n-Alkane Systems. 1. Phase Diagrams from DSC

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J. Phys. Chem. B 2000, 104, 7483-7489

7483

Symmetrical Ketone/n-Alkane Systems. 1. Phase Diagrams from DSC Keiko Nakasone Department of Physics and Earth Sciences, Faculty of Science, UniVersity of The Ryukyus, Nishihara, Okinawa 903-0213, Japan

Kohzoh Shiokawa, Yoshiko Urabe, and Norio Nemoto* Department of Molecular and Material Sciences, IGSES, Kyushu UniVersity, Hakozaki, Fukuoka, 812-8581, Japan ReceiVed: March 14, 2000; In Final Form: May 8, 2000

Binary phase diagrams have been determined for solution-crystallized samples of two long-chain symmetrical ketones (Kn) and n-alkanes (Cn) with the same carbon numbers of n ) 25 and 39 from differential scanning calorimetry (DSC) measurements. Toluene was used as the solvent for crystallization of K39/C39 samples and hexane for K25/C25. They are found to form a solid solution which exhibits the solid-solid phase transition behaviors characteristic of a crystal of the pure n-alkane component, when the molar fraction fm of Kn is lower than a critical value fm,c, being dependent on n. For fm > fm,c, K39/C39 samples have formed another type of solid solution which melts after a rather broad solid-liquid equilibrium at elevated temperatures. On the other hand, K25/C25 samples have given eutectic mixtures for fm > fm,c, and the phase diagram is found to agree with that of the same samples crystallized from melt. We apply the Flory-Huggins theory for reproducing the phase diagrams using the interaction parameter χ12 between Kn and Cn molecules and discuss effects of solvent which gave rise to these anomalous phase behaviors in terms of χ12 and χ0i (i ) 1, 2) between solvent and Kn (or Cn) estimated from their solubility curves.

Introduction Polymorphism is one of thermal characteristics of long-chain n-alkane crystals hereafter abbreviated as Cn with n being the number of carbon atoms, which is well-known as a model compound of polyethylene. Over the past few decades a considerable number of studies have been made on thermal properties and crystal structures of pure n-alkane homologues with special attention to the solid-solid phase transitions.1-6 Big efforts have been also made for determination of phase diagrams of binary n-alkane solids crystallized from melts,7-14 which have revealed that they form solid solutions when a difference between their chain lengths is small, but they make eutectic mixtures when the difference is large. Symmetrical ketone abbreviated as Kn is different only from Cn in that the former has a carbonyl group with a permanent dipole moment at the center of the chain, and they belong to the same crystal group as the odd n-alkane crystals and lattice constants of their subcells taking the orthorhombic form are almost the same. A characteristic feature of Kn crystals is that they generally melt, without showing any solid-solid phase transition, to the liquid phase at a temperature higher than the melting temperature of Cn with the same n.15-19 Binary systems of symmetrical long-chain ketones and n-alkanes with the latter as a matrix have been studied with a dielectric relaxation technique to examine molecular motions of long-chain polar ketones in nonpolar n-alkane crystals.20-24 A fruitful result is that the motion of Kn mainly occurs along its chain axis in this nonpolar field, and its activation energy increases with increas* To whom correspondence should be addressed.

ing n. In a previous differential scanning calorimetry (DSC) measurement on the same Kn/Cn system crystallized from melts with n ranging from 25 to 39,25 we found that solid solutions could be formed in a limited low range of the molar fraction fm of Kn if n of a ketone molecule is either equal to or slightly lower than that of Cn. Above the solubility limit fm,c eutectic mixtures were obtained, which indicates that the enthalpy of mixing is positive to this system. Although phase diagrams have been determined for the above systems with different combinations of carbon numbers of Kn and Cn in a subsequent paper,26 studies on crystal structures in respective phases have been left as a future task. In addition, there is an intriguing problem for crystallization of long-chain n-alkanes from their melted states. Takamizawa et al. reported that solution-crystallized samples of C51 and C52 clearly showed solid-solid transition peaks on their DSC peaks before melting, while melt-crystallized samples did not show these peaks.27 This phenomenon was interpreted, on the basis of results of X-ray measurements as well as scanning electron microscopy (SEM) observation, as being due to that the hightemperature structures were frozen during crystallization from the melt. This result may give little doubt about whether phase diagrams for Kn/Cn systems obtained by melt crystallization represent thermodynamic equilibrium properties in a strict sense. Stated in another way, there remains a possibility that we may obtain a different phase diagram for the same system prepared with a different crystallization method. From this point of view, it seems meaningful to make a similar study on Kn/Cn samples obtained with the solution-crystallization method, which is known as the best method for preparing the giant single crystal.

