9848
J. Phys. Chem. 1996, 100, 9848-9853
Crystal Structure of Modulated n-Paraffin Binary Solids Douglas L. Dorset* Electron Diffraction Department, Hauptman-Woodward Medical Research Institute, Inc., 73 High Street, Buffalo, New York 14203-1196
Robert G. Snyder Chemistry Department, UniVersity of California, Berkeley, California 94720-1460 ReceiVed: January 30, 1996; In Final Form: April 10, 1996X
The solid formed after a metastable solid solution, composed of n-C30H62 and n-C36H74, demixes, after being held within a miscibility gap, is a commensurately modulated superstructure. For a specific molar concentration, electron diffraction measurements on epitaxially oriented specimens indicate that, from the standpoint of both superlattice spacings and intensities, the same average chain packing is maintained for all microcrystalline areas. Direct crystallographic phasing methods were used to determine the structures of nearly 0.6/0.4 and 0.4/0.6 molar ratio combinations. Molecular packing models suggested from the crystal structure analyses lead to predicted 0kl intensities that are in reasonable agreement with the observed data (R ) 0.25). It is clear that, as described earlier, a longitudinal sequence of essentially pure components exists. In addition, there must be a lateral association of different chain length domains within a given layer, in agreement with earlier vibrational spectroscopic measurements.
Introduction Although polydisperse combinations of n-alkanes often have been employed as models for a variety of linear flexible molecules from lipids to polymers, they are also important themselves for the study of petroleum fractions.1-3 Recent crystallographic investigations4 have shown that the structure of multicomponent solid solutions is very similar to the simpler cosoluble binary solids.5,6 (That is to say, there is a single lamellar thickness with a fractional occupancy of atoms at molecular ends to account for the statistical distribution of two or more chain lengths in an average mixed lamella.) Further, it is apparent that the study of binary combinations of linear molecules can be used to understand, at a molecular level, how a variety of chain packing arrays, between the solid solution7 and the fully fractionated eutectic solid,8 are stabilized. Extensive work with the n-paraffins has shown that the sequence of solids formed, as the molecular volume difference increases, have crystal structures that are determined by molecular packing efficiency.9 In fact, there is no evidence for the so-called “mechanical mixtures” often assumed to constitute totally fractionated eutectics.10 High-resolution electron microscopy has shown that there is an intimate contact between the (001) faces of the pure phase-separated components.11 One of the most fascinating solids formed by two n-paraffins is the one involving a superstructure. Such a structure appears when a melt-crystallized metastable solid solution is allowed to equilibrate for an extended period at room temperature.8 This solid was first observed by Mazee12 for a n-C30H62/n-C35H72 combination that had aged for 1 year and was rediscovered in systematic electron diffraction studies of other binary n-paraffin solids,8 e.g., appearing in n-C30H62/n-C36H74 after approximately 2 days. Other pairs of paraffins exhibiting the same behavior were subsequently identified. Although extensive diffraction,8,13,16 calorimetric,8 and vibrational spectroscopic14,15 data have been obtained from the emergent crystalline form, very little is known, in terms of molecular packing, about the lamellar X
Abstract published in AdVance ACS Abstracts, May 15, 1996.
