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Official Journal of the Japan Wood Research Society

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Effect of pore conformation on dielectric anisotropy of oven-dry wood evaluated using terahertz time-domain spectroscopy and eigenvalue problems for two-dimensional photonic crystals

Journal of Wood Science volume 60, pages 194–200 (2014)Cite this article

Abstract

To clarify the effect of pore conformation on the dielectric anisotropy of wood, the relative permittivity along the longitudinal and tangential axes of flat-sawn oven-dry specimens of 12 different wood species was measured using terahertz time-domain transmission spectroscopy and compared with the values calculated using the eigenvalue problem for two-dimensional photonic crystals. The measured dielectric anisotropy, which is the ratio of the relative permittivity along longitudinal axis to that along the tangential axis, was well explained by the calculated value. It was concluded that the ratio of tangential to radial widths of wood pores affects the relative permittivity along the tangential axis, and that the dielectric anisotropy decreased with an increase in the ratio. This discussion can also be applied to the relative permittivity in frequencies below 0.15 THz. These findings show promise as a new method for evaluating the porous structure of wood.

Introduction

Wood is a natural material and imposes less environmental loading than other materials such as plastics, ceramics, and metals. However, to facilitate the usage of wood in industry (such as structural members in furniture, instruments, and buildings), it is necessary to correctly evaluate fluctuating physical properties, which include its density, moisture content, and grain direction [1]. Grain direction is especially important because cross-grain cut boards and poles have inferior strength, and it has been reported that the strength of members parallel to grain is about 20 times larger than those perpendicular to grain [2]. Thus, the nondestructive evaluation (NDE) of wood grain direction is an important consideration.

The depolarization technique that uses microwaves [3–8], millimeter waves [9], and terahertz waves [10] has attracted notice as an NDE technique for wood grain direction detection because it is contact free, noninvasive, and safe as well as nondestructive. This technique is based on the phenomenon that wood depolarizes free electromagnetic waves. This wave depolarization is caused by the dielectric anisotropy of wood, which is the difference in the complex permittivities measured along longitudinal and transverse axes.

To date, numerous researchers have studied the cause of dielectric anisotropy in wood [11–27]. For example, Norimoto and Yamada [20] regarded wood as a mixture of pores and wood substance, and proposed that its dielectric anisotropy is caused by the alignment of pores and/or the dielectric anisotropy of wood substance. Tanaka et al. [23] reported that the dielectric anisotropy of oven- and air-dry wood was primarily caused by the pore alignment and depended on density for a millimeter wave of 0.1 THz. Inagaki et al. [11] and Todoruk et al. [24, 25] stated that pore alignment was the major cause of dielectric anisotropy in the frequency range of 0.1–0.6 THz. However, the dielectric mixture models they used [28–33] cannot explain the relationship of the conformation of pores to the dielectric anisotropy, so alternative theoretical approaches have been sought.

The method of solving the eigenvalue problem for photonic crystals has been studied primarily to find the photonic band gap [34], which is the frequency range where electromagnetic waves are extraordinarily absorbed by photonic crystals. Fujimura [35] reported on the dielectric anisotropy of a two-dimensional (2D) photonic crystal whose structure is periodic in two axes. Since this structure is quite similar to the porous structure of wood [36–40], the above-mentioned solution method should be useful for evaluating the pore conformation of wood.

This study was performed to clarify the effect of pore conformation on the dielectric anisotropy of wood. The relative permittivity along the longitudinal and tangential axes of oven-dry flat-sawn specimens prepared from 12 different wood species was measured at a frequency of 0.15 THz using a transmission measurement system for terahertz time-domain spectroscopy (THz-TDS). Additionally, the relative permittivity was calculated by solving the eigenvalue problems for wood specimens by regarding them as 2D photonic crystals. Ultimately, the dielectric anisotropy of wood was obtained from the measured and calculated relative permittivities. The relations of two kinds of anisotropy to wood density were compared and analyzed based on the pore arrangement observed using a scanning electron microscope (SEM).

Theory

In general, wood is periodic along its radial and tangential axes [36–40] and uniform along its longitudinal axis. This structure is similar to that of a 2D photonic crystal, in which the behavior of an electromagnetic wave is theoretically analyzed by transforming Maxwell’s equations to the eigenvalue equation [34, 35].

