Bidirectional Transformations of n-Pentadecane between Low

Jul 13, 2016 - between the solid phases of SII and SI of n-pentadecane of which a p−T diagram has a parallel and wide gap between the coexistence li...
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Bidirectional Transformations of n‑Pentadecane between LowPressure and High-Pressure Phases Koji Shigematsu,*,† Wataru Chiba,† Seiji Sawamura,‡ and Yoshinori Takahashi†,§ †

Faculty of Education, Iwate University, Morioka 020-8550, Japan Department of Applied Chemistry, Faculty of Life Science, Ritsumeikan University, Kusatsu 525-8577, Japan

‡

ABSTRACT: We optically observed the transformation between the solid phases of SII and SI of n-pentadecane of which a p−T diagram has a parallel and wide gap between the coexistence lines between liquid-SI and between SI−SII. We were able to grow an SII crystal in the liquid by cooling or compression, and subsequently filled the high-pressure specimen vessel with the SII crystal. The transformation to SI crystals in the SII crystal started after successive cooling or compression. Final compression transformed again all the SI crystals to the SII crystal. At the growth of an SII crystal in the liquid, we measured the SII-liquid equilibrium pressures and temperatures, which were located below the coexisting lines of the SI and liquid phases in the p−T diagram. We found that the SII crystal of n-pentadecane is stable even below the coexisting line between the SI and liquid. We elucidate thermodynamically the stabilities of the SI and SII phases based on the variations of some thermodynamic functions calculated from the measurements of the p−T diagram and the molar volume variation. Finally, we proposed a new arrangement among the planes of the Gibbs energy of the SII, SI, and liquid phases of n-pentadecane in the G−p−T diagram.

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nearly all the SI crystals were first crystallized in the liquid at a temperature below 40 °C. The difference was attributed to the unparallel arrangement of the coexistence lines between SIliquid and between SI−SII in the p−T diagram of n-tridecane (The lines nearly overlap above 40 °C, and separate from each other with temperature decreasing less than 40 °C. See Figure 1b of ref 4.) Moreover, we found that an SII crystal of n-tridecane could stably coexist with the liquid on an unknown coexistence line between liquid-SII, which is even below the coexistence line between liquid-SI. We measured the equilibrium pressures between liquid-SII by the ruby fluorescence technique. As expected, the coexistence line between liquid-SII existed below that between SI-liquid. We regarded the coexistence of SII and liquid as the SII metastable region spread widely through the SI stable region to the liquid stable region. We finally found the nondestructive transformation from SII to SI crystals of n-tridecane in the low temperature region, where the pressure width of the stable region of the SI crystal is large, although the transformation accompanied expansion. We continued the research of the transformation between the solid phases of SII and SI of other odd n-alkanes. For our successive subject, n-pentadecane is suitable, because the p−T diagram of n-pentadecane has a parallel and wide gap between the coexistence lines between SI-liquid and between SI−SII

INTRODUCTION Odd normal alkane is a hydrocarbon constituted of an odd number of carbon atoms, all with single covalent bonds among carbon atoms, and with no carbon-chain branches. A few odd normal alkanes provide a pair of polymorphism structures, which comprise the low-pressure and high-temperature solid phase SI (disordered, hexagonal), and the high-pressure and low-temperature solid phase SII (ordered, orthorhombic) (Figure 1).1−3 The phase transition between them is an order−disorder transition and which currently attracts the attention of many solid-state physics researchers. The normal alkanes in any state are transparent in visible light. Moreover, their liquids and solids have characteristic refractive indices peculiar to each. Therefore, we can optically observe the boundary among their liquid and solid states. Fortunately, the morphologies of the odd normal alkane solids have clearly separate forms; for example, the SI solid grown in the liquid shows a hexagonal thin plate with very round corners, and the SII solid grown in the liquid shows a long and narrow pillar-like form.4,5 We can thus easily detect the bidirectional phase transitions between these solid phases through an optical microscope. We previously reported the pressure-induced crystal growth of the n-tridecane in the liquid4 using a diamond anvil cell (DAC) optimized for optical microscopy.6 Here, we found the epitaxial growth of the n-tridecane thin-plate-like SI crystal along the edge of its thick-pillar-like SII crystal in the liquid.4,5 The epitaxial growth was brought about by the difference of first crystallized crystals. Nearly all the SII crystals were first crystallized in the liquid at a temperature above 40 °C, and © 2016 American Chemical Society

Received: September 15, 2015 Revised: June 29, 2016 Published: July 13, 2016 4805

