Spinodal Phase Separation of Unstable Solid-State Binary n-Alkane

The measurements followed the band shapes of the scissors vibrations as a function of time. The band envelopes are analyzed as composites of a number ...
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J. Phys. Chem. B 2007, 111, 13957-13966

13957

Spinodal Phase Separation of Unstable Solid-State Binary n-Alkane Mixtures R. G. Snyder,† D. Clavell-Grunbaum,‡ and H. L. Strauss* Department of Chemistry, UniVersity of California, Berkeley, California 94720 ReceiVed: August 9, 2007; In Final Form: September 18, 2007

Two mixtures of unequal chain length n-alkanes in which one component is deuterated have been investigated by infrared spectroscopy as they demixed. The measurements followed the band shapes of the scissors vibrations as a function of time. The band envelopes are analyzed as composites of a number of reference mixtures of known concentration. The unequal-chain mixtures separate into phases that slowly change their composition toward pure alkane phases. The method of analysis, which reveals local concentrations, should be generally applicable to polymethylene systems.

I. Introduction n-Alkanes and their mixtures often serve as models for the study of the properties and structures of systems having a significant polymethylene component. Waxes, lipid membrane assemblies, and polymer mixtures are examples. Pure n-alkanes have played a critical role in our own efforts to identify infrared (IR) spectral markers that characterize polymethylene chain packing and conformation in such systems.1-3 Similarly, n-alkane mixtures and mixtures of n-alkane derivatives serve as models to determine the effect of the variations of polymethylene chain lengths that are often found in natural substances. The literature on n-alkanes, substituted n-alkanes, and their mixtures is consequently extensive. A sampling can be found in refs 4 and 5. The subject of this paper is spontaneous demixing in binary n-alkane mixtures: in particular, the time evolution of the microscopic compositional heterogeneity as determined by IR spectroscopy. Our method is based on the methylene scissors infrared band and the sensitivity of its shape to isotopic composition. From this, we can estimate the local heterogeneity of a binary mixture in terms of the distribution of local component concentration. In our studies, the distribution can be monitored at times beginning a few minutes after preparation. Spontaneous demixing in n-alkane mixtures has remained a largely unexplored subject in spite of the fact that most solidstate n-alkane mixtures prepared by melt crystallization are in some degree initially unstable. Other unstable systems such as alloys and polymer mixtures have been widely studied.6,7 For binary n-alkane mixtures, with chain-length differences greater than 2 carbons, chain layer packing is nearly always that most commonly found for pure crystalline odd-numbered n-alkanes, namely, packing with an orthorhombic-perpendicular methylene subcell.4,8 The mixture maintains this structure as the phase separation proceeds. A cartoon of the structures involved in the transformations is sketched in Figure 1. The vertical lines represent fully or nearly fully extended n-alkanes that pack in stacked layers in which the chains are aligned perpendicular to the layer plane. The one important difference between the structure of the unstable as-quenched solid solution * E-mail address: [email protected]. † Deceased. ‡ Holy Names University, Oakland, CA 94619.

Figure 1. Cartoon of an n-alkane mixture A + B at successive times: Unmixed components; melted mixture; freshly cooled solid solution; aged phase-separated crystalline solid; detail of interlamellar disorder.

and the fully phase-separated system is in the thin, interlayer regions in which the packing is disordered. The disorder involves the ends of the chains and is due to chain-length mismatch. Chain packing is otherwise essentially identical in both the unstable and phase-separated mixtures. Spinodal demixing in alloys and polymer mixtures investigated by X-ray or neutron diffraction is normally characterized in terms of lengths that relate to the spatial extent of the component concentration. In contrast, the present infrared study uses the distribution of the local component concentration to characterize phase transformation. Although each of the three types of systems can phase separate by spinodal decomposition, they are otherwise quite different. Some of the differences for typical systems studied are listed in Table 1. In the next section, we describe the unstable alkane mixtures and their phase separation in more detail. We consider the sample preparation and the measurement of IR spectra, and describe the method used to determine the concentration distribution.