Figure 1 shows a transverse section of wood regarded as a 2D photonic crystal. In this figure, rectangular virtual unit cells, which are composed of pores (air) and wood substance [23] with relative permittivities of ε′0 (=1) and ε′WS, respectively, are arranged periodically along the elementary lattice vectors, a 1 and a 2. The relative permittivity, ε′(x), along an arbitrary vector, x, perpendicular to the longitudinal axis is determined by the radial width, a, of a virtual unit cell, the dimensional ratio, r, of its tangential to radial widths of a pore, the relative permittivity of wood substance, ε′WS, and the volume fractions of the pore and wood substance, (ρ WS − ρ 0)/ρ WS and ρ 0/ρ WS [23], respectively, where ρ 0 and ρ WS are the densities of oven-dry wood and wood substance, respectively. The relative permittivity function can be written as.

Fig. 1
figure 1

Transverse section of wood regarded as a 2D photonic crystal. k represents the wave vector of an electromagnetic wave (parallel to the radial axis), \( \rho_{\text{WS}} \) the density of wood substance (=1.5 g/cm3), ε′0 (=1) and ε′WS relative permittivities of pore (filled with air) and wood substance, respectively, w the width of a pore along the radial axis, respectively, and r is the ratio of tangential to radial widths of a pore. The rectangular virtual unit cell has a radial width of a, which is related to w, \( \rho_{\text{WS}} \), and wood oven-dry density, ρ 0, as \( a = w\sqrt {{{\rho_{\text{WS}} } \mathord{\left/ {\vphantom {{\rho_{\text{WS}} } {\rho_{0} }}} \right. \kern-0pt} {\rho_{0} }}} \), and defines elementary lattice vectors a 1 and a 2. S 0 is an area of an elementary lattice

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$$ \varepsilon^{\prime}(\varvec{x} + \varvec{a}_{i} ) = \varepsilon^{\prime}(\varvec{x}) $$
(1)

where a i represents the elementary lattice vectors, a 1 and a 2 (Fig. 1).

When Eq. (1) holds, and an electromagnetic wave transmits along the radial axis, Maxwell’s equations can be transformed to the following eigenvalue equations for electric fields parallel to the longitudinal and tangential axes [34, 35].

$$ \sum\limits_{{\varvec{G^{\prime}}}} {\kappa (\varvec{G} - \varvec{G^{\prime}})\left| {\varvec{k} + \varvec{G}} \right|\left| {\varvec{k} + \varvec{G^{\prime}}} \right|\xi_{{\varvec{G^{\prime}}}} } = k_{0}^{2} \xi_{\varvec{G}} $$
(2)
$$ \sum\limits_{{\varvec{G^{\prime}}}} {\kappa (\varvec{G} - \varvec{G^{\prime}})(\varvec{k} + \varvec{G}) \cdot (\varvec{k} + \varvec{G^{\prime}})\varphi_{{\varvec{G^{\prime}}}} } = k_{0}^{2} \varphi_{\varvec{G}} $$
(3)

where k 0 and k represent the scalar wave number in a vacuum and the wave vector in wood (Fig. 1) of an electromagnetic wave, respectively, G and G′ are arbitrary reciprocal lattice vectors, ξ G , ξ G′, φ G , and φ G′ are eigenfunctions, and κ(G − G′) is expressed as follows:

$$ \kappa (\varvec{G} - \varvec{G^{\prime}}) = \frac{1}{{S_{0} }}\int\limits_{{S_{0} }} {\frac{1}{{\varepsilon^{\prime}(\varvec{x})}}\exp {\kern 1pt} \left\{ { - i(\varvec{G} - \varvec{G^{\prime}}) \cdot \varvec{x}} \right\}{\text{d}}\varvec{x}} $$
(4)

where S 0 is an area of an elementary lattice (Fig. 1).

The relative permittivity of wood can be calculated using the following steps:

  1. 1.

    ε′(x) is determined by ρ 0, ρ WS, ε′WS, r, and a (Fig. 1) and substituted in Eq. (4) to obtain κ(G − G′).

  2. 2.

    k and κ(G − G′) are substituted in Eqs. (2) and (3) to obtain eigenvalues \( k_{0}^{2} \) for electric fields parallel to the longitudinal and tangential axes, respectively.