DOI: 10.1021/acs.cgd.5b01335 Cryst. Growth Des. 2016, 16, 4805−4812

Crystal Growth & Design

Article

pressure of n-pentadecane), the liquid of n-pentadecane transformed very often to the SI phase leaping over the SII phase. Although the SII crystals of n-pentadecane rarely first crystallized in the liquid, we could find the nondestructive transformation from SII to SI crystals of n-pentadecane in the liquid to be the same as that of n-tridecane, in a wide temperature range, where the pressure width of the stable region of the SI crystal of n-pentadecane is large. Moreover, we could grow an SII crystal in the liquid, completely filling the gasket hole (specimen vessel) of the DAC with the SII crystal. After we compressed the gasket hole of the DAC, the transformation to SI crystals in the SII crystal started. When we compressed the hole further, it was filled completely with the growing SI crystals. When we finally compressed the hole, all the SI crystals transformed again to the SII crystal. We thus concluded that the SII crystal of n-pentadecane, which coexisted with the liquid, was a stable phase. It was simply because the SII crystal could grow by compression in the liquid. We thermodynamically interpreted the stability of the SII crystal in the liquid even below the coexistence line between SI-liquid in the p−T diagram of n-pentadecane. For our new interpretation, we calculated the variation of the Gibbs’ energy based on the molar entropy variation Δs calculated from the gradient dp/dT (= Δs /Δv) measured from the p−T diagram of n-pentadecane and the molar volume v simultaneously measured in the measurement of the transformation pressures of n-pentadecan by the cylinder-piston method (CPM). We finally proposed the relationships of pressure and temperature dependences of the Gibbs’ energy of the npentadecane solids and liquid. The stable region of the SII solid is separated by that of the SI solid into two areas in the p−T diagram of n-pentadecane.

Figure 1. Orthorhombic and hexagonal unit cells of n-pentadecane.1−3 The SII phase is orthorhombic, and the SI phase is hexagonal. (a) A projection of the orthorhombic unit cell onto the a-plane. (b) A projection of the orthorhombic unit cell onto the c-plane. (c) A projection of the hexagonal unit cell onto the a-plane. (d) A projection of the hexagonal unit cell onto the c-plane. A rectangle in (b) with thick black lines is a projection of the unit cell onto the c-planes. The lattice constants of the orthorhombic unit cell are a = 0.497 nm, b = 0.748 nm, and c = 4.192 nm. In all panels, an n-pentadecane molecule was denoted only with a zigzag C−C bond chain, where a thick and short black stick shows a C−C bond. In the orthorhombic unit cell shown in (a) and (b), the planes defined by the C−C bond chains are fixed. In the hexagonal unit cell shown in (c) and (d), the planes can rotate partially hindered along their chain axes within blue circles indicated in (d).12,13 The C−C bond chains of n-pentadecane, which rotate partially hindered in the hexagonal unit cell, are slightly displaced perpendicularly to the chain axes and thus cease the rotations for denser packing of the orthorhombic unit cell.

(Figure 2). The gap is nothing but the wide stable region of the SI crystal.4 Nucleation in any liquid requires an excess pressure of 20− 30% above the freezing pressure of the liquid. As the width of a part of the SII “stable” region spreading below the wide SI stable region is narrow (probably narrower than the excess

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EXPERIMENTAL APPARATUS AND PROCEDURES

We used n-pentadecane of analytical grade (purity >99%) made by Merck Schuchardt OHG. It was used as-supplied without further purification and was directly drawn from the sample bottle using a disposable microsyringe to prevent unnecessary contamination. We measured the phase transition pressures and temperatures of npentadecane by CPM, as previously reported.4 We simultaneously measured the pressure dependences of the liquid, SI and SII of the npentadecane densities by CPM. See ref 4 for the detailed description of the measurement apparatus and procedure of the CPM. The sample volume in the cylinder, which was varied by the translation of the plug inserted into the cylinder moving with a piston rod, was calculated from the plug position measured by a dial gauge (resolution: 1 μm). The initial sample volume was 3.190 cm3. The sample density was 766.8 kg/m3 at 22.5 °C,7 the temperature when the sample completely filled the cylinder. The sample mass capsulated in the cylinder was 2.446 g. We could easily obtain the pressure dependence of the molar volume of the sample. We used the DAC for pressure generation and optical observation of the crystal growth.8 We confined a small amount of the npentadecane liquid in the hole (diameter: 0.6 mm) of a Pt-5%Au gasket (thickness: 0.5 mm), interposed between the upper and lower diamond faces of the DAC. A ruby ball (diameter: 0.134 mm) was enclosed together with n-pentadecane liquid for pressure measurement by the ruby fluorescence technique (RFT).9,10 We improved the pressure measurement with a precision of less than ±0.01 GPa using a reference light of a Ne spectrum peak.11 See ref 11 for the detailed description of the measurement apparatus procedure of the RFT. We directly observed the n-pentadecane crystals in the DAC, where the crystals coexist with the liquid in most cases, using an inverted optical microscope.6 After solidification of the entire liquid by compression, a single crystal survived after several pressure adjustments and which was used for our observation.

