices (b, g, and d) are the only ones that depend on h. When 7 = 0, one of the e indices becomes a constant (6 = 5) and one of the N indices also becomes a constant (g = 0). Two of the five dependent e indices depend on u only ( a and p ) and the corresponding N indices ( a and m ) depend on n only. The 6 index is the largest e index, and d is the largest N index. Finally, there are the completely analogous sum rules: 3n = 26 g = m n and 3v = 28 y = p u and the deGennes transformation equations between "corresponding" sum rules. For example, 3n + d = (6 l ) b corresponds to 3v = (6 1)@. An important question is: Is the deviation of r from its classical value of caused by critical point fluctuations or is it due to other factors which may be amenable to a classical type analysis? The LGW model gives no clue. Muthukumar4' has developed a classical theory, in the sense that critical fluctuations are ignored, which assumes that monomer ternary interactions dominate at the critical point. H e obtained & N-'l3 and b = 2/g = 0.222. It is generally believed that both attractive binary and repulsive ternary interactions are comparable in magnitude a t the critical point, so the Muthukumar calculation is a possible lower bound on r if binary interactions are completely ignored. S t e p a n ~ w ~ ~ has already pointed out that r = 'I3is a theoretical lower bound based on the properties of the LGW model. If we adopt the physical picture that at the critical point the polymer chains are just beginning to ~ v e r l a p , ~then ~ * ~4, * must equal the average monomer concentration in a single chain at the critical point. The implication is that r is related to another critical exponent, the Flory exponent uF. The Flory exponent characterizes the size of a flexible polymer chain. If R, is the radius of gyration, or some other scalar measure of the chain size, then
+
+
+
+
+
+
+
-
R,
-
(74)
lVYF
In very dilute solution where the chains are isolated, we have ( 0.588 T >> 6' T-6'
-
Ix
(75)
T
