Chapter 2 Experimental Methods
2.4 Thermal Conductivity Measurement
23 In real cases, corrections need to be made to take into account the heat loss on the surface and heat transfer along the radial directions to the sides, as well as finite pulse wide of the laser. This can be done by choosing a proper correction model for each sample before measurement, and the detailed calculation is automated using the LFA. Different models are offered by LFA 457 including the Cowan model70 from the 1960s which considered heat loss on surface due to radiation and convection, together with square shaped laser pulse correction and the Cape-Lehman model71 from around the same time which further included heat transfer along the radial directions, both with “+”
or “-” pulse correction where “+/-” really meant “with/without”. In general, when the sample is thin, such as films and t1/2 is small, the finite width of the laser pulse should be considered while the heat transfer can be regarded one dimensional; On the other side when the sample is thick and t1/2 is large, the laser can be regarded instant; but heat transfer to the sides need to be taken into account.
For bulk thermoelectric materials, the samples are usually thick, and t1/2 on the order of hundreds of millisecond, the correction for heat loss and heat transfer sideways is more important. Also experimentally, it is better to use sample holders with bigger aperture given it is fully covered by the sample, and perform the measurements on relatively thin (~ 1 mm) samples.
The heat capacity Cp is another property needed to determine κ. This quantity can be measured using drop Calorimetry, differential scanning Calorimetry, and even relative methods comparing to a “standard” in commercial thermal diffusivity system like LFA 457. It turns out accurate Cp measurement is rather challenging and the result can be easily affected by operator errors. In fact, it is necessary to compare measured values with that from theory, i.e., the Dulong-Petit Cv value, 3kB/atom, when above Debye temperature (otherwise from the Debye integral) plus the volume expansion correction term72 9α2T/βd, α is the linear coefficient of thermal expansion, β is the isothermal compressibility, and d is the density. In normal cases, any experimental value, if noticeably different from the theory, likely contains certain error during measurement and it is indeed more accurate to use the theoretical values instead. Consider α and β is not readily available to many systems, using Dulong-Petit heat capacity at lower temperatures (roughly Debye temperature) and values ~10% above that at higher temperatures should be a reasonable estimate.
For this thesis study on Pb chalcogenides, the fitting equation Cp/kB atom-1 = 3.07 + 4.7 × 10-4 (T/K- 300) is used. This is from Blachnik’s drop calorimetry measurement73, which is consistant with theoretical calculated values within 2% error (Figure 2.4).
Figure 2.4. A comparison of Cp used in this thesis with results from drop calorimetry measurement as well as theoretical calculation and Dulong-Petit Cv, for a) PbTe, b) PbSe and c)
PbS. Error bars represent 1%, 2%, and 2% uncertainty in each plot.
Density values are from measured geometry and weight of each sample. This is much simpler than the Achimedes method, but actually gives very close results. The Achimedes method on the other hand, needs to be carried out very carefully making sure no water (or other liquid) is absorbed by the sample (expecially when density is low, a standard treatment is to weigh the sample than coat the surface with a thin layer of wax to close open pores), or no bubbles forming on surface of sample when it is immersed in the liquid, otherwise the result can be rather inaccurate. The 300 K density is used for κ calculation regardless of temperature. A more accurate value can be obtained considering thermal expansion so:
d= d300K
1+3a(T!300) Equation 2.5
For Pb chalcogenides, α ≈ 2 × 10-5 K-1, so at 900 K the density is roughly 4% less which means the thermal conductivity calculated using a constant density will be overestimated by 4%.
PbTe Blachnik, drop calorimetry
Dulong-Petit Cv Dulong-Petit Cv + volume correction Fitting equation
a
Blachnik, drop calorimetryDulong-Petit Cv + volume correction Fitting equation
Debye Cv Dulong-Petit Cv
b
PbSePbS Blachnik, drop calorimetry Fitting equation
Dulong-Petit Cv + volume correction Dulong-Petit Cv
c
25 It has been noticed that samples with low density tends to have low thermal conductivities. The values are often lower than expected from the effective median theory. One possible reason is that the pores are not spherical or have broad size distribution. However, this could also be an indication that the laser flash method underestimates κ when sample has low density. I tend to not trust results from samples with < 90% relative density.
Besides the laser flash method, many researchers also measure κ using the direct steady state method. One advantage of this is it doesn’t require knowledge on heat capacity, and the result doesn’t need further interpretation from models. In this method a constant power is generated by a heater that is in good thermal contact with the sample, the other end of sample is connected to a heat sink and two thermal couples are placed along the direction of heat transfer with known distance.
The thermal conductivity is readily obtained from Fourier’s equation Q = κΔT/Δx. At high temperature, significant amount of heat is lost through radiation to the environment and this need to be minimized or calibrated. Ioffe Institute has been using steady-state method since 1960s. Their setup27 uses a radiation shield thermally anchored to both the heater and heat sink to establish a temperature gradient similar to the gradient in the sample. The space between sample and heat shield is filled with thermally insulating powder to further reduce the radiation loss, whereas heat loss due to conduction through the powder was calibrated. Comparing the most recent Ioffe Institute steady-state setup with the laser flash method, the results are fairly consistent up to 700 K for n-type PbSe, suggesting the steady-state method as implemented by the Ioffe Institute could be as accurate.
But for a lot of their older publications, the κ tends to be overestimated at least at high temperature compared with results from laser flash method on very similar samples.
Other methods to determine thermal conductivity include the Harman method and the 3ω method, which is mostly used for thin films. These techniques are subject to more complicated model interpretation or calibration and are less accurate for materials with low thermal conductivity.
It is common in the thermoelectric community to claim each measurement has 5% uncertainty.
Unfortunately, there hasn’t been a “standard method” for most property measurements described above thus there is no way to decide the “real value” for a sample. The 5% claim is at best the statistical uncertainty reflecting the quality of data rather than the difference from the “real value”.
In fact, difference around 15% is often seen among results from different groups. Even for the simplest dimension measurement on the same sample can easily yield different result by improper use of calibers or just by using different electronic calibers that are common nowadays in labs.
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