Precise Analysis of Thermal Volume Expansion of Crystal Lattice for

Article pubs.acs.org/Macromolecules

Precise Analysis of Thermal Volume Expansion of Crystal Lattice for Fully Aromatic Crystalline Polyimides by X‑ray Diffraction Method: Relationship between Molecular Structure and Linear/Volumetric Thermal Expansion Ryohei Ishige, Toshiaki Masuda, Yukiko Kozaki, Eisuke Fujiwara, Tomohiro Okada, and Shinji Ando* Department of Chemical Science and Engineering, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8552, Japan S Supporting Information *

ABSTRACT: Coefficients of thermal linear and volumetric expansion (CTE, CVE) of crystal lattice for 13 fully aromatic crystalline polyimides (PIs) were evaluated from lattice parameters measured from variable-temperature (VT) synchrotron X-ray diffraction patterns, and the effects of chemical structure on CTE and CVE are discussed. The smallest CVE (116 ppm K−1) was observed for PMDA-PPD with the simplest rigidrod structure, and the largest CTE anisotropy was observed for PMDA-ODA containing an ether linkage with an extraordinarily negative CTEa (−44 ppm K−1). The values and anisotropy of the CTEs strongly depended on the crystalline structure, whereas the CVEs were negatively correlated with the weight density, regardless of the PI type. The correlation was explained using the Grüneisen equation, (∂V/∂T)P/V = γCv,interχ, assuming that isothermal compressibility χ dominates the equation. An increase in the weight density and/or molecular weight of repeating units effectively suppresses the CVEs of crystalline PIs. Nominal CVE can be quantified as −(∂V/∂T)P/V0, where V, V0, and T are the sample volume at temperature T, sample volume at the standard temperature, and sample absolute temperature, respectively (the rigorous definition of CVE is −(∂V/∂T)P/V). Generally, the CVE of a solid is closely related to the vibrational motion of atoms and molecules5 and is expected to be sensitive to the aggregation state (packing density) for organic molecular materials. In the case of polymeric materials, the CTE of a solid polymer film strongly depends on the orientation of the polymer’s main chain, whereas the CVE does not (CVE ≈ CTEx + CTEy + CTEz, CTEi is a CTE in the ith direction),6−15 because the thermal expansion along the main chain of the polymer is much smaller than that in the direction perpendicular to that of the main chain. This occurs because repeating units are covalently linked along the main chain, whereas in the perpendicular direction, polymer chains are weakly attracted via the van der Waals interactions, dipolar− dipolar interaction, π−π interaction, and hydrogen bonds. Therefore, the CVE of a polymer is more internal than the CTE and can potentially be used as a characteristic parameter of a polymeric material. The CVEs of polymers are expected to be controlled by controlling the molecular design and the aggregation state of solids.

1. INTRODUCTION Fully aromatic polyimides (PIs) are among the most versatile insulating materials and are used in various electronic circuits (such as flexible printed circuits and integrated circuits) and electronic devices as passivation films and interlayer insulators, owing to their high mechanical strength, flexibility, chemical and thermal stability, high processability from poly(amic acid) precursors, and electric insulation. Recently, following the development of mobile devices, such electronic circuits and devices have been undergoing increasing miniaturization and compactification. As a result, linear and volumetric thermal expansions of insulating layers have been recognized as potential problems because these processes can cause warpage in a way similar to that of residual stress, inducing failures such as increase in dielectric loss and circuit shorting.1−4 The warpage is caused by a large difference between the coefficients of thermal expansion (CTEs) of the PI insulating layer and conducting metallic wire. The CTE of the PI layer is usually much larger than that of copper (17 ppm K−1), although the CTEs of PIs are among the smallest for engineered polymers that are used for electronic packaging. Therefore, suppressing thermal expansion is one of the most critical issues for electronic and photonic applications. Consequently, it is very important to quantitatively and systematically investigate the CTEs and the coefficients of thermal volume expansion (CVEs) of various PIs for developing universal guidelines for molecular designs that would allow to reduce and control CVEs. © XXXX American Chemical Society

Received: January 17, 2017 Revised: February 14, 2017

A

DOI: 10.1021/acs.macromol.7b00095 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Scheme 1. Chemical Structures of Crystalline Polyimides, with Main Chain Axis Parallel to the Horizontal Direction

Pottiger et al.15 have investigated the CVEs of five types of commercially available PI films using the PVT technique based on the Bridgeman bellows. They reported that the CVEs of Upilex-S (sBPDA-PPD), PI-2611 (sBPDA-PPD), PI-2525 (BTDA-ODA/MPD), Kapton-H (PMDA-ODA), and PI-2540 (PMDA-ODA) were 130, 154, 150, 174, and 192 ppm K−1, respectively. On the other hand, Tong16,17 measured the in-plane and out-of-plane CTEs (CTE∥ and CTE⊥) of Kapton-H films separately by thermal mechanical analysis (TMA), capacitance change, and using the Fabry−Perot laser interferometric method. The CVE was estimated as 149 ppm K−1. Thus, significantly different values have been obtained for the same skeletal structures of PIs, suggesting that the CVEs of noncrystalline PI films depend on the preparation and curing conditions. In this study, we focused on the CTE and CVE of crystalline lattice of highly crystalline aromatic PIs in order to avoid problems of the chain orientation and the crystallinity. Owing to the remarkably high glass transition temperatures (Tg) and thermal degradation temperatures (Td) of the aromatic PIs, their crystal structures could be investigated in a very wide range of temperatures (from room temperature to 350 °C). In the crystal phase, the CVEs of the crystalline lattice were precisely evaluated based on the temperature dependence of lattice parameters using the wide-angle X-ray diffraction (WAXD) method and were not affected by the sample orientation and coexisting amorphous (noncrystalline) regions. Moreover, it is natural to consider that polymer chains exist in the most stable conformation in the equilibrium crystalline phase; thus, vigorous molecular motion is widely restricted and free volume is minimized in the crystal lattice. Consequently, the CVE of the crystalline lattice can be recognized as the smallest (limiting) value that could serve as a characteristic parameter for a focused material. In addition, the CVE of the crystalline lattice can be used as a target for designing novel semicrystalline or amorphous PIs with reduced thermal expansion. However, only a few groups have reported the CVEs of aromatic crystalline polymers evaluated using the variabletemperature (VT) WAXD method. To the best of our knowledge, the smallest CVEs for organic polymers have been reported for poly(p-benzamide) (119 ppm K−1)18 and poly(pphenyleneterephthalamide) (101−127 ppm K−1),19,20 which

