Thermal conductance between water and nm-thick WS2: extremely localized probing using nanosecond energy transport state-resolved Raman

Liquid–solid interface energy transport has been a long-term research topic. Past research mostly focused on theoretical studies while there are only a handful of experimental reports because of the extreme challenges faced in measuring such interfaces. Here, by constructing nanosecond energy transport state-resolved Raman spectroscopy (nET-Raman), we characterize thermal conductance across a liquid–solid interface: water–WS2 nm film. In the studied system, one side of a nm-thick WS2 film is in contact with water and the other side is isolated. WS2 samples are irradiated with 532 nm wavelength lasers and their temperature evolution is monitored by tracking the Raman shift variation in the E2g mode at several laser powers. Steady and transient heating states are created using continuous wave and nanosecond pulsed lasers, respectively. We find that the thermal conductance between water and WS2 is in the range of 2.5–11.8 MW m−2 K−1 for three measured samples (22, 33, and 88 nm thick). This is in agreement with molecular dynamics simulation results and previous experimental work. The slight differences are attributed mostly to the solid–liquid interaction at the boundary and the surface energies of different solid materials. Our detailed analysis confirms that nET-Raman is very robust in characterizing such interface thermal conductance. It completely eliminates the need for laser power absorption and Raman temperature coefficients, and is insensitive to the large uncertainties in 2D material properties input.


Introduction
Thermal transport across a solid-liquid interface is a topic of ongoing research due to its various applications in micro/ nanoscale thermal transport, such as evaporation cooling and energy conversion, 1-5 thermal management, 6-8 ultrafast ow delivery, 9 cancer treatment, 10 solar thermal heating, 11 and nanouids. 12,13 Continuum based interface thermal resistance (ITR) models describe this resistance as an irruption on phonon propagation in a crystalline lattice. This is due to the difference in the speed of sound between two materials which leads to a mismatch in acoustic impedance. 14 The Acoustic Mismatch Model (AMM) and the Diffuse Mismatch Model (DMM) are the main models to explain this mismatch across a solid-liquid interface and have been used widely for the theoretical calculation of interface thermal transport. 15 The AMM model neglects phonon scattering at the interface, while the DMM model considers their diffuse scattering across the interface. 16,17 AMM and DMM predict high and low interface thermal resistance, respectively, which provide upper and lower limits for the interface thermal resistance. However, these two models do not consider surface complexities and solid-liquid interaction strength. Molecular dynamic (MD) simulation is an alternative method for studying ITR theoretically without considering continuum based governing equations, and it is capable of studying several factors that can affect the ITR, such as surface wettability. Note that in some calculations the term Kapitza length l K is used to represent the ITR quantitatively. l K is dened as: l K ¼ R K k, where R K is ITR or Kapitza resistance and k is the thermal conductivity of one of the phases, usually the liquid. Barrat et al. studied the dependence of R K on wetting properties using non-equilibrium MD simulation as a function of the interaction coefficient (c 12 ) of the Lennard-Jones equation and under normal pressures. Their results showed relatively large values of R K when the liquid is not wetting the solid (small c 12 values). 18 They reported that l K decreased from 50 nm to less than 10 nm, as the c 12 coefficient increased from 0.5 to 1. Kim et al. investigated the interface thermal transport between parallel plates separated by a thin layer of liquid argon using a 3D MD simulation employing 6-12 Lennard-Jones potential interactions, and studied l K as a function of surface wettability, thermal oscillation frequency, wall temperature (from 80 to 160 K), and channel height. They assumed that the solid molecules had the same mass as the argon molecules. Their results indicated that l K varies from 1 to 10 nm under several scenarios. 19 Similar results were reported by Giri et al. and Vo et al. regarding the effect of interaction strength and thermal boundary conductance. 20,21 In another work, R K was reported in the range of 5 Â 10 À8 to 4 Â 10 À7 m 2 K W À1 using non-equilibrium MD simulations at liquid-vapor Ar mixtures adjacent to warmer Fe walls. 22 Murad et al. studied the ITR between Si and water using MD simulation, and they found that R K decreases with increasing temperature from 5 Â 10 À6 m 2 K W À1 to 3 Â 10 À9 m 2 K W À1 when temperature increases from $350 K to $550 K. 23 In the work by Shenogina et al., it is reported that the Kapitza conductance is proportional to the work of adhesion, and for a highly hydrophilic surface it can be up to $160 MW m À2 K À1 . 24 Barisik et al. performed MD simulations of heat conduction in liquid Ar that is conned in Ag nano-channels and reported that R K can vary from 0.8 Â 10 À9 to 5 Â 10 À9 m 2 K W À1 from cold to hot surface temperature, respectively. 25 In another work they utilized MD simulations to study ITR at Ar-Ag and Ar-graphite interfaces, and concluded that l K increases with increased wall temperature, and is three times larger at an Ar-graphite interface than that at an Ar-Ag interface which is due to the difference between the interaction potentials of the molecular pairs in the two cases. 16 While the last two works were conducted under generally low temperatures ($130 K), Barisik et al. conducted other MD simulations and reported that l K at Si-water in a higher temperature range (more than RT) decreases slightly with increased wall temperature, and is on average around 9 nm. 26 The pressure dependence of ITR at Au-water and Siwater interfaces was studied using MD simulations by Pham et al. 27 Their results revealed that the pressure dependence of l K depends on surface wettability. The l K of the Au-water (hydrophobic) interface was stable despite increasing water pressure, while it changed signicantly across an Si-water interface (hydrophilic). Han et al. drew the same conclusion that ITR increases with liquid pressure enhancement through an MD simulation of n-peruorohexane in contact with gold. 28 The ITRs of several linear alkane liquids in contact with gold were obtained using non-equilibrium MD by Bin Saleman et al. They found that ITR is directly proportional to the number of carbon atoms in an alkane molecule and on average is $1.5 Â 10 À7 m 2 K W À1 . 29 Past discussion was mostly focused on theoretical works, especially MD simulations. Unfortunately, there are only a few experimental works in the eld of solid-liquid ITR measurement to compare with those calculated values. In 2002, M.