10.1021/jp076414x CCC: $37.00 © 2007 American Chemical Society Published on Web 11/23/2007

13958 J. Phys. Chem. B, Vol. 111, No. 50, 2007 TABLE 1: Spinodal Decomposition in Different Systems constituents state isotropic entangled mechanism

metals

n-alkanes

polymers

atoms cryst yes no exchange

chains (20-40 units) cryst no no diffusion?

chains (>>100 units) amorph yes yes reptation

Snyder et al. TABLE 2: Values of xAi for 30H and 30D in Base-Set Mixtures 30H + 30Da

We then detail the time evolution of the concentration distribution of the minority component for some melt-crystallized n-C30H62/n-C36D74 mixtures. II. Structure of Unstable Binary Mixtures The major factor that determines the rate of demixing in meltcrystallized n-alkane mixtures at a fixed temperature is the chainlength difference, although component concentration and average chain length are also important. The components of the mixtures considered in our studies have chain lengths in the range 23-40 carbons (the melting point for pure C23 is 49 °C and for C40, 82 °C). If the chain-length difference for such a mixture is 3 or 4 carbons, it usually undergoes a small, but detectable, degree of demixing in its freshly quenched state. If the difference is greater than about 10 carbons, nearly complete phase separation will likely ensue before the melt has cooled to room temperature. For intermediate chain-length differences, the time required for demixing to approach completion varies from minutes to years. Spontaneous phase separation in n-alkane mixtures was first reported, almost 50 years ago, by Mazee9 in an X-ray powderpattern study of the time dependence of the layer spacing in a series of n-C30H62/n-C35H72 mixtures with n-C35H72 concentrations ranging from 20% to 70% mol fraction. He found that these mixtures were initially solid solutions, but after about a year had phase-separated. Some of the mixtures showed spacings longer than the layer spacing. These were interpreted to indicate longitudinal superstructure, that is, some degree of regularity in the order in which layers of different composition were stacked. Recently, interest in the subject has been rekindled due in large part to a series of detailed electron diffraction and calorimetric studies on phase separation in binary n-alkane mixtures summarized by Dorset.4 Of particular relevance to us here is the dependency of the mutual solubility on the chainlength difference. At room temperature for equimolar C30H62/ CnH2n+2 mixtures, those with n < 36 can be described loosely as solid solutions and those with n > 36 as eutectic mixtures. This leaves the C30H62/C36H74 mixture in an intermediate position that is reflected by its initial state as an unstable solidsolution that demixes fairly rapidly, but slowly enough to permit IR measurements.10 We note that, for the n-alkane mixtures, diffraction methods are much more effective than IR methods for analyzing order and disorder in the longitudinal direction (i.e., in the chain direction), whereas the reverse is true in a lateral direction (i.e., perpendicular to the chain direction). In this way, the two methods are complementary. Our previous work on the time evolution of compositional heterogeneity used infrared spectroscopy to estimate an overall degree of demixing. The IR characterization of demixing in the equimolar C30/C36 mixture11,12 generally agreed with Dorset’s measurements,13,14 but added new detail. Some separation was found to occur during or immediately after the quench. After 2 days, the mixture was mostly phase-separated, but measurable demixing was found to continue for months.

a

i

CH2

CD2

1 2 3 4 5 6 7 8 9 10 11 12 13

0 0.055 0.102 0.216 0.307 0.398 0.503 0.613 0.702 0.805 0.886 0.944 1.000

0 0.056 0.114 0.195 0.298 0.387 0.497 0.602 0.693 0.784 0.898 0.945 1.000

Note the i ) 1 mixture for CH2 is the i )13 mixture for CD2, etc.

Measurements on the 1:4 and 4:1 mixtures C28/C36, C29/C36, and C30/C36 indicated that a change in chain-length difference of one carbon resulted in roughly an order of magnitude increase in the rate of demixing.11 It was found also that the H/D isotope effect was sizable. For example, the demixing rate for a 1:4 C30D62/C36H74 mixture was found to be roughly three times faster than that for the isotopically complementary mixture12 1:4 C30H62/C36D74. The difference is due to the slightly smaller effective volume of a deuterated chain relative to that of a hydrogenated one, so that the C30D62/C36H74 mixture has components that are more dissimilar than in the C30H62/C36D74 mixture. We have also used Raman and IR spectroscopy to probe the chain-end packing disorder in the interlamellar region of the unstable mixture that results from the component chain-length difference.15 We postulate two kinds of chain packing arrangements that tend to reduce void volume and thus lower the energy of the crystal. One is interdigitation, in which the longer chain extends a bit into an adjacent layer. The other is a “bending over” of the chain ends, that involves mostly the longer chains so as to, in effect, shorten them.16 These arrangements are incorporated in the sketch of the disordered region shown at the bottom of Figure 1. III. Measurements A. Experimental. Films suitable for IR measurements were prepared by melting together two preweighed samples of n-alkanes, one hydrogenated and one deuterated. The liquid was stirred and then allowed to cool to room temperature. A small amount of the resulting solid was placed on a KBr window at room temperature on a hot plate and warmed until melted. A second preheated window was then placed on it and the resulting sandwich allowed to cool. About 5 or 10 min after the sample film was observed to crystallize, its IR spectrum was measured. Measurements were repeated usually at doubled aging times for as long as significant demixing continued. Measurements were made using a Nicolet Magna 550 interferometer equipped with a cooled MCT/InSb detector. The spectra were derived from 128 sample interferograms collected at a resolution of 1.0 cm-1. B. Component Concentration. The mole fraction, XH, of the hydrogenated component of a given sample was determined from the integrated IR intensities of the methylene scissors bands using the equation