suggests that the combination of p-phenylene linkages and intermolecular hydrogen bond networks between amide groups effectively suppresses the CVE. However, the relationship between the CVE and the primary structure of a polymer remains elusive. 18,20,21 Therefore, we systematically and comprehensively analyzed the CTEs and CVEs of 13 types of crystalline PIs, with rigid-rod, bent, and crankshaft structures. The CVE values were evaluated with very high accuracy by conducting the VT-WAXD analysis using a synchrotron X-ray source for highly crystalline powder or highly oriented crystalline PI specimens. The effects of the primary and crystalline structures of PIs on the values of CVE and CTE are presented and discussed, revealing that the CVEs of PIs are closely related to their weight densities.

2. EXPERIMENTAL SECTION 2.1. Reagents. Two types of aromatic dianhydrides were used for synthesizing PIs: pyromelliric anhydride (PMDA) and 3,3′,4,4′biphenyltetracarboxylic dianhydride (sBPDA). The PMDA sample, purchased from Mitsui-Toatsu Chemicals Inc. (Mitsui Chemicals Co., Ltd., today) was purified by drying at 120 °C for 3 h in a vacuum and subsequent sublimation before use. The sBPDA sample, purchased from Wako Pure Chemical Industries Ltd., was dried at 180 °C for 7 h and used without purification. Eight types of diamines were used: p-phenylenediamine (PPD), 4,4′benzidine (BZ), 4,4″-diamino-p-terphenyl (DATP), 2,2′-dimethyl-4,4′diaminobiphenyl (DMDB), 2,2′-bis(trifluoromethyl)-4,4′-diaminobiphenyl (TFDB), 4,4′-bis(4-aminophenoxy)biphenyl (BAPB), 4aminophynyl-4-aminobenzoate (APAB), and 4,4′-diaminobenzanilide (DABA). The PPD sample, purchased from Wako Pure Chemical Industries Ltd., was purified by sublimation under reduced pressure. 4,4′-Benzidine (BZ) was synthesized in a benzidine transition according to the literature. (Manufacturing of BZ without permission of authorities is illegal in several countries, such as Japan and UK, because BZ has been linked to bladder and pancreatic cancer. We obtained a precertification from the Tokyo Labor Bureau of the Ministry of Health, Labour, and Welfare, Japan, before synthesizing BZ. In the lab setting, BZ must be treated with protectors under a hood.22) The DATP sample was kindly provided by the Central Research Lab of Japan Carlit Co., Ltd., and was purified by recrystallization from the THF solution and subsequent sublimation. The DMDB, APAB, and DABA samples were kindly provided by Wakayama Seika Kogyo Co., Ltd., and were purified by B


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Macromolecules

Figure 1. SR-WAXD fiber photographs of (a) PMDA-BZ, (b) PMDA-TFDB, and (c) sBPDA-BZ and WAXD fiber photographs of (d) PMDA-DMDB, (e) PMDA-APAB, (f) PMDA-DABA, and (g) sBPDA-DATP. For each photograph, except for those in panels (e) and (f), the left and right sides are presented in lower and higher contrast, respectively. The drawing axis is in the meridional direction. For each photograph, Miller indices, [hkl], are shown as insets. 2.3. Measurements. Variable-temperature synchrotron wide-angle X-ray diffraction (VT-SR-WAXD) measurements were conducted at the BL-10C beamline at the Photon Factory (KEK, Tsukuba, Japan). The Xray’s wavelength was set to 0.089 nm, and the PI samples were installed in a FP90 hot stage using a FP82HT controller (Mettler Toledo Co.), heated to 350 °C, and were held for 10 min. The samples were gradually cooled from 350 °C, and for each temperature measurement, the samples were held at a preset measurement temperature (varying in steps of 20 °C) for 5 min before performing the measurement. The resulting diffraction patterns were then exposed on a PILATUS3 2M detector for 1 min. The camera length was calibrated using the 00l diffractions from lead stearate as a standard. WAXD measurements for PMDA-DMDB, PMDA-APAB, PMDADABA, and sBPDA-DATP were conducted at ambient temperature using a Bruker D8 DISCOVER equipped with a Vantec 500 detector using Cu Kα radiation.