Wilson et al. investigated thermal interface conductance between Au, Pt, and AuPd nanoparticles suspended in water or toluene. They found a thermal conductance (G) of 130 MW m À2 K À1 for a citrate-stabilized Pt nanoparticles and water interface by heating particles with a 770 nm optical laser and interrogating the decay of their temperature through time-resolved changes in optical absorption. 30 In their next work, the effect of the organic stabilizing group on the G of AuPd particle-water and AuPd particle-toluene interfaces was studied with a similar technique. 31 Two conclusions were arrived at in their work: (1) the values of G of the particle-water interface under different stabilizing groups were in the order of 100-300 MW m À2 K À1 , which means that G is large, regardless of the self-assembled stabilizing group, and (2) the G of an AuPd particle-water interface was larger than that of an AuPd particle-toluene interface, which indicates the effect of the liquid phase on ITR. In another work, Ge et al. performed a similar time-domain thermoreectance technique and studied the effects of surface wettability on l K using Au and Al based surfaces. The results indicated that l K at hydrophobic (Al) interfaces (10-12 nm) is a factor of 2-3 larger than l K at hydrophilic (Au) interfaces (3-6 nm), which is in agreement with MD simulations. 32 Park et al. reported ITR studies for a system of Au nanorods immobilized on a crystalline quartz support and immersed in various organic uids by heating the nanorods with a subpicosecond optical pulse and monitoring their cooling process by transient absorption. 33 They found the thermal conductances of the nanorod-uid interface at 36 AE 4 MW m À2 K À1 , 32 AE 6 MW m À2 K À1 , 30 AE 5 MW m À2 K À1 , and 35 AE 4 MW m À2 K À1 , for methanol, ethanol, toluene, and hexane, respectively. This indicated that G drops signicantly as water is replaced by an organic uid. Using a similar technique, it was reported that the G of Au nanodisks coated with a hydrophilic self-assembled monolayer varies over 90-190 MW m À2 K À1 , depending on the amount of water in the liquid mixture. For hydrophobic surfaces, G is in range of 70 AE 10 MW m À2 K À1 . This was attributed to the effects of the work of adhesion on interface thermal conductance. 34 Raman spectroscopy has proved to be a powerful tool for studying thermal transport at micro/nanoscales. Several works have been reported that show the potential of this tool to investigate the thermal conductivity and hot carrier diffusion coefficient of 2D materials, such as graphene 35,36 and transition metal dichalcogenides (TMD). [37][38][39][40] Raman spectroscopy is able to measure the ITR of solid-solid interfaces, as well as the aforementioned properties. Yuan et al. reported the interface thermal conductance between few-layered to multi-layered MoS 2 lms and Si, and showed that G increases with an increased number of layers of MoS 2 thin lm from 1 to 69 MW m À2 K À1 . 41 They reported other works that successfully measured the ITR between thin layers of TMD materials and a glass or Si substrate. [42][43][44] Raman spectroscopy based techniques have the advantage of being non-contact, non-invasive, and material-specic leading to higher accuracy of measured parameters.
In this work, for the rst time, the interfacial thermal conductance (G int ) between de-ionized (DI) water and WS 2 nmthick lm is measured using a novel nanosecond energy 5822 | Nanoscale Adv., 2020, 2, 5821-5832 This journal is © The Royal Society of Chemistry 2020 Nanoscale Advances Paper transport resolved Raman (nET-Raman) technique. Each WS 2 sample is suspended over a hole, and immersed in a water bath. Using this experimental structure, WS 2 lm is in contact with water from the top, while its other side is isolated thermally by air inside the hole. Interfacial thermal transport between solid and liquid is characterized here for three samples of different thicknesses. The measured G int is compared and veried with other literature values based on both experimental and MD methods. It is shown in detail that the accuracy of the measurement can be improved by using shorter laser pulses as the transient part of the Raman thermometry. Also, it is proved that uncertainties in the laser absorption coefficient, Raman temperature coefficient, and values of thermal properties of WS 2 lm in theoretical calculations do not downgrade the precision of characterization. In the following, the feasibility and capability of this method are explored in detail.

