Zero Thermal Expansion Fluid and Oriented Film Based on a Bistable

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Zero Thermal Expansion Fluid and Oriented Film Based on a Bistable Metal-Cyanide Polymer Hiroko Tokoro,*,†,‡ Kosuke Nakagawa,† Kenta Imoto,† Fumiyoshi Hakoe,† and Shin-ichi Ohkoshi*,†,§ †

Department of Chemistry, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan NEXT, JSPS, 8 Ichibancho, Chiyoda-ku, Tokyo 102-8472, Japan § CREST, JST, K’s Gobancho, 7 Gobancho, Chiyoda-ku, Tokyo 102-0076, Japan ‡

S Supporting Information *

ABSTRACT: A zero thermal expansion (ZTE) material based on plateshaped rubidium manganese hexacyanoferrate, Rb0.97Mn[Fe(CN)6]0.99· 0.3H2O, is prepared using a polyethylene glycol monolaurate (PEGM) surfactant matrix. The prepared microcrystals show a charge transfer induced phase transition between the cubic MnII−NC−FeIII and tetragonal MnIII−NC−FeII phases. The MnIII−NC−FeII phase exhibits a small negative thermal expansion (NTE) along the aLT and cLT axes with a thermal expansion coefficient of α(aLT) = −1.40 ± 0.12 × 10−6 K−1 and α(cLT) = −0.17 ± 0.13 × 10−6 K−1 over a wide temperature range of 15 K − 300 K. Such small |α| materials are classified as ZTE materials. The far-infrared spectra show that NTE originates from the transverse modes δ(Fe−CN−Mn) of the transverse translational mode around 304 cm−1, and transverse librational modes at 253 and 503 cm−1, which are assigned according to first principle calculations. Molecular orbital calculations indicate that ZTE and the charge transfer phase transition both originate from the transverse mode. Additionally, an oriented film on SiO2 glass is prepared using a microcrystal dispersive methanol solution and a spin-coating technique. This is the first example of a ZTE film that maintains a constant film thickness over a wide temperature range of 300 K. KEYWORDS: zero thermal expansion, oriented film, charge transfer induced phase transition, transverse mode

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INTRODUCTION Most materials exhibit a positive thermal expansion (PTE), which leads to an expanded lattice as the temperature increases because the higher energy levels of the lattice vibrations are populated. A PTE behavior causes flaking or destruction at a joint interface between different materials. From this viewpoint, zero thermal expansion (ZTE) and negative thermal expansion (NTE) materials are useful for space materials, precision mechanical equipment, electrical circuits, etc. ZTE and NTE effects have been reported in metal alloys and metal oxides,1−3 e.g., Fe−Ni alloys and ZrW2O8. Recently, ZTE or NTE behaviors have also been observed in molecular materials,4−7 including M(CN)2, M(CN)3, and M(CN)6 moieties. ZTE and NTE effects often occur in structural phase transition materials because structural flexibility plays an important role in these effects. From this perspective, cyano-bridged bimetal assemblies are useful candidates because they show a structural phase transition based on a charge transfer (CT) or spin crossover effect.8−10 In fact, we and co-workers have previously observed ZTE and NTE effects in Prussian blue analogs11,12 with charge transfer characteristics such as manganese hexacyanoferrate.13 Rubidium manganese hexacyanoferrate shows a structural phase transition based on the charge transfer between MnII− NC−FeIII [high-temperature (HT)] and MnIII−NC−FeII © 2012 American Chemical Society

[low-temperature (LT)] phases with a large thermal hysteresis loop.10 In the present work, we prepare plate-shaped Rb0.97Mn[Fe(CN)6]0.99·0.3H2O using a surfactant matrix. The prepared sample exhibits a ZTE behavior over a wide temperature range. In addition, using a spin-coating technique and a dispersive solution of the microcrystals, a film is realized. This film is composed of oriented plate-shaped microcrystals along the substrate and exhibits an oriented phase transition between the HT and LT phases. To the best of our knowledge, this is the first example of a film with such an oriented phase transition.