Figure 2. Temperature dependences of SII−SI, SI-liquid, and SIIliquid equilibrium pressures for n-pentadecane. The equilibrium pressures between SII−SI and SI-liquid phases were measured by both CPM and RFT. Those between SII-liquid phases were measured by only RFT. The pressure dependences of sample volume and density are measured by CPM. The liquid sample is confined in a cylinder and is compressed by a piston. The sample volume is measured from the piston movement, and the sample pressure is measured by a pressure gauge. Please see ref 4 for a description of the detailed measurement apparatus and procedure of the CPM. The sample pressure in the DAC was measured by RFT. The sample pressure in the DAC was measured using the pressure dependence of the wavelength of ruby fluorescence. The RFT thus requires a UV incident light sauce and a precise spectrometer. Please see ref 11 for a description of the detailed measurement apparatus and procedure of the RFT. 4806

DOI: 10.1021/acs.cgd.5b01335 Cryst. Growth Des. 2016, 16, 4805−4812

Crystal Growth & Design

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We regarded the compression of the DAC as almost hydrostatic, even when the sample vessel was fully filled with the solids of npentadecane, for the following reasons: (1) The volume of the sample vessel is very tiny. (2) There are two directions of the compression of the DAC, which is similar to a hydrostatic compression. One of the compressions is vertical, arising from the vertical movement of the diamond anvil faces. The other is horizontal, caused by the radial shrinking of the gasket inner cylindrical face. (3) n-Pentadecane is a soft matter. The DAC is highly optimized for high-definition microscopy using an inverted optical microscope. For this purpose, the DAC has a thin hinge-like shape, which makes it possible to use an ×16 magnification objective lens with an ultralong working distance and with optical-path compensation for diamond-thickness. The DAC thus should have large apertures on the upper diamond anvil and under the lower diamond anvil. The former is for the incident of the illumination light, and the latter is for the insertion of the objective lens. The DAC, therefore, has no aperture for other spectroscopic measurements.

Figure 3. (a) Pressure dependences of the SII, SI, and liquid densities of n-pentadecane at temperatures from 20 to 40 °C with 5 K interval measured by PCM. (b) Pressure dependences of molar volumes (v−p diagrams) of the SII, SI, and liquid phases of n-pentadecane at temperatures from 20 to 40 °C, calculated from the pressure dependences of the densities shown in (a). In both the panels, the black lines indicate the pressure dependences of the liquid phase, the blue lines indicate those of the SI phase, and the orange lines indicate those of the SII phase. Unfortunately, we could not measure the molar volume of the SII phase of high-temperature and low-pressure side during both the compression and decompression steps of PCM, as we always obtained the SI solid as the first crystal in the cylinder of PCM. We thus estimated the molar volumes of the SII phase of hightemperature and low-pressure side at the five temperatures between 20 and 40 °C by the extrapolations of pressure dependences of the molar volumes of the SII phase of low-temperature and high-pressure side to the equilibrium pressures on the coexistence line between SII-liquid measured by the RFT. The molar volumes of the liquid at the equilibrium pressures between SII-liquid were connected with the broken lines, which are extended upward and parallel to the ordinate along the equilibrium pressures between SII-liquid.

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RESULTS Figure 2 shows the temperature dependences (20−40 °C) of the SII−SI, SI-liquid, and SII-liquid equilibrium pressures for npentadecane.4 Those of SII−SI and SI-liquid were measured by both CPM and RFT, which were in correspondence. That of SII-liquid was measured by only RFT. It should be noted that the dependence of SII-liquid settled down below that of SIliquid. In other words, the SII phase was stable even below the equilibrium pressure between SI-liquid, where the SI phase was no longer stable. The data used for plotting Figure 2 are tabulated in Table 1. Figure 3a shows the pressure dependences of the SII, SI, and liquid densities of n-pentadecane at the temperatures from 20

to 40 °C with 5 K intervals measured by CPM. We can easily calculate the temperature and pressure dependences of the compressibilities and molar volumes of the SII, SI, and liquid phases from Figure 3a. The data used for plotting Figure 3a are tabulated in Table 2. Figure 3b shows the pressure dependences of the molar volume of the SII, SI, and liquid of n-pentadecane at temperatures from 20 to 40 °C with 5 K intervals. We could easily detect the liquid freezing and the transformation from the SI to SII phases. As the liquid and the SI phases coexist in the cylinder during freezing, and the SI and SII phases coexist during the transformation, the pressures measured by the load cell were kept at a constant value (equilibrium pressure) during freezing or transformation, and the resistance to move the plug driven by a piston rod decreased drastically. During the liquid freezing and the transformation between the SI and SII phases, the resistances to move the plug were kept at a large value, and the slopes of the pressure dependences of the densities of the liquid and SII phases were kept at constant values, except near the freezing and transformation (however, the slope of the SI phase was not constant, as a large part of the pressure dependence of the density of the SI phase was occupied by the near regions of the freezing and transformation, due to the narrow pressure width of the density of the SI phase). We interpret the constant slopes as the properties of single phases. Unfortunately, we could not obtain the SII phase of hightemperature and low-pressure side, but only the SI phase as the first crystal in the cylinder of PCM, during the compression steps in PCM. The following reason for the lack of the SII phase is that pressure-induced crystallization requires an excess pressure of 20−30% of the equilibrium pressure between the