Sample preparation
Two different sizes of holes are made on an Si substrate using FIB to prepare the suspended samples. One of the holes is circular with a diameter of 10 mm and the other one is square with 22 mm side length. Fig. 1 shows the cross-sectional view of the hole that is used to suspend the sample on top of it. Then, three nm-thick WS 2 akes are prepared using the mechanical exfoliation method from bulk WS 2 , which guarantees the quality and crystallinity of the layers. Mechanical exfoliation makes it possible to prepare several samples of different thicknesses depending on the force applied to the bulk sample. Finally, these samples are transferred to the holes by gel-lms and a 3D micro-stage. More details of this process can be found in our previous work. 45,46 The Si substrate with the WS 2 lm on top of it is mounted on a stage inside a glass container. This container is lled with DI water. Using this setup, the WS 2 lm is in contact with air from the bottom, while touching the water on top (Fig. 1). Comparing the heat transfer on both sides of the WS 2 layer, this design guarantees that heat transfers to water as much as possible and maximizes the effect of the water-WS 2 interface on the temperature evolution of the lm. A glass slide is placed on top of the container to prevent water evaporation and to stabilize water inside the container. It should be noted that water will not penetrate underneath the WS 2 layer in the rst few hours during which the Raman experiment is being performed. We observe that aer 24 hours or more, a few micro-bubbles are formed beneath the WS 2 layer, which shows water penetration. As will be mentioned in the next section, the nET-Raman technique is based on the ratio of the temperature rise of the sample under two different heating states; therefore, any constant parameter that contributes equally under both states will have a negligible effect on the measured interface thermal resistance. Placing the glass substrate on top of the container obviously affects the laser power irradiating the sample, but since the transmission of the glass slide under two heating states is the same, it will not affect our measurement and is not considered in the characterization process.
This method can also be applied to other materials, such as bulk ones, by constructing an appropriate geometry. For instance, for bulk silicon with a thickness in the order of 100 s Fig. 1 Cross-sectional view of the experimental sample design to measure the interfacial thermal conductance (G int ) at a water-WS 2 nm-thick film interface. The nm-thick WS 2 film is suspended over a hole in an Si layer. The hole depth is 3 mm. A graphical illustration of the effects of relative contribution of total interface resistance (R int ) and water thermal resistance (R w ) under (a) CW and (b) ns heating states. Under each state, the WS 2 film is irradiated using a specific laser and the Raman signal is collected. A sample Raman spectrum of WS 2 is shown in the inset of figure (a). Under CW laser heating, R int is $4% of R w , showing it has a weak effect on total thermal resistance between the WS 2 sample and DI water, while under the ns laser this ratio is $20%. As a result, we expect to observe the effects of R int on the temperature evolution of WS 2 film under the ns heating state. Also, these two figures represent the thermal diffusion to the water and the fact that L w,ns is much shorter than L w,CW . The red thermal contour in each figure shows this effect. Also, the time-dependent temperature evolution under laser irradiation is represented schematically in the inset of each figure. For the CW case, the temperature rise (DT CW ) is constant due to the steady-state heating of this laser. The transient temperature rise and Raman weighted average temperature rise of the ns case are shown using red and orange curves in the inset of part (b). Also, the blue curve indicates a single ns laser pulse. of micrometers, it is possible to drill/cut a hole at a micrometer dimension from the bottom of the Si, in such a way that only a thin layer of Si remains on the top, and its bottom is totally in contact with air. Again, by putting this sample inside a DI water chamber, its top surface will touch the water, and the interfacial thermal conductance between the Si layer and water could be measured.

Physical principles of nET-Raman
The temperature rise of the suspended sample under laser irradiation is directly related to the thermal conductivity of the WS 2 lm (k), the thermal conductivity of water, and the interfacial thermal resistance at the water-WS 2 interface ðR 00 int Þ: Temperature changes of the sample could be investigated by studying the frequency variation of Raman-active optical phonons under laser heating. In the nET-Raman technique, two different energy transport states are constructed to analyze the thermal response of the material. Under the rst state, the thin sample is irradiated using a continuous-wave (CW) laser to construct steady-state heating. Under this state, the temperature rise of the sample is mainly controlled by the in-plane thermal conductivity of the sample (k) and the thermal conductivity of water. The second state, which is a transient state, is a nanosecond (ns) state. This state is constructed using a 300 kHz ns pulsed laser. Under this state, the temperature rise of the lm receives more effects from R 00 int : The contribution of R 00 int to the total thermal resistance between WS 2 and water is more signicant in the ns case than in CW. For the CW heating state and under the area of laser heating, the thermal resistance of water R w could be estimated as: R w ¼ 1/(2D CW k w ), where D CW and k w are the laser spot diameter of the CW laser under a 20Â objective lens and thermal conductivity of water, respectively. Taking k w x 0.6 W m À1 K À1 for water, and D CW ¼ 3.6 mm ( Table 2, see below), R w will be around 2.3 Â 10 5 K W À1 . The total interface resistance (R int ) can be estimated as: R int ¼ ð4R 00 int Þ=ðpD CW 2 Þ: Take R 00 int x1 Â 10 À7 m 2 K W À1 ; the total interface resistance will be around 9.8 Â 10 3 K W À1 , which is 4% of the total water resistance covering the WS 2 lm. Therefore, the interfacial thermal resistance plays a negligible role compared with R w in controlling the temperature of the WS 2 lm under the CW state and it is hard to detect its effects under this heating state [ Fig. 1(a)]. It should be noted that performing the Raman experiment using a CW laser is necessary in this method, since it leads to the cancelling of the effects of several known and unknown parameters, such as laser absorption and temperaturedependent Raman coefficients, on the nal results. This idea is represented in detail in the following paragraphs.
