Macromolecules 1998, 31, 5853-5860
5853
Thermodynamic Interactions in Isotope Blends: Experiment and Theory Buckley Crist Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208-3108 Received December 23, 1997; Revised Manuscript Received June 19, 1998
ABSTRACT: Small-angle neutron scattering (SANS) studies of binary mixtures provide χNS, a measure of thermodynamic interactions between dissimilar polymer chains, one of which is usually labeled with deuterium. For polymers differing only in isotopic substitution (isotope blends), χNS is seen to diverge strongly upward (or sometimes downward) at low concentrations of either blend component. This concentration dependence seems to vanish in the limit of large degree of polymerization N. Experimental results can be described by χNS(φ,N) ) β + γ/Nφ(1 - φ), where φ is the volume fraction of deuterated polymer. For SANS from a series of blends with different φ it is shown that systematic errors in N and/or the static structure factor S(0) lead to precisely the same χNS(φ,N) when the Flory-Huggins interaction parameter χ is constant. While results for some isotope blend systems can be accounted for with reasonable error estimates, others appear to have a real dependence of χNS on φ and N. It is suggested that these “non-Flory-Huggins” effects stem from a modified entropy of mixing that is most evident in dilute blends. The concentration dependence of χNS(φ) has no practical effect on macroscopic phase behavior.
Introduction Thermodynamic interactions between polymer chains are often studied by small-angle neutron scattering (SANS), with one component of a binary blend labeled with deuterium. After the suggestion by Buckingham and Hentschel1 that isotopic substitution would lead to small but observable repulsive interactions in polymer blends, this effect was seen in SANS experiments by Bates et al.,2 in interdiffusion studies by Green and Doyle,3 and in phase separation observed by Bates and Wiltzius.4 While the “isotope effect” is known to modify interactions between chemically dissimilar polymers, that aspect of the problem is reasonably well understood.5,6 More perplexing are SANS results for blends of polymers differing only in isotopic substitution. These appear, at the present, to challenge our understanding of the thermodynamics of polymer-polymer blends. Consider the SANS results from Londono et al.7 from narrow molecular weight fractions of conventional and deuterated polyethylene (PE/d-PE) shown in Figure 1. Most striking is that the apparent interaction parameter χNS has a strong dependence on blend composition, expressed as volume fraction of d-PE. The interaction parameter is obtained in the usual way, starting with the general expression for coherent cross-section per unit volume I(q):
I(q) )
(
)
b1 b2 2 S(q) v1 v2
(1)
Here bi/vi is the scattering length density of the monomer in polymer i ()1, 2), and the static structure factor S(q) is the absolute intensity normalized by the contrast factor in parentheses. Intensity is extrapolated to zero scattering angle (q ) 0) to obtain S(0), which is written as
2χNS 1 1 1 + ) v0 S(0) N1φ1v1 N2φ2v2
(2)
Figure 1. Dependence of χNS on blend concentration φd for (b) isotope blends PE/d-PE, N ≈ 4400 (C2H4 monomers), T ) 155 °C, from Londono et al.7 and (O) model copolymer blends HPB(97)/DPB(88), N ≈ 3200 (C2H4 monomers), T ) 83 °C, from Krishnamoorti et al.10 The dashed line is a fit of PE/d-PE data to eq 5.
Ni are (weight average) degrees of polymerization, vi are monomer volumes, φi are volume fractions, and v0 ) (v1v2)1/2 is the reference volume. Equation 2 is rearranged to give the apparent interaction parameter from SANS:
χNS ≡
(
)
v0 v0 1 1 + 2 N1φ1v1 N2φ2v2 2S(0)
(3)
There is no formal restriction on the method used to extrapolate intensity or S(q) to q ) 0. This step is most often done with the q > 0 variant of eq 2 obtained with the random phase approximation.8 It should be emphasized that incompressible RPA theory, or, equivalently, Flory-Huggins lattice theory for a polymerpolymer blend, is invoked in the evaluation of χNS by eqs 2 and 3. While the simplest Flory-Huggins thermodynamic model has the interaction parameter χ being independent of blend concentration φi and component degree of polymerization Ni, this is not required. The relation
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Macromolecules, Vol. 31, No. 17, 1998
Figure 2. χNS at T ) 155 °C versus 1/Ne for PE/d-PE blends having different φd; data from Londono et al.7 The concentration dependence of χNS is proportional to N-1.