  3. 3.

    The relative permittivities along the longitudinal and tangential axes, ε′L and ε′T, are calculated using the relations of k (= |k|) to k 0 obtained in Step 2 for the corresponding axes and the following equation:

$$ \varepsilon^{\prime} = \left( {\frac{k}{{k_{0} }}} \right)^{2} $$
(5)

In Eq. (5), the effective medium approximation is assumed [35], i.e., ka/2π is less than 1/10 [32].

Experimental

Sample preparation and SEM observation

Flat-sawn specimens, 25 mm square and 3 mm thick, were cut from softwoods: hinoki (Chamaecyparis obtusa), sugi (Cryptomeria japonica), and akamatsu (Pinus densiflora); ring-porous hardwoods: kiri (Paulownia tomentosa), kuri (Castanea crenata), and keyaki (Zelkova serrata); diffuse-porous hardwoods: kusunoki (Cinnamomum camphora), tochinoki (Aesculus turbinata), buna (Fagus crenata), isunoki (Distylium racemosum), and ekki (Lophira alata); and radial-porous hardwood: shirakashi (Quercus myrsinaefolia). Five specimens were prepared for each species. All specimens were dried to a constant weight at 105 °C.

Specimens were also prepared from the blocks adjacent to the flat-sawn specimens for each species. Their transverse sections were cut by a microtome, conditioned to 0 % moisture content, covered with gold using the spatter coating process, and then observed using a scanning electron microscope (JSM-5600ED, JEOL Ltd.).

THz-TDS transmission measurement

The terahertz technique, which has been the subject of extensive research recently, makes it possible to detect the terahertz waves that penetrate through and reflect from wood specimens [10, 11, 24, 25, 41–43]. Figure 2a shows the experimental setup of the THz-TDS transmission measurement system (TAS7500SP, Advantest Co.). In operation, a THz emitter was irradiated by a femtosecond laser pulse to obtain a THz pulse. The THz pulse was then radiated to a specimen via a set of off-axis parabolic mirrors. The transmitted pulse was focused by another set of off-set parabolic mirrors with a diameter of 25.4 mm and a focal length of 101.6 mm on a THz detector, in which the pulse component was synchronized with another femtosecond laser pulse, of which the time delay, t (ps), was changed stepwise at constant intervals (2 fs), detected, and processed into a voltage, V (mV) in a lock-in amplifier. The voltage V for each time delay t was measured 4096 times and averaged to obtain the waveform V(t).

Fig. 2
figure 2

Experimental setup for terahertz transmission time-domain spectroscopy. A specimen is set in the region surrounded by the dotted line in a so that its b longitudinal or c tangential axis is parallel to the direction of polarization of the terahertz pulse

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The phase θ (rad) and amplitude A (mV) at a frequency of 0.15 THz were obtained through Fourier transformation of the waveform V(t). The relative permittivity of the specimen, ε′, was obtained using the following equation:

$$ \varepsilon^{\prime} = n^{2} - \kappa^{2} $$
(6)

where n and κ are the refractive index and extinction coefficient, respectively, and formulated as follows:

$$ n = 1 + \frac{{(\theta - \theta_{0} )}}{{k_{0} d}} $$
(7)
$$ \kappa = - \frac{1}{{k_{0} d}}\ln {\kern 1pt} \left\{ {\frac{{(1 + n){}^{2}}}{4n}\frac{A}{{A_{0} }}} \right\} $$
(8)

where θ 0 and A 0 represent the phase and amplitude without a specimen, respectively, and d represents the specimen thickness (=3 mm). k 0 is related to the wavelength in a vacuum, λ 0 (=2 mm at a frequency of 0.15 THz), as k 0 = 2π/λ 0.

In the region surrounded by the dotted line in Fig. 2a, the beam component of the THz pulse at 0.15 THz was approximately collimated and found to have a width of about 21.4 mm. The specimens were set on the measurement system so that the direction of linear polarization of the THz pulse was parallel to the longitudinal and tangential axes of wood to obtain the relative permittivity along the corresponding axes, ε′L and ε′T, respectively (Fig. 2b, c).