The laser pulse width (t 0 ) of the ns laser used in this work is 212 ns. During ns laser pulse heating, the thermal diffusion length to the water layer can be estimated as: L w;ns ¼ ffiffiffiffiffiffiffiffiffiffiffiffi ffi pa w t 0 p ; where a w is the thermal diffusivity of water. L w,ns is around 300 nm. The total thermal resistance caused by water under the ns state is estimated as: where D ns is the laser spot diameter of the ns laser under a 20Â objective lens, which is around 2.5 mm. R w under this state is $100 Â 10 3 K W À1 . While this time R int , using the same estimation as in the CW case and taking D ns as 2.5 mm (Table 2), is $20.3 Â 10 3 K W À1 , which is $20% of R w [ Fig. 1(b)]. Hence, we expect that R 00 int will play an important role under transient heating in the thermal response of the sample. Fig. 1(a) and (b) show a graphical representation of the relative effects of R w and R int under both states on total thermal resistance. Also, note that the thermal diffusion length to water under the CW state can be estimated as: L w,CW x 10D CW , which is $36 mm. This signicant difference between L w,CW and L w,ns is also schematically shown in these two gures by red thermal contours.
In both states, laser heating and Raman signal excitation take place simultaneously. Collecting this Raman signal under various laser powers could be used to track the temperature evolution of the sample. In fact, we can obtain the Raman shi power coefficient (RSC) under each state by irradiating the sample using several laser powers (P). RSC is dened as: , where a and vw/vT are the laser absorption coefficient and Raman shi temperature coefficient, respectively. Under an ns energy transport state, which is designed to probe localized heating, RSC can be obtained as: j ns ¼ vw=vP ¼ aðvw=vTÞgðk; R 00 int ; rc p Þ; where rc p is the volumetric heat capacity of each WS 2 thin lm. The thermal conductance at the water-WS 2 interface is dened as:G int ¼ 1=R 00 int : These denitions of R 00 int and G int are consistent in the rest of this article. As mentioned earlier, due to the localized heating of the ns state, the contribution of R 00 int to j CW is almost negligible in comparison to j ns ; therefore the Raman shi power coefficients are different under these two states. Note that the f and g functions depend on the thermal properties of the materials under each heating state, and are more complicated to solve analytically. Therefore, it is too complicated to show their analytical forms, and they have to be solved numerically.
Using the last two Raman shi power coefficients j CW and j ns , a new experimental parameter is dened as: Q exp ¼ j ns / j CW , which is called the normalized Raman shi power coefficient. It can easily be shown that Q exp is only a function of k, R 00 int ; and rc p . And it is no longer a function of the temperature dependent Raman shi coefficient or laser absorption coefficient. This is the beauty of the nET-Raman technique which makes it independent of the last two coefficients. a and vw/vT are generally the main sources of error in steady-state Raman thermometry. Using a 3D numerical model that calculates the temperature rise of the sample under CW (DT CW ) and ns (DT ns ) heating states, we can nd the theoretical value of the temperature rise ratio (Q th ) as: Q th ¼ DT ns /DT CW . Using known values for k and rc p for water and WS 2 , a relationship between Q th and R 00 int is found. Finally, this relationship is used to nd the R 00 int value that meets the condition: Q exp ¼ Q th . As mentioned earlier, known values of k and rc p are used here from the literature. 39,47,48 In the discussion part, it will be shown that both of these values have a negligible effect on the uncertainty and value of measured R 00 int : The rst part of the 3D heat conduction model deals with steady-state heating under a CW laser, which is governed by the following differential equation: 5824 | Nanoscale Adv., 2020, 2, 5821-5832 This journal is © The Royal Society of Chemistry 2020 Nanoscale Advances Paper where T CW (K) is the temperature in CW heating and _ q is the volumetric Gaussian beam heating, which is shown as: Here r is the radial direction that starts at the center of the hole all the way to the boundaries of the suspended area. z is the position in the thickness direction. I 0 (¼P/pr 0 2 ) and s L are laser power per unit area at the center of the laser spot and laser absorption depth, respectively. s L is calculated as: s L ¼ l/4pk L , where l and k L are the laser wavelength and extinction coefficient of WS 2 at corresponding l, respectively. In this work, l is 532 nm, and at this wavelength k L takes the value 0.903. 49,50 Therefore, s L will be $46.9 nm. Although this value of s L is used in our calculation, it should be noted that this parameter has a negligible effect on the measured R 00 int value using the nET-Raman technique, since it will be canceled out by dividing the temperature rise under two heating states. 48 Transient-state heating is generated using a 532 nm nanosecond laser with a 212 ns pulse width (t 0 ). It should be noted that t 0 should be smaller than the time needed for the sample to reach thermal equilibrium (t eq ). This time can be estimated as: t eq $ (10r 0,ns ) 2 /a water , where a water is the thermal diffusivity of water. In this work, t eq is around 25 ms, which is much larger than t 0 . Another point that is worth mentioning is the effect of hot carrier diffusion on thermal transport in this ns state. In short, as soon as the laser irradiates the WS 2 sample, electrons in the valence band gain enough energy (more than the Fermi energy) to leave this band, leaving holes behind. These hot carriers recombine within a very short period of time (t l ) which is in the order of 1 ns for WS 2 . 51 Since t l is very much shorter than t 0 , we can ignore the effects of hot carrier diffusion on thermal transport. Hot carrier transfer inside TMD materials, such as WS 2 , was well-studied in our previous work. 42,48,52 Regarding the thermal transport in the cross-plane direction of the WS 2 sample, it is assumed that the temperature distribution in this direction is uniform. In the thickness direction, heat diffusion length (L t ) under ns pulsed laser heating can be estimated as: L t x ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pk t t 0 =rc p p ; which is around 1 mm. Here, k t is the thermal conductivity of WS 2 in the cross-plane direction and is about 2 W m À1 K À1 . 