S0022-3654(96)00294-8 CCC: $12.00
architecture, aside from an approximate description of the lamellar stacking sequence of essentially pure ingredients8 (Figure 1), with an intimate contact between (001) layer faces. (This latter feature was verified by recent small angle neutron and X-ray scattering measurements.16) These superstructures may be categorized as a type of nearly commensurately modulated solid. In this paper, we describe their structures on the basis of the direct determination of crystallographic phases for observed electron diffraction amplitudes. As a result, a correlation between lateral phase separations, measured by vibrational spectroscopy,14,15 to the longitudinal changes, observed in diffraction experiments,8,16 had been found for the first time. Materials and Methods Crystallization and Data Collection. Two binary combinations of n-C30H62/n-C36H74 were investigated in this study. (Physical properties and sources of the pure paraffins used for the preparations have already been published.7,8) First, the approximate 1:1 combination, investigated originally by electron diffraction8 and recently by small angle neutron and X-ray diffraction,16 was intentionally avoided, in order to minimize ambiguities in interpretation of data from such a simple molar concentration. One mixture of pure n-paraffins was weighed as a 0.41/0.59 molar ratio of short to long chains; the other molar combination was 0.61/0.39. After they were fused in a DSC pan, these samples were scanned repeatedly to measure the appearance and disappearance of the small “mixing” endotherm13 found near 50 °C, as previously described.8,15 These measurements provided a time frame for which the superstructure would be expected to be observed in subsequent diffraction experiments. For electron diffraction studies, the sealed DSC pans were opened and a small amount of each solid was redissolved in light petroleum. After drops of the dilute solutions were evaporated on a cleaved mica sheet, the thin crystalline samples were epitaxially oriented on benzoic acid by an adaptation of the method of Wittmann, Hodge, and Lotz.17 That is to say, © 1996 American Chemical Society
Crystal Structure of n-Paraffin Binary Solids
J. Phys. Chem., Vol. 100, No. 23, 1996 9849 evaporated on the surface of a grid and photographed at the same electromagnetic lens currents used to record the paraffin diffraction data. The requisite minimization of the beam current density was achieved with the appropriate condenser lens excitations20 and insertion of a small diameter aperture at the second condenser lens. A sensitive X-ray film, Kodak DEF-5, was used to record the patterns at a direct magnification appropriate for densitometry of the diffraction maxima. The film sensitivity also facilitated the minimization of the total electron beam dose given to the specimen. After measurement of diffraction peak spacings on a filmreading device, intensities were obtained by scanning the peaks with a Joyce-Loebl Mk. III C flat bed microdensitometer. A triangular approximation was made to the peak integration to obtain the integrated intensity. No Lorentz correction was applied to the measured intensity values. (Due to the Gaussian distribution of crystalline subareas in the specimen, the characteristic scattering from bend-deformed single crystals19 ensures that the Ewald sphere intercepts all diffraction peaks near their maximal intensity for any given zonal projection.) Crystallographic Phase Determination. Two approaches were made to the determination of crystallographic phases for the superlattice patterns that emerged as the samples were aged. In both cases, the results of earlier direct crystallographic phase analyses on a solid solution5 were used to provide phase terms for the most intense reflections. The first attempt utilized the Sayre equation,21 a convolution of phased structure factors
Fh )
Figure 1. Original lamellar stacking model used to explain the superlattice-like diffraction from equilibrated n-C30H62/n-C36H74 binary solids.
carbon-covered electron microscope grids were placed facedown on the thin n-paraffin film. Excess benzoic acid was sprinkled around the grids, and the other half of the mica sheet was placed on this physical mixture to make a sandwich. This assembly was then placed on a thermal gradient, first to cosolubilize the n-paraffin in benzoic acid and then to cool the molten solution, thus crystallizing the diluent-rich eutectic solid18 that orients the paraffin chains on the benzoic acids crystals via lattice matching at the interface.17 The sandwich was opened, and the organic acid was then removed by sublimation under high vacuum. The carbon grids were then examined in the electron microscope to search for oriented binary paraffin crystals, e.g., after the preparation was allowed to age for 1 week. Selected area electron diffraction data were then obtained with a JEOL JEM-100CX II electron microscope operated at 100 kV. As usual,19 low beam dose procedures were used to minimize radiation damage to the crystalline specimens while the diffraction patterns were being photographed. The camera length was calibrated with a gold powder diffraction standard
