## 3 Results and Discussion
We assume that all energy required for evaporation at the interface is conducted through the droplet from the heated substrate. The convection within the droplet is neglected [18; 30]. The conductive heat transfer along the multi-layer, including a droplet and pNIPAM solution, is written as  
\[q_{cond}(r) = \frac{T_{1,P}(r)-T_{4}(r)}{\sum\limits_{i\ =\ 1}^{3}\frac{\delta_{i}(r)}{ \lambda_{i}}} \tag{4}\]  
where \(\delta\) is the layer thickness, and \(\lambda\) is the thermal conductivity. Subscripts 1, 2, 3, 4 denote interface between each of the multi-layer films as in Figure 1b. The evaporative heat flux, \(q_{cv}\) (W/m\({}^{2}\)), along the interface is defined as [31]  
\[q_{cv}(r,T_{s}) = \Delta H_{\text{bap}}J(r,T_{s}) \tag{5}\]  
\[J_{\text{theo}}(r,T_{s}) = \frac{D(c_{sat}(T_{s})-c(T_{a})\varphi)}{R_{CR}}f(\theta) \tag{6}\]  
\[f(\theta) = \left[\frac{1}{2}\sin\theta+\sqrt{2}(\cosh\alpha+\cos\theta_{c})^{ 3/2}\times\int_{0}^{\infty}\frac{\cosh\theta_{c}\tau}{\cosh\pi\tau}\tanh[( \pi-\theta_{c})\tau]P_{-1/2+i\tau}(\cosh\alpha)\tau d\tau\right] \tag{7}\]  
Figure 5: (**a**) Side view of a de-ionized (DI) water droplet, (**b**) SPRi image after locating a droplet at \(t=0\) s and \(T_{s}=29.0\)\({}^{\circ}\)C, and (**c**) magnified image of reflectance of blue dotted rectangular region. where \(\Delta H_{tap}\) is the latent heat of vaporization (2264.705 kJ/kg), \(J_{theo}\) (kg/m\({}^{2}\)s) is the theoretically estimated local evaporation flux of the droplet, \(D\) is diffusivity (26.1 mm\({}^{2}\)/s), \(c_{sat}\) is a saturated vapor concentration depending on the temperature, \(\varphi\) is the relative humidity (10%), \(R_{CR}\) is the droplet contact radius (0.895 mm), \(r\) is the radial coordinate described as \(r=R_{CR}\cdot\)sinh \(\alpha/(\cosh\alpha+\cos\theta)\), \(\theta_{c}\) is the contact angle of the droplet (77.2\({}^{\circ}\)), \(\tau\) is the integral constant, \(\alpha\) is the toroidal coordinate, and \(p_{-1/2+iT}\) is the Legendre function of the first kind. Figure 7 shows the evaporation flux of the droplet considering the surface temperature by calculating Equation (6). The evaporation flux increases along with the radial position. It shows the infinite value at the contact line region, which is physically unrealistic because the droplet evaporates with finite evaporation flux. [PERSON] et al. [28] suggested a new model to calculate the evaporation rate of the droplet with the finite evaporation flux at the contact line region. They visualized the evaporating thin film (ETF) of the DI-water droplet at the contact line region and measured the width and thickness of the ETF. The quantitatively measured width of the ETF was 8.414 \(\upmu\)m. However, there are no experimental results of the ETF depending on the surface temperature. The present study uses the results of [PERSON] et al. [28] taking a cutoff length of 8.414 \(\upmu\)m to solve the singularity problem and adapts the finite averaged evaporation flux at the contact line region expressed as  
\[J_{avg}\ =\ \frac{\int_{0}^{2\pi}\int_{R_{CR}-I}^{R_{CR}}J(r)\cdot rdrd\varphi}{ 1\pi(2R_{CR}-I)} \tag{8}\]  
where \(J_{avg}\) is the averaged evaporation flux, and \(I\) is the width of the ETF. This approach is also adopted for the binary mixture droplet by [PERSON] et al. [32]. The theoretical results of evaporation flux and averaged evaporation flux at the contact line region are used as the boundary condition to calculate the solid-liquid interface temperature of the droplet. Figure 6: (**a**) Reflectance and (**b**) temperature distribution of pNIPAM (\(T_{1,p}\)) along line AB. The solid-liquid interface temperature, \(T_{d}\), of the droplet is calculated with energy balance (\(q_{cond}=q_{ev}\)) at the liquid-air interface. \[T_{4}(r,T_{s})\ =\ \left[ \begin{array}{cc}T_{1,p}(r,T_{s})-\Delta H_{ cup}J_{tube}(r,T_{s})\sum\limits_{i=1}^{3}\frac{\delta_{i}(r,T_{s})}{k_{i}},&0<r<R_{ CR}-l\\ T_{1,p}(r,T_{s})-\Delta H_{ cup}J_{avg}(r,T_{s})\sum\limits_{i=1}^{3}\frac{\delta_{i}(r,T_{s})}{k_{i}},&R_{ CR}-l<r<R_{CR}\end{array} \right.