Table 1. Temperature Dependences of the Coexisting Pressures between the SII and SI, the SI and Liquid, and the SII and Liquid Phases of n-Penradecane Measured by CPM and RFT, Which Are Used To Draw Figure 2 pressure (MPa) temperature (°C)

SII−SI (CPM)

20.0 25.0 29.4 30.0 33.8 35.0 38.2 40.0

83.2 102.1

19.7 20.0 24.7 25.0 29.5 30.0 34.2 35.0 40.0

SII−SI (RFT)

121 124.1 141 145.3 157 168.3 SI−L (CPM)

SI−L (RFT) 41

42.2 59 64.8 89 90.6 105 114.3 140.0 SII−L (RFT)

24.7 29.6 34.2 38.9

47 72 93 122 4807

DOI: 10.1021/acs.cgd.5b01335 Cryst. Growth Des. 2016, 16, 4805−4812

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pressure variation involved in the 0.5 mm retreat of the piston of the CPM is larger than the pressure width between coexistence lines between SI-liquid and SII-liquid. We thus estimated the molar volumes of the SII phase of high-temperature and low-pressure side, and the liquid phase at the equilibrium pressures on the coexistence line between SIIliquid measured by the RFT, as described in the caption of Figure 3. Figure 4 shows sequential images and schematic drawings of the transformation from an SII to an SI crystal of npentadecane. The transformation started on both the a-planes of the SII, and propagated along the a-axis, with no destruction

Table 2. Pressure Dependences of the SII, SI, and Liquid Densities of n-Pentadecane at Temperatures from 20 to 40 °C with 5 K Interval Measured by PCM, Which Are Used To Draw Figure 3aa liquid pressure (MPa)

density (×103 kg/m3)

5.2 11.2 22.6 34.2 40.9

0.7843 0.7925 0.8007 0.8092 0.8177

8.7 17.2 27.0 38.1 50.7 63.0

0.7828 0.7907 0.7989 0.8073 0.8159 0.8246

13.5 23.3 34.3 46.1 58.6 72.8 87.8

0.7827 0.7908 0.7989 0.8072 0.8159 0.8246 0.8335

17.6 27.3 38.8 51.2 64.5 77.1 94.1 111.0

0.7828 0.7906 0.7989 0.8073 0.8159 0.8246 0.8336 0.8427

21.6 32.0 43.4 55.7 70.0 84.4 100.2 116.9 136.7 141.6

0.7827 0.7906 0.7989 0.8073 0.8158 0.8246 0.8335 0.8426 0.8520 0.8616

SI

SII pressure (MPa)

density (×103 kg/m3)

84.7 98.5 129.6 165.4 210.4

0.9491 0.9607 0.9729 0.9854 0.9983

103.0 107.8 134.0 169.8 211.3 257.8

0.9472 0.9585 0.9705 0.9829 0.9957 1.0087

125.2 127.9 149.9 184.9 227.3 274.3

0.9467 0.9585 0.9705 0.9827 0.9955 1.0086

Temperature (35 °C) 116.7 0.8917 118.5 0.9021 123.2 0.9130 134.8 0.9239 144.6 0.9349

146.8 161.2 194.6 235.1 282.7

0.9585 0.9705 0.9827 0.9954 1.0088

Temperature (40 °C) 144.5 0.8918 145.3 0.9022 148.3 0.9129 154.2 0.9238 163.2 0.9350

169.2 175.4 203.7 242.1 287.5

0.9585 0.9705 0.9828 0.9954 1.0084

pressure (MPa)

density (×103 kg/m3)

Temperature (20 °C) 43.9 0.8734 45.8 0.8836 49.5 0.8940 58.6 0.9045 74.3 0.9151 82.6 0.9261 Temperature (25 °C) 67.0 0.8815 69.2 0.8917 74.0 0.9020 85.0 0.9130 101.40 0.9237 Temperature (30 °C) 93.4 0.8918 96.2 0.9020 104.3 0.9129 120.4 0.9237