53 This value is much larger than the thickness of all samples (Table 1, see below), which conrms the validity of this assumption. The governing equation of the ns laser heating state is: 54 where T ns is the temperature under the ns heating state. The heat source term in this state is written as: Here, I 0 (W m À2 ) is the peak laser intensity. Additionally, the temperature distribution at the water-WS 2 interface could be shown as: R 00 int ¼ ðT WS2 À T water Þ=q 00 ; where q 00 is the interface heat ux. Note that T water and T WS 2 are the temperature of the water and WS 2 lm just close to the interface. Using the abovementioned equations, the temperature rise of the sample under two heating states could be calculated for different R 00 int values. As mentioned earlier, the ratio of these calculated temperature rises of the two states is equal to the experimental normalized RSC for the objective R 00 int : It worth noting that the experimental RSC is based on Raman intensity-weighted temperature rises in both space and time domains and this point is considered in the theoretical calculation of the temperature rise under each state. Note that the temperature at the edge of the suspended area could be considered to be room temperature under both CW and ns cases for two main reasons. First, the interfacial thermal resistance at the WS 2 -Si interface at the edge is much smaller than the in-plane thermal resistance of the WS 2 lm. Second, the thermal resistance of Si is very low due to its high thermal conductivity. Therefore, it is reasonable to consider the room temperature boundary condition at the WS 2 -Si interface.

Sample characterization
Three suspended samples are prepared using the mechanical exfoliation method. Both AFM and SEM characterizations are performed to study the thickness and roughness proles, and structure of these lms. Fig. 2(a) shows the 2D AFM image of Sample 3 at the boundary of WS 2 and the Si substrate. AFM measurements are conducted over the supported area to prevent sample damage. The thickness prole of this sample is shown in the gure using a gray 3D thickness prole and corresponds to the average thickness over the dotted rectangle in the direction of the arrow. The thickness of this sample is 22 nm. Fig. 2(b) indicates the 3D AFM image of this sample over a 10 mm Â 10 mm area close to the suspended area. The root mean square (RMS) roughness of this sample is measured using this image and is 2.44 nm. Table 1 includes the thickness and roughness values of all samples, as well as the ratio of roughness over thickness. This ratio for all samples is less than 15%, which indicates good contact between the WS 2 lm and the Si substrate.
As will be discussed in the next section, sample roughness is one of the main parameters that can affect R 00 int : In order to further study the sample's structure, we performed SEM measurement over the suspended area. Fig. 2(c) shows the SEM image of Sample 3. It shows that the suspended area is almost uniform in all directions. Also, it indicates that the sample is not totally at over the hole, and is concave toward the bottom of the hole. This will affect the laser spot radius measurement This journal is © The Royal Society of Chemistry 2020 Nanoscale Adv., 2020, 2, 5821-5832 | 5825 Paper Nanoscale Advances and alter the actual size of the suspended area, and therefore the theoretical temperature rise calculation under both states will vary to some degree. This effect is discussed in detail in the next section.

Water-WS 2 interface thermal conductance
A room temperature (RT) Raman experiment is conducted using both CW and ns lasers for all three samples to obtain the Raman shi power coefficient. For each sample, based on the WS 2 lm's structure and thickness, optimum laser power is used to nd the Raman shi power coefficient with the highest accuracy. For both lasers, a 20Â objective lens is used to focus the laser spot onto the surface of the WS 2 lm. This objective is chosen to minimize the effects of hot carrier diffusion on thermal transport. The hot carrier diffusion length (Dr HC ) is estimated as: where D WS 2 and s WS 2 are the hot carrier diffusion coefficient and electron-hole recombination time, respectively. Using s WS 2 and D WS 2 from reference values, Dr HC is $0.1 mm, which is much smaller than the laser spot radius under a 20Â objective lens. 48 Therefore, the hot carriers' effects on thermal transport are negligible in our experiment. The radius of the laser spot (r 0 ) for each Raman experiment is measured by analyzing the optical images of the laser spots based on a Gaussian tting method. Insets to Fig. 3(c) and (d) show the laser spots of both states for the third sample captured by a CCD camera. As mentioned in the previous section, knowing r 0 for each heating state is necessary to simulate the heating process since it determines the _ q in eqn (1) and (3). The laser spot size determines the laser intensity distribution while heating the sample, and, subsequently, the temperature rise and Raman shi. In this work, laser irradiation and laser spot measurement are conducted simultaneously, and the measured r 0 is used directly in our numerical method. Therefore, any effects of laser spot size on our nal result are considered precisely. The measured values of r 0 at e À1 of the center intensity for all samples are shown in Table 2. Both lasers are operating at 532 nm wavelength. The ns pulsed laser's repetition rate is 300 kHz. For the ns laser, this repetition rate yields $4.7 W power at the peak of the laser pulse, and decreasing the repetition rate will increase this peak power and can cause sample damage. As will be shown in the next section, decreasing t 0 without burning the lm can reduce the uncertainty level of this technique. More information about the lasers and Raman system can be found in our previous work. 46,55,56 Also, similar consideration should be involved in choosing the optimum CW laser power to prevent sample damage. Table 2 includes the laser power range for each sample under both heating states.
