A Computational Investigation of the “Equivalent Substrates” in the Evaporation of Sessile Droplets

Abstract

This paper investigates the coupled relationship between solid-phase temperature fields and droplet evaporation, focusing on the effects of substrate thermal conduction properties on droplet evaporation behavior. A mathematical model is developed to analyze the impacts of substrate thermal conductivity, thickness, and lower-surface temperature on evaporation rate, surface temperature, and evaporation flux. A dimensionless relative evaporation rate (HCs) is introduced to characterize the influence of substrate thermal conduction. Results show that increasing substrate thermal conductivity enhances droplet surface temperature and evaporation flux, thereby monotonically increasing evaporation rate until it approaches the rate of the evaporative cooling model. Conversely, increasing substrate thickness lengthens the heat transfer path, reducing heat conducted to the solid–liquid interface and decreasing evaporation rate. Changes in substrate lower-surface temperature significantly affect evaporation rate, but HCs remains nearly unaffected. The concept of equivalent substrates is proposed and verified through dimensionless analysis and simulations. It is found that different combinations of substrate thickness and thermal conductivity exhibit consistent effects on droplet evaporation, with minimal relative errors in evaporation rate and total heat transfer at the solid–liquid interface. This confirms the existence of the equivalent substrate phenomenon. Additionally, the effects of droplet properties, such as contact angle and evaporative cooling coefficient (Ec), on the equivalent substrate phenomenon are explored, revealing negligible impacts. These findings provide theoretical guidance for optimizing droplet evaporation processes in practical applications, such as micro/nanoscale thermal management systems.

1. Introduction

Droplet evaporation, a ubiquitous physical phenomenon, is widely observed in nature and industrial applications such as heat-pipe cooling [1,2], inkjet printing [3,4,5], additive manufacturing [6], and biomedical technologies [7,8]. In recent years, the evaporation of sessile droplets on substrates has garnered intensive attention from researchers [9,10,11,12,13,14,15].

During droplet evaporation, the thermal conductivity of the substrate plays a pivotal role. The role of substrate properties was systematically examined by Ristenpart [16], who identified the substrate-to-liquid thermal conductivity ratio as a critical parameter governing interfacial temperature distributions. Zhang [17] subsequently proposed a temperature distribution phase diagram as a function of contact angle and substrate thickness. Girard [18] investigated the impact of substrate heating on droplet evaporation and quantified the enhancement of evaporation due to substrate heating. To address transitions between isothermal and non-isothermal conditions, Girard [19,20] introduced the dimensionless parameter SB as a threshold for identifying such transitions. Xu [21] further unified vapor diffusion and thermal conduction fields into a simplified heat conduction framework, introducing the evaporative cooling number (Ec) to quantify cooling intensity. Wang [22,23] conducted an integrated analysis of the combined effects of substrate thermal conductivity and evaporative cooling on droplet evaporation. Their study elucidated the competitive relationship between heat transfer within the droplet and evaporative cooling mechanisms. Yu [24] advanced this work by integrating substrate thermal conductivity effects, experimentally confirming three distinct temperature patterns on the droplet surface using infrared thermography. Azzam [25] investigated the evaporation of two droplets on a heated substrate, examined how base temperature affects the evaporation of multiple droplets, and talked about its application in heat pipes. Extensive studies have also explored the effects of substrate thermal conductivity [26,27], roughness [28,29], geometry [30,31,32], humidity [33], and wettability [34,35,36] on droplet evaporation. While the evaporative cooling effect has received significant attention, the interplay between environmental gas, substrate heat conduction, and evaporative cooling, as well as their combined influence on droplet evaporation mechanisms, remains unclear.

This study systematically investigates the influence of substrate thermal conduction characteristics on droplet evaporation behavior. Based on a mathematical model, we analyze the effects of substrate thermal conductivity and thickness on evaporation rate, surface temperature, and evaporation flux. A dimensionless relative evaporation rate, HCs, is introduced to characterize the impact of substrate heat conduction on droplet evaporation. Furthermore, equivalent conditions for substrate thermal conductivity and thickness are derived, providing theoretical guidance for optimizing droplet evaporation processes in practical applications, such as selecting substrate materials with appropriate thermal properties for efficient thermal management in micro/nano-scale systems.