Figure 4. Sequential images of the transformation of an n-pentadecane from an SII crystal to an SI crystal in the liquid kept in the gasket hole of the DAC. Left column pictures are real crystal forms. A large bright circle in each panel is the gasket hole. The numbers at the upper left corner indicate the photographing date and time, and the numbers after “T” at the upper right corner indicate the temperatures of the DAC. The round shadow at the lower right side of the gasket hole is a ruby ball enclosed for the pressure measurement. Right column pictures are schematically drawn crystal forms. The edges of an SII npentadecane crystal are shown by orange solid lines in (a), an npentadecane crystal transforming from SII to SI is shown by a green curved line in (b), an SI n-pentadecane thin crystal is shown in a blue curved solid line in (c), and the ruby ball is shown by a black broken circle in all panels of the right column. The scale bars at the lower left corner in all panels indicate 0.1 mm. (a) A vertically arranged fine thick-pillar-like crystal, in the left side of the gasket hole, which is a part of an SII n-pentadecane crystal before the transformation. (b) The n-pentadecane SI crystal just after the transformation which started from the a-planes of both sides. (c) The n-pentadecane SI crystal which had been changing to a thin plate with very round corners, 5 m 41 s later from (b). The temperature was rapidly decreased from 27.9 to 25.4 °C between (a) and (c). The final pressure of the transformation was 61 MPa.

a

We can obtain molar volumes of n-pentadecane, shown in Figure 3b, by calculation of the inverse number of the densities and by conversion of a part of the unit of molar volume from kg to mol (0.21242 kg).

liquid and solid, although the pressure width between coexistence lines between SI-liquid and SII-liquid, which is 15−20 MPa as shown in Figure 2, is less than the excess pressure for the first crystallization. As a result, we always obtain the SI solid as the first crystal in the cylinder of the CPM. Moreover, we could not obtain the SII phase of hightemperature and low-pressure side in the decompression steps of the CPM. The reason for the lack of the SII phase is that the 4808

DOI: 10.1021/acs.cgd.5b01335 Cryst. Growth Des. 2016, 16, 4805−4812

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in the same manner as that of n-tridecane.4 Although the crystal volume during the transformation increased, the transformation cooperatively propagated in order and did not result any destruction. The transformation of n-tridecane rarely occurred when the pressure in the gasket hole of the DAC was slightly reduced with the temperature regulated at a low temperature less than 20 °C. However, that of n-pentadecane frequently occurred. Figure 5 shows sequential images and schematic drawings of the growth of the SII crystal of n-pentadecane surrounded by the liquid in the gasket hole of the DAC (a), the SII crystal growing and filling the hole (b), and the SI crystals newly growing in the SII crystal (c) during slow cooling of the DAC. The slow cooling made the (p, T) conditions in the npentadecane p-T diagram (Figure 2) move along the

coexistence line between liquid-SII, and to grow the crystal as shown in (a). The successive cooling caused the conditions to move horizontally to the left direction, and to grow the crystal as shown in (b). The simultaneous cooling and compression made the conditions move to the upper-left direction, with crossing the coexistence line between liquid-SI, finally enter the area between the SII−SI and SI-liquid coexistence lines, where the SI phase was stable, and transformed the crystal as shown in (c). The final (p, T) conditions were (26.2 °C, 116 MPa). Figure 6 shows sequential images and schematic drawings of the growth of the SI crystals of n-pentadecane in the gasket hole of the DAC filled with the SII crystal (a), the SI crystals growing in the SII and finally filling the hole (b), and then the SI crystals transformed to SII and finally the SII crystals filling

Figure 5. Sequential images of the growth of an SII crystal in the liquid and the growth of SI crystals of n-pentadecane in the SII crystal. Left column pictures are actual crystal forms. The notations of date, time, and temperature in each panel are the same as those in Figure 4. The round shadow or sphere at the lower right side of the gasket hole in each panel is a ruby ball. Right column pictures are schematically drawn crystal forms. The edges of an SII n-pentadecane crystal are shown by orange solid lines in (a) and (b), The boundary of SI npentadecane crystals are shown in blue curved solid lines in (c), and the ruby ball is shown by a black broken circle in all panels of the right column. The scale bars at the lower left corner in all panels indicate 0.1 mm. (a) An SII crystal coexisting with the liquid at 30.8 °C and 75 MPa. (b) The SII crystal growing and finally filling the gasket hole during the cooling to 29.3 °C. A black flake in the lower side of the gasket hole came off the surface of the inner wall of the gasket hole and is thus a Pt-5%Au alloy flake. (c) The growing SI crystals in the SII crystal. The SI crystals appeared in the right side of the gasket hole and grew toward the left side in the SII crystal after successive compression and cooling. The final conditions of the transformation were 26.2 °C and 116 MPa.

