Study of Microphase Separation in Solid, Binary Alkane Mixtures by

A combination of small angle neutron (SANS) and small angle X-ray scattering (SAXS) is used to study phase separation following crystallization in 1:1...
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J. Phys. Chem. 1996, 100, 1725-1730

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Study of Microphase Separation in Solid, Binary Alkane Mixtures by Small Angle Neutron and X-ray Scattering B. K. Annis,*,† J. D. Londono,† and G. D. Wignall‡ Chemical and Analytical Sciences DiVision and Solid State DiVision, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6197

R. G. Snyder Department of Chemistry, UniVersity of California, Berkeley, California 94720-1460 ReceiVed: May 4, 1995; In Final Form: October 24, 1995X

A combination of small angle neutron (SANS) and small angle X-ray scattering (SAXS) is used to study phase separation following crystallization in 1:1 mixtures of n-C30H62/n-C36D74 quenched from the melt to room temperature. The SAXS data show that the quenched mixture is a solid solution with a well-defined lamellar structure which is established on a time scale of minutes after quenching from the melt. On this initial time scale the SANS data indicate that the deuterated and protonated chains remain essentially randomly mixed. At longer times the scattering curves from both techniques show features consistent with phase separation in the form of nonrandom stacking of lamellae that are alternately enriched in one of the components. The possible effect on the neutron scattering of lateral intralamellar phase separation appears to be overwhelmed by this interlamellar phase separated structure.

Introduction Polymer blends have attained widespread commercial applications, though understanding of their properties has been handicapped by the absence of a consensus concerning the principles governing the miscibility and demixing (phase separation) of the components. Small angle neutron scattering (SANS), based on the ability to manipulate local scattering amplitudes by deuteration, is the premier technique for investigating polymer phase behavior,1-3 and isotopic mixtures have proven to be ideal model systems for probing the underlying principles governing the thermodynamics of both miscible4,5 and phase separated systems.5-7 In particular, binary paraffin mixtures have been shown to undergo microphase separation in the solid state and have been studied as model systems in a series of experiments to investigate the mechanisms that underlie such transformations. The electron diffraction and differenial scanning calorimetry (DSC) measurements of Dorset8 established that fractionation of melt crystallized samples of n-C30H62/n-C36H74 occurs on a time scale of days at room temperature. The diffraction measurements indicated the presence of a layered structure of periodicity of 4.6 nm. After a few days a superstructure was found to be imposed on the layer structure. Subsequently, Snyder et al.9,10 observed the lateral development of domains of the unmixed components in n-C30H62/n-C36D74, also at room temperature, by using vibrational spectroscopy. In this case a lateral domain structure was detectable approximately 1 h after quenching and continued to grow for the 4 day duration of the experiment. Deuteration is particularly useful in neutron scattering as well since significant contrast occurs between regions composed of hydrogenated or deuterated components. This was exploited in a study of small angle neutron scattering from a 1:4 n-C30H62/n-C36D74 mixture by White et al.11 with † Chemical and Analytical Sciences Division, Oak Ridge National Laboratory. ‡ Solid State Division, Oak Ridge National Laboratory. X Abstract published in AdVance ACS Abstracts, January 1, 1996.

