Contact Thermal Resistance between PVP-Coated AgNWs

Chapter 3. Contact Thermal Resistance Between Silver Nanowires with

3.4 Contact Thermal Resistance between PVP-Coated AgNWs

To explore the contact thermal resistance between two PVP-coated AgNWs, two samples were prepared from a single AgNW, one which would serve as a continuous reference sample (Figure 3-4a) with the other two aligned to form a sample with a point contact between the suspended membranes and referred to as the “contact sample” (Figure 3-4b). As with the bare AgNWs, the hydraulic diameter is adopted, and the reported Dh is based on the silver core size.

Figure 3-4: SEM micrographs of (a) an 84 nm diameter, PVP-coated AgNW and (b) the corresponding contact sample formed from two segments of the 84 nm PVP-coated AgNW.

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Importantly, after thermal characterization, the contact samples were transferred to a piece of Si wafer and focused ion beam (FIB) was used to cut the approximate center of the contact region, exposing the cross-section and allowing for estimation of the contact morphology. Figure 3-5a shows an SEM micrograph of the cross contact for the 91 nm sample after thermal transport property measurements. EBID was performed on the contact region in order to secure the composite nanofibers prior to transfer to a silicon wafer as shown in Figure 3-5b. The FIB was then used to make a series of section cuts approaching the center of contact. This process was simultaneously monitored via SEM with a tilted angle of 52°. A selection of cross-sectional images up to the point of closest contact are presented Figure 3-5c-f.

Figure 3-5: (a) SEM micrograph of the contact region between the contact sample with two 91 nm AgNW segments. (b) Contact region after EBID. (c-f) Successive SEM micrographs after FIB milling.

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The cross-sectional area of the samples was characterized according the procedure previously adopted for NbSe3 and silver nanowires (Yang et al. 2018, Zhao et al. 2020). Figure 3-6 shows the SEM micrographs of the 84 and 91 nm contact samples after cutting the approximate center of the contact region. Note, each sample was repositioned after FIB cutting such that the AgNWs are imaged at an angle approximately perpendicular to the resulting cross-section surface.

Figure 3-6: (a) An SEM micrograph of the approximate center of the contact region between two 84 nm AgNWs. (b) An SEM micrograph of the approximate center of the contact region between two 91 nm AgNWs.

At the contact region, two flat, PVP-coated faces of the pentagonal AgNWs were observed to contact each other which leads to a parallelogram whose area (Ac) can be calculated with 𝐴_𝑐 =

𝑤𝑠2

sin 𝜃, where ws is the width of the contact surface and θ is the contact angle. The contact angles were measured to be 55° and 54° for the 84 nm and 91 nm sample, with corresponding contact areas derived as 4,107 nm² and 4,984 nm², respectively.

Finally, Figure 3-7a and Figure 3-7b show magnified images of the PVP interlayers that correspond to Figure 3-6a and Figure 3-6b, respectively. In order to characterize the PVP thickness, two parallel lines were drawn in ImageJ to minimize the effects of local variation, and the separation distance was taken to be the PVP interlayer thickness. For the 84 nm sample, the

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thickness was measured to be 4 nm, and for the 91 nm sample, the interlayer was taken to be 6 nm thick. This suggests that the PVP layer on the individual wire is 2 and 3 nm thick, respectively.

Figure 3-7: (a) An SEM micrograph of the PVP interlayer measurement for the 84 nm diameter AgNW. (b) An SEM micrograph of the PVP interlayer measurement for the 91 nm diameter AgNW.

Through careful probe operation, the suspended AgNW lengths between the two membranes for the continuous wire sample and the contact sample for each sample set were adjusted to be approximately the same. The measured total thermal resistance for the contact and continuous samples are shown in Figure 3-8 and can be expressed as:

𝑅_𝑡,𝑠 = 𝑅_𝑀+ 𝑅_𝑤,𝑠, (3.2)

𝑅_𝑡,𝑐 = 𝑅_𝑀+ 𝑅_𝑤,𝑐+ 𝑅_𝑐. (3.3)

Here Rt,s and Rt,c denote the measured total thermal resistance of the single continuous wire and the contact sample, respectively. RM is the total thermal resistance that exists between the nanowire and the suspended membranes. Rw,s and Rw,c represent the intrinsic resistance of the nanowire in the continuous wire and the contact sample, respectively. Finally, Rc is the resistance of the contact between the two nanowires, which can be further written as 𝑅_𝑐 = 2𝑅_𝑖+ 𝑅_𝑃𝑉𝑃, where Ri is the interfacial thermal resistance between silver and PVP, while RPVP denotes the thermal resistance of the PVP layer.