sublimation. The TFDB sample was kindly provided by Central Glass Co., Ltd., and used as received. The BAPB sample, purchased from Wako Pure Chemical Industries Ltd., was purified by sublimation. 2.2. Sample Preparation. Poly(amic acids) (PAAs) for PI precursors were synthesized in the N-methyl-2-pyrrolidinone (NMP) solution. The diamine was first dissolved in the NMP solution in a glovebox, and then the corresponding equimolar dianhydride was added to the solution in the under a N2 atmosphere. The solution was stirred for 24−48 h until the viscosity increased sufficiently. Highly crystalline PI samples were synthesized via thermal imidization process in the solution state.23 An NMP solution of PAA was refluxed at 200−210 °C in the presence of N2. After the crystalline powder was precipitated, the solution was refluxed for an additional 2−4 h. The powder sample was filtrated from the hot solution, then washed with water and NMP alternatively for several times, and dried at 100 °C in a vacuum. Finally, the powder was heated at 200 °C and then at 400 °C, under a N2 atmosphere, to remove the residual NMP, to complete the imidization reaction, and to promote the crystallization. Thirteen different PIs that were used in the present study, with different structures, are listed in Scheme 1. For seven PIs (namely, PMDA-BZ, PMDA-DMDB, PMDA-TFDB, PMDA-APAB, PMDADABA, sBPDA-BZ, and sBPDA-DATP) highly oriented fibers or films were separately prepared for indexing the crystal lattices. In particular, highly oriented samples were used for the CVE analysis of PMDA-BZ, PMDA-TFDB, and sBPDA-BZ because the number of observed diffraction peaks in the powder pattern was not sufficient; the diffraction peaks were broadened and overlapped with each other. A highly extended fiber specimen of PMDA-BZ was kindly provided by the Toray Research Center, Inc. In addition, highly oriented films of PMDA-BZ were prepared by stretching the corresponding PAA film during the thermal imidization process. The NMP solution of PAA was cast on a Si wafer and dried at 70 °C for 1 h under the N2 flow. The PAA film was peeled from the Si substrate and cut into a 5 mm wide, 15 mm long, and 0.1−0.2 mm thick strip. The strip specimen with the initial length of 10 mm was stretched by applying a constant load (10−50 g) and simultaneously imidized by rapid heating from ambient temperature to 400 °C using a TMA (TMA-60/60H, Shimadzu Co.). The stretched specimen was held at 400 °C for 1.5 h in the TMA chamber to complete the imidization reaction. In the cases of PMDA-DMDB, PMDA-APAB, and PMDA-DABA, the oriented films were subsequently annealed in the NMP solvent at 200 °C for 5 h to increase crystallinity.

3. RESULTS AND DISCUSSION 3.1. Crystal Structure of PIs. The lattice parameters of some PIs have been clarified previously using WAXD and electron diffraction methods. Those of PMDA-PPD, PMDA-ODA, and PMDA-DATP, evaluated using electron diffraction and WAXD methods, were reported by Cheng et al.24 In the case of PMDABZ and PMDA-BAPB, Okuyama et al.25,26 precisely determined not only the lattice parameters but also the atom positions in the lattice by precise analyses of highly oriented WAXD fiber patterns. The lattice parameters for sBPDA-PPD, estimated for a plane-oriented thin film, were reported by Ree et al.27 The crystal lattice of sBPDA-DMDB in a highly oriented fiber was determined to be triclinic by Wu et al.28 All of the reported lattices described above had orthorhombic symmetry (α = β = γ = 90°) except for sBPDA-DMDB, which had a triclinic symmetry. For the remaining seven PIs (namely, PMDA-BZ, PMDA-DMDB, PMDA-TFDB, PMDA-APAB, PMDA-DABA, sBPDA-BZ, and sBPDA-DATP), the crystal structures were precisely analyzed in the present study, using highly oriented film specimens. The WAXD photographs of these PIs are shown in Figure 1. All of the seven analyzed PIs had orthogonal geometry, with [00l] diffractions appearing on the meridian (drawing C


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Macromolecules direction), [hkl] diffractions appearing on the lth layer line, and at least with the lattice parameters α and β being 90°. Moreover, the positions of the [hk0] diffractions on the equator were consistently explained by assuming γ = 90°. Therefore, the crystal structures were assigned to the orthorhombic system. For evaluating the lattice parameters at different temperatures, we assumed that the crystal lattices maintained their symmetry at different temperatures; i.e., α = β = γ = 90° for all of the studied PIs, except for sBPDA-DMDB, whose crystal lattice was α = 90°, β = 45.9°, and γ = 78.9°. The lattice parameters evaluated from the WAXD powder profiles or fiber photographs are listed in Table 1. The former six

ing powder WAXD photographs were used. The lattice parameters a, b, and c and volume, V, normalized to those evaluated at 110 °C, were linear in temperature, for temperatures in the 110−310 °C range, as shown in Figure 3. The thermal expansion coefficients, CTEa, CTEb, CTEc, and CVE, were defined as 1 ⎛⎜ ∂a ⎞⎟ 1 ⎛ ∂b ⎞ , CTEb = ⎜ ⎟ , a0 ⎝ ∂T ⎠ P b0 ⎝ ∂T ⎠ P 1 ⎛ ∂c ⎞ CTEc = ⎜ ⎟ , c0 ⎝ ∂T ⎠ P

CTEa =

CVE =

Table 1. Lattice Parameters, Molecular Weight of Repeating Unit, and Weight Density of PI Crystals samples

Mw (g mol−3)

a (Å)

b (Å)

c (Å)

Z

ρ (g cm−3)

PMDA-PPD PMDA-BZ PMDA-DATP PMDA-DMDB PMDA-TFDB PMDA-APAB PMDA-DABA PMDA-ODA PMDA-BAPB sBPDA-PPD sBPDA-BZ sBPDA-DATP sBPDA-DMDBa

290 368 444 396 502 410 409 384 553 366 444 521 473

8.41 8.56 7.97 19.8 13.8 8.40 8.40 6.25 8.16 8.40 12.3 12.3 16.4

5.52 5.37 5.58 10.1 9.9 5.25 5.21 3.92 6.32 5.80 7.20 7.08 9.38

12.3 16.4 20.9 16.4 15.8 18.5 18.7 32.0 24.8 30.5 20.5 24.6 40.2

2 2 2 8 4 2 2 2 2 4 4 4 8

1.68 1.63 1.59 1.60 1.55 1.67 1.66 1.63 1.44 1.64 1.63 1.62 1.47

+

b c a c 1 ⎛⎜ ∂V ⎞⎟ = CTEa + CTEb ⎝ ⎠ V0 ∂T P b0 c0 a0 c0

a b CTEc ≅ CTEa + CTEb + CTEc a0 b0

(1)