Figure 3. χNS(φ) for PE/d-PE, N ≈ 4400, at two temperatures above the critical temperature Tc ) 132 °C (405 K); data from Londono et al.7 Concentration dependencies of χNS are indistinguishable at T/Tc ) 1.02 (b) and at T/Tc ) 1.17 (O).
between the thermodynamic interaction parameter χ and the SANS result is given by
χNS ≡
2 1 ∂ (φ1φ2χ) 2 ∂φ1∂φ2
(4)
If and only if χ is independent of concentration, then χNS ) χ. Any concentration dependence of the thermodynamic χ will lead to a larger change in the SANS quantity χNS. It was shown9 that the ca. 4-fold increase in χNS seen in Figure 1 corresponds to a much smaller, ca. 30%, increase in χ. Another observation is that the concentration dependence of χNS is a function of N (or molecular weight) and appears to vanish in the limit of infinite N (Figure 2). Following Krishnamoorti et al.,10 these effects can be summarized by the following empirical expression:9
χNS ) β +
γ Neφ1φ2
Figure 4. Temperature dependence of χNS(T) ) A + B/T for PE/d-PE of different φd, N ≈ 4400; data from Londono et al.7 The increase in χNS at low φd seen in Figures 1 and 3 are associated with larger “A”. Table 1. Concentration Dependence of χNS in Polymer Isotope Blends
(5)
Here Ne ) N1N2/(φ1N1 + φ2N2) is the appropriate average degree of polymerization for blends in which N1 * N2. The dashed line in Figure 1 is a fit of the PE/ d-PE data to eq 5. It is emphasized that there is no theoretical foundation for eq 5, but the second “nonideal” term has a structure reminiscent of that deriving from the usual combinatorial entropy of mixing. Also instructive is the temperature dependence of χNS. First, note that the upward curvature of χNS(φ) is virtually unaltered by proximity to the critical temperature (Figure 3). Blends of different φd each display the conventional temperature dependence χNS(T) ) A + B/T, but the coefficient A (implying an entropic contribution to the interaction) has by far the greatest sensitivity to concentration (Figure 4). Although the data of Londono et al.7 on PE/d-PE are the most comprehensive in terms of φd, N, and T dependence, similar results are seen with most isotope blends. The first report of upward curvature of χNS(φ) was by Bates et al.11 for PEE/d-PEE and PVE/d-PVE; those data closely resemble the effect shown in Figure 1. (PEE is equivalent to atactic poly(butene-1), having an ethyl branch on every second backbone carbon; PVE has unsaturated vinyl branches.) Krishnamoorti12 has recently studied isotope blends of 1,4-polyisoprene, poly(ethylene oxide), and poly(methyl methacrylate); in each
χNS(0.1) blend
T (K)
10-3Ne
PEE/d-PEE PVE/d-PVE PE/d-PE PI/d-PI PEO/d-PEO PMMA/d-PMMA PS/d-PS PS/d-PS
299 310 428 300 347 381 433 439
1.50 1.50 4.36 1.18 2.39 1.58 9.91 9.47
a
NeχNS(0.5) χNS(0.5) 1.3 1 1.9 0.9 1.1 0.5 1.8 1.6
1.78 1.67 1.88a 1.31 1.24a 1.32 0.80b -0.65a
c
ref
1.4 1.3 1.5 1.1 0) alters neither the thermodynamics nor scattering of a binary blend in a significant manner. Any comparison of analyses by compressible and incompressible models should account properly for differences in structure factors S(0) and cell or reference volumes v. When this is done, the concentration independent interaction energy embodied in χ (or χNS) is affected only by a constant factor (1 - φv). There is no divergence of χNS(φ) that can be attributed to compressibility of an
5860 Crist
Macromolecules, Vol. 31, No. 17, 1998
Figure 10. Entropic contribution to the free energy of mixing as a function of blend concentration for ideal mixing (s) and the empirical eq 12 for γ ) 0.167 (- - -) and for γ ) -0.055 (‚-‚).
experimental blend. It is obvious, however, that mixing elements of the two approaches, e.g., using a compressible vc/Sc(0) on the right-hand side of eq 3, when the combinatorial entropy terms are based on in incompressible model, will cause a meaningless divergence of “χNS” at low concentrations. In fact, this is just another example of a systematic error (from theory, not experiment) affecting S(0) and hence χNS(φ). The need for consistent application of a particular model has been emphasized by Taylor et al.26 Appendix 2 The empirical expression for χNS(φ,N) in eq 5 can be integrated twice over φ1 to obtain the following expression for the thermodynamic χ consistent with the FloryHuggins model:9,10
χ)β-
2γ (φ ln φ1 + φ2 ln φ2) Neφ1φ2 1
(18)
The second term can be assigned to the departure from ideal entropy of mixing, leading to a phenomenological equation for symmetric blends (N1 ) N2 ) N):
-N∆Sm ) (1 - γ)(φ1 ln φ1 + φ2 ln φ2) k
(19)
This modified version of the entropy of mixing is plotted in Figure 10 for γ ) 0.17 (equivalent to c ) 1.5 for PE/ d-PE) and γ ) -0.055 (equivalent to c ) 0.9 for PS/dPS). Be reminded that there is no theoretical basis for the particular form of eq 19, which derives from experimental measurements of χNS(φ,N). For γ > 0 (upward curvature) this model reflects the reduced entropy of mixing expected from general consideration of chain connectivity. Downward curvature is here associated with γ < 0, implying that ∆Sm > ∆Sid m, which seems impossible if the ideal entropy is indeed the maximum. This is another reason for believing that negative curvature of χNS(φ) has causes other than thermodynamic ones (e.g., systematic errors). References and Notes (1) Buckingham, A. D.; Hentschel, H. G. E. J. Polym. Sci., Polym. Phys. 1980, 18, 853. (2) Bates, F. S.; Wignall, G. D.; Koehler, W. C. Phys. Rev. Lett. 1985, 55, 2425. (3) Green, P. F.; Doyle, B. L. Phys. Rev. Lett. 1986, 57, 2408. (4) Bates, F. S.; Wiltzius, P. J. Chem. Phys. 1989, 91, 3258.
