A novel theoretical approach for characterizing pore size distribution of wood cell wall using DSC technique


 The pore size distribution of wood cell wall is crucial for wood modification and drying in practice. This study proposed a novel approach to characterize continuous cell wall pore size distribution by solving the differential equation of the water amount melting inside wood obtained from the differential scanning calorimetry (DSC) technique. Compared with the method using the isothermal process, this approach gained a more accurate pore size distribution, and overcome the disadvantages of long-time measurement. Chinese fir (Cunninghamia lanceolata (Lamb.) Hook.) was investigated with this novel method to analyze the cell wall pore size distribution at the fiber saturation point (FSP). The results showed that most pores of the cell wall were below 10 nm. With the increase of pore size, the pore volume decreases sharply, and the cell wall pores above 20 nm hardly existed. The relation between cell wall pore size and volume had the similar trend between the discontinuous and continuous methods. However, there was a quantity gap in pore volume of 9.4-52.4 times between the two methods.


Introduction
Wood is a natural cellulosic material with excellent mechanical properties, environmental, acoustic, and aesthetic characteristics. However, wood has inevitable disadvantages such as dimensional instability, ammability, and susceptibility to biodeterioration. These inherent defects need to be overcome by modi cation before wood utilization.
Wood modi cation is generally achieved by physically lling the wood porous structure with the modi er or chemically reacting with wood cell wall components, particularly cellulose and hemicellulose (Hill, 2006). In these cases, the pore distribution of wood, especially the pores in the cell wall, plays an important role during the treatment.
However, the cell wall pore distribution is not always constant. Studies have shown that there are huge differences in the cell wall pores with different moisture content (Fengel 1970;Stamm 1967). The water in wood can be divided into two categories, namely free water in the cell lumen and bound water in the cell wall. The interaction between water and cell wall pores mainly comes from the latter, which corresponds to the moisture content below the ber saturation point (FSP). Since FSP is the turning point of the physical properties of wood, the cell wall pore distribution at this speci c moisture content has crucial research signi cance. results in the study of the cell wall pores at different moisture contents including FSP. The DSCT is based on the phenomenon of the melting-point depression for liquid. That is, in porous materials, the lower pressure at the curved inner surface of the pores results in a decrease in the melting point of the liquid in these pores (Burghoff and Pusch 1979). On the hypothesis that all pores are cylindrical, the relationship between the pore diameter and the melting-point depression could be described by the Gibbs-Thomson equation (Furó and Daicic 1999;Zauer et al. 2014b): where T m is the melting temperature, γ is the surface tension, θ is the contact angle, ρ is the density and H f is the speci c heat of fusion of water. According to the above principles, Park et al. (2006b) proposed a method based on the isothermal process and used it to discuss the pore distribution of cellulose.
However, the method held an additional assumption that the water inside the pores did not melt at very low temperatures, which lead to a relatively large calculation deviation. In addition, the time cost and accuracy of the pore size distribution depend on the setting of the isothermal process. To obtain results in high precision, the testing period of the experiment should be greatly extended.
In this paper, we proposed a novel approach based on differential equations to calculate the pore distribution in wood. This method could greatly improve the accuracy of the results and shorten the test time. At the same time, it needs no additional assumption and thus is more reliable.

Materials And Method
Experimental materials Compared with hardwood, the anatomical structure of softwood is simpler and the variability is less obvious. Therefore, in this study Chinese r (Cunninghamia lanceolata (Lamb.) Hook.) was utilized to compare the two analytical methods. Chinese r was sourced from Hebei Province, China. The samples with the size of 5mm(R)×5mm(L)×1mm(T) were cut from the green sapwood with straight texture and no defect.
The samples were placed in a desiccator in which the relative humidity was kept at 100% by distilled water at 25°C. A thermometer was used to detect the relative humidity and temperature. The samples were weighed every day. When the mass change of the samples between two weights was less than 0.02%, the samples were deemed to have reached moisture equilibrium (Li and Ma 2021).