10.1021/jp0009655 CCC: $19.00 © 2000 American Chemical Society Published on Web 07/14/2000

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In this paper, we shall describe results of DSC measurements on two Kn/Cn systems with the same carbon numbers of n ) 39 and 25 crystallized in solution using toluene and hexane as solvent, respectively. The results show that, for fm < fm,c, they form a solid solution which exhibits the solid-solid phase transitions characteristic of the single crystal of the corresponding Cn component irrespective of the solvents used. For fm > fm,c, K39/C39 samples prepared in toluene form another type of solid solution, whereas K25/C25 crystallized from hexane solution gave eutectic mixtures in this fm region. The phase diagrams and effects of solvent on them are extensively discussed using the Flory-Huggins theory.28 Experimental Section Materials. Two symmetrical ketones, 13-pentacosanone (K25) and 20-nonatriacontanone (K39), and corresponding n-alkanes C25 and C39 were used for this investigation. The ketones were synthesized through a ketene dimerization reaction from the corresponding carbonic acid chlorides which were free from other homologues. Details of the purification technique of the starting materials and the synthetic method are reported elsewhere.29,30 The ketones were finally purified with recrystallization and elution through a silica column. C39 was synthesized by reducing the corresponding ketone K39 using the Wolff-Kishner method and purified by column elution after treatment with hot concentrated sulfuric acid. C25 was purchased from Sigma Chemical Co. and used without further purification. The sample purity was determined with a capillary gas chromatograph (GC-14A, Shimazu) equipped with a column of CBP1-M25-025. The purity for homologues was up to 99.7% for all samples. After solubility tests of pure K39 and C39 using several organic solvents, we chose toluene as the solvent for solution crystallization of the K39/C39 binary system, because it is a good solvent for both of K39 and C39, and temperatures at which crystallization took place were around room temperature over the whole range of fm from 0 to 1, as will be shown later. The latter made it possible to determine a crystallization temperature of each solution by direct observation. Twelve samples including pure K39 and C39 crystals were crystallized from 0.85% toluene solutions containing weighed amounts of K39 and C39 with a slow cooling rate of 0.02-0.05 K/min. Molar fractions fm of K39 in binary crystals thus obtained were determined using the capillary gas chromatograph and were found to be close to mixing molar ratios of K39 to (K39 + C39) in respective solutions except at low fm, where the former was much lower than the latter. Hexane was used as solvent for solution crystallization of the K25/C25 system. Solubility tests for pure K25 and C25 in hexane indicated that their crystallization temperatures were different by nearly 20 °C at the same concentration. In crystallization from two 2.0% hexane solutions with high mixing ratios of fm by slow cooling either in a refrigerator or a freezer compartment, we noticed a subtle change in shapes of crystals sedimenting during a long time period for crystallization. Then, we separately collected crystals deposited from each solution by dividing into two parts, one being rich in K25 and the other rich in C25, whose fm determined by the capillary gas chromatograph were found to be considerably different to each other. Eight samples were examined for DSC measurements. Method. DSC measurements were performed using a calorimeter (DSC 8240B, Rigaku) equipped with a data acquisition system (Rigaku TAS-100). The standard heating rate was 0.5

Figure 1. DSC curves of K39/C39 samples crystallized from toluene solution: (a) pure C39, (b) molar fraction fm of K39 ) 0.029, (c) fm ) 0.204, (d) fm ) 0.276, (e) fm ) 0.509, (f) fm ) 0.848, and (g) pure K39. The weight of the samples used is 1.0 mg, and the scanning rate, 0.5 K/min.