2. Theoretical Model

In this study, we analyze a small, pinned liquid droplet situated on a flat surface. The droplet’s shape is primarily influenced by the Bond number and the capillary number [37]. The Bond number is defined as

Bo = ρgRh₀/σ

(1)

where ρ is the liquid density, g is the gravitational constant, R is the contact line radius, h₀ is the droplet initial height, and σ is the liquid surface tension, governs the balance of the surface tension and the gravitational force on the droplet shape. The capillary number is defined as

Ca = µu_r/σ

(2)

where µ is the liquid viscosity and u_r is the average radial velocity, defines the ratio of viscous to capillary forces. Owing to its small size and slow evaporation rate, the droplet typically exhibits low Bond and capillary numbers, allowing its shape to be approximated as a spherical cap (see Figure 1a). Due to the axisymmetric configuration, a cylindrical coordinate system with radial coordinate r and axial coordinate z is used here. The height of the droplet can be expressed as

h(r) = (R²/sin²θ − r²)^1/2 − R/tanθ

(3)

where θ is the contact angle and R is the contact line radius. While Marangoni flow alters the droplet’s surface temperature, its effect on the evaporation rate is insignificant [38,39]. Thus, the Marangoni flow in the droplet is neglected in the present model. In the context of quasi-steady, diffusion-limited evaporation [37,40], the concentration of vapor within the air, denoted as c, along with the temperature distribution T_S within the solid phase and the temperature distribution T_L within the liquid phase, all adhere to Laplace’s equation:

∇² c = 0, for z ≥ h(r),

(4)

∇² T_L = 0, for 0 ≤ z ≤ h(r),

(5)

∇² T_S = 0, for −h_s ≤ z ≤ 0,

(6)

The boundary conditions are:

c = Hc₀, for z = ∞, r = ∞,

(7)

c = c₀ + b (T_L − T₀), k_L ∇ T_L • n = H_L D ∇ c • n, for z = h(r), r ≤ R,

(8)

T_L = T_S, k_L ∇ T_L • n = k_S D ∇ c • n, for z = 0, r ≤ R,

(9)

∂ c/∂ n = 0, ∂ T_s/∂ n = 0, for z = 0, r > R,

(10)

∂ T_s/∂ n = 0, for r = ∞, −h_s ≤ z ≤ 0,

(11)

T_s = T_a, for z = −h_s,

(12)

where D is the diffusion coefficient of vapor in the atmosphere, k_L and k_S are the thermal conductivities of the liquid and the substrate, T_a is the temperature of the lower surface of the substrate, h_s is the thickness of the substrate, H is the relative humidity of the ambient air, T₀ is the room temperature, T_a is the temperature at the lower surface of the substrate, H_L is the latent heat of evaporation, c₀ is saturation vapor concentration at T₀, and n is the normal vector.

By using the combined field approach induced by Xu and Ma [21], the equations can be written in a dimensionless form as

∇² T₁ = 0, for z ≥ h(r),

(13)

∇² T₂ = 0, for 0 ≤ z ≤ h(r),

(14)

∇² T₃ = 0, for −h_s ≤ z ≤ 0,

(15)

The boundary conditions are

T₁ = −1, for z = ∞, r = ∞,

(16)

Ec (∂T₁/∂ n) = (∂T₂/∂ n), for z = h(r), r ≤ R,

(17)

T₂ = T₃, ∂T₂/∂ n = k_RS (∂T₃/∂ n), for z = 0, r ≤ R,

(18)

∂ T₁/∂ n = 0, ∂ T₃/∂ n = 0, for z = 0, r > R,

(19)

∂ T₃/∂ n = 0, for r = ∞, −h_s ≤ z ≤ 0,

(20)