where a0, b0, c0, and V0 are the lattice parameters and volume at 110 °C, respectively. The values of CTEa, CTEb, CTEc, and CVE correspond to the slopes of the plots. For PMDA-TFDB, the slope was estimated for temperatures in the 110−250 °C range because a peak of tan δ, corresponding to β relaxation, was observed at 260 °C by dynamic mechanical analysis, and the slope deviated from linearity above 260 °C.29 The CTE values along the a, b, and c directions and the CVEs evaluated for the PIs from the graphs are listed in Table 2. In addition, a measure of CTE anisotropy in the ab plane (η), defined as |CTEa − CTEb|/ CVE, was calculated and the results are listed in the table. 3.3. Effect of Flexible Linkage on CVE. The CVE values of PMDA-PPD, PMDA-ODA, and PMDA-BAPB, which contain none, one, and two phenylene−ether linkages in their repeating units, respectively, were cross-compared for confirming the effect of flexible linkages on CVE. The CVE values gradually increased with increasing the number of flexible phenylene−ether linkages in the backbone structure; the values were 116 ppm K−1 for PPD, 144 ppm K−1 for ODA, and 196 ppm K−1 for BAPB, as listed in Table 2. It is interesting to note that the CVE of PMDA-ODA (with one ether linkage) was significantly smaller than those of the other PIs, whereas that of PMDA-BAPB (with two ether linkages) was the largest among the PIs. This trend was apparently reflected in the density of the crystals; the densities were 1.68 g cm−3 for PMDA-PPD, 1.63 g cm−3 for PMDA-ODA, and 1.44 g cm−3 for PMDA-BAPB. The CVE of PMDA-BAPB was the largest, while the density was the smallest even among the 13 PIs. Another remarkable point is that only PMDA-ODA exhibited negative CTE along the a direction (CTEa) as well as the largest CTE anisotropy in the ab plane (η = 1.16) among all the PIs. The CTE and CVE values of PMDA-ODA and PMDA-BAPB are compared in Figure 4. One of the characteristic features of PMDA-ODA is its planar conformation in the lattice,30 in which the p-phenylene linkages in the ODA moiety are almost parallel to the ac plane. One of the probable crystal structures of PMDAODA is drawn in Figure 5 for a reference, although the precise atomic positions have not been clarified yet by WAXD analysis because of the small number and broadening of the diffraction peaks. In this model, two sequential repeating units of PMDAODA were accommodated in a P1 orthorhombic lattice (a = 6.25 Å, b = 3.92 Å, and c = 32.0 Å). The molecule was geometrically optimized in the unit cell (Materials Studio version 8.0, Accelrys Software Inc., United States). When vigorous rotational motion of the phenyl rings was activated around the PI-chain axis at

A triclinic lattice with α, β, and γ equal to 90°, 45.9°, and 78.9°, respectively.

a

PIs showed the same values as those reported previously. For PMDA-DMDB and sBPDA-DATP among the latter seven PIs, fewer diffraction spots were observed compared with the powder samples, and the diffraction spots were broadened; that is, for these oriented films, the degree of crystallinity was much smaller than that of the powder, even though the degree of orientation was sufficiently high for analyzing. As a result, the lattice parameters of these two PIs were slightly different between those of the oriented film and the powder. The reason for the lower crystallinity of the films may be owing to the lower molecular mobility of PI chains during crystallization that proceeds simultaneously with thermal imidization. In the case of powders, the PAA chains in the solution are sufficiently mobile to reach higher crystallinity during imidization, whereas the PAA chains in the films slightly swollen with a little amount of NMP are not sufficiently mobile; thus, in the latter case, the orientation induced by uniaxial stretching is fixed during imidization. Hereafter, fiber photographs were used for evaluating the CVE values of PMDA-BZ, PMDA-TFDB, and sBPDA-BZ, whereas those of highly crystalline powders were used for the other ten PIs. 3.2. Temperature Dependence of WAXD and Lattice Parameters. The WAXD intensity profiles, acquired every 20 °C when cooling the samples from 350 to 50 °C, are presented in Figure 2. For PMDA-BZ, PMDA-TFDB, and sBPDA-BZ, sectoraveraged intensity profiles around the meridian and the equator of the corresponding fiber WAXD photographs were used for evaluating the positions of diffraction peaks. For the other samples, circular-averaged intensity profiles of their correspondD


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Figure 2. WAXD intensity profiles for the 13 PIs observed at every 20 °C during cooling from 350 °C. For PMDA-BZ, PMDA-TFDB, and sBPDA-BZ, intensities in the meridional and equatorial directions of the fiber WAXD photographs are separately presented. Miller indices [hkl] are shown as insets.

temperatures above the β-transition,31 the intermolecular distance along the a direction likely decreased, whereas that along the b direction likely increased, resulting in the extraordinary negative CTEa, as observed in this experiment. It should be noted that the conformation of the ether linkage, “C− C−O−C”, in the ODA moiety had to be almost fixed in the trans conformation during rotational fluctuation of planar phenyl rings; otherwise, the molecular axis would not become parallel to the c-axis. The relatively large shrinkage of PMDA-ODA in the c direction during heating (CTEc = −5.0 ppm K−1) is consistent

with the rotational motion of the planar phenylene rings because the intramolecular steric hindrance in the ac plane can be relaxed by the rotational motions and also the bond angles and bond lengths in the main chain can be partially relaxed by the shrinkage along the c direction. In addition, it should be noted that such a rotational mode seems to contribute less to the CVE because the center of mass of the PI’s repeating unit is not changed by the rotational fluctuation, even though the rotational mode may induce a large characteristic anisotropy in CTE. E


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Figure 3. Thermal expansion measures of the lattice parameters, defined as (a/a0 − 1) × 100 [%], (b/b0 − 1) × 100 [%], and (c/c0 − 1) × 100 [%], and that of the lattice volume, for each PI, vs temperature. The parameters a0, b0, and c0 are the lattice parameters evaluated at 110 °C.