Results and discussion

Relation of relative permittivity to wood density

The relations of relative permittivity along the longitudinal and tangential axes, ε′L and ε′T, to the oven-dry density, ρ 0, for all the wood specimens are shown by plots in Fig. 3a. Here, it can be seen that ε′L is consistently larger than ε′T. This indicates that wood is a dielectric anisotropic material. Furthermore, ε′L and ε′T increased with density, ρ 0, and there was little effect of wood species. This corresponds well with the results of previous studies [11, 23] and indicates that the density dependence of the dielectric permittivity can be explained by a simple dielectric mixture composed of air and wood substance [11, 12, 20, 23, 24]. It was also indicated, from the linear relation of ε′L to ρ 0, that the relative permittivity along the longitudinal axis of wood can be described as a simple rule of mixture, ε′L = ε′0 + (ε′WS – ε′0)ρ 0/ρ WS [23], where the relative permittivities of air and wood substance are ε′0 (=1) and ε′WS, respectively, and where ρ WS is the oven-dry density of wood substance (=1.5 g/cm3 [44]). The regression line of this equation, shown by a gray line in Fig. 3a, fitted well with the plots of ε′L (ε′L = 1 + 1.63ρ 0, R 2 = 0.991), and the relative permittivity of wood substance was estimated as ε′WS = 3.45. This agrees with the results obtained using a 100 GHz-millimeter wave [23].

Fig. 3
figure 3

Relations of a relative permittivity and b dielectric anisotropy along the longitudinal and tangential axes, ε′L and ε′T, to the density, ρ 0, of oven-dry wood at a frequency of 0.15 THz. The gray line in a represents the regression line (ε′L = 1 + 1.63ρ 0, R 2 = 0.991) based on the rule of mixture for the relative permittivity along the longitudinal axis [23]

Full size image

Numerical calculation of relative permittivity

The relative permittivity along the longitudinal and tangential axes, ε′L and ε′T, were numerically calculated using the process described in Theory with the parameters ρ 0 = 1–1.5 g/cm3, ρ WS = 1.5 g/cm3, ε′WS = 3.45, r = 1/2, 1, 2, and 8, and ka/2π = 1/1000–1/10. In the process, Eq. (4) was calculated in a discrete form using ε′(x) of a 121 × 121 matrix, in which the elements are required to be the real number to deduce Eqs. (2)–(4) from Maxwell’s equations. The dielectric loss of wood, which was estimated using ε′ = 2nκ and Eqs. (7) and (8), can probably be ignored because the ratio of the dielectric loss to the relative permittivity was at most 0.15 in the measurement. The relative permittivity was constant over the range from 1/1000 to 1/10 of ka/2π and was reported to be constant when ka/2π was less than 1/1000 [35]. Thus, this confirms that the effective medium approximation is valid for ka/2π less than 1/10 [32]. In general, the dimension of the transverse section of the wood cell lumens is about less than 200 μm. Accordingly, the calculated relative permittivity is expected to be valid in wavelengths above 2 mm, or frequencies below 0.15 THz.

Comparison of relations of relative permittivity to wood density with calculated ones

The calculated relations of ε′L and ε′T to ρ 0 are shown by black lines in Fig. 3a. The calculated lines for ε′L coincided well with each other, and also matched the regression line based on the rule of mixture, while the lines for ε′T shifted slightly upwards with an increase in r. This indicates that the ratio, r, of the tangential to radial widths of a pore (Fig. 1), affects the relative permittivity along the tangential axis, but does not affect the relative permittivity along the longitudinal axis. The calculated lines fitted well with the measured values of ε′L and ε′T. This indicates that the relative permittivity of wood is explained by the eigenvalue problem for 2D photonic crystals.

Comparison of measured and calculated relations of dielectric anisotropy to wood density

The relations of the dielectric anisotropy, ε′L/ε′T, to ρ 0 for the measured and calculated permittivities are shown in Fig. 3b. SEM photographs of the transverse sections for all the examined wood species are shown in Fig. 4. The dielectric anisotropy, ε′L/ε′T, decreased with an increase in r and the majority of the measured values were distributed in the area surrounded by the calculated lines for r = 1/2 and 8 (Fig. 3b). This probably results from the effect of r on the relative permittivity along the tangential axis (Fig. 3a).