θ V
∑k FkFh-k
following the recommendations of Fan et al.22 for determining the crystallographic phases for emergent superlattice reflections in diffraction patterns from similar inorganic structures. It was assumed that the known phases of the most intense reflections would provide a good approximation, via the Sayre convolution, for the unknown phase angles of the weaker superlattice reflections. Unfortunately, as will be shown below, the unit cell length along the molecular chain axes, needed to find a nearly commensurate set of Miller indices, was very large. Moreover, the index connectivity by the convolution operation, between “main” intense reflections and the weaker superlattice reflections, was not very good so that many of the unknown values could not be accessed directly by the vector sums of the Miller indices. For this reason, this approach was abandoned. The second attempt merely used the phase envelope of the strongest 00l reflections in the parent solid solution structure5 to assign values to the weaker superlattice reflections. (Although this average model structure, determined in earlier work,5 was actually composed of a nearly 1:1 combination of n-C32H66/ n-C36H74, it would have the same features as the metastable form observed at the beginning of our experiments, since they both could be indexed as the same average lattice; see below.) From the reverse Fourier transform of the phased 00l data set, a structural model for the lamellar repeat was obtained and used to construct a model to phase the complete 0kl set, based on the layer packing of even chain n-paraffins23,24 in the orthorhombic form. The success of a model was judged by the fit of the calculated structure factors Fc to the observed diffraction amplitudes |Fo| via the usual kinematical crystallographic residual, R ) ∑||Fo| - k|Fc||/∑|Fo|. Results Electron diffraction patterns from the 0.41/0.59 molar combination of n-C30H62/n-C36H74, epitaxially oriented on benzoic
9850 J. Phys. Chem., Vol. 100, No. 23, 1996
Dorset and Snyder TABLE 1. Indices of 00l Lamellar Superlattice Reflections from an Equilibrated Binary Solid Formed from a 0.41/0.59 Molar Combination of n-C30H62 and n-C36H74; c ) 445.2 Å
Figure 2. Electron diffraction patterns from epitaxially oriented 0.41/ 0.59 n-C30H62/n-C36H74. (top) Soon after growth from comelt with benzoic acid. Note the low resolution of the 00l row. (middle) After equilibration. The increase of resolution is readily seen for the 00l row. (bottom) View of the lamellar reflection row after equilibration. Note the broadening of weak superlattice reflections.
acid, revealed the presence of the metastable solid solution, soon after the sample crystallized (Figure 2, top). The major feature of such a pattern was the abbreviated row of lamellar reflections (i.e., only three or four orders of the lamellar repeat are ordinarily observed, compared to more than 10 orders found in diffraction patterns from pure n-paraffins). On the basis of the spacing of this 00l row, indices of the most intense 01l reflections, using the rules for identification of the average layer
l
〈d*〉obs
d*calc
10 20 27 29 37 39 46 49 56 66
0.0221 0.0437 0.0604 0.0657 0.0821 0.0870 0.1036 0.1105 0.1256 0.1483
0.0225 0.0449 0.0606 0.0652 0.0831 0.0876 0.1033 0.1101 0.1258 0.1482
packing given previously,9 could be shown most frequently to match the average n-C34H70 layer packing. The lamellar packing was ascertained to be in the orthorhombic space group Pca21, with unit cell dimensions a ) 7.42 Å, b ) 4.96 Å, c/2 ) 45.2(3) Å. (Nyburg and Potworowski24 predicted a lamellar spacing of c/2 ) 45.0 Å for this paraffin.) As stated above, the crystal structure of such a solid solution structure already has been published.5 Within the average layer, therefore, the occupation parameter for outer methylene groups was the most important factor to account for the restricted number of observed 00l reflections. This is because an approximate Gaussian distribution of group occupancies in the crystal Fourier-transforms to another Gaussian function in diffraction space that sifts out the part of the diffraction pattern corresponding to the lamellar chain interfacial order.19 Restricting the number of inner 00l reflections to a few orders of the lamellar repeat, therefore, corresponds to fractional occupancies of lamellar atomic positions near the interlamellar interface.4-6 Upon standing for 1 week, the lamellar reflections from epitaxially ordered samples were observed (a) to show a dramatic increase