Similar large anisotropy in the deformation of the a- and b-axes

linear compressibility and the linear thermal expansion (CTE) in the ab plane. The details are discussed in section 3.7. On the other hand, for PMDA-BAPB, the phenyl rings in the BAPB moiety do not adopt planar conformations and are parallel to neither the ab plane nor the bc plane, as in the model reported by Okuyama et al.,26 resulting in the relatively isotropic expansion in the ab plane with a much smaller CTE anisotropy.

for PMDA-ODA was observed in our previous high-pressure WAXD measurements, with pressures reaching 7 GPa. The aand b-axis were compressed by 6% and 15%, respectively, at 7 GPa; that is, the b-axis was much more easily deformed than the a-axis.32 This result suggests a close relationship between the F


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Macromolecules Table 2. CTEa, CTEb, CTEc, CVE, and Anisotropy of CTE in the ab Plane, |CTEa − CTEb|/CVE, for Crystalline PIs

a

samples

CTEa (ppm K−1)

CTEb (ppm K−1)

CTEc (ppm K−1)

CVE (ppm K−1)

|CTEa − CTEb|/CVE

PMDA-PPD PMDA-BZ PMDA-DATP PMDA-DMDB PMDA-TFDB PMDA-APAB PMDA-DABA PMDA-ODA PMDA-BAPB sBPDA-PPD sBPDA-BZ sBPDA-DATP sBPDA-DMDBa

85 98 91 75 67 108 95 −44 118 99 4.0 72 63

31 56 49 71 108 37 48 196 79 15 149 86 94

−0.4 −4.6 −0.4 7.0 −8.0 −2.0 −2.0 −5.0 2.0 5 −8.8 −3.0 34

116 146 140 154 167 143 142 144 196 126 144 156 185

0.46 0.28 0.30 0.03 0.25 0.50 0.33 1.67 0.20 0.66 1.01 0.09 0.17

CTE values were evaluated for the angles α, β, and γ fixed at 90°, 45.9°, and 78.9°, respectively.

incorporated imide and aromatic rings are expected to increase the CVEs of PIs by expanding their corresponding free volumes. Thereby, the CVEs of the PIs consisting of linear para-linked multiphenylenediamine (namely, PPD, BZ, and DATP units) were cross-compared to clarify the effect of the number of phenyl rings on the thermal expansion behavior. First, the CVEs and CTEs of PMDA-PPD, PMDA-BZ, and PMDA-DATP (with fully linear rigid-rod structures) are discussed (Figure 6a). Compared with the very small CVE (116 ppm K−1) and a relatively large CTE anisotropy (η = 0.46) of PMDA-PPD, the PMDA-BZ and PMDA-DATP samples exhibited larger CVEs (146 and 140 ppm K−1, respectively) and smaller CTE anisotropy (η = 0.28 and 0.30, respectively). Even though the crystal density gradually decreased with increasing

Figure 4. Comparison of CTEa, CTEb, CTEc, and CVE between PMDA-ODA and PMDA-BAPB.

Figure 5. (a) One of the probable crystal structures for PMDA-ODA. Projections of the lattice on the (b) ab plane, (c) ac plane, and (d) bc plane are shown on the right side.

3.4. Effect of the Number of Phenyl Rings in Repeating Units on CVE. Rigid-rod fully aromatic PIs without any flexible linkages have very large persistence lengths; as a result, only a localized rotational fluctuation around the chain axis is allowed in the ordered phase. However, rotational fluctuations of the

Figure 6. Comparison of thermal expansion coefficients (CTE and CVE) for the PIs consisting of linear multiphenyldiamine moieties, PPD, BZ, and DATP: (a) PMDA series and (b) sBPDA series. G


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Macromolecules the number of phenyl rings, that is, the crystal densities of PMDA-PPD, PMDA-BZ, and PMDA-DATP were 1.68, 1.63, and 1.59 g cm−3, respectively, the CVE values did not monotonically decrease with increasing the density. PMDA-BZ exhibited the highest CVE among the three. Obata reported that the 4,4′-benzidine (BZ) unit in the PMDA-BZ crystal adopted a characteristic coplanar conformation.25 Based on the quantum chemical calculation, the coplanar conformation is less stable than the twisted one. The most stable dihedral angle (θ) of BZ for an isolated PMDA-BZ was evaluated as 40° (Supporting Information). If the dihedral angle can fluctuate toward the most stable conformation (θ = 40°) from the coplanar conformation (θ = 0°), the mirror symmetry perpendicular to the c-axis will be broken in the BZ moiety and additional CVE will be possibly induced in PMDA-BZ. It should be noted that the planes of PMDA and BZ aromatic rings are relatively twisted by 58° in the lattice, and the rotational fluctuation of the phenyl rings contributes less to the CTE anisotropy, which is significantly different from the case of the p-phenylene ether linkage, PMDAODA (η = 1.66). Second, the CVEs and CTEs for sBPDA-PPD, sBPDA-BZ, and sBPDA-DATP (with an sBPDA moiety with a kinked structure) are discussed (Figure 6b). For these three PIs, the CVE significantly increases with the number of phenyl rings in the diamine unit and with decreasing the density (the CVEs are 126, 144, 156 ppm K−1, and the densities are 1.64, 1.63, 1.62 g cm−3, respectively). Of note is that a significantly large CTE anisotropy is observed for sBPDA-PPD (η = 0.66) and sBPDA-BZ (η = 1.01). These could be related to the facts that (1) sBPDA has a kinked (crankshaft) structure, which makes the longer axis of the sBPDA unit (the Ph−Ph bond direction) tilt from the c-axis of the orthorhombic lattice, and (2) the biphenlydiimide moiety in sBPDA has a coplanar conformation, similar to PMDA-ODA, as discussed in section 3.3. We have recently analyzed the conformations of PI films derived from sBPDA dianhydride using far-infrared (far-IR) spectra compared with the calculated vibration spectra of a model compound and confirmed that the dihedral angle between biphenyldiimide moieties tends to be coplanar in the highly ordered region.33 In other words, the rotational axis of phenyl rings in sBPDA is not parallel to the chain axis (c direction), strongly suppressing the rotational motion in the crystal lattice; otherwise, the chain direction changes and the orthorhombic symmetry is broken. If the biphenyldiimide plane of sBPDA is closely parallel to the ac plane, similar to the case of PMDA-ODA, the CTEa should become much larger than the CTEb because the side groups (C− H, CO), which are parallel to the ac plane, can induce thermal expansion. The CTE anisotropy is expected to decrease with increasing the number of p-phenylene groups in the diamine moiety, which are not necessarily parallel to the sBPDA plane. However, the largest anisotropy is observed for sBPDA-BZ, which presumably implies that the BZ moiety also adopts a characteristic coplanar conformation similar to that in PMDABZ25 and the BZ plane is closer to the sBPDA plane than sBPDAPPD. In contrast, a small CTE anisotropy was observed for sBPDA-DATP (η = 0.17), in which the phenyl rings in the DATP diamine moiety may not be parallel to the biphenyl plane of the sBPDA unit and contribute equally to both CTEa and CTEb, resulting in a small CTE anisotropy. 3.5. Effect of Side Groups on CVE. The CVEs for the PIs consisting of BZ, DMDB, and TFDB diamine unit were crosscompared to discuss the effects of the side groups (−H, −CH3, and −CF3) on the thermal expansion of the PIs (Figure 8).