Figure 6. Sequential images of the transitions from an SII crystal to SI crystals and from the SI crystals to an SII crystal, during the temperature regulation of the DAC at 14.8 °C, and under continuously increasing pressure. Left column pictures are actual crystal forms. The notations of date, time, and temperature in each panel are the same as those in Figure 4. The round sphere at the lower right side of the gasket hole in each panel is a ruby ball. Right column pictures are schematically drawn crystal forms. The crystals mixed with SI and SII phases are shown as purple area in (a), the SI crystals filling up the gasket hole are shown as blue area in (b), and the SII crystal is shown as orange area in (c). The black broken circle in each panel of the right column is the ruby ball. The scale bars at the lower left in all panels indicate 0.1 mm. (a) Growing SI crystals in an SII crystal kept at 14.8 °C with increasing pressure. (b) The SI crystals filling up the gasket hole after successive compression. (c) An SII crystal filling up the gasket hole after successive compression. The SII crystal shown in (a) was completely transformed to the SI phase as shown in (b). The SI crystals shown in (b) were completely transformed again to the SI phase. The final pressure of the gasket hole shown in (c) was 190 MPa. 4809

DOI: 10.1021/acs.cgd.5b01335 Cryst. Growth Des. 2016, 16, 4805−4812

Crystal Growth & Design

Article

again the hole (c), during continuous compression of the DAC under temperature regulation. The (p, T) conditions in the p− T diagram (Figure 2) translated vertically to the upward direction at the regulated temperature and finally entered the area, above the SII−SI coexistence line, where the SII phase was stable. The temperature of the sample in the gasket hole of the DAC was continuously monitored using a copper-constantan thermocouple of thin wires tightly inserted between the lower diamond anvil and the gasket of the DAC. Therefore, the temperature was always superimposed on the monitored images, as shown at the upper right corner of each panel in the left columns of Figures 4−6. The pressure measurement by the RFT requires a measurement of the ruby fluorescence spectra of 692.0−694.5 nm. As the spectra were scanned with a very slow speed (0.15 nm/ min) for our precise measurement, one time of the pressure measurement by the RFT required at least 17 min (usually over 20 min). We thus measured the sample pressures at the initial and final stages of our observations, shown in Figures 4, 5, 6.

Table 3. Differences of the Gibbs’ Free Energy between the SII and SI, the SI and Liquid, and the SII and Liquid Phases of n-Penradecanea dp/dT (MPa/K)

SII−SI

4.27

SI−L

4.90

SII−L

5.21 20.0

25.0

30.0

35.0

40.0

SII−SI

−5.58

−5.70

−5.58

−5.58

−5.55

SI−L SII−L SII−SI SI−L SII−L SII−SI SI−L SII−L SII−SI SI−L SII−L

−16.6 −36.0 −23.8 −81.2 −187 83.6 42.4 21.7 −6.51 −23.1 −54.2

−16.6 −32.9 −24.4 −81.5 −171 102 65.0 47.8 −6.68 −23.2 −49.5

−16.7 −29.8 −23.8 −81.6 −155 123 90.6 73.9 −6.54 −23.2 −44.8

−13.9 −29.2 −23.8 −67.9 −152 146 114 99.9 −6.53 −19.3 −44.0

−8.34 −26.6 −23.7 −40.9 −139 166 143 126 −6.50 −11.6 −40.1

temperature (°C) Δv (×10−6 m3/mol)

Δs (J/K mol)

p (MPa)

ΔG (kJ/mol)

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DISCUSSION The transformation from the SII to SI phases only occurs when the pressure and temperature are regulated in the SI stable region, which is situated between the SII−SI and SI-liquid equilibrium lines as shown in Figure 2. The lines of n-tridecane are very close to each other in the high temperature region greater than 40 °C, and they gradually separate with reducing temperature. The transformation of n-tridecane thus occurred only at a low temperature below 20 °C where the area between the separated lines was large; in other words, the SI stable region was wide. On the other hand, the transformation of npentadecane frequently occurred because of the widely separated lines. The n-pentadecane p−T diagram shown in Figure 2 exhibits the pairs of widely separated lines not only between the SII−SI and SI-liquid equilibrium lines, but also between the SI-liquid and SII-liquid equilibrium lines. The SII crystals could grow sufficiently under wide temperature and pressure ranges between the SI-liquid and SII-liquid equilibrium lines. We could thus completely fill the gasket hole of the DAC with the SII crystal of n-pentadecane. Moreover, the SII crystal transformed to the SI crystals both by cooling as shown in Figure 5c and by compressing as shown in Figure 6b. Finally, the SI crystals, which had transformed from the SII crystal by compression, transformed again to the SII crystal by successive compression as shown in Figure 6c. Therefore, we conclude that the SII crystal is a stable phase even below the coexisting line between the SI and liquid. We thus can examine thermodynamically the stability of the SII phase of n-pentadecane under the differences of the Gibbs energy ΔG between the SII and SI, the SI and liquid, and the SII and liquid phases (Table 3). The ΔG between the SII and SI, the SI and liquid, and the SII and liquid phases of n-pentadecane under the coexisting (p, T) conditions of the SII and SI, the SI and liquid, and the SII and liquid phases are tabulated in Table 3. We use the incomplete definition of ΔG, ΔG = −TΔs + pΔv, which excludes the differences of the internal energy ΔU, as the internal energy U is impossible to evaluate (here, Δs: molar entropy variation, Δv: molar volume variation, T: coexisting temperature, p: coexisting pressure). The Δs and Δv are calculated by (the value of higher ordered phase) − (that of