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Figure 3-8: Measured total thermal resistance of the contact and continuous samples.

To extract Rc from the above equations, RM is first considered. Recently, the thermal conductivity of individual bare AgNWs of different diameters was characterized with careful confirmation that RM in those measurements is reduced to a negligible level (Zhao et al. 2020). As such, the thermal conductivity of the bare AgNW from that study can be treated as the intrinsic wire property. Now the measured thermal conductivity of PVP-coated AgNWs here is lower than the intrinsic value as shown in Figure 3-9, and the difference can be attributed to the non- negligible RM in the current measurement. Therefore, RM can be solved based on Eqn. (3.2) through calculating Rw,s with the intrinsic wire thermal conductivity. For the 84 nm diameter sample, a bare AgNW of the same diameter has been previously measured, and RM is estimated from Eqn. (3.2) as 9.82 × 10⁵ K W^-1, which is ~6% of Rt,s. For the 91 nm sample, the intrinsic thermal conductivity of a 89 nm bare AgNW is used in the calculation, and the contact thermal resistance is found to be 1.93 x 10⁶ K W^-1, representing ~13% of Rt,s.

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Figure 3-9: Measured thermal conductivities of the 84 and 91 nm samples compared against data previously reported for bare AgNWs (Zhao et al. 2020). The suspended lengths of the 84 and 91 nm samples from this work are 28 and 27 µm, respectively. The suspended lengths of the 84 and 89 nm bare AgNW samples are 42 and 44 µm, respectively.

The contact between the AgNW and suspended membranes occurs through one surface of the PVP-coated, pentagonal nanowire, with the 84 nm diameter wire having a 2 nm thick PVP layer.

As such, the thermal conductance for the Ag-PVP-Pt composite interface can be estimated as ℎ = (^{2 𝑛𝑚}

𝜅𝑃𝑉𝑃+ 2𝑅_𝑖^′′)⁻¹. Note that it is assumed that the interfacial thermal resistance for unit area (𝑅_𝑖^′′) between silver and PVP and that between platinum and PVP are approximately the same, which is reasonable considering that electron-phonon coupling on the metal side dominates the interfacial thermal resistance (Majumdar et al. 2004, Stevens et al. 2005). While 𝑅_𝑖^′′ for PVP and metal is not available, a typical value of 1×10^-8 m² K W^-1 is taken, which is based on the value recently reported for metal-polymethyl methacrylate (PMMA) interfaces (Sandell et al. 2020). For the PVP layer, the thermal conductivity measured in this study is used, which yields ℎ=3.48 × 10⁷ W m^-2 K^-1 at room temperature. Now, the contact width of the 84 nm diameter wire is 58 nm and using the intrinsic thermal conductivity of AgNWs from Zhao et al.’s measurement, Lc,min is estimated as 2.03 µm. For the 84 nm sample, the minimum contact length is 3.37 µm, which is beyond Lc,min.

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Then, RM calculated from Eqn. (3.1) is 9.75 × 10⁵ K W^-1, which compares very well with the 9.82

× 10⁵ K W^-1 as previously derived using Eqn. (3.2).

Similarly, for the 91 nm sample the conductance for the Pt-PVP-Ag composite interface can be estimated as ℎ = (^{3 𝑛𝑚}

𝜅𝑃𝑉𝑃+ 2𝑅_𝑖^′′)⁻¹. Again, 𝑅_𝑖^′′ is assumed to be equal for both interfaces and a typical value of 1×10^-8 m² K W^-1 is taken. The contact width for the 91 nm sample is 63 nm, and using the intrinsic thermal conductivity of AgNWs as reported by Zhao et al., Lc,min is found to be 2.18 µm. Note that the minimum observed contact across all samples is 2.20 µm as measured from the contact sample in sample set 2.

The above analysis indicates that RM is approximately the same for both the continuous wire and contact samples, which allows for extraction of Rc through subtracting Eqn. (3.2) from Eqn.

