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This paper presents the temperature dependence of in-plane thermal diffusivity and anisotropy distribution for pitch-based carbon-fiber-reinforced polymers (CFRPs). The measurement was performed using the laser-spot periodic heating method. The samples were unidirectional (UD) and crossply (CP) CFRPs. All carbon fibers of the UD samples ran in one direction, while those of the CP samples ran in two directions. In both UD and CP CFRPs, from -80 ° ;C to +80 ° ;C, temperature dependence of thermal diffusivity values increased as temperature decreased. In this temperature range, the anisotropic ratio between the fiber direction and its perpendicular direction of the UD CFRP was 106 - 124. During the anisotropy distribution measurement, it was found that thermal anisotropy can be visualized by scanning the laser in a circle on the sample. The thermal diffusivity of the UD CFRP in the fiber direction was 17 times larger than that in the 15 ° ; direction, and the thermal diffusivity in the other directions was lower than that in the 15 ° ; direction. The anisotropy distribution for the CP CFRP reflected its inhomogeneous structure.

High-thermal-conductive carbon-fiber-reinforced polymers (CFRPs) have recently been proposed as base material for thermal control devices in spacecraft [

Many studies have been conducted on the thermal conductivity of PAN-based CFRPs [

CFRP samples were fabricated by laminating prepreg sheets and curing with an autoclave. The prepreg consisted of pitch-based carbon fibers (Mitsubishi Plastics Inc., HYEJ16M90D) and epoxy resin (the final resin content was 32% by weight and the remaining was fibers). The diameter of the fibers was about 8 - 11 μm.

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 [

where T_{0} 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) [

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) [

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).

^{−4} Pa. The temperature of the copper heat exchanger under the sample is controlled from −190˚C to +110˚C by a heater and liquid nitrogen. The temperature of the sample is controlled by radiation from the heat exchanger. The heat exchanger and sample are covered by a radiation shield, which has a hole only around the heating position in order to reduce the radiational heat loss. The 25 μm E-type thermocouple is attached to the center of the sample’s rear surface by a silver paste to measure the AC and DC temperatures of the sample. The lock-in amplifier measures the phase difference between the AC signal and reference signal. The lock-in amplifier model is SR830, from Stanford Research Systems. The DC signal is measured by the digital voltage meter to monitor the sample temperature. The temperature of the system is measured using a thermistor, which has an error of 0.5˚C. The heating position is controlled by the XY-stage; the error in the measurement of the position is 1 μm. The measurement uncertainties are within ±2.6%, as verified using stainless-steel (SRM141) and pure copper as the reference samples [

Following three types of measurements were conducted.

1) Measurement of temperature dependence of thermal diffusivity over the temperature range from −80˚C to +80˚C.

2) Evaluation of repeatability in measurement at room temperature.

3) Measurement of anisotropy distribution at room temperature.

The laser was scanned linearly through the thermocouple and the distance dependence of the phase lag was measured as shown in _{d} and D_{L}, were obtained by preliminary measurement. For UD CFRP, the thermal diffusivity parallel to the fiber direction, D_{L}_{//}, was 297

mm^{2}/s; that perpendicular to the fiber direction, D_{L}_{⊥}, was 2.1 mm^{2}/s; and that in the depth direction, D_{d}, was 2.1 mm^{2}/s. For CP CFRP, the thermal diffusivity in the fiber direction, D_{L}_{//}, was 171 mm^{2}/s and that in the depth direction, D_{d}, was 2.1 mm^{2}/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.

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).

The laser was scanned in a circle in a regular pattern of varying radius and angle, as shown in

_{//} 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_{⊥}.

_{//}, and _{//} 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

Sample | UD ([0˚], 69 × 22 × t0.12 mm) | CP ([0/90˚], 74 × 52 × t0.24 mm) | |
---|---|---|---|

Direction | UD_{//} | UD_{⊥} | CP_{//} |

Frequency [Hz] | 7, 10 | 2, 3 | 1, 2 |

Scanning range [mm] | −0.5 to 8.5 | −0.1 to 1.5 | −0.5 to 8.5 |

Scanning pitch [mm] | 0.5 | 0.1 | 0.5 |

Temperature [˚C] | −80 to 80 |

Sample | UD ([0˚], 69 × 22 × t0.12 mm) | CP ([0/90˚], 74 × 52 × t0.24 mm) | ||
---|---|---|---|---|

Direction | UD_{//} | UD_{⊥} | CP_{//} | CP_{⊥} |

Frequency [Hz] | 2, 4, 7, 10, 12 | 1.5, 2, 3, 4 | 1, 2, 3 | 2, 3 |

Scanning range [mm] | −10 to 10 | −1.6 to 1.6 | −8.5 to 8.5 | −8.5 to 8.5 |

Scanning pitch [mm] | 0.5 | 0.1 | 0.5 | 0.5 |

Number of times | 20 | |||

Temperature [˚C] | 25 |

Sample | UD ([0˚], 69 × 22 × t0.12 mm) | CP ([0/90˚], 74 × 52 × t0.24 mm) |
---|---|---|

Frequency [Hz] | 2 | 2 |

Radial range [mm] | 0.1 to 2.5 | 0.5 to 7.5 |

Radial pitch [mm] | 0.1 | 0.5 |

Angular pitch [deg.] | 15 | 15 |

Temperature [˚C] | 25 |

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. _{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. _{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.

