## 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 [PERSON] [16], who identified the substrate-to-liquid thermal conductivity ratio as a critical parameter governing interfacial temperature distributions. [PERSON] [17] subsequently proposed a temperature distribution phase diagram as a function of contact angle and substrate thickness. [PERSON] [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, [PERSON] [19, 20] introduced thedimensionless parameter \(SB\) as a threshold for identifying such transitions. [PERSON] [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. [PERSON] [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. [PERSON] [24] advanced this work by integrating substrate thermal conductivity effects, experimentally confirming three distinct temperature patterns on the droplet surface using infrared thermography. [PERSON] [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.