in the number of diffraction orders (Figure 2, middle) and (b) to include superlattice-like reflections that were not simply related to either pure component lamellar repeat or that of the parent solid solution (Figure 2, bottom). (However, the solid solution pattern could be restored by reheating, the sample through the “mixing” DSC endotherm, as described earlier.8) The d00l* value was measured for all reflections in the lamellar row in four separate 0kl diffraction patterns, and their intensities were also averaged (showing them to have a good internal agreement). On the basis of a layer spacing, c ) 445.2 Å, corresponding to a 0.4/0.6 combination of n-C30H62 with n-C36H74 in 10 successive lamellar layers, indices were assigned to all reflections that were close to expected commensurate values (Table 1) after the multiplication c‚d*00l. (Given the spacing of a first-order lamellar reflection measured in SANS experiments16 on a 1:1 mixture, it was also clear that this search for a commensurate indexing scheme might not be unique.) Using the phase angles for the solid solution model,5 the superlattice reflections were assigned values consistent with the defined continuous envelope of crystallographic phase. (See papers in ref 23 for a discussion of these phase relationships for lamellar reflections in diffraction patterns from pure n-paraffins.) Using the amplitudes and phases for all of the 00l reflections, a one-dimensional Fourier transform was calculated to seek, in the resultant one-dimensional potential map, the features of the lamellar packing within the defined superstructure repeat, assuming the model to be pseudocentrosymmetric. (The approximate centrosymmetry for nparaffin layer structures is also justified in earlier papers.23) The appearance of this one-dimensional transform is shown in Figure 3. Depending on the weight given to the outer intense 00l reflections (corresponding to a temperature factor) two possible
Crystal Structure of n-Paraffin Binary Solids
J. Phys. Chem., Vol. 100, No. 23, 1996 9851 TABLE 2. Observed and Calculated 0kl Structure Factors for 0.41/0.59 n-C30H62/n-C36H74 Superlattices
Figure 3. Experimental one-dimensional potential map obtained after the phased 00l superstructure reflections were assigned phases for an approximate 2:3 C30/C36 combination. (Only one half of a repeat is shown. The other half is generated by a mirror operation perpendicular to the chain axis repeat.) Note that two kinds of interlamellar density minimum are found. Deep and sharp minima were interpreted as a general methyl plane interface at the same level. Wider and shallower minima were interpreted as interfaces where domains of unequal chain length shared terminii but at slightly different levels. The derived model A in Figure 4 is shown in relation to this transform.
0k
l
|Fo|
|Fc| (A)
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 01 01 01 01 02 02 02 02 02 03 03
10 20 27 29 37 39 46 49 56 66 68 336 346 356 366 158 168 178 188 0 10 20 346 356 168 178 R
0.57 0.49 0.19 0.45 0.21 0.22 0.25 0.17 0.26 0.22 0.14 0.43 1.34 1.19 0.35 0.53 1.05 1.66 0.47 3.45 0.35 0.30 0.62 0.52 0.66 1.10
0.36 0.25 0.15 0.16 0.14 0.12 0.32 0.04 0.36 0.34 0.03 0.33 1.32 0.92 0.25 0.26 0.70 1.56 0.25 3.57 0.14 0.10 0.69 0.48 0.76 1.71 0.25
phase (deg) 180.2˚ 181.2 180.4 178.2 176.5 178.1 182.2 176.9 181.8 178.1 181.4 7.2 6.7 186.2 188.4 94.5 93.4 -86.7 -85.6 0.0 0.2 1.2 186.7 6.2 -86.6 93.3
|Fc| (B) 0.37 0.27 0.09 0.10 0.09 0.06 0.33 0.01 0.36 0.34 0.00 0.34 1.33 0.92 0.26 0.26 0.71 1.56 0.25 3.56 0.14 0.10 0.70 0.49 0.76 1.71 0.26
phase (deg)
initial phase (rad)
180.1˚ 181.1 180.3 176.7 174.0 175.6 182.1 164.7 181.7 178.0 1.6 7.2 6.7 186.2 188.3 94.4 93.4 -86.7 -85.5 0.0 0.1 1.1 186.7 6.2 -86.6 93.3
π* π* π π π π π π π π π 0* 0* π* π* π/2* π/2* -π/2* -π/2* 0* 0* 0* π* 0* -π/2* π/2*
* phases from solid solution structure.5
Figure 4. Models for the superlattices found from the reverse Fourier transform of phased 00l structure factors. (One half of the c-unit cell repeat is shown and the other half is assumed to be related by a mirror.) Individual longitudinal sequences of chains will pack as shown in Figure 1. However, at some transverse distance, another sequence is aligned with this original structure. Different sequences of lamellae may share methyl endplane surfaces at various points along c, but, otherwise, the interfaces will appear at different points. (This conclusion was reached after examining one-dimensional potential maps of the kind shown in Figure 3.) Laterally, the blocks indicate approximate concentrations in a layer for a given chain length, and longitudinally, the lamellar sequence of chain lengths is indicated. On the left are two possible models for the 0.41/0.59 combination of the two paraffins (see Table 2). On the right is the single model found for the 0.61/0.39 combination.
