# Theoretical and Experimental Analysis for Cleaning Ice Cores from EtisolTM 140 Drill Liquid
## 2 Theoretical Section and Calculations  
### General Theoretical Aspects  
The state diagram of water shows that ice and snow do not evaporate (sublimate) below the freezing temperature at standard pressure. Indeed, the direct solid-vapor transition can happen only below 611.73 Pa (which is 6 \(\times\) 10\({}^{-3}\) atm) at temperatures below 0.01 \({}^{\circ}\)C. Yet, it is a common observation that ice and snow can disappear rapidly in external environments especially when exposed to wind and sunlight. An analysis of ice sublimation was presented by [PERSON] et al. [6] and provides a quantitative description for the diffusion of ice molecules at the ice-air boundaries. Other theoretical analyses are available on the analogous problem of water evaporation in the coexistence of liquid and gas phases [7; 8], an issue that was investigated for long [9; 10; 11]. The first pioneering studies of [PERSON] on the evaporation of mercury [9], further developed by [PERSON] [10] set the basis for understanding the physics of evaporation (and sublimation). The partial pressure of water vapor over water (or ice) plays a major role as well as the relative ambient humidity, determining a dynamic equilibrium between the rate of evaporation (or sublimation) and the rate of condensation. In the framework of the classical kinetic theory, the calculations are based on a statistical approach, considering a Maxwellian distribution function for the molecular velocities. The Hertz-Knudsen formula for the net evaporation rate, corrected for mass transportation across the interface in non-equilibrium conditions [12], is expressed by the following formula:  
\[\mathcal{M}=\frac{2}{2-\sigma_{c}}\ \sqrt{\frac{m}{2\pi k_{B}}}\left(\sigma_{e} \frac{P_{sat}}{\sqrt{T_{L}}}-\sigma_{c}\frac{P_{V}}{\sqrt{T_{V}}}\right) \tag{1}\]  
where \(M\) is the net mass flux, \(m\) is the molecular mass, \(k_{B}\) is the Boltzmann constant, \(P_{sat}\) and \(P_{V}\) are the saturation vapour pressure and the partial pressure of the vapor phase, respectively, \(T_{L}\) and \(T_{V}\) are the temperatures of the liquid and the vapor. Further developments were proposed by [PERSON] [13], [PERSON] [14], and [PERSON] [15; 16], but Equation (1)represents the concept well. A major difficulty is knowing the values of the evaporation and condensation coefficients and the conditions that exist at the interface. These coefficients must be determined experimentally, but values obtained in several experiments show large discrepancies. For experimental and theoretical purposes, \(\sigma_{e}\) and \(\sigma_{c}\) are often assumed as equal (\(\sigma_{e}=\sigma_{c}\)). An additional critical element is the effective temperature in the liquid/vapor phase at the interface, where mass transfer takes place. Indeed, \(P_{sat}\) and \(P_{V}\) are dependent on the temperature. Moreover, evaporative cooling reduces the local liquid temperature below that in the bulk, with a minimum temperature observed at the liquid-vapor interface, where a discontinuity can be predicted. Direct experimental measurements have demonstrated that the temperature gap at the interface can be as large as 4 \({}^{\circ}\)C [17]. The equations can be directly transferred to the ice-vapor system for studying the case of sublimation. A different approach to describe mass-transport phenomena is based on the diffusion process. The paper by [PERSON] et al. [6] is straightforward for ice sublimation. The diffusion equation at the interface allows calculating the local evaporative flux. It is shown that this equation is a Laplace equation; therefore, there is a strong analogy with electrostatics. The strong curvature dependence is analogous to the known electrostatic tip effect, indicating that the flux of sublimated material is maximum at the edges. Few mathematical steps lead to the mass transportation equation:  
\[\mathcal{M}=-H\left(\frac{P_{sat}}{T_{ice}}-\frac{P_{V}}{T_{V}}\right) \tag{2}\]  
with \(H\) a mass transfer coefficient and \(T_{ice}\) the ice temperature.