XH ) IH/[IH + 1.53(RH/RD)ID]

(1)

Separation of Alkane Mixtures

J. Phys. Chem. B, Vol. 111, No. 50, 2007 13959

Figure 2. Infrared CH2 and CD2 scissors bands in the spectra of a series of C30H62/C30D62 mixtures at the component concentrations listed in Table 2. These bands represent the two basis sets used in the present analysis. There are thirteen basis mixtures, but of course, the mixtures with xAi ) 0 show no spectrum. Note that the integrated intensities of the basis-set bands are scaled so that the total numbers of chains (both hydrogenated and deuterated chains) represented are the same for each.

where IH and ID are the intensities of the CH2 and CD2 bands, and RH and RD are the number of carbons in each component. The numerical factor 1.53 was an average evaluated from the IH and ID measured for 11 30H/30D (n-C30H62/n-C30D62) mixtures with known gravimetrically based values of XH. The 11 mixtures studied had XH values ranging from 0.05 to 0.95. In addition to these 11 mixtures, the 2 pure samples containing either all hydrogen or all deuterium were considered, making 13 basis reference samples in all. The mole fractions of the basis mixtures are listed in Table 2 and their spectra are shown in Figure 2. The measured integrated intensities were virtually independent of aging time. C. Determination of the Concentration Distribution. The shape of an IR methylene scissors band for a randomly mixed solid solution composed of two n-alkanes, one hydrogenated

and one deuterated, is a function of component concentration, provided that the chain-packing subcell is orthorhombicperpendicular. Thus, the shapes of the CH2 and CD2 methylene scissors bands in the IR spectrum of a mixture of n-alkanes of equal length (for which we can assume random mixing) depend on component concentration or, equivalently, on the H/D chain isotope ratio. Scissors band shapes for partially phase-separated mixtures of chains of different lengths are complex, since the local composition varies at different locations within the sample. The observed CH2 or CD2 band envelope therefore consists of the superposition of a continuum of differently shaped bands, each of which represents some specific local component concentration. The intensity of each contributing band is weighted by the number of chains with the specified local concentration. The

13960 J. Phys. Chem. B, Vol. 111, No. 50, 2007 number of chains over all concentrations can be determined from the observed shape of the scissors band of a mixture, if the relation between band shape and the H/D concentration ratio is known. We show how this relation can be approximated and then used to estimate the concentration distribution. The scissors band shape dependence on component concentration is due to interchain coupling of the scissors vibrations. Lateral interchain coupling in pure n-alkanes, whose chain packing subcell is orthorhombic-perpendicular, has been extensively studied and is usually modeled using an exp-6 potential. The exponential term models the repulsion and the r-6 term, the van der Waals attraction between the atoms. This potential is short-ranged, and its effect varies with the size of the region that can be considered as having a crystalline packing. In the pure crystalline solid, the splitting becomes the factorgroup splitting, which produces a doublet. The CH2 and CD2 scissors band doublets are centered, respectively, near 1467 and 1088 cm-1 with separations of the component bands of 10.4 and 7.6 cm-1, virtually independent of chain length. The doubling results from vibrational coupling between the two chains that traverse the subcell. Selection rules based on the orthorhombic-perpendicular subcell symmetry predict IR activity only for those modes that are in-phase or 180° out-of-phase, and this yields two narrow and well-defined bands. The concentration dependence of the scissors band shape is due to the difference between the lateral short-range interchain vibrational coupling between isotopically alike and isotopically different neighboring chains. This leads to the complex band shapes shown in Figure 2 for the CH2 and CD2 scissors bands of the series of 30H/30D basis mixtures. The scissors band of either component at a low concentration is narrow as a result of its isotopic isolation. In going to higher concentrations, it broadens and the doublet band associated with high concentration begins to appear, its bands broader and their separation smaller than for the same bands in the spectrum of the neat n-alkane. The presence of both the singlets and doublets make a triplet. With increasing concentration, the singlet disappears and the doublet bands sharpen and their separation increases. In earlier studies,11,12 we used these bands to estimate the degree of demixing. The relation between scissors band shape and component concentration for unequal chain length mixtures can be assumed to be virtually the same as that observed for the 30H/30D base mixtures, because chain-chain vibrational coupling is lateral with no significant longitudinal component. This assumption is supported by the observation that, for pure n-alkanes, the scissors band splitting is independent of chain length. For example, we have found that the relation between scissors band shape and component concentration for the mixture 36H/36D is very nearly the same as that for 30H/30D. The effect of the chain-end methylenes that do not have a full set of nearestneighbor methylenes to interact with is negligible for reasons discussed in section IV.C. The scissors band for a mixture of n-alkanes with different chain lengths is shown in Figure 3. The mixture is 30H/36D with 30H at a concentration of 26%. The CH2 band consists of a main singlet or doublet band. There are some shoulders that are considered below. The main scissors band is initially a singlet and, as such, represents regions of low 30H concentration. The doublet that grows in at the expense of the singlet represents regions of high 30H concentration. At long enough times, only the doublet will be present. We define fi, as the fraction of the total number of chains associated with the A-chain concentration, xAi. The function