Fig. 4
figure 4

SEM photographs of transverse sections of a hinoki, b sugi, c akamatsu, d kiri, e kuri, f keyaki, T and R axis labels show tangential and radial directions, respectively. Scale bars 200 μm, g kusunoki, h tochinoki, i buna, j isunoki, k ekki, and l shirakashi

Full size image

The following discussion on the relation of the dielectric anisotropy at 0.15 THz (Fig. 3b) to the porous structure of wood (Fig. 4) is limited to the pores with sizes less than about 200 μm, because ka/2π < 1/10 was assumed in the theory [32]. For softwood species (hinoki, sugi, and akamatsu), the plots of ε′L/ε′T were in the range of r ≥ 1. This is because the transverse section of their tracheids is rectangular and elongated along the tangential axis in latewood while it is almost square in earlywood (Fig. 4a–c). For kusunoki, ekki, and keyaki, the plots were around the line for r = 2. This is because the tangential widths of their fibers are larger than the radial ones for kusunoki and ekki (Fig. 4g, k) and the tangential widths of the aggregation of vessels in the latewood of keyaki are larger than the radial ones (Fig. 4f). For kuri, buna, isunoki, and shirakashi, the plots were roughly along the line for r = 1. This results from the fact that the tangential and radial widths of vessels and fibers are almost the same for these species (Fig. 4e, i, j, and l). For kiri and tochinoki, the plots of ε′L/ε′T were in the range of r ≤ 1, which is due to the radially elongated rectangles of the transverse sections of fibers and vessels (Fig. 4d, h). These findings show that the wood dielectric anisotropy is explained by the eigenvalue problem for 2D photonic crystals, and that the dielectric anisotropy undoubtedly depends on the ratio of tangential to radial widths of the wood cells.

Conclusion

To clarify the effect of pore conformation on the wood dielectric anisotropy, the relative permittivity of flat-sawn specimens of 12 wood species along their longitudinal and tangential axes was measured using the transmission measurement system for THz-TDS at a frequency of 0.15 THz. The relative permittivity was also calculated by solving eigenvalue problems for two-dimensional photonic crystals composed of wood substance and pores with the ratio of tangential to radial widths, r. The measured relative permittivity along the longitudinal axis showed a linear relation to wood density, and the relative permittivity of wood substance, ε′WS, was estimated to be 3.45 by fitting the line based on a rule of mixture. The dielectric anisotropy, which is the ratio of relative permittivity along the longitudinal axis to that along the tangential axis, was well explained by the calculated permittivity with the values of ε′WS = 3.45 and r = 1/2, 1, 2, and 8. It was concluded that the ratio of the tangential to radial widths of pores in wood affects its relative permittivity along the tangential axis, and that the dielectric anisotropy decreased with an increase in the ratio. The relative permittivity calculated in this study can be applied to that in the frequency range below 0.15 THz if the effective medium approximation is valid. These findings show a promising new method for evaluating the porous structure of wood.

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Acknowledgments

The authors would like to express their gratitude to Motoki Imamura and Akiyoshi Irisawa (Advantest Corporation, Japan) for their technical support, and to Ryota Kodama (Laboratory for Complex Energy Processes, Kyoto University) for his advice regarding the application of an eigenvalue problem for 2D photonic crystals to the analysis of dielectric anisotropy of wood.

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Authors and Affiliations

  1. Materials Research Institute for Sustainable Development, National Institute of Advanced Industrial Science and Technology, Nagoya, 463-8560, Japan

    Soichi Tanaka

  2. Graduate School of Agriculture, Kyoto University, Kyoto, 606-8502, Japan

    Keiichiro Shiraga, Yuichi Ogawa & Yoshihisa Fujii

  3. Formerly Graduate School of Agriculture, Kyoto University, Kyoto, 606-8502, Japan

    Shogo Okumura

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Correspondence to Soichi Tanaka.

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Part of this report was presented at the 38th International Conference on Infrared, Millimeter, and Terahertz Waves, Mainz, Germany September 2013.

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Tanaka, S., Shiraga, K., Ogawa, Y. et al. Effect of pore conformation on dielectric anisotropy of oven-dry wood evaluated using terahertz time-domain spectroscopy and eigenvalue problems for two-dimensional photonic crystals. J Wood Sci 60, 194–200 (2014). https://doi.org/10.1007/s10086-014-1390-6

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  • DOI: https://doi.org/10.1007/s10086-014-1390-6

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