Figure 7. (a) Crystal structure of PMDA-BZ, provided by Obata et al.25 Two sequential lattices in the c direction are presented, allowing to easily identify the planar conformation of the BZ moiety. Projections of the lattice on the (b) ab plane, (c) ac plane, and (d) bc plane are presented on the right side.

Figure 8. Comparison of CTEs and CVEs for the PIs consisting of BZ, DMDB, and TFDB units: (a) PMDA series and (b) sBPDA series.

The CVE increased and the density decreased gradually with the size of the substituted side group in the order of −H, −CH3, and −CF3 for both of the PIs derived from PMDA and sBPDA having orthorhombic crystal lattices, while sBPDA-DMDB showed a triclinic symmetry. The bulky side groups of −CH3 and −CF3 probably interfere with closer packing of adjacent PI chains. Actually, in the order of PMDA-BZ, PMDA-DMDB, and PMDA-TFDB, the crystalline volume occupied by one repeating unit significantly increased (376.9, 410.0, and 539.6 Å3), and the density decreased (1.63, 1.60, and 1.55 g cm−3). The density decreased in a similar manner as sBPDA-BZ > sBPDA-DMDB H


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Macromolecules (1.63 and 1.47 g cm−3). In particular for PMDA-TFDB, the −CF3 group significantly reduced intermolecular interactions owing to the low polarizability of the C−F bond and potentially increased the interchain distance compared with the −CH3 group, owing to its larger van der Waals volume (1.9, 18.7, and 33.0 Å3 for −H, −CH3, and −CF3, respectively). These results imply that the CVE is related to the crystal density, similarly to the other PIs discussed above. Among these PIs, PMDA-BZ, PMDA-TFDB, and sBPDA-BZ exhibited larger shrinkage along the c direction: −4.6, −8.0, and −4.6 ppm K−1, respectively. This phenomenon can be explained in the same manner as PMDA-ODA. The BZ moiety in PMDABZ adopts a characteristic coplanar conformation in the crystal lattice. When vigorous rotational motion around the biphenyl group in the BZ moiety is activated, steric hindrance among the four H atoms in the neighboring benzene rings in BZ can be released, and the C−C bond length relaxes to a shorter stable state. In the case of the PMDA-TFDB film, β-relaxation, which is related to the rotation of benzene rings,31 occurs around 260 °C,29 and then a similar relaxation of the intramolecular steric hindrance is induced. On the other hand, the β-relaxation of the PMDA-DMDB film occurs at much lower temperatures (ca. 140 °C), and the bond length is already relaxed in lower temperature region, resulting in the positive CTEc in the temperature measurement region (temperature dispersion of the dynamic mechanical analysis is provided in the Supporting Information). Finally, we should discuss the remarkably large CTEc of sBPDA-DMDB. The lattice parameters a, b, and c for sBPDADMDB were evaluated by assuming the same angles, α = 90°, β = 45.9°, and γ = 78.9°, at all temperatures because it was difficult to evaluate the angles accurately from the powder patterns. The actual angle between the c-axis and the ab plane, however, can vary with temperature, dramatically affecting d00l. Therefore, the observed large CTEc may be attributed not to the change in the repeating unit length but rather to the change in the angle. 3.6. Effects of the Hydrogen Bond on CTE and CVE. The effects of strong molecular interactions on CVEs can be examined by comparing two PIs that have similar structures, PMDA/DABA and PMDA/APAB. The former has an intermolecular hydrogen-bonding ability with the amide (−NHCO−) group, though the latter does not with the ester (−COO−) group. As the corresponding WAXD profiles in Figures 1 and 2 show, these two PIs have very similar crystal structures. Interestingly, the PI with hydrogen bonds, PMDADABA, has a slightly smaller density (1.66 g cm−3) than PMDAAPAB (1.67 g cm−3). For these PIs, the CTEa of PMDA/DABA is smaller than that of PMDA/APAB, while the CTEb of the former is larger than that of the latter (Figure 9). We assume that the hydrogen bonds in PMDA/DABA are relatively parallel to the ac plane, which may suppress the CTEa, although the reduction in the CTEa is compensated by the increase in the CTEb.34 As a result, the CTE anisotropy of PMDA/DABA (η = 0.33) is lower than that of PMDA/APAB (η = 0.50), even though their CVEs are almost the same (142 and 143 ppm K−1). For PMDA-DABA, hydrogen bonds are assumed to be formed along a uniaxial direction, yielding two-dimensional pleated sheet networks, such as β-sheet structures that are frequently found in polypeptides. This structure is not effective for suppressing the CTE in the direction perpendicular to that of the hydrogen-bond networks. If hydrogen bonds are formed both along the a and b directions, yielding a three-dimensional network, both the CVE and the CTE anisotropy could be more effectively reduced by incorporating hydrogen bonds.

