Measurement Details

Chapter 3. Contact Thermal Resistance Between Silver Nanowires with

4.2 Measurement Details

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proportional to the volume fraction of embedded AgNWs. While individual layers were more difficult to discern at higher volume fractions due to the enhanced proportion of aligned AgNWs, the layers were estimated as the minimum observed distance between two distinct regions of layer

“welding” and ranged from ~4 µm for the sample with a volume fraction of 0.2 to ~25 µm for those with a volume fraction of 0.025. Therefore, while the AgNWs were aligned primarily to the in-plane direction across all measured samples, the decreasing layer thickness with increasing AgNW volume faction could potentially result in a greater degree of in-plane alignment (anisotropy) for the higher AgNW concentration samples.

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As noted in Section 2.2, by ensuring good thermal contact and minimizing radiation losses, the temperature profile of the suspended sample can be described according to (Zhang et al. 2006):

𝑇(𝑥) = 𝑇₀ + 𝑉𝐼

2𝐴𝜅_𝑠𝑥 − 𝑉𝐼

2𝐿𝐴𝜅_𝑠𝑥², (4.1)

where T is the temperature, V is the DC voltage, I is the DC current, A is the sample area, κs is the combined thermal conductivity of the sample and metal layer, L is the sample suspended length, and x is the distance along the suspended sample. Further, because the heating power (VI) is selected such that the temperature rise of the sample is small (< 5 K), the average temperature profile of the sample is given by (Zhang et al. 2006):

∆𝑇 = 𝑉𝐼𝐿

12𝐴𝜅_𝑠. (4.2)

This temperature rise of the sample will result in a corresponding increase in the resistance of the patterned gold layer such that (Zhang et al. 2006):

𝑅_𝑚 = 𝑅₀+ 𝛽𝑅₀∆𝑇, (4.3) where Rm is the resistance of the metal layer after heating, R0 is the resistance of the metal layer with no heating power applied, β is the temperature coefficient of resistance (TCR), and ∆T is the temperature rise of sample. By combining Eqns. (4.2) and (4.3), a relationship can be obtained between the applied heating power and the resistance of the metal layer as described by Eqn. (4.4) (Zhang et al. 2006).

𝑅_𝑚 = 𝑅₀+𝛽𝑅₀𝑄_ℎ𝐿

12𝐴𝜅_𝑠 . (4.4)

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Here, the value of Rm can be obtained through the application of Ohm’s Law, and by applying a sweeping DC current at each set temperature, R0 can be extrapolated from a linear fit of the measured resistance data Figure 4-3.

Figure 4-3: Typical fitting of heater resistances for the extraction of R0

The TCR of the gold layer is then determined by calculating the slope of R0 over the entire measured temperature range (290-310 K). For the samples in this work, the calculated TCR of the gold heater was found to have an average value of 0.00091 K^-1 at 300 K which is in good agreement with TCR data for 10 nm gold thin films deposited on polyimide (Oliva et al. 2017). Finally, the thermal conductivity of the samples, κs, including the combined thermal conductance of the layered composite, insulating PVP layer, and gold heater layer, can be determined according to Eqn. (4.5) (Zhang et al. 2006).

𝜅_𝑠 = 𝛽𝑅₀𝐼𝑉𝐿

12𝐴(𝑅_𝑚− 𝑅₀), (4.5)

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In order to accurately quantify the thermal conductivity of the samples, it is important to consider the effects of the contact thermal resistance which exists between the sample and sample holder. This contact thermal resistance can be estimated from a fin model according to (Bifano et al. 2012):

𝑅_𝑐 = 2

√ℎ𝑤_𝑐𝜅_𝑠𝐴 tanh (𝐿_𝑐√ℎ𝑤_𝑐 𝜅_𝑠𝐴)

(4.6)

where Rc is the contact thermal resistance between the sample and sample holder, h is the contact conductance per unit area, wc is the contact width, and Lc is the contact length. In order to ensure good thermal contact and minimize the contact thermal resistance, a commercial, electrically insulating thermal paste with a reported thermal conductivity of 15.7 W m^-1 K^-1 (Shenzhen Tensan Co,.Ltd, SYY-157) was applied to the copper sample holder, and the samples were pressed firmly into the holder surface. The contact conductance per unit area, h, can be estimated from the reported thermal conductivity of the paste as ℎ = ( ^𝑡

15.7)⁻¹ where t is the thickness of the contact layer. Using the plastic applicator supplied with the thermal paste, a thin layer of paste (t < 200 µm) could be readily achieved, and without considering the effects of pressing the samples into the paste, the resulting value h is 78,500 W m^-2 K^-1.

For the highest thermal conductivity samples, the combined thermal conductivity of the composite thin films, κs, and insulating PVP layer is 20 W m^-1 K^-1, the contact width is 0.6 mm, the sample cross-sectional area is 0.038 mm², and the contact length is 20 mm. From Eqn. (4.6), the worst-case contact resistance is then found to be 334 K W^-1, which is small as compared to the intrinsic sample resistance of 8,355 K W^-1. Therefore, the maximum contact thermal resistance contributes < 4% to the total resistance and is taken to be negligible.

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Also, due to the negligible radiation assumption in the one-dimensional heat conduction model, an important consideration when implementing the steady-state DC thermal bridge measurement technique is designing the sample dimensions such that the radiation heat transfer from the sample to the surroundings is minimized. For temperatures below ~1000 K, Cahill notes that the effects of radiation losses, Qrad, can be approximated by comparing the linearized radiation coefficient, hrad, to the conduction of the sample, Qc, as given in Eqn. (4.7) (Cahill 1990):

𝑄_𝑟𝑎𝑑

𝑄_𝑐 =ℎ_𝑟𝑎𝑑𝐿²

4𝜅_𝑠𝑡 , (4.7)

where ℎ_𝑟𝑎𝑑 = 4𝜀𝜎𝑇³, L is the sample suspended length, t is the thickness of the sample, ε is the emissivity, and σ is the Stefan – Boltzmann constant.

From Eqn. (4.7) it becomes apparent that radiative heat losses are most critical for the lowest thermal conductivity samples. For the 2.5% samples, the sample length is 6.35 mm, the sample thickness is 295 µm, and the combined thermal conductivity of the composite film and the PVP insulation layer has a measured value of 3.1 W m^-1 K^-1. Therefore, by taking the emissivity to be equal to unity at 300 K, the ratio of radiation to the sample conduction has a maximum value of 0.067. From this the one-dimensional heat transfer model is considered valid for the PVP-AgNW composite samples.

Finally, while the thermal conductance of the metal layer may be considered negligible due to the relative size and increased electrical resistivity relative to bulk gold (Gilani and Rabchuk 2018), the thermal conductance of the PVP insulating layer must be considered. Here, a simple parallel model can be used to calculate the thermal conductivity of the composite according to 𝜅_𝑐 = ^{𝜅𝑠− 𝐴𝑟𝜅𝑃𝑉𝑃}

1−𝐴𝑟 where κc is the thermal conductivity of the composite, Ar is the area ratio of

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the insulating layer relative to the total, and κPVP is the thermal conductivity of PVP. The thermal conductivity of PVP was measured to be 0.23 W m^-1 K^-1 as noted in Chapter 3. Thus, by measuring the PVP cross-section via high-resolution SEM imaging, the composite thermal conductivity can be extracted.