On the applicability of continuum scale models for ultrafast nanoscale liquid-vapor phase change

Highlights

• Continuum simulations can accurately model liquid-vapor phase change processes at nanoscopic length and time scales.

• Schrage equation, used as a constitutive law, is required to describe interfacial mass transfer during phase change.

• Prescribing temperature jump at liquid-vapor interface is necessary to obtain correct temperature profile.

Abstract

Continuum methods are efficient in modeling multi-phase flow at large time and length scales, however, their applicability to nanoscale systems and processes is questionable. When mean free path and average time between atomic collisions are comparable to the characteristic length and time scales of interest, the continuum hypothesis approaches its spatial and temporal limit. Here we discuss the implications of modeling such a limiting problem involving liquid-vapor phase change using continuum equations of mass, momentum, and energy conservation. Our results indicate that, continuum conservation laws can correctly represent the dynamics of the specific problem of interest provided appropriate constitutive relations are used at liquid-vapor interfaces. We show that with the Schrage relation for phase change rates and a physically motivated expression for temperature jump, interfacial phenomena can be described quite accurately.

Introduction

The advent of micro/nano technology and miniaturization of traditional electronic devices have led to an increased interest in understanding transport processes at the micro/nano scales. Liquid-vapor phase change (evaporation/condensation) is one such process which has been extensively used in electronics cooling (Hanks et al., 2018), spray cooling (Aamir et al., 2014), inkjet printing (Park and Moon, 2006), DNA chip manufacturing (Dugas et al., 2005), water desalination (Al-Shammiri and Safar, 1999; Aly and El-Figi, 2003), agriculture (Valiantzas, 2006), self-assembly of colloidal particles (Sefiane, 2010) etc. Utilizing evaporation for powering engineered systems (Cavusoglu et al., 2017; Chen et al. (2015), solar steam generation Yang et al., 2017) and electricity generation (Xue et al., 2017) are a few promising applications which are being currently researched. Some of the aforementioned applications are so complicated that simple models cannot describe the physics accurately; thus numerical simulations are often necessary.

Molecular Dynamics (Liang et al., 2017; Liang and Keblinski, 2018), Direct Simulation Monte Carlo (Jafari et al., 2019), and Continuum scale simulation techniques (Zhang et al., 2019; Chandra et al., 2019; Kharangate and Mudawar, 2017) are some popular methods used to analyze liquid-vapor phase change processes across varied length and time scales. At the nano/micro meter length scale, the applicability of continuum techniques is often questionable. Numerical methods based on the continuum hypothesis (Finite element based methods) are often computationally less expensive than atomistic methods as the number of degrees of freedom are much smaller when a continuum description of matter is used. Moreover, finite element based methods are deterministic and do not require statistical averaging like most methods which utilize an atomic description of matter. Thus, continuum scale numerical methods are most suitable for rapid prototyping and analysis of small scale devices.

Majority of continuum numerical methods which aim to model liquid-vapor phase change processes, assume continuity of temperature at phase interfaces (Kharangate and Mudawar, 2017; Gibou et al., 2007) – liquid-vapor interfaces are assumed to be in local thermal equilibrium. In other words, value of the interfacial temperature is usually determined by the saturation pressure of the liquid (Welch and Wilson, 2000). However, in certain small scale applications (Hardt and Wondra, 2008; Stierle et al., 2020), such as micro-scale droplet evaporation (Cao et al., 2011; Hardt and Wondra, 2008; Rana et al., 2019), the assumption of temperature continuity at the liquid vapor interfaces might lead to non-negligible errors. Evaporation experiments in low pressure environments have also shown that there is a temperature discontinuity (temperature jump) at the liquid-vapor interfaces (Fang and Ward, 1999; Badam et al., 2007). In addition to this jump, a prescription of phase change rates is also essential for an accurate description of interfacial phase change phenomena. Hertz and Knudsen (Persad and Ward, 2016) developed one of the first models for describing evaporation/condensation rates in near equilibrium conditions. This relationship is popularly known as the Hertz-Knudsen relation. Later on, Schrage (Schrage, 1953) formulated the Hertz-Knudsen-Schrage relationship by accounting for non-zero macroscopic velocity of the vapor and evaporation/condensation probabilities. Although the validity of Schrage relations is still a subject of active research (Polikarpov et al., 2019; Chandra and Keblinski, 2020; Liang et al., 2017) and debate, we will use this model in our continuum simulations and discuss its applicability with respect to a micro-scale phase change problem.

When mean free path is comparable to the characteristic length scale (0.1$<$Kn$<$10), consideration of both diffusive and ballistic transport mechanisms are important (Jafari et al., 2019), and the continuum hypothesis approaches its spatial limit. This flow regime is regarded as transitional flow, and is typically studied using the Boltzmann transport equation (BTE). In certain micro/nano scale phase change processes (Liang et al., 2017; Liang and Keblinski, 2018; Park et al., 2017) or in the Knudsen layer (Bird and Liang, 2019), characteristic length scale is often comparable to the mean free path of the molecules in vapor phase; thus, these are transitional flow problems. In this paper, we discuss the implications of modeling a liquid-vapor phase change problem in the transitional regime using continuum equations of mass, momentum, and energy conservation. The characteristic timescale of our problem is an order of magnitude larger than the average time between atomic collisions; thus, we are near the temporal limit of the continuum hypothesis wherein the condition of local thermal equilibrium might not hold. We show that, nanoscopic systems in the transitional regime can be accurately modeled using the continuum conservation laws provided appropriate constitutive relations are used at liquid-vapor interfaces. The necessity of imposing thermal boundary resistances (TBR) at solid-liquid interfaces, while modeling nanoscopic systems, is also discussed. Remainder of this manuscript is structured as follows. In the next section we present details of our numerical method. Description of our model problem is given in Section 3. Results are discussed in Section 4. Finally, in Section 5 we summarize our findings.