Figure 9. Comparison of CTEs and CVEs values for hydrogen-bonding PI, PMDA-DABA, and non-hydrogen-bonding PI, PMDA-APAB.

3.7. Determinants of CVE and CTE of PIs. Based on the above discussion, the weight density of a PI crystal is expected to be strongly related to its CVE. On the other hand, it is plausible to expect that the volume fraction of the crystal’s free volume will be correlated to some extent with its CVE.35,36 In Figures 10a and 10b, the CVEs of all of the studied PIs are plotted against their volume fraction and absolute weight density, respectively. Herein, the volume fraction was calculated based on the van der Waals volume of the repeating unit (Supporting Information). It should be noted that a clear negative linear correlation is seen between the CVE and the absolute weight density, regardless of the primary structure and the imidization condition, while a poor correlation is seen between the CVE and the volume fraction. In particular, the uniaxially drawn films show a trend similar to that of the crystalline powders, even though the imidization reaction of the films occurs in a quasi-bulk (slightly swollen) state and the second kind (paracrystalline) disorder for the crystalline lattices could be larger than that for the powder samples.

Figure 10. CVEs vs (a) volume fraction of the repeating unit in the lattice and (b) absolute weight density, for all of the PIs in the present study. I


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(Figure 10b). Based on eq 3, a first-order differential equation (eq 4) and the solution of eq 5 are readily obtained:

The trend in Figure 10b strongly suggests that the absolute weight density is a more useful parameter (compared with the volume fraction) for predicting the CVE of a PI crystal. Herein, the weight density dependence of the CVE is discussed in terms of the Grüneisen equation.37,38 For the case of an isotropic (or cubic symmetry) system, the Grüneisen equation of a polymeric material is given by CVE ≈ γCv,interχ, where the parameters γ, Cv,inter, and χ are the Grüneisen coefficient, the interchain heat capacity at a constant volume per unit volume, and the isothermal volume compressibility, respectively. The parameter Cv,inter reflects the contribution of interchain vibrations to the heat capacity and is much smaller than the bulk heat capacity.39 The coefficient γ is related to the degree of anharmonicity of the localized motion of polymer chains, and its value has been evaluated as γ ≈ 4 for high-density polyethylene (PE), poly(methyl methacrylate) (PMMA), and poly(styrene) (PS).39−41 The anharmonicity of the localized motion corresponds to the deviation from a harmonic oscillation (i.e., asymmetricity of the potential), and the thermal expansion is induced by an increase in the oscillation amplitude and deviation from harmonic oscillations. In the case of the orthorhombic crystal, the set of Grüneisen equations for thermal expansion in each direction (α1, α2, α3) is given by

(3)

⎛ ∂V ⎞ −⎜ ⎟ = k(V − vo) ⎝ ∂P ⎠T

(4)

V = (Vo − vo) exp( −kP) + vo

(5)

The solution of eq 5 can be interpreted as follows: vo is a volume occupied by a PI chain per unit weight, and only the free volume, Vo − vo, is exponentially compressed in the lower pressure region. The volumes of PMDA-PPD, PMDA-DATP, and PMDA-ODA were measured at variable pressures, up to 7 GPa, and similar compression behavior was actually observed for pressures below 1 GPa.32 Therefore, the observed negative linear relationship between the CVE and the weight density can be attributed to the fact that the exponent of −k and the limiting value of vo in eq 5 hardly depend on the type of PI for a low pressure, in which the volume occupied by the PI chains is not compressed and only the free volume is compressed. In other words, the compressibility of the free volume per unit weight is almost independent of the PI type. The weight density also possibly affects the Grüneisen coefficient because shorter inter- and intramolecular distances in the equilibrium can induce a deeper intermolecular potential and smaller anharmonicity. However, as mentioned above, the Grüneisen coefficients are nearly the same even for totally different types of polymers, and it would be natural to assume that γi does not depend on the type of PI. The several negative values of CTEc can also be discussed in terms of the Grüneisen equation. The elastic compliance, sij (i ≠ j), is usually negative for polymeric materials. Particularly along the c direction, γ3 is much smaller than γ1 and γ2; the term |s33γ3| can be smaller than |s31γ1 + s32γ2| in eq 2, owing to which the CTEc possibly becomes negative, even though all γi are positive. If a certain characteristic conformation, e.g., a planar conformation, induces higher CTE anisotropy in the ab plane and |sii| becomes smaller than |sij| (i ≠ j), CTEa or CTEb can also become negative. As another possibility, if the inter- and intramolecular attractive interactions are enhanced and the potential becomes deeper during heating by a relaxation of the steric hindrance between side groups, accompanied by a slight conformational change, e.g., rotation of an aromatic ring, the anharmonicity is possibly decreased at higher temperatures, to yield a negative γi. However, the values of sij are fairly difficult to measure explicitly for powder crystals, and it is difficult to clearly distinguish the effects of γi and/or sij. Finally, let us discuss the contribution of Cv,inter to the CVE. Hereafter, we assume that the compressibility χ is represented by k(1 − voρ) and γ1 does not depend on the type of PI. The interchain heat capacity Cv,inter above the Debye temperature is 3kBρ/m*, where m* is the mass of a coherently moving segment.39 Assuming that the segment’s length corresponds to that of a repeating unit and m* is equal to Mw, a linear relationship can be expected between Cv,inter and ρ/Mw. In Figure 11, CVE/{k(1 − voρ)} = γ1Cv,inter is linearly correlated with ρ/ Mw, implying that Cv,inter actually contributes to the CVE and the repeating unit can be identified as an oscillator associated with intermolecular motion, for the PIs examined in this study.