(3.3). For the 84 nm diameter sample, the lengths of the suspended AgNWs for the single wire and contact sample are both ~ 28 m, which leads to 𝑅_𝑐 = 𝑅_𝑡,𝑐− 𝑅_𝑡,𝑠. However, for the 91 nm sample the two AgNW segments in the contact sample are slightly longer than that of the continuous sample, 29 µm versus 27 m. In this case, Rt,s is scaled to account for the length difference and 𝑅_𝑐 = 𝑅_𝑡,𝑐− 𝑅_𝑡,𝑠× 29/27. This inevitably introduces some error because the scaling also applies to RM; however, since RM for the 91 nm sample is ~13% of the total resistance, the above approach only introduces a small error.

At 300 K, Rc is found to be 6.55 × 10⁶ K W^-1 and 7.71 × 10⁶ K W^-1 for the 84 nm and 91 nm samples, respectively. Interestingly, despite the presence of the PVP interlayer, these Rc values are lower than the ~1.3 × 10⁷ K W^-1 reported for point contacts between two 68 nm diameter bare MWCNTs (Yang et al. 2014), which suggests silver nanowires could be more effective for enhancing the thermal conductivity of polymer composites. To further understand thermal

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transport at the contact, the contact thermal resistance for unit area is calculated by multiplying the total thermal resistance by the contact areas derived above. The area normalized contact thermal resistance (𝑅_𝑐^′′) values at 300 K are then found to be 2.69 × 10^-8 m² K W^-1 for the 84 nm sample and 3.84 × 10^-8 m² K W^-1 for the 91 nm sample, and the temperature dependent trends are plotted in Figure 3-10.

Figure 3-10: Area normalized contact resistance for the 84 and 91 nm contact samples.

As mentioned previously, Rc contains contributions from the thin PVP layer, RPVP, and the interfacial thermal resistance between PVP and silver, Ri. For unit contact area, the resistance of the PVP layer can be solved by 𝑅_𝑃𝑉𝑃^′′ = 𝑡 𝜅⁄ _𝑃𝑉𝑃, where t is the thickness of the PVP, and 𝑅_𝑖^′′ = (𝑅_𝑐^′′− 𝑅_𝑃𝑉𝑃^′′ ) 2⁄ . Here a factor of 2 is introduced as there are two PVP-silver interfaces. The derived 𝑅_𝑖^′′ is shown in Figure 3-11, and at 300 K, 𝑅′_𝑖^′is found to have an average value of 5.50 × 10^-9 m² K W^-1. It is important to note here that because the thicknesses of the PVP interlayers are approaching the resolution of the SEM, there is considerable uncertainty associated with the determination of 𝑅_𝑖^′′ as indicated by the red shaded region in Figure 3-11 and discussed in the

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appendix. Nevertheless, the data is still able to shed light on aspects important to the design of polymer composites.

Figure 3-11: Area normalized interfacial thermal resistance between PVP and silver. Note that the shaded region indicates the measurement uncertainty, and for clarity, nearest neighbor averaging is used to smooth the upper and lower bounds.

Firstly, 𝑅_𝑐^′′ for contacts between AgNWs with a PVP interlayer is much higher than the 8.26

× 10^-11 m² K W^-1 recently reported for Ag-Ag interfaces (Zhao et al. 2020). Moreover, despite ~40 times larger contact area, Rc for the PVP coated AgNWs is still approximately 10 times higher than the value of 7.70 × 10⁵ K W^-1 reported for contacts between bare AgNWs of 65 nm diameter.

This indicates that it is critical for AgNWs to reach direct contact to most effectively enhance the composite thermal conductivity.

Moreover, even when considering the upper bound of the uncertainty, the derived interfacial thermal resistance between PVP and silver, 𝑅_𝑖^′′, is lower than the typically reported values (1-8 × 10^-8 m² K W^-1) for CNT-polymer systems as suggested by a number of experimental and theoretical studies (Huxtable et al. 2003, Shenogin et al. 2004, Nan et al. 2004, Haggenmueller et al. 2007).

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This difference in the thermal boundary resistance in polymer-AgNW and polymer-CNT composites could contribute to an improved thermal conductivity enhancement.