Snyder et al.

Figure 3. Time evolution of the CH2 and CD2 scissors bands of a (30H)36D mixture with a 30H component concentration of 26.0%. The log(aging times in hours) values are indicated in the CH2 picture and are the same for the CD2 picture. The CD2 spectra are very similar and difficult to tell apart in the picture.

SA(ν) represents the shape of the A-chain scissors band of a concentrationally inhomogeneous A/B mixture in terms of relative absorptivity at frequency ν. The function SAi(ν) is the A-chain band shape associated with homogeneous mixture i whose A chain concentration is xA. Then, approximately Nf

SA )

fiSAi ∑ i)1

(2)

where Nf is the number of 30H/30D base mixture bands used in the basis set. We take SA to represent the A-chain band shape of the mixture expressed as a column vector of absorptivities measured at regular frequency intervals of about 0.25 cm-1. The SAi are similarly defined column vectors. The values of the xAi for the 13 basis mixture spectra used are listed in Table 2. The integrated intensities of the basis-set bands as well as the unequal chain mixture bands are scaled so that the total numbers of chains (both hydrogenated and deuterated chains) represented are the same for each.

Separation of Alkane Mixtures

J. Phys. Chem. B, Vol. 111, No. 50, 2007 13961

The fi coefficients in eq 3 are normalized, so that Nf

fi ) 1 ∑ i)1

(3)

The values of the fi are those determined to best reproduce the observed band shape. We have used a least-squares method to evaluate the fi that suppresses negative values. There is one problem, however. The i ) 1 value of the coefficient fi, which represents the fraction of the mixtures that consists only of B-chains, cannot be determined directly, because it does not contribute to the observed A-chain band shape. In principle, f1 could be determined from the B-chain band shape. However, this approach is impractical, because the shapes of bands representing the given B component are insensitive to change in the B concentration at concentrations higher than about 65%. The value of f1 can, however, be estimated by utilizing the B-chain conservation equation. If, in the leastsquares calculation, gi is substituted for fi so that the minimization is for Nf

[SA -

giS(30H)i]2 ∑ i)1

(4)

g1 ) 0

(5)

fi ) Kgi

(6)

D. Measures of Concentration. The concentrations can be expressed in two ways, one in terms of the fi and the basis mole fractions, xAi, and the other in terms of the overall mole fractions in the unequal chain mixture. The latter uses the A- and B-chain numbers of the mixture, NA and NB, whose sum is NT. In contrast, we define nAi and nBi as the numbers of A and B chains in base fraction i. Their sum is nTi. The definitions that follow are written to apply to A-chains, and of course also apply to the B-chains. The bulk A-chain mole fraction of the mixture, which is determined from eq 1, is given by

XA ) NA/NT

(12)

Similarly, for A-chains in base fraction i

xAi ) nAi/nTi

(13)