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RESULTS AND DISCUSSION Synthesis. Plate-shaped microcrystals of Rb0.97Mn[Fe(CN)6]0.99·0.3H2O were synthesized via the following method. A polyethylene glycol monolaurate (PEGM) matrix containing MnCl2 (aqueous, (aq)) (0.2 mol dm−3) and RbCl (aq) (1 mol dm−3) solutions, and PEGM containing K3[Fe(CN)6] (aq) (0.2 mol dm−3) and RbCl (aq) (1 mol dm−3) solutions were mixed (Figure 1a). A precipitate was obtained by Received: December 17, 2011 Revised: March 5, 2012 Published: March 9, 2012 1324

dx.doi.org/10.1021/cm203762k | Chem. Mater. 2012, 24, 1324−1330

Chemistry of Materials

Article

from each water droplet, and S is related to the effect of the interferences between water droplets.14 SAXS analyses indicate that a water droplet of MnCl2 in a PEGM matrix measures ca. 1.6 nm (Figure 2b, left), whereas a water droplet of K3[Fe(CN)6] in a PEGM matrix is ca. 1.7 nm (Figure 2b, right). Because surfactant PEGM has a high viscosity, crystal growth is suppressed, resulting in small microcrystals compared to the previously reported synthesis in water (ca. ∼3 μm).10b Magnetic susceptibility. Figure 3 shows the product of the molar magnetic susceptibility (χM) and the temperature (T)

Figure 1. (a) Schematic illustration of the synthetic procedure and (b) SEM image of a Rb0.97Mn[Fe(CN)6]0.99·0.3H2O microcrystal. Green, red, blue, gray, dark gray, purple, and white balls indicate Rb, Mn, Fe, C, N, O, and H atoms, respectively.

centrifuging, washing in methanol, and drying in air. Elemental analyses and the infrared (IR) spectrum confirm that the formula of the present compound is Rb0.97Mn[Fe(CN)6]0.99· 0.3H2O (Experimental Section). The scanning electron microscope (SEM) image indicates the morphology of the obtained microcrystals is not cubic, but plate-shaped with dimensions of X = 690 ± 250 nm, Y = 250 ± 125 nm, and Z = 94 ± 37 nm (Figure 1b). Small-Angle X-ray Scattering. Small-angle X-ray scattering (SAXS) spectra were measured to understand the states of MnCl2 (aq) and K3[Fe(CN)6] (aq) solutions in a PEGM matrix. Figure 2a (left) shows the scattering intensity (I) vs

Figure 3. Observed χ M T−T plots of the Rb 0 . 97 Mn[Fe(CN)6]0.99·0.3H2O microcrystals under 5000 Oe with cooling (Δ) and warming (○) processes.

vs T plots of the plate-shaped microcrystals. The χMT value of the HT phase at 300 K is 4.43 cm3 K mol−1. As the sample is cooled, a phase transition from the HT phase to the LT phase occurs at T1/2↓= 176 K. Conversely, warming the sample in the LT phase increases the χMT value at T1/2↑= 294 K, and the value reaches that of the HT phase. The width of the thermal hysteresis loop (ΔT= T1/2↑ − T1/2↓) is 118 K. Temperature Dependence of Crystal Structures. The X-ray diffraction (XRD) patterns of Rb 0. 97 Mn[Fe(CN)6]0.99·0.3H2O microcrystals at 300 K in the range of 10−40° show that the crystal structure of the HT phase is cubic (space group: F43̅ m) (Figure 4 and Table S1 in the Supporting Information). As temperature decreases, the XRD pattern of the HT phase disappears and a different XRD pattern appears below 220 K. This new pattern is assigned to the LT phase with a tetragonal structure (I4m ̅ 2) (Figure 5 and Table S2 in the Supporting Information). Figure 4 shows the temperature dependence of the XRD patterns of the HT phase in the temperature range of 230 K − 300 K. At each temperature, the XRD pattern shows a cubic structure. The aHT value remains almost constant, e.g., 10.55952(9) Å at 230 K and 10.56065(10) Å at 300 K. Figure 6a plots the lattice constant of aHT as a function of temperature. The thermal expansion coefficient (α= (dl)/(ldT) defined by the differential of the lattice constant as the temperature increases where l is the value of lattice constant) of the HT phase is estimated as α(aHT) = +1.17 ± 0.05 × 10−6 K−1, which is a small PTE coefficient. In general, materials with |α| < 2 × 10−6 K−1 are classified as ZTE materials.1b Hence, the present HT phase is a ZTE material in the temperature range of 230− 300 K. Figure 5 shows the temperature dependence of the XRD patterns of the LT phase in the temperature range of 15− 260 K. The XRD pattern for each temperature indicates a tetragonal structure. The values of aLT and cLT remain almost constant, e.g., aLT = 7.0874(3) Å and cLT = 10.5112(12) Å at