As shown in Table 2, the laser spot under the ns laser is smaller than that under the CW laser. This is caused by the collimation difference between the two laser beams. Also, the slight difference between r 0 under each heating state is induced by a variation in focusing level. Note that all of these r 0 values are more than the phonon mean free path (MFP) of the WS 2 samples ($15 nm); 51,57,58 therefore, it is reasonable to assume that thermal transport is diffusive and under local-equilibrium. Additionally, these laser power ranges ensure the linear decrement of Raman shi against increased laser power with minimal local heating effects. Local heating effects induced at higher laser powers can alter the thermal properties of the WS 2 lm and reduce the quality of the experimental data. Note that these laser power ranges are for the laser beam before it reaches the glass cap on top of the substrate. The amount of laser power absorbed by each sample is even less than this and is $60%. All of these details are considered in the numerical calculation.
Sample 3 is used here to detail the data processing and the results. Fig. 3(a) and (b) show the Raman spectra of this sample under both heating states by varying the laser power. During the Raman experiment, we did not observe any signicant auto-uorescence in the background while collecting the Raman signal under both lasers. Each spectrum has two main Raman modes: E 2g and A 1g . E 2g relates to in-plane vibrations and A 1g represents the out-of-plane vibrations. Two dashed lines in this gure indicate the decrease in Raman shi of the E 2g mode with increased laser power. The E 2g mode is used in this work to nd the Raman shi power coefficient, because it is stronger and more suitable for Raman peak tting. Note that considering the A 1g peak and performing the Raman experiment to nd the RSC values will not affect the nal results. This is shown by conducting nET-Raman using another sample (Sample 4), and the results are reported in ESI. † Ihe insets of these two plots show the 3D Raman intensity contour of this sample under CW and ns states. Also, they indicate that the Raman intensity of both E 2g and A 1g peaks increases linearly with increased laser power. It can be seen from both contours that both Raman peaks are red-shied with increased laser power. 2D representations of these two contours are shown in Fig. S2 of ESI. † Note that each point's value in the 3D contour of the ns state follows the contour bar of the inset of Fig. 3(a). All representative Raman spectra of WS 2 , as shown in Fig. 3, are tted using the Lorentzian function to nd the exact Raman shi of the E 2g peak at each laser power. The results of this peak tting are shown in Fig. 3(c) and (d) for CW and ns heating states, respectively. The tting quality depends on the quality of the experimental data and the Raman peak intensity. Generally, for intensities larger than a certain amount, the tting quality will be almost intact. In this work, the integration time and laser power are chosen in such a way as to guarantee that the peak tting uncertainty for low and high power cases are similar and less than 0.02 cm À1 . Since this value is negligible, it is not included as the uncertainty of measured interface thermal conductance. As mentioned in the previous section, the slope of this line in the low power range indicates the RSC (j) value as: Du ¼ jDP, where u is the Raman shi and P is the laser power. The j of the E 2g mode under a CW laser is À(0.49 AE 0.01) cm À1 mW À1 , and under the ns laser is À(1.30 AE 0.04) cm À1 mW À1 . Similar results for all samples are included in Table 3. Note that j under the ns state is generally higher than the steady-state value. This is because for the same average power, the laser peak power of the ns laser is very high and induces a greater temperature rise. Also, the thermal diffusion length under this Fig. 3 Raman spectra of WS 2 nm-film (Sample 3) under (a) CW, and (b) ns heating states. Both plots show that the Raman intensity of the E 2g and A 1g modes increases with increased laser power, and the peak position redshifts with the increased laser power. Here the E 2g peak is used to perform the analysis and measure the interfacial thermal resistance R 00 int : Two dashed lines in both figures indicate the redshift of the E 2g peak. The insets of these two figures represent the 3D contour of Raman intensity as a function of peak position (u) and laser power (P). These two contours confirm the aforementioned trends, as well as the linear increase in Raman intensity (I) with increased P. Note that the I value of the 3D contour of the ns state corresponds to the contour bar that is shown in the inset of part (a). The Raman shift power coefficient (j) corresponding to the E 2g peak of WS 2 under (c) CW and (d) ns laser of Sample 3. Black dots indicate the experimental position of the E 2g peak at different laser powers, and the red line on each plot shows the fitted line to find the j value under each state. Note that the x-axis of both plots is the laser power just after the objective lens and before the laser beam enters the container. Hence, the absorbed laser power under each case is even lower. Since in the nET-Raman technique the ratio of these two RSCs is used to measure R 00 int ; the laser absorptions of the glass layer, DI water, and WS 2 sample for each sample are identical for both heating states. This will not affect the determined R 00 int : Ihe inset of each plot shows the laser spots that irradiate Sample 3 under a 20Â objective lens for both CW and ns cases. This journal is © The Royal Society of Chemistry 2020 Nanoscale Adv., 2020, 2, 5821-5832 | 5827 Paper Nanoscale Advances state is much smaller than the CW value. These two phenomena lead to the higher temperature rise under pulsed laser heating. It can be seen from this table that j generally increases with decreasing lm thickness under each heating state. This is due to the fact that the temperature rise of each sample depends on the amount of absorbed laser energy, k, and thickness. The thickness affects the heat conduction in the sample and the laser absorption (multiple reections in 2D samples and the optical interference effect). Note that for TMD materials, k increases gradually with increased thickness for samples of more than $5 nm and it reaches the bulk k value at larger thicknesses. 