Its value equals the concentration of the one isotopic component, 30H or 30D, of our ith 30H/30D base mixture. The base set concentrations are listed in Table 2. The ith base-fraction fi is the mole fraction of the unequal-chain mixture that has a molar concentration equal to xAi

fi ) nTi/NT

(14)

then

whereas for i > 1

where K is a proportionality constant. The unknowns, f1 and K, can be evaluated using the fi normalization eq 3 together with the B-chain concentration conservation equation

Its value is evaluated in the least-squares band fitting calculation. CAi is the number of A-chains in the base set i that is in the unstable mixture divided by the total number of A-chains in the unstable mixture

CAi ) nAi/NA

and thus has units of a fraction, fraction of chains or fraction of moles. It can be written

CAi ) (nTi /NT)(nAi/nTi)(NT/NA)

Nf

fixBi ) XB ∑ i)1

(7)

CAi ) fixAi/XA

Nf

gi ) 1 ∑ i)2

and is normalized as

(8) Nf

CAi ) 1 ∑ i)1

Writing eq 8 as Nf

f1(1 - xA1) + K

gi(1 - xAi) ) XB ∑ i)2

(16)

or

From eq 3 expressed using eq 6 as

f1 + K

(15)

(9)

(17)

A measure of the separation at time, t, of the unstable mixtures is Nf

we have

R(t) ) Nf

K ) X A[

gixAi]-1 ∑ i)2

(10)

(18)

or Nf

And then rearranging eq 8

R(t) ) Nf

f1 ) 1 - K

xAiCAi ∑ i)1

gi ∑ i)2

(11)

The value of f1 obtained in this way is subject to the uncertainties in the values of XA and xAi. As a result, f1 is sometimes determined as a small negative number, which we then set to zero.

fixAi2/XA ∑ i)1

To illustrate the meaning of R, consider an unstable mixture with XA ) 1/3, which has not begun to separate. Then, fi is 1, XA and xAi are both 1/3, and the sum is for only one value of i (we assume that we have a base mixture at the appropriate value of XA) and is 1/3. At infinite time, the unstable mixture will have separated into 2/3 of regions with xA of

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Snyder et al.

Figure 5. The observed (solid line) CH2 scissors band and the corresponding base spectra fitted scissors band (dashed line) of the (30H)36D mixture. Only alternate spectra from Figure 3 are shown.

Figure 4. Satellite combination bands, whose intensities are derived from the CH2 or CD2 scissors bands. The spectra are of the pure C30 alkanes. The CH2 band is much broader, but unchanged in intensity, in this highly pure perpendicular-orthorhombic crystal form. This may be seen by comparison to the same band in a mixture such as that shown Figure 3.

0, and 1/3 of regions with xA of 1. XA still remains 1/3. The R sum is then 1. IV. Band Shape Fitting A. Base Spectra. The base-mixture spectra are of the CH2 and CD2 scissors bands measured for a series of 30H/30D mixtures with mole fractions ranging from about 5% to 95% and displayed in Figure 2, as already mentioned. The shape dependence is similar for the H and D components. The small differences are due to the degree to which the CH2 and CD2 scissors modes are mixed with other types of polymethylene chain vibrations. This mixing is greater for the CD2 scissors mode, because the frequency of this mode is markedly closer to the C-C stretching and other types of methylene modes that are of the correct symmetry to mix with the scissors modes. There is an additional problem with the basis set. At high 30H concentrations, an increasing fraction of the 30D/30H mixture assumes another crystal structure. This is not surprising, since pure 30H is known to be polymorphic. The change in crystal structure causes the high-frequency band of the CH2 scissors doublet to be about 0.7 cm-1 higher than any band in the other 12 members of the basis set as can be seen in Figure 2. The problem was minimized by substituting for the 13th

Figure 6. Residuals. The observed minus calculated absorptivity curves of Figure 5 are plotted for the spectra of the (30H)36D mixture. The residuals are less than 5% of the spectra of Figure 5.

member of the basis set, a spectrum that represents 100% 30H in the orthorhombic phase. The added spectrum is that of the CH2 band of a highly phase-separated nD/(n′H) mixture. In this notation, n′ represents the longer chain of an unstable mixture and the parenthesis enclose the component whose spectrum is being studied. Using this corrected spectrum as one of the basis elements markedly improves the fits. B. Band Fitting. We will illustrate band fitting for the mixture C30H62/C36D74, which we abbreviate (30H)36D. The