Figure 2. (a) Small-angle X-ray scattering (SAXS) curves for a water droplet of MnIICl2 and K3[FeIII(CN)6] aqueous solutions. Black dots are the observed values and red lines are the fitted curves. (b) Calculated size distribution of the water phases in MnIICl2 and K3[FeIII(CN)6] aqueous solutions.

scattering angle (2θ) of MnCl2 (aq) in PEGM. I monotonously decreases as 2θ increases. Similarly, I monotonously decreases as 2θ increases in the I vs 2θ plots of K3[Fe(CN)6] (aq) in PEGM (Figure 2a, right). The observed I vs 2θ plots can be fitted by I = nPS (Figure 2a, red lines), where n is the number density of water droplets, P is related to the scattering intensity 1325



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Figure 4. XRD patterns and Rietveld analyses of randomly oriented Rb0.97Mn[Fe(CN)6]0.99·0.3H2O plate-shaped microcrystals in the HT phase during the cooling process. Impurity signal due to manganese(II) hexacyanoferrate(II) is eliminated.15 Red dots, black lines, and green dots are the observed plots, calculated pattern, and their difference, respectively. Red bars represent the calculated positions of the Bragg reflections in the cubic structure (F43̅ m). Miller indices are reflections.

Figure 5. XRD patterns and Rietveld analyses of randomly oriented Rb0.97Mn[Fe(CN)6]0.99·0.3H2O plate-shaped microcrystals in the LT phase during the warming process. Impurity signal due to manganese(II) hexacyanoferrate(II) is eliminated.15 Blue dots, black lines, and green dots are the observed plots, calculated pattern, and their difference, respectively. Blue bars represent the calculated positions of the Bragg reflections in the tetragonal structure (I4̅m2). Miller indices are reflections.

15 K and 7.0843(6) Å and 10.5107(11) Å at 260 K, respectively. Figures 6b and 6c plot the lattice constants of aLT and cLT as functions of temperature, respectively. The thermal expansion coefficients are α(aLT)= −1.40 ± 0.12 × 10−6 K−1 and α(cLT) = −0.17 ± 0.13 × 10−6 K−1, which are small NTE

coefficients. The LT phase is also classified as a ZTE material over a wide temperature range of 15−300 K. Mechanism of the Zero Thermal Expansion. The mechanism for the observed ZTE behavior is related to the structural flexibility of the material (Figure 7). The LT phase of 1326



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Figure 7. Schematic illustrations of phonon modes: (a) Stretching mode ν(CN) at 2100 cm−1 present in the medium-IR spectrum. (b) Soft phonon modes δ(Fe−CN−Mn) of the transverse translational mode at 304 cm−1 and the transverse librational modes at 253 and 503 cm−1 observed in the far-IR spectrum. First principle calculations give the calculated frequencies as: stretching mode at 2110 cm−1 (symmetry: B2 (R, I)), transverse translational mode at 296 cm−1 (E (R, I)), and transverse librational modes at 257 cm−1 (E (R, I)) and 506 cm−1 (E (R, I)). R and I indicate Raman and IR activities, respectively. Thermal expansion originates from the stretching mode, whereas the transverse translational and transverse librational modes are the origin of thermal contraction. Thermal contraction counteracts the thermal expansion, resulting in ZTE.

Figure 6. Temperature dependence of the lattice constants of (a) aHT in the HT phase, and (b) aLT and (c) cLT in the LT phase. Lines are to guide the eye. Error bars for aHT and aLT are included in the plots.