a

The differences were obtained with the following steps: (1) The direct measurement of the gradient dp/dT (= Δs/Δv) in the p−T diagram of n-pentadecane shown in Figure 2 (here, Δs: molar entropy variation, Δv: molar volume variation). (2) The direct measurement of the molar volume variation Δv between these phases within the temperature range of 20.0−40.0 °C with 5 K interval from the p−V diagram shown in Figure 3b. However, the molar volume variation Δv between the SII and liquid phases were estimated from the extrapolated molar volumes of the SII phase as described in the caption of Figure 3. (3) The calculation of molar entropy variation Δs these phases within the temperature range of 20.0−40.0 °C with 5 K interval from the gradient (dp/dT)s and the molar volume variation (Δv)s. (4) The direct measurement of the coexisting pressure ps between these phases within the temperature range of 20.0−40.0 °C with 5 K interval from the p−T diagram shown in Figure 2. (5) The calculation of differences of the molar Gibbs energy ΔG between these phases of n-pentadecane under the coexisting conditions of the SII and SI, the SI and liquid, and the SII and liquid phases using the incomplete definition of ΔG = −TΔs + pΔv. The Δs and Δv are calculated by (the value of higher ordered phase)(that of lower ordered phase). The values of the Δs and Δv thus are negative.

lower ordered phase). The values of the Δs and Δv thus have negative values. Any absolute value of TΔs is several increments of 10 degrees larger than the corresponding absolute value of pΔv. We can evaluate the order of the internal energy of the SII, SI, and liquid phases as U(SII) < U(SI) ≪ U(liquid). We should note that all Us have negative values (the order of the absolute values of U is converse to that of the negative (real) values). Although all the ΔGs tabulated in Table 3 show positive values, we can regard the true ΔGs as nearly zero, based on the qualitative evaluation of the ΔUs of the SII, SI, and liquid phases. This is because the ΔUs also have negative values, as the ΔUs are calculated by (the ΔU of higher ordered phase, which has a negative large number) − (the ΔU of lower ordered phase, which has a negative small number), the same as those of the values of the Δs and Δv. We can easily expect that the SII crystal, on the coexisting line between the SII and liquid phases in the p−T diagram, has a nearly equal Gibbs energy G to that of the liquid on the line. Moreover, we can propose a new arrangement among the planes of the Gs of the SII, SI, and liquid phases of n4810

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slightly decrease) near the center of the stable area of the G(p, T) plane of the SI phase, because of the following reason: The slight increase of the decreasing rate of the G(p, T) of the SI phase is brought about by the large compressibility of the SI phase than that of liquid, as shown in Figure 3b. Moreover, the large compressibility of the low-pressure side near the liquid stable region varies more than that of the high-pressure side near the SII stable region with decreasing pressure. The compressibility of the SI phase of the low-pressure side near the liquid stable region is very large even compared with that of the liquid. The hexagonal unit cell of the SI phase can be easily compressed along the radial directions on the a−b plane of the unit cell. This is because the cylinders, which are formed by the hindered rotation of n-pentadecane molecule with the C−C bond chain axis of the molecule as a rotation axis, are easily compressed along the radial directions.12,13 The compression of the SI phase in the low-pressure side near the liquid stable region is easier to accomplish than that that in the highpressure side near the SII stable region. The quite easy compression along the radial directions on the a−b plane in the unit cell of the SI solid provides the large value and variation of the compressibility of the SI phase. The large compressibility of the SI phase provides the large variation of pv, which increases with decreasing pressure. We should notice that the Gibbs energy Gs have always negative values. In the complete definition of G, G = U − Ts + pv, the negative value variations of Us, which decrease (the absolute values of Us increase) with increasing temperature and with decreasing pressure, nearly cancel the negative value variations of the (−Ts)s, which decrease (the absolute values of (−Ts)s increase) with increasing temperature. We can say, from the common sense of thermodynamics, that the increasing/decreasing variations of (−Ts) and pv cancel the decreasing/increasing variation of U in the transitions between stable states. Since the positive values of pvs occupy only a few percent of the absolute values of (−Ts)s, the increasing/decreasing variations of (−Ts) nearly cancel the decreasing/increasing variation of U in any transition between stable states, including different phase states. Accordingly, the variations of the positive values of pvs, which increase with decreasing pressure, affect strongly to lower the decreasing rate of Gs with decreasing pressure. The increasing of pv of the SI phase, at the high-pressure and low-temperature side, near the SII stable region, has a small positive value due to the relatively small compressibility with decreasing pressure. As a result, the G(p, T) of the SI phase near the SII stable region decreases rapidly, with increasing temperature and with decreasing pressure. This is because the small increase of the pvs with decreasing pressure hardly accelerates the decrease of the Gs with increasing temperature and with decreasing pressure. Inversely, the increase of pv of the SI phase, at the lowpressure and high-temperature side, near the liquid stable region, has a large positive value due to the very large compressibility with decreasing pressure. As a result, the G(p, T) of the SI phase near the liquid stable region decreases slowly, with increasing temperature and with decreasing pressure. This is because the large increase of the pvs with decreasing pressure strongly accelerate the decrease of the Gs with increasing temperature and with decreasing pressure. Therefore, the large difference between the variations of pv at high-pressure side and that at low-pressure side in the SI stable