K/min in a flow of dry nitrogen for a sample mass of about 1.0 mg. The heating rate dependence of thermal behaviors was examined for several samples of the K39/C39 system chosen randomly. Temperature calibration was performed using indium (In) and shorter n-alkanes with known equilibrium melting temperatures.27 The heat of transition or fusion was calibrated using In and galium. DSC measurements were also conducted for bulk-crystallized samples which were obtained by melting solution-crystallized samples and subsequently cooling slowly to room temperature. Results K39/C39 System. Figure 1 shows five typical DSC curves chosen from those of 10 solution-crystallized samples of the K39/C39 system for clarity of the figure along with those of pure C39 and K39 samples. The curve for pure C39 unambiguously exhibits the presence of the B and the C solid-solid phase transitions at 333 and 341 K, respectively, and sharp melting at 353 K, while there appears to be no such transition in the curve for K39 but melting at 366 K. It is likely that thermal behaviors of the K39/C39 binary system may be classified into two different ones with fm,c ) 0.20 as a boundary molar fraction from the figure. Namely, small peaks ascribed to the B and C transitions were observed in the range of 0.20 g fm g 0. On the other hand, a pretty broad endothermic peak starts to appear for fm > 0.20 at a temperature T just exceeding the melting point of pure C39 with a shoulder at a higher temperature side, and then the main peak gradually approaches the melting peak of pure K39 with increasing fm. It is to be noted that no peak corresponding to solid-solid phase transitions is observed at low T in the latter region.

Phase Diagrams of Ketone/n-Alkane Systems from DSC

Figure 2. Scanning rate dependence of the peak temperature in the DSC curves of the K39/C39 system. Symbols are (O) pure C39, (9) fm ) 0.73, and (b) pure K39.

Because of a dynamical nature of the DSC technique, it seems legitimate to define the melting temperature or the phase transition temperature as the point at which an extrapolated line of each endothermic peak front intersects with a baseline and then to extrapolate the values to zero scanning rate for obtaining properties of the system in thermodynamic equilibrium. Furthermore, we proposed one method in an earlier work,25 for determination of temperatures of solidus and liquidus lines of melt-crystallized Kn/Cn systems with low fm, of which the solid-liquid coexistence region is limited in a narrow T range. Temperatures of the solidus line were assigned as the point at which the DSC curve departs from the baseline and those of the liquidus line as an inflection point just before of the peak in the peak front based on model calculation on heat-flux type DSC curves.31,32 The above procedure may be successfully applicable for a single peak without any shoulder. For a broad peak with a shoulder as observed for K39/C39 samples with high fm in Figure 1, however, decomposition into two peaks overlapping each other is, first of all, necessary, which results in difficulty of accurate determination of transition temperatures especially for a shoulder type of a peak. In taking into account this difficulty, we simply took a peak temperature as the characteristic temperature for each endothermic peak or shoulder in this study. Melting temperatures Tm or phase transition temperatures Tp thus determined were found to be slightly dependent on the scanning rate and, thus, were linearly extrapolated to obtain values at zero scanning rate, as illustrated in Figure 2. For some peaks, we could estimate Tm(or Tp) as the point at which an extrapolated line of each endothermic peak front intersected with a baseline. Extrapolation to zero scanning rate gave a considerable uncertainty, but extrapolated values were about 1 K lower in average than the corresponding peak temperature values. Figure 3 shows the phase diagram for the K39/C39 samples prepared with the solution crystallization method. In the figure, solid curves are calculated ones using the Flory-Huggins theory, whose validity shall be extensively discussed later. The dashed lines are empirically drawn for connecting Tp corresponding to the solid-solid phase transitions. The diagram indicates that two different types of solid solutions (S1 and S2) are formed at room temperature above and below fm ) 0.20 indicated as the dotted line. The B and the C transitions characteristic of the n-alkane system with an odd number of carbon atoms n are observed for S1 with fm e 0.20, and the solid-solid phase transition temperatures appear to become

J. Phys. Chem. B, Vol. 104, No. 31, 2000 7485

Figure 3. Binary phase diagram for K39/C39 samples crystallized from toluene solution. Two different types of solid solutions are formed for the molar fraction fm of K39 above and below fm ) 0.20. Solid lines are calculated from eq 4 with the interaction parameter pχ12 ) 1.5. Dashed lines are empirically drawn for a guide of the eye.