Pore size distribution calculation
Two pore size distribution calculation methods were used and compared in this study.
( ) The continuous method proposed in this paper solves the endothermic differential equation of the freezethaw process of water in the wood to obtain the frozen water content of the samples at different temperatures, and then calculate the pore size distribution. During the heating process of the samples in the low-temperature state, the heat absorbed by the samples is divided into two parts: sensible heat and latent heat. Sensible heat is the heat required to increase the temperature of the substance, including the substance of wood cell wall, the frozen water in the pores, and the unfrozen water. Note that the air in the sample was not included in this study. Latent heat is the heat required for the phase change of frozen water in pores of different sizes. Based on the above, the heat absorption can be expressed by Eq. (2): where ΔT is the difference in temperature (℃); ΔH is the heat absorption corresponding to the temperature change (mJ); m cw , m fw , and m w are the mass of wood cell wall substance, frozen water, and total water in the samples respectively (mg); c cw , c w and c i are the speci c heat capacities of wood cell wall, water and ice, respectively (J ℃ −1 g −1 ); Δm fw is the melting amount of frozen water corresponding to temperature changes (mg); H f is the speci c heat of fusion of pure water (J g −1 ).
Frozen water in wood was assumed to behave like pure water, thus the accepted value of 333.6 J g −1 was where λ is the rate of temperature change, that is, dT /dt, and t represents the time; dH dt refers to the time rate of change of heat absorption, that is, heat ow, which can be expressed as a function of temperature; dm fw dT represents the rate of change of frozen water content concerning temperature and it is also a function of temperature. Thus Eq. (4) can be obtained by organizing Eq. (3): Obviously, Eq. (4) is a non-homogeneous rst order linear differential equation for frozen water content m fw with respect to temperature T. Solving Eq. (4) can obtain the frozen water content as a function of temperature m fw (T): Since the heat ow function is not clear, the DSC measurement data can be used to calculate the numerical solution of the equation instead of the analytical solution. And it is acceptable that when the temperature is 0°C, there is no frozen water in the wood sample (i.e., when T = 0, then m fw (T) = 0).
Considering this condition into the Eq. (5) can easily obtain the value of the constant C: There is m cw c cw + m w c w in both Eq. (5) and Eq. (6), which just represents the heat capacity of the samples at a speci c state when no water is frozen, and thus it can be easily measured at a temperature above 0°C. Finally, the pore size distribution dV dD of the samples is derived from the relationship between the content of frozen water and the temperature in Eq. (7).
where V fw is the volume of frozen water (cm 3 ); ρ fw is the density of frozen water (g cm −3 ), the value of 0.9 g cm −3 (the density of ice) was used in this study; dm fw dT and dT dD can be derived from the Eq. (5) and (1), respectively.
The discontinuous method was performed as a comparison which took an isothermal step method according to Park et al. (2006b) as well as Zauer et al. (2014b). Pore size distribution was determined by measuring the amount of water that has its melting temperature depressed at each isothermal step. If the frozen water does not melt in the rst heating step, the sensible heat can be calculated based on the heat absorption of this step, and the latent heat of melting in the subsequent steps is calculated by subtracting the sensible heat from the total heat: where H i is the latent heat of water at each step (J); H t , i is the total heat absorption at each step (J); c s is the speci c heat capacity of the samples (J ℃ −1 ); m fw , i is the mass of frozen water at each step (g).
In summary, there are the following differences between the continuous method and discontinuous method: 1) The continuous method uses a continuous measurement procedure, during which no isothermal stage is required, thus a continuous pore size distribution could be acquired, while the discontinuous method uses an intermittent measurement procedure, and the obtained pore size distribution is discontinuous. Therefore, the former can make better use of the measured data, and greatly reduce the measuring time.
2) The discontinuous method assumes that the speci c heat of the samples does not change in the early stage of the experiment, which may cause errors. In the contrast, the continuous method does not require this assumption, thus it is possible to achieve more theoretical accuracy.
DSC measurement TA Company's DSC-Q2000 instrument was employed for the freeze-thaw test. The samples were cut to a chip with a diameter of about 4mm and put in an aluminum pan for sealing. Each sample was cooled to -30°C and maintained for 5 minutes to ensure that the water in the sample was fully frozen. Due to the different principles of continuous method and discontinuous method as mentioned above, two different DSC heating programs were taken: Continuous method: The temperature rose to 0°C at a constant rate of 2°C min −1 , maintained for 3 minutes, and then continued to rise to 10°C at the same rate. This measurement took 23 minutes in total.
Discontinuous method: The temperature rose step by step to -21, -11.6, -7.3, -5.4, -4.2, -2.8, -2, -1.3 -0.8°C at a rate of 2 ℃ min −1 , and maintained to return the heat ow to the baseline in each step, and nally continued to rise to 10°C at the same rate. This process took about 45 minutes. The pore diameters corresponding to different temperatures calculated by the Eq. (1) are shown in Table 1. Generally, a slower heating rate and lower sample mass can lead to more accurate enthalpy measurements (Simpson and Barton 1991). However, if the heating rate is too low, the measured signal will show noise and it is di cult to perform the repeatable results. Therefore, a heating rate of 2 ℃ min −1 was used in this study (Zauer et al. 2014b).
