Introduction

Natural cellulose often forms a rigid crystal of microfibrils in wood cell walls. Its framework accounts for the majority of the cell walls. Especially in the middle layer of secondary walls, they align relatively parallel to the fiber direction. This structure is considered to contribute to the better mechanical properties of wood in longitudinal direction [1, 2].

It is well known that X-ray diffraction (XRD) techniques provide valuable information about molecular conditions of crystalline cellulose in wood cell walls (e.g., microfibril angle (MFA) [3, 4], crystallinity [5,6,7], crystal lattice spacing [8,9,10,11,12,13]). Some researchers found a relationship between stress loading in the longitudinal direction of wood and changes in the crystal lattice spacing [11, 12, 14,15,16,17,18,19]. In our previous work, we tried to investigate the relationship between changes in the crystal lattice strain of (004) plane and the longitudinal macroscopic strain of Japanese cypress (Chamaecyparis obtusa Endl.) under cyclic tensile stress [19]. It was found that the crystal lattice strain was always smaller than the macroscopic strain, regardless of the cyclic loading. Moriizumi and Okano [17] also found a similar relationship under some conditions of moisture content. They also investigated crystal lattice strains during creep and stress relaxation and proposed a mechanical model based on a series of crystalline and amorphous substances. Zabler et al. [11] tried to understand the changes in crystal lattice spacing in (004), (200), (110), and (1\(\bar {1}\)0) plane of crystalline cellulose in spruce (Picea abies) cell walls under stress loading during continuous moisture adsorption and desorption using synchrotron radiation techniques.

Indeed, the mixed structure of rigid crystal and amorphous regions like in wood cell walls often makes it complex to understand the relationship between wood strain and microscopic strain in cellulose molecular chains during stress loading. Therefore, more information based on additional and precise case studies are required to understand the mechanical behavior or spatial configuration of wood cell walls at microscopic level.

The purpose of this study is to investigate the effect of MFA of wood cell walls on the creep and relaxation process from the viewpoint of both microscopic and macroscopic levels. Thus, we collected detailed information of wood cell walls using XRD techniques under continuous longitudinal loading and unstressed condition.

Theory

The crystal lattice strain was obtained based on the Bragg’s equation, as follows:

$$n\lambda =2\overline {d} \sin \theta ,$$
(1)

where \(\overline {d}\) is the crystal lattice spacing (nm) for each diffraction plane, λ is the incident X-ray wavelength (0.1542 nm), and θ is the Bragg angle (°) for each plane on the basis of the peak position obtained by the X-ray diffraction diagram. In this study, \(\Delta \overline {d}\) is regarded as a result of loading on a wood specimen, as well as the corresponding diffraction peak angle of incident X-ray (\(\Delta \theta\) ) varying with it. Then, following equation is obtained when \(\overline {d}\) and θ are replaced by \(\overline {d}\) + \(\Delta \overline {d}\) and \(\theta +\Delta \theta\), respectively.

$$n\lambda =2(\overline {d} +\Delta \overline {d} )\sin (\theta +\Delta \theta ).$$
(2)

The Eq. (2) is expanded, as follows:

$$n\lambda =2(\overline {d} +\Delta \overline {d} )\left\{ {\sin \theta +\cos \theta \cdot \Delta \theta - \frac{1}{2}\sin \theta {{(\Delta \theta )}^2}+ \cdots } \right\}.$$
(3)

When the high-ordered terms \({(\Delta \theta )^{\text{2}}}\) and \(~\Delta \theta \cdot \Delta d\) are ignored, we obtained:

$$n\lambda =2\overline {d} \sin \theta +2\overline {d} \cos \theta \cdot \Delta \theta +2\sin \theta \cdot \Delta \overline {d} .$$
(4)

Combining (1) and (4) yields:

$$\frac{{\Delta \overline {d} }}{{\overline {d} }}= - \cot \theta \cdot \Delta \theta .$$
(5)

Therefore, the crystal lattice strain can be calculated by substituting the declination of peak angle value of diffraction intensity before and after stress loading.

Experimental

Materials

The test specimens were prepared from Japanese cypress (Chamaecyparis obtusa Endl.). Average values of air-dried density, width of annual ring, and moisture content were 0.33 g/cm3, 1.5 mm, and 12%, respectively. The MFA of wood specimens were in the range of 9–21°. The crystallinity calculated from Segal method [20] was 54%. The dimensions and shape of test specimen were the same as in our previous study [18, 19]; quarter-sawn specimens with dimensions of 600 µm × 10 mm × 60 mm (R × T × L) were used. The middle part of all specimens was sawn to make dumbbell shape, and pieces of beech wood were glued on both edges of specimen using epoxy resin as a reinforcement material. A hole 5 mm in diameter was drilled at both ends to allow the bolt fixing on the tensile load attachment in XRD device. A strain gauge (gauge length: 5 mm, Tokyo Measuring Instruments Laboratory Co., Ltd., Japan) was pasted on the surface of center part using cyanoacrylate adhesive.