0022-3654/96/20100-1725$12.00/0

the objective of detecting superstructure. In that work a series of measurements extending over 12 h after quenching found changes in the form of inflections corresponding to Bragg spacings of 18 nm and (possibily) 6.5 nm. Neither feature has an obvious correlation with the layer structure indicated by the electron diffraction work.8 For example, the periodicity for a repeat unit consisting of one layer of n-C30H72 and four layers of n-C36H74 is estimated to be about 23 nm. It might be thought that the peaks could arise from regular spacing of the lateral domains, however, a domain size of approximately 2 nm2 after 12 h from quenching can be inferred from the spectroscopic measurements, and for a 1:4 mixture with regular lateral domains a spacing of 3-3.5 nm would be expected. In an effort to clarify the structural characteristics of the fractionation process at approximately 25 °C, we report here on SANS and SAXS (small angle X-ray scattering) measurements with similar time scales for a 1:1 mixture of n-C30H62/ n-C36D74. In addition SANS data for a 1:1 mixture containing 5% of n-C36D74 are discussed. Experimental Section The deuterated alkane was obtained from MDS Isotopes with a deuterium concentration specified as >99.0 at. %. The C30H62 was obtained from ICN Pharmaceuticals and had a purity >99%. The SANS measurements were made from samples in fused quartz cells of 2 mm thickness. The SAXS samples were 1 mm thick, molded in washers. In both cases the samples were maintained for several hours in the molten state and were quenched to approximately 25 °C. The SANS measurements typically lasted 20 min, and the SAXS measurements took 45-60 min. The reported times correspond to the time lag between the quench and the midpoint of the data collection period. The SANS data were obtained on the 30 m instrument12 at the W. C. Koehler small angle scattering facility of Oak Ridge National Laboratory (ORNL) via a 64 × 64 cm2 area detector with element size ∼1 cm2. The range of the momentum transfer, Q, was 0.05-3.5 nm-1, but the bulk of the data was © 1996 American Chemical Society

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Figure 1. SANS from 1:1 n-C30H62/n-C36D74. The times after removal from the oven are to the midpoint of the data collection period. In order from the bottom most curve: O, 0.3 h; b, 3.3 h; 3, 7.5 h; 1, 12.2 h; 0, 16.7 h; 9, 28.9 h; 4, 1163 h.

obtained for a range of momentum transfer Q of 0.2-1.8 nm-1. Here Q ) (4π/λ) sin(θ/2) with λ ) 0.475 nm, the neutron wavelength, and θ, the scattering angle. Standard corrections for data were then converted to an absolute scale by comparison with secondary standards13 and reported as differential cross sections per unit volume, (dΣ/dΩ), with units of (sr‚cm)-1. The SAXS measurements were undertaken at the 10 m facility14 also at ORNL. The incident radiation was copper KR (λ ) 0.154 nm) from a rotating anode source equipped with a monochromator, and different distances were used which resulted in a Q range of 0.05-4.8 nm-1. Here also standard corrections were applied and the data converted to an absolute differential scattering cross section.15 Procedures for rigorously removing the instrumental broadening are not available. However, estimates of the effect on the peak widths are included in the subsequent discussion.

Annis et al.

Figure 2. SANS from 1:1 n-C30H36/n-C36H74, D74, where the fraction of deuterated material is 0.05. The dashed line represents the estimate of the contribution from incoherent scattering. Annealing time was 1166 h.

Results

Figure 3. 0, Lorentz corrected SANS data for an annealing period of 398 h. The incoherent background of 0.5 cm-1 has been subtracted. The solid line is from a model curve normalized to the height of the first peak in the data.

SANS results for a series of annealing times between 0.3 and 1163 h are shown in Figure 1 for a 3 m sample to detector distance (0.2 < Q < 2 nm-1). The sample was remelted, and measurements were made with a 15 m distance (0.04 < Q < 0.4 nm-1) for times of 0.3 and 16.7 h. The sets of data corresponding to similar annealing times were spliced together and have been plotted as a single curve. A final data set was obtained at 3 m after an annealing period of 1163 h. A peak at 0.75 nm-1 corresponding to a Bragg spacing of 8.4 nm can be seen to develop with time. The scattered intensity in the region 0.25-0.50 nm-1 can also be seen to increase. By using the data at 0.3 h as a base line, it was found that the ratio of the peak height to the minimum at about 0.4 nm-1 increased by a factor of 2 during the duration of the experiments. As may be seen in Figure 2, a sample composed of 1:1 n-C30H62n-C36H74,D74 where the fraction of deuterated material is 0.05 also shows a peak at 0.75 nm-1 after annealing for 1166 h. Data which were taken at lower resolution (and therefore not directly comparable with that shown in Figure 2) also indicated the presence of a small peak after 88 h. Due to the high hydrogen atom content the data for this sample are dominated by incoherent scattering (dΣ/dΩ ∼ 1 (sr‚cm)-1). Data from the 50:50 H/D sample taken 398 h after removal from the oven are shown in Figure 3. In this case the data are plotted as Q2 dΣ/dΩ(Q) vs Q, which is sometimes referred to