TABLE 1: Heats of Transition for the K39/C39 System ∆H/(J g-1) K39 molar fraction

A′a

Bb

Cc

Ld

overall

0 0.029 0.139 0.204

0.34

4.67 4.53

13.22 11.52 13.95 4.01

250.20 247.20 254.55 258.75

268.42 263.25 268.50 263.16

1.67

∆H/(J g-1) K39 molar fraction

solidus

liquidus

overall

0.276 0.419 0.509 0.728 0.845 0.907 1

250.12 238.83 203.96 116.34 45.55 27.78

16.10 21.86 54.83 144.08 218.44 231.48 261.86

266.22 260.70 258.79 260.42 263.99 259.26 261.86

a A f A′, solid-solid phase transition. b A′ f B, solid-solid phase transition. c B f C, solid-solid phase transition. d Melting.

higher with an increase in molar concentration of K39. The melting temperature Tm is almost independent of fm, and the solid-liquid coexistence region is, if it may exist, limited to a very narrow temperature range. These results strongly suggest that C39 molecules can accommodate K39 in the crystal lattice, while keeping quite the same crystal structures as that of pure C39 up to fm ) 0.20. In fact, this has been confirmed by X-ray diffraction measurements on the samples whose details will be soon reported in a forthcoming paper. The samples with fm > 0.20 also form a solid solution S2 at room temperature, which does not show any solid-solid phase transition with elevating temperature. Instead, the solid-liquid coexistence region is present over a broad range of temperature higher than Tm of pure C39. This is in harmony with direct observation with a polarizing microscope for the samples in the coexistence region, which have shown that bright small particles are floating in a dark liquid phase. We attempted to calculate enthalpy ∆H accompanied by melting or phase transitions from an area of respective endothermic peaks in DSC curves. It was difficult to decompose a broad peak with a shoulder into two independent peaks in the region of 0.20 e fm e 1.0 so that values listed in Table 1 may be subjected to a considerable uncertainty. Figure 4 shows fm dependence of ∆H, and solid curves in the figure are empirically

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Figure 4. Heat of transitions ∆H for the K39/C39 system as a function of fm. Symbols are (4) overall, (2) melting of the solid solution for fm e 0.20, (b) solidus line for fm > 0.20, (O) liquidus line for fm > 0.20, (]) A′ phase transition, ([) B phase transition, and (0) C phase transition.

Figure 6. DSC curves of K25/C25 samples crystallized from hexane solution: (a) pure C25, (b) molar fraction fm of K25 ) 0.032, (c) fm ) 0.196, (d) fm ) 0.242, (e) fm ) 0.574, (f) fm ) 0.755, (g) fm ) 0.959, and (h) pure K25. The weight of the samples used is 1.0 mg, and the scanning rate, 0.5 K/min.

Figure 5. Binary phase diagram for K39/C39 samples crystallized from melt. Symbols are (b) this work and (O) earlier data.25 Solid lines are theoretical predictions of eqs 4 and 5 for fm e 0.20 and fm > 0.20 with pχ12 ) 1.5 and 0, respectively. Dashed lines are empirically drawn for a guide of the eye.

drawn for a guide of the eye. The sum of ∆H for each sample given by the unfilled triangle symbol in the figure was found to be independent of the scanning rate, which may guarantee that it represents an equilibrium property of the system. Also the sum is independent of a mixing ratio of C39 and K39, as expected from a fact that the heat of fusion is the same between pure C39 and K39 in magnitude. In the region of 0e fm e 0.20, ∆H appears to decrease with fm slightly for the solid-solid phase transition, while it increases for melting. As mentioned above, the total enthalpy remains constant to an experimental accuracy of 2 J g-1 in this low-fm region. For fm > 0.20, ∆H decreases along the liquidus line (O) and increases along the solidus line (b) with increasing fm, respectively. It is to be noted that they are not directly proportional to fm. The phase diagram was also constructed for bulk-crystallized C39/K39 samples from DSC measurements, and the results are shown as filled circles in Figure 5 where earlier data25 obtained for fm < 0.50 are also reproduced as unfilled symbols for comparison. In the figure, the solid curve is the theoretical prediction and dashed ones are empirically drawn for guidance of the eye. As is evident from the figure, new data not only agree with earlier ones in the low-fm region but also reconfirm

