In case of a heat source that heats a point on a thin opaque sample with a frequency f, the AC temperature at the

point distant r from the heat source is expressed as follows [7] .

where T₀ is constant, and k is the inverse of the thermal diffusion length, l, expressed by Equation (2).

At the point r, the phase lag, θ, with respect to the heat source is expressed as

Thermal diffusivity, D, is calculated from the spatial dependence of the phase lag, dθ/dr, i.e.

For accurate measurement of thermal diffusivity, the heating frequency must satisfy two restrictions, depending on the sample size. First, the acceptable heating frequency is limited by the sample thickness, d. The conditional equation is expressed as Equation (5) [8] .

where l_d is the thermal diffusion length in the out of-plane direction. In Equation (1), one-dimensional thermal conduction is assumed. However, the actual phenomenon is two-dimensional because of the finite sample thickness. If the thermal diffusion length, l_d, is much longer than the sample thickness, d, the phenomenon can be treated as one-dimensional. From Equation (5), the frequency, f, must satisfy Equation (6).

where D_d is the thermal diffusivity in the thickness direction of the sample.

Second, the acceptable heating frequency is also limited by the sample length and width. The conditional equation is expressed as Equation (7) [9] .

where L is the distance between the heat source and sample edge, and l_L is the in-plane thermal diffusion length. In Equation (1), the sample area is assumed to be infinite. However, the actual sample area is limited; therefore, the heat flow is reflected from the sample edge. If the distance, L, is much longer than the in-plane thermal diffusion length, l_L, the reflected heat flow decays and can be neglected. From Equation (7), the frequency f must satisfy Equation (8),

where D_L is the in-plane thermal diffusivity of the sample.

From Equations (6) and (8), the heating frequency must satisfy Equation (9).

The laser was scanned linearly through the thermocouple and the distance dependence of the phase lag was measured as shown in Figure 4(a). The value of the heating frequency was selected within the permitted range, as calculated from Equation (9). For the calculation, the value of thermal diffusivities, D_d and D_L, were obtained by preliminary measurement. For UD CFRP, the thermal diffusivity parallel to the fiber direction, D_L//, was 297

mm²/s; that perpendicular to the fiber direction, D_L⊥, was 2.1 mm²/s; and that in the depth direction, D_d, was 2.1 mm²/s. For CP CFRP, the thermal diffusivity in the fiber direction, D_L//, was 171 mm²/s and that in the depth direction, D_d, was 2.1 mm²/s. For UD CFRP, the allowed frequencies in the fiber direction were 1.5 - 12.8 Hz, and those in the direction perpendicular to the fiber direction were from 0.7 - 12.8 Hz. For CP CFRP, the allowed frequencies in the fiber direction were 0.61 - 3.1 Hz, and those in the direction perpendicular to the fiber direction were 1.61 - 3.1 Hz.

The measurements were conducted at temperatures from −80˚C to +80˚C. For UD CFRP, scanning directions are parallel (UD_//) and perpendicular (UD_⊥) to the fiber direction. For CP CFRP, scanning directions are parallel (CP_//) to the fiber direction of the surface. In order to evaluate the frequency dependence of thermal diffusivity, each measurement was conducted with more than two frequencies that satisfied Equation (9). Each measurement with a certain frequency was taken two times. Table 1 lists the measurement conditions.

All measurements were conducted at room temperature. In order to evaluate repeatability, each measurement at a certain frequency was conducted 20 times. In order to evaluate the frequency dependence of thermal diffusivity, each measurement was conducted with more than two frequencies that satisfied Equation (9). Table 2 lists the measurement conditions.

The laser was scanned in a circle in a regular pattern of varying radius and angle, as shown in Figure 4(b). Each measurement at a certain frequency was conducted two times. Table 3 lists the measurement conditions.

Figure 5 shows the temperature dependence on thermal diffusivity of the UD and CP CFRP. The anisotropic ratio of the UD CFRP is also shown in Figure 5. It was found that thermal diffusivity increases as temperature decreases. This is because when temperature increases, the mean free path of the materials diminishes the length. Comparing the thermal diffusivity of UD_// at −84˚C and +84˚C, the thermal diffusivity at −84˚C was 1.7 times larger than that at +84˚C. Comparing the thermal diffusivity of UD_⊥ at −88˚C and +81˚C, the thermal diffusivity at −88˚C was 1.5 times larger than that at +81˚C. The anisotropic ratio, UD_///UD_⊥, was 106 - 124. Comparing the anisotropic ratio at −84˚C and +81˚C, the anisotropic ratio at −84˚C was 1.4 times higher than that at +81˚C. The thermal diffusivity of the CP CFRP at −88˚C was 1.7 times larger than that at +84˚C. The thermal diffusivity of the CP CFRP was larger than the average of the thermal diffusivity of UD_// and UD_⊥.