as Lorentz corrected data. In addition to the peak at 0.75 nm-1, there is a small shoulder at Q of about 1.4 nm-1 and a second peak at Q ) 2.2 nm-1. The latter is approximately correct for a third order reflection due to an 8.4 nm spacing. The SAXS data for a 50:50 H/D sample over a Q range similar to that in Figure 1 is shown in Figure 4. The major feature is a peak at 1.40 nm-1 corresponding to a Bragg spacing of 4.5 nm. For purposes of comparison, the peak for pure n-C36H74 gave a spacing of 4.8 nm in good agreement with the X-ray results of Teare (4.76 nm)16 and the electron diffraction work of Dorset (4.86 nm).8 A spacing of 4.0 nm was found for n-C30H62, which agrees with the value given by Broadhurst.17 Only a relatively small change occurs in the 15.5 h time period as compared to the changes in the SANS data for 16.7 h. By 808 h clear changes have occurred to the left and right of the peak. This aspect was further explored by remelting the sample and extending the Q range to 5 nm-1. These data are shown in Figure 5. The three major peaks were clearly visible in the SAXS data 10 min after removal of the sample from the oven, and after 28.3 h of annealing the scattered intensity in the regions between the peaks has clearly increased. By 808 h, if not before, a pair of secondary peaks has developed. SAXS measurements were also made on n-C30H62/n-C36H74 and gave similar results.

Microphase Separation in Binary Alkane Mixtures

J. Phys. Chem., Vol. 100, No. 5, 1996 1727 Figure 5 are predominantly determined by instrumental broadening. In Figure 4, the instrumental broadening contributes about 50% to the full width at half-maximum (fwhm). In Figure 4 (SAXS) there appears to be comparatively little difference between the positions and widths of the first peak as the phase separation indicated by the SANS measurements occurred. This is in accord with model calculations for a solid solution (assumed to have sharp interlamellar boundaries) and a model of regularly alternating lamellae of similar thicknesses. The development of the intermediate peaks seen in the SAXS data in Figure 5 are a more sensitive indication of structural change. These are analogous to the observations of the electron diffraction work8 and would not occur in a solid solution. The solid curve in Figure 5 was calculated from

I(Q) ) I0(sin2((NL/2)Q)/sin2(QL))(F12 + F22 + 2F1F2 cos(QL)) (1) Figure 4. SAXS results from 1:1 n-C30H62/n-C36D7. The times after removal from oven are to the midpoint of the data collection period; O, 0.8 h; b, 15.5 h; 3, 808 h; 1, 1173 h.

Figure 5. Lorentz corrected SAXS data: O, 0.3 h; 3, 28.3 h; (, 814 h. The solid line is from a model curve normalized to the height of the first peak in the data for 814 h.

Discussion In the SAXS data for values of Q less than about 0.4 nm-1 a rapid increase occurs which is typical of a polycrystalline material and is generally attributed to the presence of cracks or voids on the scale of a few tens of nanometers. The SANS data for the shortest time are similar in shape below a Q of about 0.25 nm-1, and the visual appearance of both samples was somewhat cloudy which indicates that even larger inhomogenites of this type are present. The SAXS measurements show that a lamellar structure with a long period of 4.5 nm was present after 10 min of annealing. This compares with the value of 4.56 nm found earlier by Dorset8 by electron diffraction and is intermediate between the spacing for the individual components. The positions of the three major peaks in Figure 5 for times of 0.3 and 28.3 h (roughly 1.4, 2.8, and 4.2 nm-1) and the monotonic decrease in their intensities with Q can be understood as the first and higher order reflections from an array with single periodicity, in which there is a significant difference between the length of the scattering element (molecular chains) and the gap between the chain ends. At short annealing time this would be the case for crystalline lamellae formed from a solid solution of the two molecules. The SAXS peak widths of the first two peaks in