T₃ = T_a1, for z = −h_s,

(21)

where r₁ = r/R, z₁ = z/R, h₁ (r₁) = h (r)/R, T₁ = (c − c₀)/[c₀ (1 − H)], T₂ = b (T_s − T₀)/[c₀ (1 − H)], T₃ = b (T_s − T₀)/[c₀ (1 − H)], T_a1 = b (T_a − T₀)/[c₀ (1 − H)], c₀ = c_sat(T₀), b = dc_sat(T)/dt, Ec = H_LDb/k_L, h_R = h_s/r, k_RS = k_S/k_L. After the nondimensionalization, the dimensionless contact line radius of the droplet is 1. Thus, in this dimensionless model, the effect of droplet size on evaporation is negligible. In the meantime, the effect of ambient humidity on droplet evaporation is also negligible.

3. Finite Element Method

To investigate the heat and mass transfer during the droplet evaporation, the dimensionless Equations (13)–(21) are numerically solved by using the commercial software COMSOL 5.0 Multiphysics. The liquids and the substrates used here and their physical dimensionless parameters for each droplet/substrate pair are listed in

Table 1. Dimensionless parameters for droplet/substrate pair.

Droplet	Substrate	kRS	Ec
Water	PS	0.13245	0.10986
	PU	0.29801	0.10986
	PTFE	0.41391	0.10986
	Glass	1.5927	0.10986
	Ti	36.258	0.10986
	Si	246.69	0.10986
	Al	392.38	0.10986

. To obtain an accurate solution, the mesh in the model is refined continuously until the criterion

ε = |J_i+1 − J_i|/J_i < 0.005

(22)

is satisfied, where J_i is the dimensionless evaporation rate J at the computation domain for the nth finite element mesh refinement. The meshing details are showed in

Table 2. Mesh refinement details.

Refinements	Numbers of Elements
1	219,745
2	791,510
3	1,127,187
4	1,788,891

.

4. Results and Discussion

4.1. The Influence of Substrate Heat Transfer on Droplet Evaporation

In this study, the impact of substrate thermal conduction on droplet evaporation behavior was investigated, building on the mathematical model established in previous works [21]. Water droplets with a 30° contact angle were used as a model to study the variations in dimensionless surface temperature, evaporation flux, and evaporation rate across substrates of differing thermal conductivities, with results presented in Figure 2. Across all substrates, the droplet’s surface temperature gradually increased from the center to the contact line, then rose sharply near the contact line, aligning with the substrate’s upper surface temperature. Higher relative substrate thermal conductivity (k_RS) was correlated with elevated droplet surface temperatures and evaporation fluxes, which also increased monotonically from the center to the contact line. The surface temperature distribution thus served as a key indicator of evaporation flux and rate.

In the model, the liquid–gas interface acted as a low-temperature heat source due to latent heat, while the substrate’s lower surface functioned as a high-temperature heat source. As k_RS increased, heat transfer from the substrate’s lower surface to the solid–liquid interface accelerated, raising the interface temperature and leading to monotonic increases in droplet surface temperature and evaporation flux. Consequently, the evaporation rate increased until it approached that of the evaporative cooling model (Figure 2c). To quantify the substrate’s thermal conduction effect, a dimensionless relative evaporation rate (HC_s) was defined as the ratio of the evaporation rate in the substrate heat conduction model (J) to that in the constant-temperature substrate evaporative cooling model (J_Ec). A smaller HC_s indicated less heat transfer from the substrate to the droplet and a lower evaporation rate, whereas an HC_s closer to 1 signified greater heat transfer, aligning the substrate’s thermal conduction with the constant-temperature substrate evaporative cooling model.