Separation of Alkane Mixtures

Figure 7. Top diagram: Plots of base fractions fi vs log t for 30H in the (30H)36D mixture. The fi values for the unstable mixture at a fixed time sum to 1 (eq 4). Each fi is plotted on a scale from 0 to 1, and values for successive times are displaced up by one. Individual base plots are identified by their A-chain concentration xAi. Bottom diagram: Plots of A-chain concentration CAi vs log t, but otherwise the same as the top plots.

component studied is usually the minority component whose concentration in the present case is 26% (XH ) 0.26). Its spectrum was measured at 15 aging times, starting at 10 min after the quench and subsequently at doubled time intervals for 99 days. The CH2 spectra are shown in Figure 3. The lower diagram of Figure 3 shows the corresponding CD2 spectra. The CH2 scissors band has a triplet shape, the center band representing regions of low 30H concentration. The outside bands belong to the doublet band that is associated with regions of a high 30H concentration. As the demixing proceeds, the doublet grows at the expense of the center band. That part of the CH2 scissors band used in the band fitting calculation normally extends from about 1460 to 1475 cm-1. For CD2 bands, it is from about 1060 to 1100 cm-1. The integrated intensities of the scissors band measured at each aging time are scaled to the same (arbitrary) value to account for the variation in the observed intensities due to such factors as changes in sample density and differences in the position of the sample relative to the IR beam. A weaker, but significant, band appears on the lower frequency side of both the CH2 and CD2 scissors bands at about 1455 and 1056 cm-1, respectively, as shown in Figure 4. The 1455 cm-1 band for the pure n-alkane is broad due to interchain

J. Phys. Chem. B, Vol. 111, No. 50, 2007 13963

Figure 8. Top graph: Bar graph of the base fraction fi vs log t for 30H in the (30H)36D mixture. The values of xAi are marked along the x-axis, and the 30H bulk concentration is indicated by the vertical dashed line. Bottom graph: Same bar graphs but for A-chain concentration CAi instead of fi.

interaction and sharpens on dilution by a deuterated component. The satellite bands represent overtones of a closely spaced series of the CH2 and CD2 rocking modes at 720 and 530 cm -1, respectively. The intensity of these side bands, about 15% of that of the scissors bands, is derived from the scissors bands through resonance interaction. However, we have excluded these bands in the band-fitting calculation. This calculation used the MATLAB suite of programs and provided a least-squares fit of each spectrum of the separating mixture as the sum of the spectra of the basis set. The observed CH2 scissors band and the calculated best fit are shown in Figure 5 at selected aging times. The residuals, defined as calculated minus observed absorptivities, are displayed in Figure 6. The first derivative-like patterns centered near some of the band maxima reflect small frequency shifts in band position. The magnitudes of the residuals at the band centers are small, usually less than 5% of the band peaks. Spectral comparisons between separately prepared mixtures showed that reproducibility was excellent. C. Factors Affecting Band Shape. Randomness of Mixing. As noted, we have assumed that mixing in the 30H/30D mixtures is essentially random. To our knowledge, there is no evidence for significant nonrandomness at least at the level of our measurements. Frequency Shifts. Shifts in band frequency are a significant source of error in band fitting. The shifts result primarily from differences in the molecular environment of chains in the

13964 J. Phys. Chem. B, Vol. 111, No. 50, 2007

Figure 9. Same as Figure 8 for 36D in the 30H(36D) mixture.

reference and studied mixtures. The entire composite scissors band may shift more or less as a unit, because the bands that determine its shape tend to be similar in character. To cover this possibility, the curve fitting calculation was carried out with different overall shifts (the increment being 0.25 cm-1) until the best fit was located. Normally, no such shift was required. Longitudinal Displacement Effects. It is likely that some limited longitudinal displacement among neighboring chains occurs in unstable mixtures in order to consolidate chainend packing in the interlamellar region (Figure 1). Whereas the local environment of most of the methylene groups is left unchanged, a few nearest the end will be without a normal set of neighboring methylenes. If the chains neighboring such a chain are of the same isotope, the interchain coupling will be affected, and this will affect the scissors band shape. Fortunately, the effect is predicted to be small. The reason is the methylenes nearest the chain ends are minimally involved in the IR active scissors vibration because the normal coordinate associated with the fundamental vibration involved has its nodes located at ends of the chain.20 Hence, the participation of the methylenes nearest the chain ends is minimal. Background Absorption. In general, this can be a serious problem in measuring the scissors band shape for components at low concentrations, i.e., below 5%. It is commonly encountered when measuring the CD2 scissors band, since this band is located in the same frequency region as that of the CH2 rocking progression bands associated with the other component. If, for example, the CD2 component concentration is 1%, the intensity of a typical CH2 rocking band in this

Snyder et al.