pentadecane, depending on pressure p and temperature T, which we call G−p−T diagram (Figure 7a). As the G(p, T)

Figure 7. Two types of G−p−T diagram of n-pentadecane. Here, G has negative values. The bottom planes of the diagrams correspond to the p-T diagram shown in Figure 2. Purple planes are the G (p, T) planes of the liquid phase, blue planes are those of the SI phase, and orange planes are those of the SII phase. The G(p, T) planes form convex surfaces of the second order. As expected from their convex surfaces, both the first and second differential coefficients of the planes are always negative. (a) A newly proposed G−p−T diagram of the special arrangement based on the p−T conditions of the transformations between the SI and SII phases shown in Figures 5 and 6. The negative values of the second differential coefficient of the G(p, T) plane of the SI phase slightly decreased with increasing temperature and with decreasing pressure at the SI phase stable region, due to the larger compressibility of the SI phase compared to that of the liquid. (b) A G−p−T diagram of an ordinary arrangement of the G(p, T) planes of the three phases. (c) The crossing lines between the G(p, T) planes and the cross section A in the G−p−T diagram (a). This panel is drawn to show clearly the complex behavior of the G(p, T) plane of the SI phase as described above. All the crossing lines are drawn as the convex lines of the second order. However, the crossing line of the SI phase has a shallow convex part in the SI phase stable region. (d) The crossing lines between the G(p, T) planes and the cross section B in the G−p−T diagram (b). All the crossing lines are drawn as the convex lines of the second order.

plane of the SI phase twice intersects the G(p, T) plane of the SII phase, the plane of the SII phase is separated into two areas. As a result, the two areas of the separated G(p, T) plane of the SII phase interpose the stable area of the G(p, T) plane of the SI phase. All the G(p, T) planes of the G−p−T diagram of ordinary arrangement shown in Figure 7b monotonically decrease with increasing temperature and with decreasing pressure. The G(p, T) planes of SII and liquid phases in Figure 7a also monotonically decrease with increasing temperature and with decreasing pressure. Only the decreasing rate of the G(p, T) of the SI phase shown in Figure 7a varies from a slightly high rate to a slightly low rate near the center of the stable area of the G(p, T) plane of the SI phase with increasing temperature and with decreasing pressure. That is, the first and second differential coefficients of the G(p, T) of the SI phase with respect to increasing temperature and decreasing pressure, which are retaining negative values according to common sense of thermodynamics, slightly increase (their absolute values 4811

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area brings the change of the decreasing rate of the G(p, T) of the SI phase near the center of the stable area of the G(p, T) plane of the SI phase, as shown in Figure 7a. Thus, the stable region of the SI phase separates the SII phase into two areas, as shown in the p−T diagram of npentadecane (Figure 2).

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CONCLUSIONS The SII crystal of n-pentadecane is stable even below the coexisting line between the SI and liquid. The newly proposed G−p−T diagram provides the interpretation of the stability of the SII crystal of n-pentadecane and bidirectional phase transitions of n-pentadecane between SII and SI phases only with increasing pressure. However, an insufficiency remained in detecting the other stable SII crystal, which is located below the coexisting line between the SI and liquid in the p−T diagram of npentadecane, in the measurement by the CPM. We need not only to improve the apparatus and measurement procedures of the CPM, but also to interpret the first nucleation in compressed liquid, which is a difficult problem to solve.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Fax: +81 19 621 6560. Telephone: +81 19 621 6548. Present Address §

(Y.T.) Department of Agriculture, Forestry and Fisheries, Iwate Prefecture Government. Notes

The authors declare no competing financial interest.

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REFERENCES

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DOI: 10.1021/acs.cgd.5b01335 Cryst. Growth Des. 2016, 16, 4805−4812