After the measurement, the samples were taken out from the aluminum pan, placed in an oven to be dried at 103 ℃ for 8 hours, and the oven-dry mass was weighed, and the moisture content of the samples was calculated.
Results And Discussion Figure 1 shows the change process of the frozen water content calculated by the Eq. (5) with increasing temperature. The gure indicates that as the temperature increases, the frozen moisture content inside the wood samples decreases linearly until around 0°C. In the range very close to 0°C, since the temperature is close to the melting point of bulk water, the water inside the samples melts faster, so the frozen water content shows an accelerated downward trend. When the temperature rises to 0°C, the calculated frozen water content is equal to zero, which depends on two idealized conditions: 1) There is no free water in the cell cavity or other large voids inside the samples, that is, all water is present in the cell wall; 2) There is no signal delay in the endothermic data measured by the DSC instrument. However, these conditions are di cult to fully meet in reality, which will lead to an error.
The dashed horizontal line in Fig. 1 represents the total water content of the test samples obtained by weighing after drying. Therefore, the difference between the dashed line and the solid line shows the unfrozen water (or melted bound water) content inside the samples. The unfrozen water increases as the temperature rises, and when the temperature reaches 0°C, the value is equal to the total water content of the samples.
Due to the limitation of the Gibbs-Thomson equation, the measurement of sample pores is effectively applicable to the pore size range above 2 nm, corresponding to the temperature higher than -20°C (Jähnert et al. 2008). However, even at -20 ℃, it is observed that the frozen water content is not equal to the total water content, which means that at this temperature, there is indeed non-freezing water in the wood samples, which agrees with Telkki, Ylinemi and Jokisaari (2013).
The mesopore pore size distribution of wood cell walls calculated according to Eq. (7) is shown in Fig. 2.
In the small pore diameter range, dV/dD has a very high value. Within the pore diameter range below 10 nm, the curve shows a great uctuation, which may be attributed to the variability of cell wall pores in this range and the noise of the DSC signal. According to Eq. (1) and (7), in this small pore range, even a slight difference in pore diameter will result in a large change in the melting temperature, which makes dV/dD highly sensitive to signal noise. As the pore diameter increases, dV/dD decreases rapidly, and dV/dD is very close to zero with subtle variability in the pore diameter range above 20 nm. The pores within 10 nm account for most of the cell wall pores, which is in consistent with the results of previous studies (Hill 2006;Yin et al. 2015).
The discontinuous method was conducted as a comparison, and the result is shown in Fig. 3. Since this method used piecewise integration to calculate the sample enthalpy changes at different temperature ranges, the results are demonstrated in histograms. To compare the results of the two methods intuitively, data in Fig. 2 are grouped and integrated with the same pore diameter range, and the results are also displayed in Fig. 4 as histograms. Comparing the pore size distribution between Fig. 3 and Fig. 4, it is evident that the trends of these two methods are generally similar. The pore volume is the largest in the range of 1.9-3.4 nm. With the increase of pore diameter, the pore volume gradually decreases. Only a tiny increase appears in the range of 9.4-14.1 nm.
However, the overall pore volume obtained by the method of discontinuous integration (Fig. 3) is signi cantly smaller than that of the continuous method (Fig. 4). For better comparison of the pore volume obtained by the two methods, the ratio of the continuous method and discontinuous method was calculated and shown in Fig. 5. In the range of 30.5-49.5 nm, the pore volume obtained by the continuous method is 9.4 times that of the discontinuous method, while in the range of 1.9-3.4 nm, is 52.4 times. The discontinuous method assumes that no water in the sample melts at the initial stage of the measurement, but the water behavior at this stage is actually not clear. Since the latent heat of water melting is much higher than the speci c heat capacity of frozen water, even if only a small amount of frozen water melts, it will result in higher heat absorption. This will cause the measurement of sensible heat to be too high, and underestimate the latent heat in each subsequent stage, therefore undervalue the amount of melting water. In addition, another consequence of this assumption is that the melting water in large pores is overestimated. As the temperature increases, the water in the samples continues to melt, and since the speci c heat capacity of water is twice that of ice, the sensible heat of the samples also continues to increase. However, when calculating the latent heat in later stages, only the lower sensible heat value in the initial stage is considered and subtracted from total heat absorption, which leads to an overestimation of the latent heat, and this error will become more serious as the temperature rises. Therefore, the decrease in pore volume in the range of larger pore diameter in Fig. 3 is not as obvious as in Fig. 4, which can also be re ected in Fig. 5.
Replace c w in the Eq. (4) with c i to obtain a simpli ed equation to roughly simulate the impact of the above assumption. The results are still given in the form of a histogram in Fig. 6. The pore volume in Fig. 6 and its change trend has become close to the result in Fig. 3, which con rms that this assumption is an important factor for the difference in the calculation results between the two methods.