To further explore the effect of substrate thickness on droplet evaporation through simulations of water droplet evaporation on PS, PU, PTFE, glass, Ti, Si, and aluminum substrates with varying thicknesses and a constant contact angle of 30°. PS, PU, PTFE, and glass, characterized by their extremely low relative thermal conductivities, exhibited low heat transfer rates from the substrate’s lower surface to the solid–liquid interface. As the substrate thickness increased, the heat transfer path lengthened, causing a delay in heat delivery to the solid–liquid interface. This inability to compensate for the heat loss due to evaporation resulted in a rapid decrease in droplet surface temperature, as illustrated in Figure 3a. According to Fick’s Law, the heat transfer quantity is inversely proportional to the length of the heat transfer path, leading to a corresponding decrease in the droplet’s evaporation rate. Under these conditions, the relative evaporation rate (HC_s) showed a significant decrease with an increase in the relative substrate thickness (h_R).

In contrast, aluminum substrates, with a high relative thermal conductivity of 392.38, demonstrated highly efficient heat transfer. The droplet’s surface temperature remained largely unaffected by changes in substrate thickness, as shown in Figure 3c. For these substrates, changes in relative substrate thickness (h_R) had minimal effect on HCs, with the relative evaporation rate (HCs) consistently approaching 1.

4.2. Equivalent Substrates

To investigate the combined effects of substrate thermal conductivity and thickness on droplet evaporation, this study selected a set of substrates with varying relative thicknesses (h_R) and adjusted their relative thermal conductivity (k_RS) to achieve the same dimensionless evaporation rate for water droplets with an initial contact angle of 30°. As shown in Figure 4a, for droplets with the same dimensionless evaporation rate, higher substrate relative thermal conductivity required greater relative thickness. The trends for different dimensionless evaporation rates were consistent. Using a substrate with h_R0.1 = 0.1 as the reference (denoted as k_RS0.1), normalization of k_RS for different evaporation rates was performed (Figure 4b). The overlapping of the normalized curves indicates the existence of a functional relationship between k_RS and h_R that maintains a constant evaporation rate. By fitting substrates in Figure 4b, the relationship between the substrate thermal conductivity and substrate thickness required to achieve identical droplet evaporation rates can be approximated by the following equation:

k_RS/k_RS0.1 = 6.9183 [1 − exp (−0.14993 h_R/h_R0.1)]

(23)

The equivalent relationship between two distinct substrates that satisfy the equation, with thermal conductivities k_RS1 and k_RS2 with corresponding substrate thicknesses h_R1 and h_R2, can be expressed as follows:

k_RS1/k_RS2 = [1 − exp (−1.4993 h_R1)]/[1 − exp (−1.4993 h_R2)]

(24)

This relationship was defined as “equivalent substrates”. This means that the substrates of different materials will show identical evaporation kinetics if their thickness and thermal conductivity are balanced.

4.3. Verification of Equivalent Substrates

The comparison of evaporation fluxes on the surfaces of droplets placed on a set of equivalent substrates, as depicted in Figure 4, is presented in Figure 5. The evaporation flux distributions on the surfaces of droplets across different substrates are found to be nearly identical, indicating that the evaporation characteristics of droplets on a set of equivalent substrates are essentially uniform.

The comparison of evaporation rates between the equivalent substrates and the reference substrate is shown in Figure 5. To better characterize the equivalence between the equivalent substrates and the reference substrate, a relative error was introduced, defined as the ratio of the absolute difference between the evaporation rate on the reference substrate and that on the equivalent substrate to the evaporation rate on the reference substrate, or the absolute difference between total heat transfer on the reference substrate and that on the equivalent substrate to total heat transfer on the reference substrate:

ε = |Q_s − Q_s′|/Q_s

(25)

To further investigate the properties of equivalent substrates, this study calculates the evaporation rates of droplets and the total heat transfer at the solid–liquid interface for a set of equivalent substrates, as shown in Figure 5. The relative errors in evaporation rates between the equivalent substrates and the reference substrate are found to be less than 0.0001, while the relative errors in total heat transfer at the solid–liquid interface are less than 0.02, even as the substrate thickness and relative thermal conductivity vary. Based on these results, it can be concluded that the thermal characteristics of the entire system during droplet evaporation on this set of equivalent substrates are similar. Therefore, the thickness and thermal conductivity of these substrates can be considered thermally equivalent.