valence states (the LT phase). In contrast, the charge distributions for the distorted configuration are 22% for Fe and 58% for Mn, indicating that the valence state is close to Fe3+−CN−Mn2+ (the HT phase). Because of the distortion by the transverse translational mode, the valence states of the metal ions interchange. These results suggest that thermal excitation of the transverse translational mode can cause a CT phase transition. Although thermodynamic analysis based on Slichter−Drickamer model has been reported for RbMnFe Prussian blue analogs,10c,18 the driving force of the CT phase transition remains unclear at the atomic level. The present molecular orbital calculations indicate that the driving force of the CT phase transition is the transverse translational mode. If the translational phonon mode of the HT phase is calculated, then the quantum mechanical coupling between the HT and LT phases may be also quantitatively clarified. The ZTE or NTE effect often occurs in phase transition materials. In the present compound, the transverse mode is the origin of both the ZTE and the CT phase transition, i.e., the hypothesis is correct, at least in the present CT system. Oriented Film-Form HT and LT Phases. To prepare a ZTE film, Rb0.97Mn[Fe(CN)6]0.99·0.3H2O plate-shaped microcrystals were dispersed in methanol, and then the fluid was spin-coated onto a glass substrate (Figure 9a). The SEM image shows that the film is composed of stacked plate-shaped microcrystals with thicknesses of 820 ± 40 nm (Figure 9b). The large plane (XY-plane) of the plate-shaped microcrystal is parallel to the substrate plane. Figure 10 shows the XRD patterns of this film sample. The XRD pattern at 300 K displays

the present compound has a stretching mode ν(CN) around 2100 cm−1 in the medium-IR spectrum, whereas the compound displays transverse modes δ(Fe−CN−Mn) of the transverse translational mode at 304 cm−1 and transverse librational modes at 253 and 503 cm−1 in the far-IR spectrum.16 First principle calculations confirm the assignments of these phonon modes.17 The NTE effect over a wide temperature range is due to the following two reasons. (i) The transverse modes of δ(Fe−CN−Mn) are energetically separated from the stretching mode ν(CN) at 2100 cm−1. Hence, a lattice contraction effect is available over a wide temperature range. (ii) Water molecules in the interstitial sites lead to steric dampening of the transverse modes, which limits the contraction coefficient. However, the present compound does not have water molecules. Consequently, the thermal contraction due to δ(Fe− CN−Mn) counteracts the thermal expansion due to ν(CN), resulting in a very small NTE (i.e., ZTE). Relationship between NTE and CT. Here, let us consider the influence of the transverse modes δ(Fe−CN−Mn) on the electronic states in the present CT compound. The discrete variable (DV)-Xα method is used to calculate two types of Fe− CN−Mn molecular orbitals: a Fe−CN−Mn linear configuration and a distorted configuration due to the transverse translational mode (Figure 8). The charge distributions of the metal ions for the Fe3dzx−C2pxN2px−Mn3dzx magnetic orbital with a linear configuration are 49% for Fe and 26% for Mn, which are close to Fe2+−CN−Mn3+ in the classical 1327



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Figure 8. Fe3dzx−C2pxN2px−Mn3dzx magnetic orbitals for the (a) linear configuration with ∠Fe−CN−Mn angle of 180° and (b) distorted configuration due to transverse translational mode with ∠Fe−CN = ∠Mn−NC = 168° calculated by the DV-Xα method. Charge distributions on metal ions with a linear configuration are 49% for Fe and 26% for Mn, which are close to the values for Fe2+−C N−Mn3+ with classical valence states (the LT phase). In contrast, the charge distributions for the distorted configuration are 22% for Fe and 58% for Mn, indicating that the valence state is close to Fe3+−C N−Mn2+ (the HT phase). Due to the distortion by the transverse translational mode, the valence states on the metal ions interchange.

Figure 10. XRD patterns and Rietveld analyses of a Rb0.97Mn[Fe(CN)6]0.99·0.3H2O film at (a) 300 K and (b) 100 K. Impurity signal due to manganese(II) hexacyanoferrate(II) is eliminated.15 Red (or blue) dots, black lines, and green dots are the observed plots at 300 K (or 100 K), calculated patterns, and their differences, respectively. Red (or blue) bars represent the calculated positions of the Bragg reflections of cubic (or tetragonal) phases. Miller indices are reflections. Insets schematically depict the crystal structure alignment.

spin-coated on the substrate, the XY-plane of the plate-shaped microcrystals is parallel to the substrate plane (Figure 11).