X-ray stress measurement

The tensile load attachment was mounted on the turntable of a goniometer according to the same method as our previous work [18, 19]. The tensile load, controlled by a desktop-type material testing machine, was applied parallel to the fiber direction through a stainless wire. An X-ray diffractometer (XD-D1w, Shimadzu Co., Japan) was used to measure the crystal lattice spacing of crystalline cellulose in wood cell walls under the following conditions. All XRD measurements were made with the transmission method. The incident X-ray radiation was a Cu Kα characteristic X-ray passing through a nickel filter with a power of 40 kV and 40 mA. Both the air-scattering prevention slit and the divergence slit were 1°. The width of the detection slit was 0.1 mm. Continuous scanning was made with the driving shaft of θ –2θ. Scanning speed and integration time were 0.125°/min and 4.0 s, respectively. The diffraction intensity profile of the peak of (004) plane was measured in the range of 2θ = 32.5–36.5°. The determination of diffraction peak angle was made using Gaussian approximation.

The load during tensile creep test and tensile stress relaxation test was determined based on the following procedure. First, 10 specimens with the dimensions mentioned above were subjected to a static tensile test; 60% of the load at the proportional limit was adopted as the constant load (σ0 ≃ 33.7 MPa) in the tensile creep test. This load corresponding to a longitudinal stress of σ0 was also adopted as the initial load of tensile stress relaxation test. Tensile load in stress relaxation test was gradually decreased to keep a constant macroscopic strain of specimen with the progress of stress relaxation. Three specimens were subjected to each X-ray stress measurement in this study.

All measurements were performed in the experimental laboratory where the temperature and relative humidity were kept at 20 °C (± 0.5 °C) and 60% (± 1%), respectively. Control regulation of X-ray diffractometer with the tensile load attachment was made by a personal computer. Output values from the load cell, which was set for tensile load attachment, and measured values of macroscopic strain from the strain gauge were collected by a data logger (DE-1200IF, San-ei Instruments, Ltd., Japan) and stored in the personal computer.

Results

Tensile creep properties

Temporal changes in creep response

Figure 1 shows the temporal change of macroscopic strain and of (004) crystal lattice strain, fitted as functions εs(t) and εc(t), respectively, for each specimen having a different MFA. The macroscopic creep compliance Js(t) was defined by the following formula:

Fig. 1
figure 1

Temporal changes in tensile creep response. εs(t); macroscopic strain (filled triangle), εc(t); crystal lattice strain of (004) plane (open circle), Js(t); ratio of macroscopic strain (εs(t)) to constant stress (σ0 ≃ 33.7 MPa) (tensile creep compliance) (filled triangle)

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$${J_{\text{s}}}(t)=\left\{ {\begin{array}{*{20}{c}} {{J_0}\quad (t=0)} \\ {{\varepsilon _{\text{s}}}(t)/{\sigma _0}\quad (0<t<{t_r})} \end{array}} \right.,$$
(6)

where J0 is a glass compliance calculated by back extrapolating εs(t) to t = 0, tr is the unloading time and σ0 is the loading stress. εs(t) for both MFAs gradually increased with elapsed time, then an instantaneous elastic recovery occurred, followed by creep recovery after unloading. A part of macroscopic strain was still remaining 30 min after unloading. The extent was similar in both specimens (ε = 0.02–0.03%). On the other hand, εc(t) for both MFAs remained constant after instantaneous elastic deformation due to loading. Then, it began to show a gradual increase after 70 min. Both specimens also expressed strain recovery like for εs(t) after unloading, however, little permanent strain remained in comparison with macroscopic strain. It was found that εs(t) was always larger than εc(t) almost throughout the creep process in both specimens. However, εs(t) and εc(t) almost coincided each other in the specimen with small MFA at the latest stage of loading.

Relationship between ε s(t) and ε c(t)

Figure 2 shows the relationship between εs(t) and εc(t). To analyze their relationship, the ratio of strains at each elapsed time (εc(t)/εs(t)), was plotted. It was found that the ratio kept almost a constant value and then began to increase after Log t = 1.8 (t = 70 (min.)) in both specimens. This means that the creep strains proceeded in the same degree at macroscopic and microscopic levels for the initial period (t < 70), and then the creep strain of crystal lattice began to significantly increase. This increasing trend of the ratio was expressed until the end of loading. Moreover, the ratio at 70 min was larger in the specimen with small MFAs.

Fig. 2
figure 2

Temporal changes in εc(t)/εs(t) in tensile creep processes. Dashed line showed the elapsed time at t = 70 min

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Changes in the diffraction intensity in creep process

Figure 3 shows the ratio of diffraction intensity of (004) plane of crystalline cellulose I(t) to its initial value (I0). It was found that I(t)/I0 slightly decreased, then seemed to increase around Log t = 1.8. A more significant effect was found in the specimen with large MFA (21.2°) compared to the smaller one. It was also found that I(t)/I0 was smaller than 1.0 throughout the creep process.