Figure 5. Evaporation of a droplet on a set of the equivalent substrates in

Table 3. The thickness and the relative thermal conductivity of an equivalent substrate.

hR	kRS	kRS	kRS
0.05	0.06142	0.52417	0.36541
0.1	0.01	1	0.69712
0.15	0.016897	1.4345	1
0.3	0.029963	2.5366	1.7693
0.45	0.040213	3.4028	2.3722
1	0.06308	5.2318	3.7239
1.5	0.07349	6.2244	4.3391

. (a) Evaporation flux distribution of the equivalent substrates in column 1 of Table 2, (b) evaporation flux distribution of the equivalent substrates in column 2 of Table 3, (c) dimensionless evaporation rate of the equivalent substrates in column 1 of Table 3, (d) heat transfer at the solid–liquid interface of the equivalent substrates in column 2 of Table 3, (e) heat transfer at the solid–liquid interface of the equivalent substrates in column 1 of Table 3, and (f) heat transfer at the solid–liquid interface of the equivalent substrates in column 2 of Table 3.

Figure 5. Evaporation of a droplet on a set of the equivalent substrates in Table 3. (a) Evaporation flux distribution of the equivalent substrates in column 1 of Table 2, (b) evaporation flux distribution of the equivalent substrates in column 2 of Table 3, (c) dimensionless evaporation rate of the equivalent substrates in column 1 of Table 3, (d) heat transfer at the solid–liquid interface of the equivalent substrates in column 2 of Table 3, (e) heat transfer at the solid–liquid interface of the equivalent substrates in column 1 of Table 3, and (f) heat transfer at the solid–liquid interface of the equivalent substrates in column 2 of Table 3.

To verify the applicability of the equivalent substrate phenomenon to different droplet materials, Figure 6 shows the evaporation characteristics of water, methanol, and acetone droplets with an initial contact angle of 30° on a set of equivalent substrates. The result indicates that different kinds of liquid droplets suit the equivalent substrate phenomenon.

Figure 6. Evaporation of a droplet on a set of the equivalent substrates in

Table 4. The parameters of different droplets.

Parameters	Unit	Water	Methanol	Acetone
Density	kg·m−3	998	790	788
Thermal conductivity	W·m−1·K−1	0.604	0.203	0.161
Heat capacity	J·kg−1·K−1	4180	2530	2170
Diffusion Coefficient	m2·s−1	2.44 × 10−5	1.5 × 10−5	1.06 × 10−5
Specific heat	m2·s−2	2.45 × 106	1.2 × 106	5.49 × 105
Ec	1	0.10986	0.83970	1.0265

. (a) Evaporation flux distribution of methanol evaporating on the equivalent substrates in column 3 of Table 4, (b) evaporation flux distribution of water evaporating on the equivalent substrates in column 3 of Table 4, (c) dimensionless evaporation rate of acetone evaporating on the equivalent substrates in column 3 of Table 4, (d) dimensionless evaporation rate of water evaporating on the equivalent substrates in column 3 of Table 4, (e) dimensionless evaporation rate of methanol evaporating on the equivalent substrates in column 3 of Table 4, (f) dimensionless evaporation rate of acetone evaporating on the equivalent substrates in column 3 of Table 4, (g) heat transfer at the solid–liquid interface of water evaporating on the equivalent substrates in column 3 of Table 4, (h) heat transfer at the solid–liquid interface of methanol evaporating on the equivalent substrates in column 3 of Table 4, and (i) heat transfer at the solid–liquid interface of acetone evaporating on the equivalent substrates in column 3 of Table 4.