45,59,60 A 3D numerical calculation based on the nite volume method is conducted to nd R 00 int and consequently interfacial thermal conductance (G int ). The thermal properties of WS 2 are held constant at: k ¼ 32 W m À1 K À1 , and rc p ¼ 1.92 Â 10 6 J m À3 K À1 . 47,48 Also, the thermal properties of DI water and air are taken from reference values. It will be shown in the following part of this work that uncertainties in these parameters have negligible effects on the determined R 00 int or G int and their uncertainties. Using this simulation, the Raman intensity weighted average temperature rise over both space and time domains for the ns state (D T ns ), and only over space for the CW state (D T CW ), are calculated as: Ð V 0 I e Àz=sL dv; respectively. These two temperature rises are shown schematically in the insets of Fig. 1. The exponential terms (e Àz/s L ) in these equations are related to the attenuation of the Raman signal as it leaves each scattering location. In these equations, I, V and DT represent the laser intensity under each state, sample volume, and temperature rise of each point, respectively. To match the laser intensity with experimental laser heating, the real laser spot radius, as shown in Table 2, is used to perform the simulation. This calculation is conducted for a range of R 00 int values and Q th is calculated for each R 00 int : Finally, the resultant R 00 int is deduced by equating Q th to Q exp . This process in shown in Fig. 4(a) for Sample 3. Also, the green area represents the uncertainty of the measured R 00 int based on the uncertainty of Q exp , as indicated in Table 3. Measured R 00 int values, as well as G int , for all samples are summarized in Table 4.
It can be seen from this table that the G int values of the three samples are almost in the same order, especially for samples 1 and 3. The larger resistance at the WS 2 -water interface of the second sample compared with the other samples could be caused by several factors. First, although the roughness of this sample is in the same order as that of the other two (Table 1), R q is over the supported region close to the suspended area and the surface roughness of samples over the suspended region could be different than R q , especially for the second sample.

Discussion
The measured G int values in this work are in good agreement with the reference values of solid-water interface thermal transport measurements. Results from other work as well as the current work are summarized in Table 5.
Comparing our result with other experimental work, it is obvious that the G int of the WS 2 -water interface is an order of magnitude smaller than the G int at AuPd-water or Pt-water interfaces, as shown in Table 5. The main factor that could  int of the default case where t 0 is 212 ns. This is due to the higher contribution of R 00 int to the total thermal resistance between WS 2 and DI water during shorter laser pulse heating.
5828 | Nanoscale Adv., 2020, 2, 5821-5832 This journal is © The Royal Society of Chemistry 2020 Nanoscale Advances Paper contribute to this is the difference between the surface wettability of these three solids. Generally, Au and Pt possess smaller water contact angles than WS 2 , which means that these surfaces are more hydrophilic than a WS 2 surface. For clean Au and Pt surfaces, the room temperature contact angle (q CA ) at atmospheric pressure is in the range of 5-40 . [61][62][63][64][65] While the q CA of multilayer WS 2 at RT is around 50-80 . 66,67 Also, in these works, the solid surfaces are more uniform and are in form of nanoparticles, and are smoother compared with the WS 2 samples used in our experiment. q CA depends signicantly on surface microscale roughness. As discussed in the introduction, hydrophobicity is one of the main parameters that affects the thermal transport at a solid-water interface and a lower q CA leads to stronger solid-water contact. A similar argument is valid regarding the G int of ref. 27. Regarding the MD simulation results, it should be noted that ref. 26 reports G int at several temperatures from 350 to 550 K, and at temperatures closer to RT, G int is of the same order as our results.
As mentioned earlier, one parameter that affects the accuracy of our measurement is ns laser pulse width t 0 . As t 0 takes smaller values, the thermal diffusion length in water will be shorter, and R 00 int contributes more to the thermal transport under the ns state compared with longer t 0 cases. To show this fact, the temperature rise of the 22 nm sample under ns is calculated versus R 00 int for several t 0 cases ranging from 10 to 212 ns, and subsequently, Q th is calculated for each t 0 case. Fig. 4(b) shows the result of this calculation. It is reasonable to see that Q th increases with decreased laser pulse width, since shorter pulses means higher pulse peak power that leads to a higher temperature rise. Also, as shown in Fig. 4(b), Q th is plotted for each case. This gure shows that the slope of each Q th À R 00 int curve increases with decreased t 0 . Now, considering R 00 int ¼ 1:02 Â 10 À7 m 2 K W À1 ; as indicated in Table 4, and assuming constant 5% uncertainty for each hypothetical Q exp value, we can nd the uncertainty in R 00 int for each t 0 case. This is shown by the shaded areas in Fig. 4(b) for two extreme cases when t 0 takes 212 ns and 10 ns. It is obvious that this area for smaller t 0 values is narrower that for larger t 0 values, which means higher accuracy in the measurement of R 00 int : As mentioned earlier, the R 00 int =R w ratio is $20% when t 0 ¼ 212 ns. Similar calculation shows that when t 0 is 10 ns, this ratio is $60%, which indicates a higher contribution of interfacial thermal resistance to total thermal resistance between the WS 2 lm and water under ns laser heating. Note that under the ns pulsed laser that is used in this work, when t 0 is 10 ns, the peak power of each laser pulse is $12 kW, and could damage the suspended lm. Another note worth mentioning is that depending on the increment in laser intensity, the light absorption could be linear or non-linear. As long as the laser intensity is not so high as to make the light absorption nonlinear, the laser pulse width could be decreased to increase the sensitivity of G int measurement. An alternative way to implement this experiment with smaller pulse widths is using an amplitude modulated frequency laser, with appropriate frequency and narrow pulse. Under such conditions, the pulse width can be short enough to measure R 00 int more accurately, while the laser power is kept below the damage threshold.