that eutectic mixtures are formed for bulk-crystallized samples with fm higher than fm,c ) 0.23. The sample with fm ) 0.907 gave a very small endothermic peak at T ) 333 K, to which we have no explanation. K25/C25 System. Figure 6 shows six DSC curves of the K25/ C25 binary system crystallized in hexane with those of pure C25 and K25. To the sample with fm ) 0.755, we still observe the hexagonal transition at T ) 319 K characteristic of pure C25, which indicates that the crystal structure, quite same as that of pure C25, is formed to some extent even in the high-fm region. Peak temperatures are plotted against fm as filled circles in Figure 7 where unfilled circles are earlier data25 which were obtained for bulk-crystallized K25/C25 samples. As is evident from the figure, those two sets of data are in good agreement with each other despite a difference in crystallization method of the samples. The phase diagram shows such characteristic features that the system forms a solid solution for fm < 0.1 and an eutectic mixture for fm > 0.14. We calculated enthalpy ∆H accompanied by melting or phase transition for this system as given in Table 2, from which we have found that the sum of ∆H values related to two transitions at T ) 320 and 326 K decreases almost in proportion to fm, while ∆H corresponding to melting of the system linearly increases with fm in the region of 0.14 < fm < 1.0. These thermal behaviors are consistent with the phase diagram that the K25/C25 system obtained by the solution-crystallization method forms an eutectic mixture for fm > fm,c ) 0.11. We show solubility curves of K39 and C39 in toluene and those of K25 and C25 in hexane in Figures 8a,b, respectively. Because of dipole-dipole interactions between carbonyl residues, a melting temperature of pure Kn has been found to be

Phase Diagrams of Ketone/n-Alkane Systems from DSC

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Figure 7. Binary phase diagram for K25/C25 samples solution crystallized in hexane (b) and bulk crystallized (O). Solid lines are theoretical predictions of eqs 4 and 5 for fm e 0.11 and fm > 0.11 with pχ12 ) 0.6 and 0, respectively.

TABLE 2: Heats of Transition for the K25/C25 System ∆H/(J g-1) K25 molar fraction

A′a

Hb

Lc

overall

0 0.032 0.071

1.18

84.11 83.71 89.37

170.13 169.48 176.91

255.42 253.19 266.28

K25 molar fraction

Hb

solidus

liquidus

overall

0.196 0.219 0.242 0.574 0.755 0.959 1

51.23 54.19 56.30 21.11 16.81 0.23

175.26 163.65 140.45 76.02 54.77 1.15

27.36 33.57 51.46 157.06 183.77 256.12 262.28

253.85 251.41 248.21 254.19 255.36 257.50 262.28

∆H/(J g-1)

a A f A′, solid-solid phase transition. b B f H, solid-solid phase transition. c Melting.

always higher than that of Cn with the same n. Nevertheless, the solubility curve of K39 is located below that of C39, while K25 starts crystallization at temperatures higher by nearly 20 °C than C25. This is a key factor for interpretation of why a solid solution has been obtained for the K39/C39 system crystallized in toluene and an eutectic mixture for the K25/C25 system crystallized in hexane in the high-fm region. Discussion We here apply the Flory-Huggins (F-H) theory for prediction of the phase diagrams shown in Figures 3, 5, and 7 as well as solubility curves of Kn and Cn in Figure 8 using the χ parameter as an adjustable one, simply because the χ parameter is, in general, complicatedly dependent on polymer concentration and molecular weight as well as temperature and cannot be calculated accurately from any molecular theory. First, we consider the liquid-solid phase equilibrium of the solid solution with volume fraction φi (i ) 1, 2, φ1 + φ2 )1) for the liquid and φi′ for the solid at temperature T and constant atmospheric pressure. Since the molecular weight of the two components is almost the same for the present system, the degree of polymerization pi may be assumed to take the same number p to a good approximation. The F-H theory gives the following expression for the difference ∆µi(l,T) in the chemical potential of the component i between µi(l,φ,T) in the solution and

Figure 8. Solubility curves for pure Kn (b) and Cn(O). Solid lines are calculated from eqs 3 and 6 by putting either φ1 or φ2 equal to zero with (a, top) p ) 7, χ01 ) 0 for K39 and χ02 ) 0.36 for C39 and (b, bottom) p ) 4, χ01 ) 0.29 for K25 and χ02 ) 0.24 for C25, respectively.