models could be constructed, as schematized in Figure 4. (The sequence was indicated by the positions of density minima found at the interfaces between successive lamellae.) If the y,z atomic coordinates were provided for chains in these layers, based on the successive layer structure of orthorhombic even-chain n-alkanes,24 then the observed 0kl amplitudes could be compared to the values predicted from structure factor calculations. The packing models thus constructed were strict diffraction gratings composed of carbon atoms at ideal positions. Specifically, for a given layer,24 the y coordinates of the first two carbon atoms were, respectively, assigned values of 0.92 and 1.56 Å. These values alternated along the zig-zag chains. The associated z-coordinates were, respectively, 1.56 and 2.86 Å, incrementing by 1.27 Å intervals along the chain. At the beginning of a new layer, the initial z-coordinate of the first carbon was advanced
by 3.08 Å from the position of the last carbon of the preceding layer and again incremented by 1.27 Å to generate a new chain. The sequential y-coordinates for the new layer were then -0.96, -1.56 Å, etc. Although there were 348 unique atoms in one model and 360 in the other (the difference due to assigned weights for different chain lengths occurring on the same layer), maintaining an overall lattice plane group pm, their positions were dictated solely by the placement of the zig-zag chains as rigid units. Obviously, the strict (z/c + 0.5) translational element of successive monolayers for the original sold solution bilayer unit cell (in space group Pca21) was lost as the superstructure forms. In addition, no partial atomic occupancy due to residual solid solution was assumed, in order to minimize the actual number of variables used. (Thus, in this model, blocks of one chain length, as shown in Figure 4, are assumed to contain essentially pure monolamellar paraffin domains with no conformational disorder at the chain ends. However, there is also a limited lateral width to each chain length domain within any layersso that there must be as tight a lateral molecular packing boundary for these domainssactually an exact epitaxial relationshipsas there is in the stacking of sequential chain lengths in successive monolayers.) While both models seemed to provide similar agreements to the measured data (Table 2), one model (labeled A in Figure 4) was slightly more successful in predicting the relative amplitudes of the superlattice reflections. Although the somewhat low overall isotropic temperature factor (B ) 2.0 Å2) may have indicated some secondary scattering perturbation,18 the final R-factor values (Table 2) were typical for electron diffraction determinations of such solids.4,5,19 A qualitative model was obtained for the 0.61/0.39 combination of the two alkanes in a similar way. Measurements of the parent solid solution patterns indicated that the average layer structure was originally close to that of pure n-C32H66. Six 00l patterns were measured to find lamellar spacings and intensities for the superlattice reflections, again demonstrating that the internal agreement of the data was quite good. Assuming a
9852 J. Phys. Chem., Vol. 100, No. 23, 1996
Dorset and Snyder
TABLE 3. Indices of 00l Lamellar Reflections from an Equilibrated Binary Solid Formed from 0.61/0.39 Molar Combination of n-C30H62 with n-C36H74; c ) 425.6 Å l
〈d*〉obs
d*calc
|Fo|
assigned phase (rad)
10 20 26 29 34 39 44 49 54 63 73 331 341
0.0235 0.0460 0.0607 0.0693 0.0790 0.0917 0.1028 0.1160 0.1258 0.1486 0.1709 0.7787 0.8024
0.0235 0.0470 0.0611 0.0681 0.0799 0.0916 0.1034 0.1151 0.1269 0.1480 0.1715 0.7777 0.8012
0.45 0.42 0.18 0.32 0.20 0.23 0.26 0.14 0.29 0.24 0.18 0.52 0.44
π π π π π π π π π π π 0 π
layer spacing of c ) 425.6 Å, i.e., a sequence of 10 layers composed of 0.6/0.4 combination of n-C30H62 and n-C36H74, all of the reflections could be indexed in a nearly commensurate fashion (Table 3). After using the phase envelope expected for the solid solution, the reverse Fourier transform was utilized to construct a model for the average lamellar repeat (Figure 4). This layer structure was more complicated than the ones proposed for the other binary models considered above but, nevertheless, was conceptually consistent with them. Discussion As shown by vibrational spectroscopic measurements,14 the formation of a modulated structure from a metastable n-paraffin solid solution