Figure 10. Top map : Pseudo-phase map for 30H in the (30H)36D mixture in terms of the base mixtures. The A-chain concentrations xAi within the enclosed regions are indicated. The dashed heavy line separates the A- and B-chain dominated regions. Bottom map: Pseudophase map for 36D in the 30H(36D) mixture.

region is about 20% of that of the CD2 scissors. Usually, the offending band can be subtracted out using the spectrum of the undiluted n-alkane that corresponds to the majority component. Sometimes the interfering band can be excluded by narrowing the frequency region used in the least-squares calculation. V. Application to Two 30H36D Mixtures We consider further two mixtures, (30H)36D and 30H(36D), whose minority component concentrations (30H and 36D) are 26% and 33.4%, respectively. Their demixing behavior is representative of that for C30/C36 mixtures with minority concentrations in the 15-40% range. The bulk concentrations of the two minority components are close enough together to permit a meaningful comparison between the demixing of shortand long-chain components. A. Graph Formats. We are interested in the time evolution of the mixtures and this involves three independent variables: the time, t, the concentrations of the basis set, xi, and the fractions, fi, of each component, i, in the aging mixture. 1. Plots of fi as a Function of Log Aging Time. The fi values come directly from the least-squares band fitting calculation. The xAi values are predetermined by the selected base set. The upper graph of Figure 7 shows these plots for the (30H)36D mixture. There are 13 separate curves, 1 for each base fraction. The x ) 0 fraction is not determined directly, but by difference, as explained in section III.C. Curves for some values of i are not apparent because the fi values are zero. Most of the others

Separation of Alkane Mixtures

Figure 11. The sum of the local concentrations squared R(t) vs log time. R(t) goes from the initial concentration of a sample to a maximum of one for the completely phase-separated sample. The concentrations of the samples are listed to the right, and the curves are in the same order and indicated by symbols starting from the bottom (1.4%): box, diamond, upright triangle, tilted triangle, inverted triangle, star.

are identified on the plots by their xAivalue. Note that the fi values for the unstable mixture at a fixed time sum to 1 (eq 3). In the lower graph of Figure 7, the A-chain concentrations CAi are plotted against log t. This graph is identical to the fi vs log t graphs except that each fi is weighted by xAi in accordance with eq 16. The CAi values also sum to 1 (eq 17). 2. Bar Graph Displays of fi and CAi Distributions. Bar graphs provide a more immediate overall view of the phase separation. Figures 8 and 9 show such graphs for the (30H)36D and 30H(36D) mixtures. Both fi and CAi summed over i add up to 1 (eqs 3 and 17). Each set of fi and CAi are graphed on a scale from zero to one and the curves for successive times are displaced on the y-axis by one. The A-chain concentration associated with each base fraction is marked along the x-axis by a triangle. The vertical dashed line marks the bulk A-chain concentration. The logarithms of the aging times are placed at the right. B. Characterization of Demixing in the Two 30/36 Mixtures. The observed scissors band can be well fit by a relatively small number of base-set spectra. This suggests that the A-chain and consequently the B-chain local concentration distributions are similar on the length scale of the entire mixture. The mixture may be heterogeneous on a smaller scalesfor example, it likely consists of at least two phases, but these same phases are likely present throughout the sample. The aging-time period during which most of the demixing takes place for (30H)36D from log t ) -1 to 3.4, or from about 6 min after the quench (the earliest we were able to observe) to about 14 weeks after. During this period, the A-chain concentration distribution consists almost entirely of contributions from 1 or 2 base spectra, specifically those associated with A-chain concentrations xAi of 40%, 61%, and 81% (i ) 6, 8, and 10). The concentration distribution starts at a uniform 26% and then splits into regions of both higher and lower concentration that eventually become two phases that approach 0% and 100%, respectively. Note that the two mixtures separate at different rates with the 30H(36D) mixture separating more rapidly.