Figure 6. Evaporation of a droplet on a set of the equivalent substrates in Table 4. (a) Evaporation flux distribution of methanol evaporating on the equivalent substrates in column 3 of Table 4, (b) evaporation flux distribution of water evaporating on the equivalent substrates in column 3 of Table 4, (c) dimensionless evaporation rate of acetone evaporating on the equivalent substrates in column 3 of Table 4, (d) dimensionless evaporation rate of water evaporating on the equivalent substrates in column 3 of Table 4, (e) dimensionless evaporation rate of methanol evaporating on the equivalent substrates in column 3 of Table 4, (f) dimensionless evaporation rate of acetone evaporating on the equivalent substrates in column 3 of Table 4, (g) heat transfer at the solid–liquid interface of water evaporating on the equivalent substrates in column 3 of Table 4, (h) heat transfer at the solid–liquid interface of methanol evaporating on the equivalent substrates in column 3 of Table 4, and (i) heat transfer at the solid–liquid interface of acetone evaporating on the equivalent substrates in column 3 of Table 4.

4.4. The Influence of Droplet Properties on Equivalent Substrates

To better understand the equivalent substrate phenomenon, this study further examined the effects of contact angles (5–90°) (see Figure 7). It is acknowledged that the contact angle significantly impacts the droplet evaporation rate [41]. An increased contact angle lengthens the heat transfer path from the solid–liquid interface to the liquid–gas interface, thereby reducing the efficiency of heat transfer from the substrate to the liquid–gas interface. Moreover, for droplets with the same contact line radius but different contact angles, the surface area of the liquid–gas interface also varies. However, in this model, droplets with different contact angles all meet the equivalent substrate condition. This is because within a set of equivalent substrates, there is a balance between the substrate thermal conductivity and substrate thickness, resulting in nearly identical heat transfer from the substrate lower surface to the solid–liquid interface.

The droplets with different evaporative cooling coefficients (Ec = 0.01–5) on droplet evaporation on a set of equivalent substrates are also numerically studied (see Figure 8). Ec is determined by four factors: the latent heat of vaporization, the diffusion coefficient in air, the temperature coefficient of saturated vapor concentration in air, and thermal conductivity. Thus, the variation in the evaporation cooling coefficient Ec can represent both the changes in liquid material properties and ambient pressure. While these parameters significantly influence droplet evaporation, the heat transfer from the substrate’s lower surface to the solid–liquid interface remains consistent across equivalent substrates. Consequently, the evaporation rate of the same droplet on a given set of equivalent substrates is essentially uniform, indicating that the properties of the droplet have negligible effects on the equivalent substrate phenomenon.

5. Conclusions

In this study, the coupled relationship between the solid-phase temperature field and droplet evaporation was investigated, with a systematic exploration of the effects of substrate thermal conduction characteristics on droplet evaporation behavior. Based on the mathematical model established in the preceding sections, the impacts of substrate thermal conductivity and thickness on the evaporation rate, surface temperature, and evaporation flux of the droplet were analyzed. A dimensionless relative evaporation rate, HCs, was introduced to more accurately characterize the influence of substrate thermal conduction on droplet evaporation. The findings revealed that an increase in substrate thermal conductivity enhances the surface temperature and evaporation flux of the droplet, thereby monotonically increasing the evaporation rate until it approaches the evaporation rate of the evaporative cooling model. Conversely, an increase in substrate thickness extends the heat transfer path, reducing the heat ultimately conducted to the solid–liquid interface and thereby decreasing the evaporation rate of the droplet.

Furthermore, the concept of equivalent substrates was proposed, wherein distinct combinations of substrate thickness and thermal conductivity yield nearly identical evaporation behaviors. Through dimensionless analysis and numerical simulations, the consistency of evaporation rates and heat transfer at the solid–liquid interface across equivalent substrates was validated, with relative errors below 0.01% for evaporation rates and 2% for interfacial heat fluxes. This confirms the existence of the equivalent substrate phenomenon.

Finally, the influence of droplet-specific properties, such as contact angle and evaporative cooling coefficient Ec, on equivalent substrates was examined. The results demonstrate the negligible effects of these parameters on the equivalence relationship, underscoring the robustness of the proposed concept. These insights provide a foundational framework for optimizing substrate design in applications requiring precise thermal management, such as microelectronics cooling and advanced manufacturing processes.