where

Fi ) 2 sin(WiQ/2) I0 is an arbitrary constant; N is the number of pairs of lamellae; Wi is the thickness of lamellae i; L is the sum of the average of the lamellae thicknesses and the gap between the chain ends. The calculated curve was normalized to the height of the first peak in the data for 1163 h, and 10 pairs of alternating electron rich regions with lengths of 4.45 and 3.68 nm and a gap of 0.31 nm were used. These dimensions were inferred from the work of Teare16 and Broadhurst.17 The presence of the two periodicities results in the intermediate peaks. The highfrequency oscillations at the base of the peaks are highly dependent upon N and would in any case be smoothed by instrumental broadening and imperfections in the actual lamellae structure. The peak widths and positions are also somewhat dependent on the number of lamellae chosen, and the model is only intended to demonstrate qualtitative agreement with measurements. Variation of the lengths and gap size over a few percent did not improve the match with experiment, and exceeding this range did increase the discrepancies. The lamellar peak at short times in Figure 4 for the SAXS data is only just discernible in the SANS data if the first curve of Figure 1 is highly magnified. This is due to two reasons. The X-ray and neutron cross sections are proportional to the product of the volume fractions and the square of the difference of the electron (X-rays) or scattering length (neutrons) densities, of the regions containing molecular chains and the regions between the chain ends.18 An estimate of the ratio of squares of the appropriate density differences indicates that the X-ray scattering should be greater by a factor of 5.5. As the X-ray peak has a height of ∼2 cm-1,we would expect a corresponding SANS peak of ∼2/5.5 ∼ 0.4 cm-1 on an incoherent (flat) background of similar magnitude. Thus, the lamellae peak is much more easily observed in the SAXS (as compared to the SANS) data. There are two additional sources of neutron scattering which affect the data. The first is incoherent scattering arising principally from the wavelength dependent cross section18 of H1 atoms (σinc ∼ 90 × 10-24 cm2 at λ ) 0.475 nm-1). Due to multiple scattering effects, the (flat) incoherent background is not a true cross section and depends on the sample thickness, transmission, etc. However, it may be estimated to a good approximation by scaling19 from 100% protonated “blanks”, and for 50/50 samples this amounted to subtracting a flat background ∼0.5 cm-1. The second arises in any homogeneous mixture of deuterated and protonated chains due to the difference in the