Figure 6 shows the variation in phase lag with distance for UD_//, and Figure 7 shows this variation for CP_// and CP_⊥. The origin of the horizontal axis is the center of the thermocouple junction. The linearity of the phase lag worsens near origin because of the depth effect described previously. The thermal diffusivity was calculated by

the rectilinear part of the line. The dispersion of the phase lag, which was evaluated by the double standard deviation, 2σ, was found to be within 1.3% for the UD CFRP and 2.7% for the CP CFRP.

The distance and phase difference between the laser and thermocouple were recorded at 20 points. The thermal diffusivities, D, were calculated from the corresponding phase lag, θ. The average thermal diffusivity, D_ave, and double standard deviation, 2σ, were calculated from the thermal diffusivities, D. Table 4 lists D_ave and 2σ of the UD_// measurements. The 2σ percentages tended to be higher at high frequencies, and were within 2.6%. The

D_ave values varied with frequency. The thermal diffusivities of low frequency measurements were lower than those of high frequency measurements. In the low frequency measurements, it is possible that the reflection of heat flow from the edge of the sample to the fiber direction has a large influence, although the heating frequency is selected within the allowed range. Table 5 lists the D_ave and 2σ values of the UD_⊥ measurements. The 2σ percentages were within 2.2%. The D_ave values varied with frequency, although the selected heating frequency was within the allowed range.

Table 6 and Table 7 list the D_ave and 2σ values of the CP_// and CP_⊥ measurements, respectively. The maximum 2σ percentage was 14%. Compared with UD CFRP, the 2σ percentages were relatively high. The D_ave values varied with frequency, although the heating frequency was selected within the allowed range. For a frequency of 2 Hz, the average thermal diffusivity of CP_⊥ was 167.7 × 10⁻⁶ m²∙s⁻¹ and that of CP_// was 153.1 × 10⁻⁶ m²∙s⁻¹. Hence, the thermal diffusivity of CP_⊥ was about 91% of that of CP_//.

Two causes for the modulation frequency dependence of the measured thermal diffusivity are conceivable. First, if we measure an isotropic sample, the modulation frequency dependence of thermal diffusivity does not occur if we use an allowable modulation frequency. However, the CFRPs are composite materials and have an interfacial thermal resistance between fibers and polymer. The interfacial thermal resistance could influence the phase lag of the temperature. Second, the heat that gets conducted through the fibers could be reflected at the sample edge because the thermal conductivity of the fibers is very high. These issues require further study, and are to be addressed in the next stage of this research.

Figure 8 and Figure 9 show the anisotropy distribution for UD and CP CFRP, respectively. The horizontal and

vertical axes were the concurrent phase lags. The fiber direction of the UD CFRP was horizontal, and those of the CP CFRP were horizontal and vertical. It was found that the anisotropy distribution can be visualized by

scanning the laser in a circle on the sample. The estimated directional dependence of the anisotropic ratio, which is based on the fiber direction, is shown in Figure 8 and Figure 9, and is compared with a calculation. Figure 10 is a schematic illustration of the calculation model. The thermal resistance in each direction can be obtained from following equations:

It was assumed that heat is transferred from the origin to a point on the circle through the x-directional thermal resistance R_x and y-directional thermal resistance R_y. The θ-directional thermal resistance R_θ is obtained from R_x and R_y.

Equations (10)-(13) represent the calculation formulae. Equation (14) is obtained by substituting Equations (11)-(13) into Equation (10), and assuming D_x = D_// and D_y = D_⊥.

Figure 11 and Figure 12 show comparisons of results from experiments and calculation for the directional dependence of the anisotropic ratio of UD and CP CFRP.

The experimental results indicate that the thermal diffusivity of UD CFRP was extremely high only in the fiber direction, and low in the other directions. The thermal diffusivity of the CP CFRP exhibited a large dispersion depending on the direction. It is possible that the inhomogeneous structure of the composite causes the dispersion. The calculation results predict the trend of the directional dependence of the anisotropic ratio. However, the calculation results did not correspond to the experimental results exactly. For example, for UD CFRP, the anisotropic ratio predicted by the calculation was lower than that of the experimental value around the 15˚

direction, and for CP CFRP, the calculation did not reproduce the large dispersion. These results indicate that it is important to evaluate the thermal diffusivity for each component.