µi°(l,T) in the pure liquid state of component i

∆µi(l,T) ) µi(l,φ,T) - µi°(l,T) ) RT[ln φi + pχ12(1 - φi)2] (1) Here R is the gas constant. A difference ∆µi(s,T) in the chemical potential µi(s,φ′,T) of the solid solution referenced to the same standard state µi°(l,T) is decomposed into two parts as

∆µi(s,T) ) µi(s,φ′,T) - µi°(l,T) ) [µi(s,φ′,T) - µi°(s,T)] + [µi°(s,T) - µi°(l,T)] (2) We can apply the F-H theory for an estimate of the first term employing the value of pχ12 the same as that used in eq 2, since molecules 1 and 2 are randomly distributed on the crystal lattice for the solid solution. For the second term, eq 3 is known to be applicable.

µi°(s,T) - µi°(l,T) ) T∆Hfus,i(1/Tm,i° - 1/T)

(3)

Here Tm,i° and ∆Hfus,i are the melting temperature and the molar heat of fusion of the pure component i. Substituting eqs 1-3 into the equilibrium condition that ∆µi(l,T) ) ∆µi(s,T), we derive

ln φi - ln φi′ + pχ12[(1 - φi)2 - (1 - φi′)2] ) (∆Hfus,i/R)(1/Tm,i° - 1/T) (4) Equation 4 was calculated for various values of pχ12 to reproduce the phase diagram shown in Figure 3 as closely as possible using Tm,1° ) 365.8 K and ∆Hfus,1 ) 147.2 kJ/mol for K39 and Tm,2° ) 353.2 K and ∆Hfus,2 ) 137.1 kJ/mol at the melting point for C39. The best fit was obtained for pχ12 ) 1.5, and the results are shown as solid curves in Figure 3 where a liquidus and a solidus line drawn below fm < 0.20 appear to

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be superposed to each other because of their very narrow temperature gaps. Reproducibility looks fairly good. For eutectic mixtures observed for the bulk-crystallized K39/ C39 system shown in Figure 5 and also for the K25/C25 system in Figure 7 in the range of fm > fm,c, the thermodynamic relationship for the boundary line between liquid and the pure Kn crystal is given by eq 5.

ln φi + pχ12(1 - φi)2 ) (∆Hfus,i/R)(1/Tm,i° - 1/T)

(5)

The solid curve in Figure 5 is the prediction of eq 5 and eq 4 for fm > fm,c and fm e fm,c respectively, in which we adopted the same values of pχ12, Tm,i°, and ∆Hfus,i used in the calculation of the liquid-solid equilibrium of the solid solution of K39/ C39. The boundary line intersects to the liquidus line of the solid solution formed in the low-fm region at fm ) 0.15. Except this deviation, agreement between theory and experiment looks good. Similar calculation was also made for the K25/C25 system using pχ12 ) 0.6 and 0 for fm e fm,c and fm > fm,c, respectively, and experimental values for Tm,i° and ∆Hfus,i. Though the value pχ12 ) 0.6 is a little bit smaller than a value calculated from pχ12 of the K39/C39 system, the theory shown as the solid curve in Figure 7 not only reproduces the volume fraction at the intersection point to the liquidus line of the solid solution as fm ) 0.11 but also gives a good fitting to experimental values. The F-H theory has originally been proposed for thermodynamic properties of polymers in either semidilute or concentrated regime and is supposed to be invalid in the dilute regime where the excluded volume effect plays an essential role for chain conformation. However the samples used in this study belong to oligomers which take the unperturbed dimension in dilute solution. Thus it may not be a false idea to apply the F-H theory to solubility curves in Figure 8. Taking into account later discussion, we start from the expression for the chemical potential difference of the component i, ∆µi(l,φ1,φ2,T), of ternary solutions in which two different polymers (subscripts 1 and 2) are mixed with a single solvent (subscript 0) at constant pressure and constant temperature. Here µi°(l,T) in the pure liquid state is again taken as the standard state.