J. Phys. Chem. B, Vol. 111, No. 50, 2007 13965 In Figure 10, we present pseudo-phase maps of the separating systems as a function of time (30H)36D (top map) and 30H(36D) (bottom map). They plot the cumulative fraction of the system in terms of the base mixtures. Thus, the amount of the pure components (0% and 100% in both cases) grows at the expense of the mixed basis concentrations. On both graphs, we plot the 50% line to provide a reference. Demixing via regions of varying composition is indicative of spinodal decomposition. Our analysis indicates that the thermodynamically stable phases are the pure components. Even complex multicomponent mixtures such as waxes are thought to decompose to multiple pure solid phases.21 Finally, we plot our measure of the amount of separation, R, vs the logarithm of time in Figure 11. As pointed out in the discussion of R (eq 18 and following), R is initially a fraction that depends on the original composition of the mixture. It then is expected to increase toward one, if the mixture completely phase-separates into pure components. Figure 11 shows curves for six (30H)36D mixtures with a variety of concentrations. The plots of R versus log time show S-shaped curves as expected (Figure 11). The curves start at a value near the bulk concentration and then increase. The higher concentration mixtures have values of R that increase more rapidly. Presumably, the components in higher-concentration mixtures need not move as far to form distinct regions, that is, for the mixture to separate. Our description is in terms of local concentrations. We provide a detailed description of the time development of these concentrations. The local regions constitute parts of more extensive phases that for binary alkane mixtures develop the long-range order of the crystalline components. The largerscale decomposition has usually been studied by diffraction methods and is often modeled by a spin 1/2 system with an appropriate interaction potential, which separates into spinup and spin-down phases.22 Diagrams of the intermingled phases as a function of time can be obtained from simulations of these systems.22 Our measurements provide an new opportunity for a detailed comparison of the experiments with the simulations. Acknowledgment. We are grateful to Dr. Douglas Dorset, who provided insights into and encouraged numerous aspects of this paper. We acknowledge the support of the U.S. Department of Energy through grant no. DE-FG02-01ER45912. References and Notes (1) Maroncelli, M.; Qi, S. P.; Strauss, H. L.; Snyder, R. G. J. Am. Chem. Soc. 1982, l04, 6237-6247. (2) Snyder, R. G. J. Chem. Phys. l967, 47, l3l6-1360. (3) Snyder, R. G. and. Poore, M. W. Macromolecules l973, 6, 708715. (4) Dorset, D. L. Crystallography of the Polymethylene Chain; Oxford University Press: Oxford, 2005; Chapter 5. (5) Small, D. M. Handbook of Lipid Research 4: The Physical Chemistry of Lipids: From Alkanes to Phospholipids; Plenum Press: London, 1986. (6) Wayman, C. M. Annu. ReV. Mater. Sci. 1971, 1, 185-218. (7) Bates, F. S.; Wiltzius, P. J. Chem. Phys. 1989, 91, 3258-3274. (8) Smith, A. E. J. Chem. Phys. 1953, 21, 2229-2231. (9) Mazee, W. M. Prep.-Am. Chem. Soc. DiV. Pet. Chem. 1958, 3, 35-47. (10) White, J. W.; Zhu, P.-W.; Epperson, J. E.; Wosniak, D.; Snyder, R. G. Mol. Phys. 1997, 91, 1017-1024. (11) Snyder, R. G.; Goh, M. C.; Srivatsavoy, V. J. P.; Strauss, H. L.; Dorset, D. L. J. Phys. Chem. l992, 96, l0008-10019. (12) Snyder, R. G.; Srivatsavoy, V. J. P.; Cates, D. A.; Strauss, H. L.; White, J. W.; Dorset, D. L. J. Phys. Chem. 1994, 98, 674-84. (13) Dorset, D. L. J. Phys. Chem. 1990, 23, 623-633. (14) Dorset, D. L. Macromolecules 1986, 19, 2965-2673.

13966 J. Phys. Chem. B, Vol. 111, No. 50, 2007 (15) Clavell-Grunbaum, D.; Strauss, H. L.; Snyder, R. G. J. Phys. Chem. 1997, 101, 335-343. (16) Devlin, M. T.; Levin, I. W. J. Raman Spectrosc. 1990, 21, 44151. (17) Snyder, R. G. J. Chem. Phys. l979, 7l, 3229-3235. (18) Barnes, J.; Fanconi, B. J. Phys. Chem. Ref. Data 1978, 7, 130921.

Snyder et al. (19) Kobayashi, M.; Tadokoro, H. J. Chem. Phys. 1977, 66, 125865. (20) Snyder, R. G. J. Mol. Spectrosc. l960, 4, 4ll-434. (21) Lira-Galeana, C.; Firoozabadi, A.; Prausnitz, J. M. AIChE J. 1996, 42, 239-48. (22) Amar, G. J., Sullivan, F. E.; Mountain, R. D. Phys. ReV. 1988, B37, 196-208.