1728 J. Phys. Chem., Vol. 100, No. 5, 1996 scattering lengths of the hydrogen isotopes and, in the absence of other sources of scattering, has frequently been used to provide a quantitative characterization in terms of the radius of gyration of the chains in mixtures of equal length components.18 The slope of the bottom curve in Figure 1 for 0.4 nm-1 < Q < 1.0 nm-1 corresponds to an Rg of approximately 1.3 as expected for a random mixture of H- and D-labeled extended chains of length 4-5 nm. During the first 29 h of annealing, dramatic changes occur in the SANS data. At 0.3 h the monotonically decreasing curve indicates that the deuterated and protonated chains are in the main, randomly mixed. At the same time the SAXS data demonstrate that a lamellar structure has developed. Subsequently, a SANS peak develops at Q ) 0.75 nm-1, corresponding to a period of 8.4 nm. Since there is no indication of a similar peak in the SAXS data, this must arise from the fractionation of the deuterated and protonated chains and is consistent with the idea of Mazee20 and Dorset8 of a superstructure of alternating lamellae of the two molecules. For example, in first approximation, if it is assumed that the lamellae are equivalent to those in the pure components, then the long period is the sum of 4.8 nm for hexatriacontane and 4.0 nm17 for triacontane or 8.8 nm which is close to the observed value. There is a small shift to lower Q as time increases, possibly indicating increased extension of the chains. Secondly, Figure 3, albeit for a longer annealing period, indicates that the second order peak in the vicinity of 1.4 nm-1 is suppressed relative to the third at 2.2 nm-1, and this is typical of diffraction from alternating lamellae of approximately the same size (cf. ref 21, p 655). A calculated curve based on a one dimensional model22 for 10 pairs of alternating lamellae in which it is assumed that the long period is split between a deuterated region 4.6 nm in length and a protonated region of 4.2 nm is also shown in Figure 3. The functional form of this model can be obtained from eq 1 by setting W1 ) W2 ) W, where W is the deuterated chain and the gap between chain ends is the region of null scattering as discussed below. These dimensions were chosen by the following rationale: (a) deuterium atoms on the ends of the C36 chains scatter neutrons with nearly the same efficiency as the carbon atoms, and consequently the effective chain length is sightly longer than the carbon chain; (b) because of the negative scattering length of hydrogen, the scattering from the C30H62 chains is essentially null and indistinguishable from the interlamellar gap. The effective spacing between the ends of the C36D74 chains is thus slightly greater than the interlamellar thickness of pure C30H62. The results are not too dependent upon the values used for the lengths but are sensitive to the sum of them. This was chosen to be that expected for the sum of the lamellae for the pure components. As was the case for the SAXS data, the calculated peak locations and relative magnitudes are in qualitative agreement with experiment. However, the mismatch of the observed and calculated first peaks suggests that the chains are not fully extended. The instrumental broadening for the SANS geometry of Figure 3 is estimated to be about 80% of fwhm, so the difference between the peak widths of the calculated and observed curves is not meaningful. The appropriate model for a random distribution of an equal but arbitrary number of C36 and C30 lamellae is not known to us. However, a calculation for 4 of each lamellae for both the random and perfectly alternating cases is shown in Figure 6. From this it is apparent that the interlamellar phase separation produces a structure dominated by the latter, a schematic illustration is shown in Figure 7.

Annis et al.

Figure 6. Model calculations for four lamellae each of 4.2 and 4.6 nm thicknesses. The solid curve is for alternating lamellae, and the dashed curve is for random stacking.

Figure 7. Schematic illustrations of the structures of the 1:1 binary n-alkane mixture A/B, where A ) C30H62 and B ) C36D74. (a) One layer of the unstable solid solution obtained immediately after quenching the melt to room temperature. The solid, in which the chains are randomly mixed, is highly crystalline except in the interfacial region, where the ends of the chains tend to be conformationally disordered in order to minimize packing voids. (b) A layer of the same mixture after it has aged for approximately a day at room temperature. During this period microphase separation has taken place leading both to the formation of lateral domains of A and B and to a significant reduction of chain-end disorder. Domain growth and conformational ordering can be monitored spectroscopically.9,10,23 (c) The crystal structure of an aged mixture in which layers of pure A and B alternate. The experimentally determined layer enrichment of ∼4:1 is not shown for simplicity.

The data for a sample containing a fraction of only 0.05 of the deuterated material are shown in Figure 2 and are also consistent with this model. The only discernible feature is a peak also at Q ) 0.75 nm-1 with a much lower magntiude than that for the fully deuterated sample as expected. It is difficult to make direct comparisons with previous SANS studies11

Microphase Separation in Binary Alkane Mixtures

J. Phys. Chem., Vol. 100, No. 5, 1996 1729

because the scattering from the superstructure was superimposed on a rapidly varying background that decreased by several orders of magnitude over the observed Q range. Thus, only changes in slope were visible in the measured cross section (see ref 11, Figure 1). By comparison, the peaks in this investigation are resolved directly (Figure 1). At present it is not known whether this difference arises from the different C30/C36 ratios of the samples or is due to instrumental differences between the two types of neutron sources. The extent to which fractionation has progressed can be estimated from the SANS data by the use of a quantity Q0, known as the invariant18 which for a two-phase system is given by