Fig. 3
figure 3

Temporal changes in I(t)/I0 in tensile creep processes. I(t) and I0: diffraction intensities of (004) plane of crystalline cellulose in tensile creep process of wood specimens under the loaded and unloaded conditions, respectively

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Tensile stress relaxation properties

Behaviors of relaxation response

Figure 4 shows the temporal change of crystal lattice strain εc(t), the imposed macroscopic strain ε0, and the macroscopic relaxation modulus defined as the ratio of loading stress σ(t) to ε0 according to the following formula:

Fig. 4
figure 4

Temporal changes in tensile relaxation response. εc(t); see legend in Fig. 1 (open circle), Es(t); ratio of stress to constant strain (ε0 ≃ 3000 × 10− 6) (tensile relaxation modulus) (filled triangle)

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$${E_{\text{s}}}(t)=\left\{ {\begin{array}{*{20}{c}} {{E_0}\quad (t=0)} \\ {\sigma (t)/{\varepsilon _0}\quad (0<t<{t_{\text{r}}})} \end{array}} \right.~,$$
(7)

where E0 is the static Young’s modulus calculated by back extrapolating Es(t) to t = 0, tr is the unloading time and ε0 is a constant value. Es(t) for both MFAs gradually decreased with the progress of relaxation time. However, it recovered its initial value soon after unloading. Moreover, εc(t) in the specimen with small MFA (10.4°) was always larger than that with small MFA (14.0°).

Relationship between ε s(t) and ε c(t)

Figure 5 shows the relationship between εs(t) (= ε0) and εc(t). The ratio remained almost constant for both MFAs throughout the stress relaxation processes unlike in the case of creep test.

Fig. 5
figure 5

Temporal changes in εc(t)/εs(t) in tensile stress relaxation processes

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Changes in the diffraction intensity in stress relaxation process

Figure 6 shows the ratio of the diffraction intensity of (004) plane of crystalline cellulose (I(t)) to that without loading (I0). The values of I(t)/I0 showed random variation below 1.0 throughout the stress relaxation test for both MFAs.

Fig. 6
figure 6

Temporal changes in the value of I(t)/I0 in tensile stress relaxation processes

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Discussion

The evaluation of strains in cellulose crystal lattice of (004) plane (functions εc(t)) until unloading indicates a viscoelastic behavior at the microscopic level, dependent on MFA like in previous work [17]. Indeed, the obtained results suggest that εc(t) is dependent on the MFA; εc(t) decreases with increasing MFAs for both tensile creep and stress relaxation (Open circles in Figs. 1, 4). In our previous work, the wood cell walls with large MFAs under longitudinal tensile stress showed smaller deformation in the cellulose crystalline region than wood cell walls with small MFAs, while macroscopic strains in both specimens were of the same amount [18]. In other words, the smaller values of εc/εs were found in specimens with larger MFA. The values of εc(t)/εs(t) in this study were 0.79 (MFA = 9.6°) and 0.52 (MFA = 21.2°) at Log t = 1.8 (t = 70 (min.)), while those were 0.74 (MFA = 10.4°) and 0.66 (MFA = 14.0°) throughout relaxation stress processes. This dependency on MFA is considered to coincide with our previous work [18]. Thus, it is considered that the increasing MFA prevents cellulose microfibril from stress loading parallel to the molecular chains in microscopic level, regardless of macroscopic deformations.

It is known that the changes in diffraction intensity are proportional to the changes in orientation of the target material, because the certain diffraction intensity in the reciprocal lattice space is proportional to the function of orientation distribution [21]. We hypothesized that the repeated tensile loading and unloading irreversibly induced a certain convergent value of MFA based on the results of our previous works [18, 19]. This hypothesis is supported by the tensile creep result shown in Fig. 3, although a clear trend was never observed in stress relaxation as shown in Fig. 6.

Previous studies proposed some models of native cellulose microfibrils including wood cell walls based on the combination of the ratio or special configuration of crystalline and amorphous substances [17, 22,23,24,25,26]. These models could conveniently express the overall mechanical properties of target materials. Indeed, a model consisting of an amorphous region of lignin and hemicellulose connected in parallel to a series of crystal and amorphous cellulose could well express the results of this study, if we daringly propose the most convenient and simple model. However, we consider that such model is effective in the case of polymer materials with a known molecular structure, but not in the case of wood cell walls where the microstructure remains unclear. The information of crystalline structure of native cellulose collected by XRD techniques is averaged, and the high-ordered structure of the wood cell wall must have a disarray between average and actual conditions. Therefore, we believe that it is essential to accumulate the common recognition based on numerical measurement data of various mechanical phenomena both from macroscopic and microscopic viewpoints for grasping the whole picture of realistic microstructure of wood cell wall rather than seeking to understand them only based on the simple two-phase model of crystal and amorphous substances.