Figure 9. (a) Photographs of Rb0.97Mn[Fe(CN)6]0.99·0.3H2O films in the HT phase (left) and LT phase (right). (b) SEM images of the surface (left) and cross-section (right) of a sample film.

two peaks at 16.68° and 33.80°, which are identified as the (200) and (400) peaks in the cubic structure (F4̅3m) of the HT phase with aHT = 10.56070(17) Å. The observation of these two peaks indicates that the (100) plane of the HT phase is oriented along the substrate plane. In contrast, the XRD pattern of the LT phase at 100 K exhibits two peaks at 17.44° and 35.58° in the region of 2θ = 10−40°. These peaks are identified as (110) and (220) of the tetragonal structure (I4̅m2) with aLT = 7.0862(5) Å and cLT = 10.5110 Å. The presence of these peaks indicates that the direction of the cLT axis in the tetragonal structure is oriented parallel to the substrate plane. The orientation is due to the anisotropic shape of the microcrystal. When a dispersive solution of the microcrystals is

Figure 11. Schematic illustration of the alignment of plate-shaped Rb0.97Mn[Fe(CN)6]0.99·0.3H2O microcrystals on the substrate plane. 1328



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Rietveld Analyses. In Rietveld analyses for powder XRD patterns, the reported atomic coordinates of rubidium manganese hexacyanoferrate10d,13a were used as the initial structures. Lattice constants were refined at various temperatures using the relative atomic coordinates. For the oriented films, the crystallographic data obtained by Rietveld analyses of the powder sample were used as the initial structure. The lattice constants were refined with the orientation parameters based on March−Dollase function.20

Next, let us consider why the Jahn−Teller distortion in the LT phase occurs along the X (or Y) axis, i.e., cLT // substrate plane. The Gibbs free energy (G) of the present plate-shaped sample is expressed by the sum of the chemical potentials of bulk (μ) and surface (σ) as G = μXYZ + σ(2XY + 2YZ + 2ZX )

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If the σ value of the surface oriented vertically against Jahn− Teller axis (σ⊥JT) is assumed to be higher than the σ value of the surface oriented parallel along Jahn−Teller axis (σ//JT), then area of σ⊥JT should be narrowed as much as possible during the phase transition from the HT phase to the LT phase in order to reduce the surface energy, resulting that Jahn−Teller elongation occurs parallel to the plate plane, i.e., X- (or Y-) axis. Currently, we are investigating the detailed mechanism.

Additional tables (PDF). This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

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*E-mail: [email protected] (S.O.); tokoro@chem. s.u-tokyo.ac.jp (H.T.).

CONCLUSIONS In this work, we prepared a ZTE fluid composed of a bistable metal cyanide assembly of Rb0.97Mn[Fe(CN)6]0.99·0.3H2O. The plate-shape Rb0.97Mn[Fe(CN)6]0.99·0.3H2O microcrystals were synthesized using a PEGM matrix, and spin coating realized oriented films with HT and LT phases. The microcrystal orientation originates from the anisotropic shape of microcrystals. This is the first example of a ZTE film that maintains a constant film thickness over a wide temperature range of 300 K. A ZTE fluid should be very useful in a variety of practical applications such as preparing thin film materials and infilling at joint interfaces, etc. It has been hypothesized that phase transition materials can display a ZTE or NTE effect. In the present system, the ZTE (a small NTE) and the CT phase transition are stimulated by the transverse phonon mode. The coincidence of these two origins shows the hypothesis that phase transition materials can display a ZTE or NTE effect is accurate, at least in the present compound.

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ASSOCIATED CONTENT

S Supporting Information *

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to Ms. Yukiko Namatame (Rigaku Corporation) for the SAXS measurements. Additionally, we acknowledge the CREST project of JST, Grant-in-Aid for Young Scientists (S) from JSPS, NEXT program from JSPS, the Izumi Science and Technology Foundation, and Asahi Glass Foundation. We also recognize the Cryogenic Research Center, The University of Tokyo, and the Center for Nano Lithography & Analysis, The University of Tokyo, which are supported by MEXT Japan.