Conclusions
This study proposed a continuous method based on the differential equation combined with the DSC technique to calculate the pore size distribution of the cell wall of Chinese r at FSP. The results showed that most wood cell wall pores were less than 10 nm, and as the pore diameter increased, the differential pore volume decreased sharply. The pore size distribution calculated by the continuous method was more reliable than the discontinuous method in the earlier report because it required no additional assumptions. In addition, since there was no need to set an isothermal step during the experiment, this method could greatly reduce the testing time while obtaining a continuous pore size distribution. The decreasing trend of pore volume with pore diameter obtained by the two methods was similar. However, the quantitative analysis of the two methods was quite different. Speci cally, the pore volume calculated by the continuous method was 9.4-52.4 times larger than that derived by the discontinuous method.
To simplify the calculation, several parameters in this study adopt theoretical values (such as the speci c heat capacity of water and ice in the cell wall pores), which may lead to subtle calculation errors. In future research, the exploration of the physical properties of the internal substances of wood could be considered to obtain more accurate calculation parameters. In addition, further study on pore distribution of wood cell wall at different moisture content by the continuous method could be expected.

Declarations Con ict of interest
The authors declare that they have no known competing nancial interests or personal relationships that could have appeared to in uence the work reported in this paper. Figure 1 The water content of the samples with increasing temperature. The solid line and the dashed line indicate the frozen water content and the total water content, respectively.

Figure 2
The mesopore pore size distribution of wood cell walls calculated according to the continuous method.

Figure 3
The pore size distribution measured by the discontinuous method.

Figure 4
The pore size distribution measured by the continuous method using histograms.

Figure 5
The ratio of the pore volume of the discontinuous method and continuous method in each pore diameter range.

Figure 6
The pore size distribution measured by the simpli ed continuous method using histograms.