Q0 ) ∫0 Q2 (dΣ/dΩ) dQ ) 2π2φ1φ2[a1 - a2]2 ∞

(2)

where the φi and ai are the volume fractions and scattering length densities, respectively. The integrand is plotted in Figure 3 for an annealing period of 398 h. The Q range on the final data set shown in Figure 1 for 1163 h is insufficient for an accurate evaluation of the invariant; however, by comparing the integrals for 0.3 mm-1 < Q < 1.8 nm-1 for the two data sets, it is found that the additional 765 h of annealing increases the contribution to the invariant by only 11%. For a sample with complete fractionation, the right hand side of eq 2 can be evaluated as follows: the scattering length density of the deuterated region is taken to be

a1 ) 2[bC + 2(bD)]/V

(3)

where bC ()0.665 × 10-12 cm-1) is the neutron scattering length for carbon, bD ()0.667 × 10-12 cm-1) is the scattering length for deuterium, and V is the volume of two CD2 units ()4.67 × 10-23 cm3 16). The quantity, a2, for the protonated region is calculated in the same way with bH ) -0.374 × 10-12 cm. The volume fractions are assumed to scale with the chain lengths. As mentioned previously, the effective length of the deuterated chain is slightly longer than the length of the carbon atom chain and an estimate of 4.6 nm is used. With the experimental value of the long period of 8.4 nm, a value of 0.248 for the product of the volume fractions results, and finally, a value of 38.9 × 1021 cm-4 for the right hand side of eq 3 may be obtained. Numerical integration under the curve of Figure 3 produces a value of 17 × 1021 cm-4 and indicates that complete separation into the respective lamellae has not occurred. The work of Snyder et al.9,10 suggests that the lamellae are subdivided laterally into protonated and deuterated regions, as indicated in Figure 7b, and the invariant can be used to estimate the number fractions of the species in a lamellae. First, it is assumed that a deuterium-rich lamellae has the same total number of molecules as an adjacent proton-rich lamellae. Then for a 1:1 mixture, the expression for a1 - a2 takes the form

a1 - a2 ) (2/V)[(φ1D - φ2D)(bC + 2bD) + (φ1H - φ2H)(bC + 2bH)] (4) where φ1D and φ2D are the number fractions of deuterated molecules in the deuterium-rich or -poor lamellae, respectively. The φiH are the analogous quantities for protonated molecules. Substitution of eq 4 into the expression for the invariant leads to a number fraction φ1D of 0.83 in the enriched lamellae and 0.17 in the other. The idea that fractionation is not limited to interlamellar separation may be supported by the observation that substantial neutron scattering occurs to the left of the principal peak in