Another study is conducted to show that the nET-Raman technique does not depend on the known values of k and rc p of the WS 2 lm. To do so, the temperature rise of Sample 3 under both heating states, and consequently Q th , are calculated for a range of k and rc p . The results of this calculation are shown in Fig. 5. Using the Q exp of this sample (Table 3), R 00 int and its uncertainty are found for each case and represented by black solid line in each plot of Fig. 5. Two dashed lines show the uncertainty of measured R 00 int corresponding to the uncertainty of Q exp . These two contours indicate that if k and rc p of WS 2 change by 10% independently, the resulting values of R 00 int change by less than 2% and 4%, respectively. Also, the uncertainty of the measured R 00 int will be almost intact, since the dashed lines and black solid line in each contour are almost parallel regardless of k and rc p values. This gure indicates a critical fact that the effects of k and rc p are almost canceled out by introducing Q in this technique, and the three lines in each contour stay almost horizontal while k or rc p is varied.
As shown in Fig. 1 and 2(c), we can see that the suspended lm is slightly concave toward the hole. In all the aforementioned theoretical calculations that are used to determine R 00 int ; it is assumed that the suspended sample over the hole is completely at. To check the uncertainty caused by this assumption, a more realistic case is considered. Here, we assume that the center of the sample is concaved 1.5 mm inward, which is an exaggerated case. The new length of the This journal is © The Royal Society of Chemistry 2020 Nanoscale Adv., 2020, 2, 5821-5832 | 5829 Paper Nanoscale Advances sample (l arc ) which is the length of the WS 2 arc over the hole is $10.6 mm for a 10 mm hole. This value is used in our theoretical calculation to nd the corresponding interface resistance. Fig. 6(b) shows the results of this study. The measured R 00 int with similar Q exp for Sample 3 (Table 3) is $1.1 Â 10 À7 m 2 K W À1 . The uncertainty caused by this elongation in R 00 int is $7% (Table  4). Therefore, this lm elongation has a negligible effect on the determined R 00 int : The temperature rise of water at each point in the close vicinity of the WS 2 lm is calculated and plotted in Fig. 6(a), for both CW and ns cases. Here the normalized local temperature rise (DT*) is reported. To nd DT* at each point, the local temperature rise at that point (DT) is divided by the maximum local temperature rise under ns laser heating (DT ns ). This plot shows that DT* is mostly increased under the laser spot area, and at higher radii close to the boundaries of the suspended region it is a minimum, and in the case of the ns heating state it is almost zero. This further proves the fact that a minimal increase in the sample length will not affect the Raman weighted average temperature rise of the sample, since the thermal transport mostly occurs under the heating region and not in further away areas.
As mentioned in the main text of the paper, the sensitivity of our technique is mostly controlled by the ns state. The contributions of the thermal resistance of water (R w ) and interfacial resistance (R int ) under this state were elaborated in Section 2.2. The ratio of these two values could be written as: ; where k w , a w , and t 0 are the thermal Fig. 5 Effects of (a) in-plane thermal conductivity (k) and (b) volumetric heat capacity (rc p ) of WS 2 thin film on measured R 00 int in this work. Each contour shows the calculated Q th for a range of k and rc p of Sample 3, and the solid black line indicates the Q exp of this sample corresponding to Table 3. The two dashed lines on each figure are related to the uncertainty in measured R 00 int caused by uncertainty in Q exp . Both plots validate the idea that each of these parameters has a negligible effect on the measured R 00 int and DR 00 int in the nET-Raman method. Fig. 6 (a) Normalized local temperature rise under CW (left contour) and ns (right contour) cases. These contours show that the local temperature rise at the edge of the suspended area, especially in the ns case, is almost zero. And the area under the laser spot contributes most to the Raman weighted average temperature rise that is used in nET-Raman to find interfacial thermal conductance. (b) Determined R 00 int using the assumption that the suspended sample is not totally flat and is concave inward 1.5 mm toward the bottom of the hole. Under this situation, the heating area domain and r 0 under both states are altered, and updated values are used in the 3D numerical calculation to find R 00 int for Sample 3. The green dashed arrow in this plot shows the measured R 00 int for flat WS 2 film, as reported in Table 4. The error caused by this change in the sample diameter on measured R 00 int is less than 8%.