∆µi(l,φ1,φ2,T) ) µi(l,φ1,φ2,T) - µi°(l,T) ) RT[ln φi + 2

2

(1 - pi/pj)φj + pi{(1 - φi) ∑ χijφj ∑ j*i j*i 2

(1/2)

2

∑ ∑χjkφjφk}], j*i k*i

i ) 0, 1, 2 (6)

Equation 6 contains three χ parameters such as χ01, χ02 for polymer-solvent interaction and χ12 for polymer-polymer interaction. The last one multiplied by p ) pi was estimated earlier from the phase diagram of the Kn/Cn binary system. The first two are related to respective solubility curves and can be estimated by making use of eqs 3 and 6, where either φ1 or φ2 is put equal to zero and by fitting curves calculated for various χ0i values to data in Figure 8. In the calculation, p was put equal to 7 for the K39/C39 system and 4 for the K25/C25 system from the ratio of the molar volume of Cn to the corresponding solvent. Though fitting is rather poor for the K39 and C39 solutions as shown in Figure 8a, we obtain χ01 ) 0 for K39 and χ02 ) 0.36 for C39. On the other hand, good fitting has been obtained for the K25 and C25 solutions with χ01 ) 0.29 and χ02 ) 0.24. Now we can estimate the crystallization temperature Tc of the respective components in the K39/C39

TABLE 3: Crystallization Temperature for K39/C39 Solutions in Toluene Tc/K K39 molar fraction

obsd

calcd from K39

calcd from C39

0.113 0.216 0.276 0.394 0.469 0.665 0.773 0.868

303.1 303.2 301.3 299.8 298.4 299.1 298.9 300.4

300.2 300.9 298.8 298.2 299.1 299.9 299.9 300.3

303.4 303.3 303.5 302.5 303.5 301.8 300.9 299.9

solutions which form the solid solution over an entire range of fm from 0 to 1 using eqs 2 and 6. We list results in Table 3 with those obtained from direct observation. It appears that experimental Tc values are close to Tc calculated for C39 in the low-fm region, where the solid solution exhibited the solidsolid phase transitions, while experimental Tc values gradually approach Tc calculated for K39 with increasing fm. This strongly suggests that solid solutions of K39/C39 formed by solution crystallization in toluene are in the thermodynamic equilibrium state from applicability of the F-H theory and also that when the samples are recrystallized from melt, eutectic mixtures are obtained as stable ones following the conventional theory of melting point depression. For the K25/C25 system, on the other hand, the crystallization temperature of K25 in hexane is always higher than that of C25 at the same concentration. Unless the concentration of K25 is very low, then it is likely that crystals almost pure in K25 are first formed from the K25/C25 solution, leading to formation of eutectic mixtures for considerably low fm as observed. Calculation of Tc roughly supports this conjecture. Concluding Remarks In this study, we determined phase diagrams for solutioncrystallized samples of two long-chain symmetrical Kn and Cn with the same carbon numbers of n ) 25 and 39 from DSC measurements. We found that the K25/C25 sample crystallized in hexane formed eutectic mixtures above fm ) 0.14, while the K39/C39 sample formed two different types of solid solutions above and below fm ) 0.20. The F-H theory was successfully applied for explanation of the results using the χ parameter as an adjustable parameter. The main conclusion is that when solubility curves of respective components are located closely enough to each other despite a pretty large difference in the melting temperatures of two components in the pure state, solid solutions can be formed over an entire range of mixing ratio. Along this conclusion, we looked for a solvent which has a smaller χ value to K25 in comparison with that to C25 and found that ethyl alcohol might be used. Preliminary DSC measurements on K25/C25 samples crystallized in ethyl alcohol indeed have shown that solid solutions are obtainable over the whole fm region. In the middle fm region, however, thermal behaviors attributed to the solid-solid transition as well as the melting characteristic of almost pure C39 crystal was observed. It is not unlikely that the poor solvent nature of ethyl alcohol to both K25 and C25 may induce liquid-liquid phase separation before or during slow crystallization of the sample. Details will be reported in a forthcoming paper. Acknowledgment. K.N. is greatly indebted to the Department of Physics and Earth Sciences, University of the Ryukyus, for allowing her to study in Kyushu University for 1 year for partial fulfillment of the doctoral dissertation.

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