α1 = Cv ,inter{χ1 γ1 + s12(γ2 − γ1) + s13(γ3 − γ1)} = Cv ,inter(s11γ1 + s12γ2 + s13γ3) α2 = Cv ,inter{χ2 γ2 + s23(γ3 − γ2) + s21(γ1 − γ2)} = Cv ,inter(s21γ1 + s22γ2 + s23γ3) α3 = Cv ,inter{χ3 γ3 + s31(γ1 − γ3) + s32(γ2 − γ3)} = Cv ,inter(s31γ1 + s32γ2 + s33γ3) CVE ≅ α1 + α2 + α3 = Cv ,inter(γ1χ1 + γ2χ2 + γ3χ3 )

v⎞ ⎛ χ = k(1 − voρ) = k ⎜1 − o ⎟ ⎝ V⎠

(2)

where αi and χi are the thermal expansion coefficient (∂ai/∂T)P/ ai and linear compressibility in the ith direction, respectively, and sij and γi are the elastic compliance and the Grüneisen coefficient, respectively (the subscripts in sij and γi denote tensor contraction).42,43 The directions 1, 2, and 3 correspond to the a, b, and c directions, respectively, and the linear compressibility χi is equal to si1 + si2 + si3. It is plausible to assume that the linear compressibility χi is the dominant factor in eq 2 because PI structures mainly consist of C, H, N, and O elements, and the other terms, Cv,inter and γi, are expected to have similar values among different PIs.44 Furthermore, the terms γχ3 should be much smaller than γ1χ1 and γ2χ2 because the anharmonicity of the intermolecular vibrations along the a and b directions of polymers, i.e., γ1 and γ2, should be much larger than that of the intramolecular vibration along the c direction. This is because the intermolecular potential, which is generally represented by the Lennard-Jones potential, is much shallower and much more asymmetric than the intramolecular potential. By the same token, γ1 and γ2 would be close to each other because the PI chains interact via the van der Waals and dipole−dipole interactions similarly, both along the 1 and 2 directions. Consequently, the CVEs of the PIs may be approximated as γ1Cv,interχ, where χ is the volume compressibility. If the correlations of Cv,inter and γ1 with the weight density are much smaller than that of χ, the volume compressibility χ is assumed to have a negative linear relation with the weight density ρ presented by eq 3, where the parameter V is the specific volume J


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interchain cross-linking or introducing three-dimensionally rigid structures could be a promising venue.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b00095. Temperature dispersion of loss tangent, tan δ, for PMDADMDB, PMDA-TFDB, sBPDA-DMDB, and sBPDATFDB films, and conformational energy of rotational isomeric states for biphenyl derivatives (PDF)

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Figure 11. Values of γ1Cv,inter estimated from CVE/{k(1 − voρ)} vs ρ/ Mw.

AUTHOR INFORMATION

Corresponding Author

*(S.A.) E-mail: [email protected].

4. CONCLUSION The directional linear CTE and CVE values were systematically investigated for 13 crystalline PIs with fundamental structures. The analysis was performed based on variable-temperature synchrotron wide-angle X-ray diffraction measurements, and the relationships between the thermal expansion behaviors and the primary structures of PIs were discussed. The smallest CVE of 116 ppm K−1 was observed for PMDA-PPD having the simplest structure, and in general, the CVEs of PIs increased with increasing the number of p-phenylene linkages, p-phenylene ether linkages, and bulky side groups in the diamine moiety of the PIs. It should be noted that the largest CTE anisotropy was observed for PMDA-ODA (η = 1.66) along with the extraordinarily negative CTEa (−44 ppm K−1). Although the CTE in each direction and its anisotropy strongly depend on the chemical structure, packing structure in the crystalline lattice, and chain conformation, i.e., the coplanarity of the biphenyl group, the CVEs of PIs were clearly negatively linearly related to the weight density ρ, regardless of the primary structures. This general trend in CVEs could be explained in terms of the Grüneisen equation, CVE ≈ γiCv,interχi. The compressibility χi appeared to dominate the CVE by considering the clear negative linear relationship between the CVEs and ρ, while the intermolecular heat capacity Cv,inter appeared to contribute less to CVE, and the Grüneisen coefficients γi were almost the same among the crystalline PIs. This general trend should be useful for predicting the CVEs of crystalline PIs. Generally, a second-kind disorder of the crystalline lattice is inherent to polymeric materials, and such a disorder makes it difficult to predict physical properties such as volumetric thermal expansion because the extent of disorder depends on the crystallization process, and the effect of the crystal’s size is often overlap the effect of the disorder. The present results, however, suggest that the CVE can be essentially predicted from the weight density, not only regardless of the chemical structure but also regardless of the extent of the second-kind disorder. Furthermore, these results indicate that an effective way to suppress the CVE of crystalline PI materials is to enhance the absolute weight density ρ for suppressing χ and to increase the molecular weight of the repeating unit Mw for decreasing Cv,inter without increasing its volume. This concept can be applied not only to the microscopic crystalline lattice but also to the macroscopic semicrystalline film. Additionally, although it appears to be not easy to achieve a CTE of 51 ppm K−1 (the CTE of metallic copper used for wiring) by molecular and crystal design of PIs, combination of the above conditions and suppressing localized molecular motion via

ORCID

Ryohei Ishige: 0000-0002-0875-8962 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported in part by Grants-in-Aid for Scientific Research, Japan Society for the Promotion of Science (25288096, 15K13782, and 15K20992). The synchrotron radiation experiments were performed at the BL40B2 of SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2013A1077) and at BL-10C of High Energy Accelerator Research Organization with the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014G708 and 2016G544). We are grateful to Prof. Nobutaka Shimizu (KEK) and Dr. Noboru Ohta (JASRI) for kind support related to WAXD measurements at BL10C in PF and those at BL40B2 in SPring-8, respectively. We gratefully thank Prof. Kunio Kimura (Graduate School of Environmental and Life Science, Okayama University, Japan) for kindly providing highly crystalline powder specimen of PMDA-ODA. We thank Prof. Masatoshi Tokita for his help with measuring the WAXD patterns of uniaxially oriented PI films. The authors also thank Mr. Tomoya Murakami, Mr. Masahi Mizoroki, and Dr. Kei Shirata for their support with the preparation of samples and WAXD measurements.

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Precise Analysis of Thermal Volume Expansion of Crystal Lattice for

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