_{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} m^{2}∙s^{−1} and that of CP_{//} was 153.1 × 10^{−6} m^{2}∙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.

Frequency [Hz] | Analysis range [mm] | Diffusivity, D_{ave} × 10^{−6} [m^{2}/s] | SD, 2σ × 10^{−6} [m^{2}/s] | Percentage of 2σ [%] | |
---|---|---|---|---|---|

2 | − | 4.5 → 8.0 | 257.3 | 1.9 | 0.7 |

+ | 240.0 | 1.8 | 0.8 | ||

4 | − | 4.5 → 8.0 | 267.6 | 2.0 | 0.7 |

+ | 248.3 | 2.1 | 0.8 | ||

7 | − | 4.5 → 8.0 | 274.8 | 3.6 | 1.3 |

+ | 254.2 | 4.1 | 1.6 | ||

10 | − | 4.5 → 8.0 | 278.2 | 7.1 | 2.6 |

+ | 256.2 | 6.0 | 2.3 | ||

12 | − | 4.5 → 8.0 | 280.2 | 7.1 | 2.5 |

+ | 256.7 | 3.3 | 1.3 |

Frequency [Hz] | Analysis range [mm] | Diffusivity, D_{ave} × 10^{−6} [m^{2}/s] | SD, 2σ × 10^{−6} [m^{2}/s] | Percentage of 2σ [%] | |
---|---|---|---|---|---|

1.5 | − | 0.4 → 0.9 | 1.33 | 0.013 | 1.0 |

+ | 1.61 | 0.018 | 1.1 | ||

2 | − | 0.4 → 0.9 | 1.35 | 0.018 | 1.3 |

+ | 1.67 | 0.021 | 1.2 | ||

3 | − | 0.4 → 0.9 | 1.42 | 0.011 | 0.8 |

+ | 1.81 | 0.026 | 1.4 | ||

4 | − | 0.4 → 0.9 | 1.50 | 0.026 | 1.7 |

+ | 1.94 | 0.044 | 2.2 |

Frequency [Hz] | Analysis range [mm] | Diffusivity, D_{ave} × 10^{−6} [m^{2}/s] | SD, 2σ × 10^{−6} [m^{2}/s] | Percentage of 2σ [%] | |
---|---|---|---|---|---|

1 | − | 4.5 → 8.0 | 144.8 | 20.3 | 14.0 |

+ | 144.7 | 19.7 | 13.6 | ||

2 | − | 4.5 → 8.0 | 171.2 | 20.2 | 11.8 |

+ | 164.2 | 20.4 | 12.4 | ||

3 | − | 4.5 → 8.0 | 173.6 | 8.6 | 5.0 |

+ | 179.6 | 11.5 | 6.4 |

Frequency [Hz] | Analysis range [mm] | Diffusivity, D_{ave} × 10^{−6} [m^{2}/s] | SD, 2σ × 10^{−6} [m^{2}/s] | Percentage of 2σ [%] | |
---|---|---|---|---|---|

2 | − | 4.5 → 8.0 | 149.2 | 8.0 | 5.4 |

+ | 157.0 | 15.8 | 10.1 | ||

3 | − | 4.5 → 8.0 | 151.1 | 5.2 | 3.4 |

+ | 171.8 | 9.0 | 5.2 |

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

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_{⊥}.

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.

The temperature dependence of the in-plane thermal diffusivities of pitch-based CFRP was measured using the laser-spot periodic heating method. It was found that the thermal diffusivity increased as temperature decreased. An anisotropic ratio of more than 100 was observed between UD_{//} and UD_{⊥}. The anisotropy distributions for the UD and CP CFRP were measured. It was found that the anisotropy distribution can be visualized by scanning the laser in a circle on the sample.

The authors are deeply grateful to Dr. Hideyuki Kato and Dr. Takashi Yagi from the National Institute of Advanced Industrial Science and Technology for their assistance, and to Ms. Takayo Kinoshita from Mitsubishi Plastics Inc. for supplying prepreg. A part of this research was supported by KAKENHI (23686122).