Figure 1, i.e., in the range 0.25 mm-1 < Q < 0.5 nm-1. There the intensity increased by as much as an order of magnitude at the smallest angles over the duration of the measurements. Changes also occur in the SAXS data but to a lesser extent. Consequently, the neutron scattering in this region must be primarily due again to separation of the two molecular species and could be consistent with scattering from the intralamellar domains found with vibrational spectroscopy.9 In the case of a 1:1 mixture the spectroscopic work23 found that the domains were highly irregular, extended shapes. In general it would be expected that the neutron scattering from this type of structure would simply decrease with increasing Q. However, as shown in the model calculations of Figure 6, contributions from random mixing of lamellae can also contribute to this region, and separation of the intralamellar effect is not feasible. Conclusions The combination of SANS and SAXS measurements demonstrate that fractionation in a 1:1 mixture of n-C30H62/n-C36D74 occurs by interlamellar transport which predominantly results in a pattern of alternating lamellae which, on the average, are enriched in one of the components. In other words, there is evidence for nonrandom compositional correlation between neighboring lamellae in the longitudinal direction. This is in agreement with the model proposed by Dorset8 from electron diffraction. However, simple calculations based on this picture give only qualitative agreement with the peak heights and locations for both the SANS and SAXS data. The models assume that the lamellae have perfectly sharp interfaces, which is unlikely for less than complete separation. More complete modeling should account for a degree of smearing caused by the intralamellar domain structure found by Snyder et al.,9,10 but that is beyond the scope of this report. The SANS observations suggest that the intralamellar domain structure can be present, but this feature cannot be even qualitatively assessed for 1:1 mixtures. Acknowledgment. The authors are grateful to P. Butler and R. Triolo for facilitating the SANS mseaurements and to S. J. Henderson and J. S. Lin for assistance with the SAXS measurements. R.G.S. gratefully acknowledges support by the National Institites of Health (Grant GM 27690), and the work at ORNL was supported by the Division of Materials Sciences, Office of Basic Energy Sciences, U.S. Department of Energy, under Contract DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc. References and Notes (1) Lohse, D. J. Rubber and Chemistry Technology 1994, 67, 367. (2) Wignall, G. D. In The Physical Properties of Polymers; Mark, J. E., Eds.; ACS Books: Washington, DC, 1993; Chapter 7, p 313. (3) Higgins, J. S.; Benoit, H. Neutron Scattering from Polymers; Oxford University Press: Oxford, U.K., 1994. (4) Bates, F. S.; Muthukumar, M.; Wignall, G. D.; Fetters, L. J. Macromolecules 1988, 89, 535; Macromolecules 1988, 21, 1086. (5) Wignall, G. D.; Bates F. S. Makromol. Chem. 1988, 15, 105. (6) Bates, F. S.; Dierker, S. B.; Wignall, G. D. Macromolecules 1986, 19, 1938. (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) Dorset, D. L. Macromolecules 1986, 19, 2965. (9) Snyder, R. G.; Kim, Y.; Strauss, H. L.; Goh, M. C. Prepr.sAm. Chem. Soc., DiV. Polym. Chem. 1989, 30, 295. (10) Snyder, R. G.; Goh, M. C.; Srivtsavoy, V. J. P.; Strauss, H. L.; Dorset, D. L. J. Phys. Chem. 1992, 96, 10008. (11) White, J. W.; Dorset, D. L.; Epperson, J. E.; Snyder, R. G. Chem. Phys. Lett. 1990, 166, 560. (12) Koehler, W. C. Physica (Amsterdam) 1986, 137B, 320. (13) Wignall, G. D.; Bates, F. S. J. Appl. Crystallogr. 1986, 20, 28.

1730 J. Phys. Chem., Vol. 100, No. 5, 1996 (14) Wignall, G. D.; Lin, J. S.; Spooner, S. J. Appl. Crystallogr. 1990, 23, 241. (15) Russell, T. P.; Lin, J. S.; Spooner, S.; Wignall, G. D. J. Appl. Crystallogr. 1988, 21, 629. (16) Teare, P. W. Acta Crystallogr. 1959, 12, 294. (17) Broadhurst, M. G. J. Res. Natl. Bur. Stand. (U.S.) 1962, 66A, 241. (18) Wignall, G. D. In Polymer Properties Handbook; Mark, J. E., Ed.; American Institute of Physics: New York, in press. (19) Dubner, W. S.; Schultz, J. M.; Wignall, G. D. J. Appl. Crystallogr. 1990, 23, 469.

Annis et al. (20) Mazee, W. M. Prepr.sAm. Chem. Soc., DiV. Pet. Chem. 1958, 3, 35. (21) Crist, B. J. Polym. Sci., Polym. Phys. Ed. 1973, 11, 635. (22) Hosemann, R.; Bagchi, S. N. Direct Analysis of Diffraction by Matter, North Holland: Amsterdam, 1962; p 419. (23) Snyder, R. G.; Conti, G.; Strauss, H. L.; Dorset, D. L. J. Phys. Chem. 1993, 97, 7342.

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