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REFERENCES

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EXPERIMENTAL SECTION

Characterization. The obtained precipitate shows IR peaks at 2153 cm−1 and 2073 cm−1 at 300 K. The former is assigned to the CN stretching frequencies of MnII−NC−FeIII, while the latter is assigned to the CN stretching frequencies of MnII−NC−FeII. Because the ratio of the oscillator strength of the MnII−NC−FeIII peak to that of the MnII−NC−FeII peak is 0.17, the estimated ratio of MnII−NC−FeIII and MnII−NC−FeII in the precipitate is 1:0.03. Elemental analyses, which were performed by an induced coupled plasma mass spectrometry and a standard microanalytical method, and the IR peak intensities indicate that the precipitate consists of 97% of RbI0.97MnII [FeIII(CN)6]0.99·0.3H2O and 3% impurity of RbI2MnII[FeII(CN)6]· 3.5H2O. Calculated values of the sum formulas: Rb, 23.91; Mn, 15.37; Fe, 15.47; C, 19.96; N, 23.28%: Found; Rb, 24.00; Mn, 15.44; Fe, 15.61; C, 20.18; N, 23.19%. Measurements. The morphologies of the compounds were measured with a JEOL JSM-7000F SEM with a 5 kV accelerating voltage. IR spectra were recorded on a Shimadzu FT-IR 8200PC spectrometer. SAXS measurement was conducted by a Rigaku RINTTTRIII using Cu Kα (λ= 1.5418 Å). SAXS data was analyzed by the Nanosolver program. The magnetic properties were measured using a Quantum Design MPMS superconducting quantum interference device (SQUID) magnetometer. The XRD patterns were measured with a Rigaku Ultima-IV instrument (Cu Kα) and the sample was placed on a Cu plate. The XRD patterns were calibrated using Si powder.19 The temperature during the XRD measurements was controlled by a RIGAKU R-CRT-105 cryostat. Rietveld analyses were performed using the Rigaku PDXL program. 1329



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P=

4πΔρ ⎡ ⎛ 2θd ⎞ 2θd ⎛ 2θd ⎞⎤ ⎟ − ⎟ cos⎜ ⎢sin⎜ ⎝ 2 ⎠⎥⎦ 2 2θ3 ⎣ ⎝ 2 ⎠

c=−

24 (1 − η)4 (2θd)3

{(1 + 2η)2 (sin 2θd − 2θdcos 2θd)

⎛ η ⎞2 ⎡ 2(1 − cos 2θd) ⎤ − 6η⎜η + ⎟ ⎢2sin 2θd − 2θd cos 2θd − ⎥ ⎝ ⎠ ⎦ 2 ⎣ 2θd ⎡⎛ ⎞ 1 24 ⎟⎟sin 2θd + η(1 + 2η)2 ⎢⎜⎜4 − ⎢⎣⎝ 2 (2θd)2 ⎠ ⎫ ⎛ 24(1 − cos 2θd) ⎤⎪ 12 ⎞⎟ ⎥⎬ − ⎜2θd − cos 2θd+ ⎝ ⎥⎦⎪ 2θd ⎠ (2θd)3 ⎭ where η is the packing density of the water droplet. (15) Rb2IMnII[FeII(CN)6]·3.5H2O, the impurity, is cubic (Fm3̅m) with a lattice constant of a = 10.22 Å at 15 K. a continuously increases upon warming, e.g., 10.33 Å at 300 K. (16) To observe transverse modes δ(M−CN−M′) in the lowfrequency IR, a far-IR spectrometer, Bruker Optics Vertex80v, was used with a T222 beam splitter. (17) First principle calculations for the phonon modes were performed by the density functional theory (DFT) within the GGA using Vienna ab initio simulation package (VASP). The calculations give the calculated frequencies as: stretching mode at 2110 cm−1 (symmetry: B2 (R, I)), transverse translational mode at 296 cm−1 (E (R, I)), and transverse librational modes at 257 cm−1 (E (R, I)) and 506 cm−1 (E (R, I)). R and I indicate Raman and IR activities, respectively. (18) (a) Slichter, C. P.; Drickamer, H. G. J. Chem. Phys. 1972, 56, 2142−2160. (b) Ohkoshi, S.; Matsuda, T.; Tokoro, H.; Hashimoto, K. Chem. Mater. 2005, 17, 81−84. (c) Tokoro, H.; Ohkoshi, S. Appl. Phys. Lett. 2008, 93, 021906/1−3. (19) Touloukian, Y. S.; Kirby, R. K.; Taylor, R. E.; Lee, T. Y. R. Thermal Expansion- Nonmetallic Solids. In Thermophysical Properties of Matter; Touloukian, Y. S., Ho, C. Y., Eds.; The TPRC Data Series 13; IFI/Plenum: New York, 1977; pp 154−161. (20) Dollase, W. A. J. Appl. Crystallogr. 1986, 19, 267−272.

2

where Δρ is the difference in the electron densities inside and outside of the water droplet, and d represents the diameter of the water droplet. Representation of S is obtained by the Percus−Yevick closure relation within the hard-sphere approximation as

S=

1 1−c 1330


Zero Thermal Expansion Fluid and Oriented Film Based on a Bistable

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