appears to be caused by the decreasing concentration of chain end conformational defects, a maximum number of which are initially present at the lamellar interface when the binary solid is grown from the melt. This increase in order at the lamellar interface is also observed in electron diffraction and vibrational spectroscopic experiments when n-paraffins, heated near the melting point, are recooled to room temperature.25 From extensive calorimetric studies of binary combinations,8,26 it is known that solid solutions are stable only if the chain length difference is less than some sharply-defined value. In that case, stability is achieved through a combination of mixing entropy and an increase in interfacial packing density. The latter occurs via the conversion, from trans to gauche, of some C-C bonds nearest the chain ends. (Gauche bonds are intimately associated with the presence of conformational disorder.) When the chain length difference reaches a critical value, which is concentration-dependent, increased chain-end disordering no long results in increased packing density. At this point, chain aggregationsan alternative way to restructure the assembly to reduce void volumesbegins to occur spontaneously, provided the structure has sufficient interfacial voids to allow diffusion to occur.8,14,15 The great sensitivity of the metastable state to even slight changes in the molar volume of its components is manifested by a large isotope effect on microphase separation. For example, there are dramatic differences in phase behavior between the various isotopic combinations of the mixtures C30D,H/C36D,H, that is, between the H/D, H/H, D/D, and D/H mixtures.28 Vibrational spectroscopic measurements on the metastable n-paraffin solid solutions have demonstrated spontaneous microphase separation of the two components, in which intralayer aggregates of the more or less pure n-alkanes evolved with time.14 Diffraction studies,8,13,16 on the other hand, indicate structural changes in the longitudinal direction; namely, an average mixed lamellar chain packing is transformed to a
complicated superstructure, each successive layer composed of nearly pure ingredients (Figure 1). These respective changes are literally orthogonal to each other and therefore tend to be independent of one another. In binary mixtures, it is expected that phase separation proceeds preferentially by longitudinal diffusion. This is because its activation energy for this mode is much lower than that for lateral diffusion. An activation energy of about 15 kcal/ mol, estimated for C30/C36,29 was derived from the temperature dependence of the demixing rate. This value is much smaller than the 80 kcal/mol determined for the self-diffusion in a pure n-C21H44 crystal,30 which may be taken as a lower limit for both longitudinal and lateral diffusion. This value is in agreement with the 60-80 kcal/mol calculated for lateral diffusion on the basis of a vacancy model.31 Stated another way, since the structure of a pure n-alkane is not much different in the lateral direction than that of a mixturesthe relatively thin interlayer region being of minor importancesthe respective activation energies for lateral diffusion must be expected to be nearly the same, around 80 kcal/mol, for both pure and mixed n-alkanes. On the other hand, for longitudinal diffusion, a large difference would be expected, since the partial voids in the interfacial region, which are necessary for reptation, are present only for mixtures. Longitudinal transport is easily envisioned to occur by chains creeping across an interface into the partial voids in the next monolayer. There is ample evidence to support this picture. This mechanism is similar to the one employed when a “nematically” disordered metastable n-paraffin structure, formed by deposition of a vapor onto a cold surface that also epitaxially orients the lateral chain packing, is annealed at a higher temperature.32,33 For example, in solid solutions of n-C50H102/ n-C60H122 (or in the pure components),33 chains with this longitudinal disorder will creep toward the stable lamellar layer packing when the solid is annealed between 60 and 80 °C. Such temperatures are well below the melting transition of either pure component and well within the temperature range of the orthorhombic methylene subcell packing.34 Given the difference in melting points for the shorter n-C30H62 and n-C36H74 components of