(5) Rhee, J.; Crist, B. J. Chem. Phys. 1993, 98, 4147. (6) Graessley, W. W.; Krishnamoorti, R.; Balsara, N. P.; Lohse, J. D. Macromolecules 1993, 26, 1137. (7) Londono, J. D.; Narten, A. H.; Wignall, G. D.; Honnell, K. G.; Hsieh, E. T.; Johnson, T. W.; Bates, F. S. Macromolecules 1994, 27, 2864. (8) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (9) Crist, B. J. Polym. Sci., Part B: Polym. Phys. 1997, 35, 2889. (10) Krishnamoorti, R.; Graessley, W. W.; Balsara, N. P.; Lohse, D. J. J. Chem. Phys. 1994, 100, 3894. (11) Bates, F. S.; Muthukumar, M.; Wignall, G. D.; Fetters, L. J. J. Chem. Phys. 1988, 89, 535. (12) Krishnamoorti, R. Unpublished results. (13) Schwahn, D.; Hahn, K.; Streib, J.; Springer, T. J. Chem. Phys. 1990, 93, 8383. (14) Lin, C. C.; Jonnalagadda, S. V.; Balsara, N. P.; Han, C. C.; Krishnamoorti, R. Macromolecules 1996, 29, 661. (15) Maurer, W. W.; Bates, F. S.; Lodge, T. P.; Almdal K.; Mortensen K.; Fredrickson, G. H. J. Chem. Phys.1998, 108, 2989. (16) Taylor-Maranas, J. K.; Debenedetti, P. G.; Graessley, W. W.; Kumar, S. K. Macromolecules 1997, 30, 6943. (17) Schwahn, D. Personal communication. Mu¨ller, G.; Schwahn, D.; Eckerlebe, H.; Rieger, J.; Springer, T. J. Chem. Phys. 1996, 104, 5326. (18) Crist, B.: Nessarikar, A. Macromolecules 1995, 28, 890. (19) Scheffold, F.; Eiser, E.; Budkowski, A.; Steiner, U.; Klein, J. J. Chem. Phys. 1996, 104, 8786. (20) “Reference Standard Polyethylene Resins and Piping Materials”, GRI-86/0070, Gas Research Institute, Chicago, IL, 1987. (21) “Reference Standard Polyethylene Resins and Piping Materials”, GRI-87/0326, Gas Research Institute, Chicago, IL, 1988. (22) Olvera de la Cruz, M.; Edwards, S. F.; Sanchez, I. C. J. Chem. Phys. 1988, 89, 1704. (23) Equation 11 of ref 11 adds to the standard Flory-Huggins free energy of mixing a term that reduces to 2x2/πN3/2 for φ ) 0 or φ ) 1. Hence the free energy of mixing does not vanish in the limits of zero concentration, a thermodynamically impossible result that leads to the upturn of χNS at low or high φ. (24) Freed, K. F. J. Chem. Phys.1988, 88, 5871. (25) Bidkar, U. R.; Sanchez, I. C. Macromolecules 1995, 28, 3963. (26) Taylor, J. K.; Debenedetti, P. G.; Graessley, W. W.; Kumar, S. K. Macromolecules 1996, 29, 764. (27) Dudowicz, J.; Freed, K. F. Macromolecules 1990, 23, 1519. (28) Dudowicz, J.; Freed, M. S.; Freed, K. F. Macromolecules 1991, 24, 5096. (29) Dudowicz, J.; Freed, K. F.; Lifschitz, M. Macromolecules 1994, 27, 5387. (30) Kumar, S. K.; Veytsman, B. A.; Maranas, J. K.; Crist, B. Phys. Rev. Lett. 1997, 79, 2265. (31) Singh, C.; Schweizer, K. S.; Yethiraj, A. J. Chem. Phys. 1995, 102, 2187. (32) Melenkevitz, J.; Curro, J. G. J. Chem. Phys. 1997, 106, 1216. (33) Melenkevitz, J. Macromolecules, in press. (34) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953; Chapter 12. (35) Equation 12 is not exact, even for the case of constant volume fraction of voids considered here. The complete expression for the inverse structure factor in a compressible blend (see eq 2.11 of ref 27) contains additional terms that may be written as
2χcorr )
[1 - N2φ2(χ12 - χ1v + χ2v)]2 N2φ2[(2χ2v - fs) - 1]
With values appropriate for SANS experiments on isotope blends (Nχ12 ≈ 1, χ1v ) χ2v ≈ 2, and fs )1/φv ≈ 10) the correction term χcorr is of order χ12/N and may safely be neglected.
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