the modulated structures, considered above, from those of the longer paraffins, a temperature near 50 °C should be reasonable for inducing the remixing of components, as has been characterized quite well by vibrational spectroscopy.15 The current results indicate that, for a given bulk concentration of two chains, the resultant structure formed after phase separation is, itself, uniquely characteristic of that molar concentration, both in terms of spacing and intensity of the new superlattice reflections. (It was already known that the spacing of the superlattice reflections was dependent on the relative concentrations of the two components.8) If the crystallographic phase envelope for the 00l row is used to assign values for the newly appearing reflections, the sequence of chain layers in the large unit cell is again consistent with the earlier-held notion8 that a sequence of nearly pure layers must occur in the longitudinal direction. However, as demonstrated by the analyses of one-dimensional Fourier transforms (e.g., Figure 3, above), there must also be a copacking of blocks composed of the pure short and long components in some of these layers, corresponding to the lateral aggregation found in spectroscopic experiments.14 This is because closely paired minima in the one-dimensional potential maps are observed as somewhat broad, shallow minima (Figure 3). The dimensions of the minimal densities (observed in the original computer-generated displays at higher spatial resolution) closely correspond to the lengths of two different lamellae, as predicted by Nyburg and
Crystal Structure of n-Paraffin Binary Solids Potworowski24 for the pure ingredients. This density profile can only occur if two lateral domains with differing chain length are sharing the same lamellar surface. Such a potential map would not be expected if only one chain length were packing in any given layer of the superlattice, since the occurrence of a constant methyl end-plane results in a deeper, sharper density minimum between lamellae. The growth of islands with limited lateral dimension is also supported by the occasionally observed elongation of emergent superlattice reflections in the b* direction, e.g., in Figure 2, bottom. (If the elongation is taken to be the usual sin2 x/x2 intensity transform of a limiting rectangle function,19 then the island radius is near 30 Å for this example.) Thus, a model for fractionated metastable solid solutions, consistent with all physical measurements, can be formulated. It is a modulated structure that seeks to minimize the volume difference between the two components as the lamellar interface becomes more highly ordered. Although the form of the structure cannot, as yet, be described exactly, a copacking of short and long chains must be maintained on average in a single lamella. When the equilibrated solid is reheated to induce the inclusion of chain end defects, the metastable solid solution can be formed efficiently with slight longitudinal shifts of the shorter component. When these defects vanish to form a more perfect lamellar interface, then there must be a more efficient packing of lamellae as pure components in the longitudinal direction but also as separate domains in a lateral direction. To conclude, the analysis and arguments presented above are also quite in accord with descriptions of inorganic solids. Amelinckx and Van Dyck35 have described how a phase transformation, starting with a high-temperature average crystal structure, can lead to a low-temperature commensurately modulated lattice. As also seen in our work, the transformation can be reversible. Incommensurate modulations are known to occur between these two endpoints in some inorganics, but it is not known yet if this intermediate form can be identified for the n-paraffin binaries considered in this paper. Acknowledgment. Research was supported by grants from the National Science Foundation (CHE94-17835), to the Hauptman-Woodward Institute, and the National Institute of General Medical Sciences (GM-27690), to the University of California at Berkeley, both of which are gratefully acknowledged. References and Notes (1) Bennema, P.; Liu, X. Y.; Lewtas, K.; Tack, R. D.; Rijkema, J. J. M.; Roberts, K. J. J. Cryst. Growth 1992, 121, 679. (2) Srivasta, S. P.; Tandon, R. S.; Verma, P. S.; Saxena, A. K.; Joshi, G. C.; Phatak, S. D. Fuel 1992, 71, 533.
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