# Sublimation

When the pressure and temperature of ice are above the triple point pressure and temperature of water is heated, melting occurs as discussed in Chapter 6. However, when the ice is exposed to moist air with a partial pressure of water below its triple point pressure, heating of the ice will result in a phase change from ice directly to vapor without first going through the liquid phase. Spacecrafts and space suits can reject heat by sublimating ice into the vacuum of space. Another application for sublimation of ice is the preparation of specimens using freeze-drying for a scanning electronic microscope (SEM) or a transmission electronic microscope (TEM). This type of phase change is referred to as sublimation. The opposite process is deposition, which describes the process of vapor changing directly to solid without going through the condensation and freezing. The phase-change processes related to solids can be illustrated by a phase diagram in Fig. 1. Sublimation and deposition will be the subjects of this chapter. When a subcooled solid is exposed to its superheated vapor, as shown in Fig. 2(a), the vapor phase temperature is above the temperature of the solid-vapor interface and the temperature in the solid is below the interfacial temperature. The boundary condition at the solid-vapor interface is

${{k}_{s}}\frac{\partial {{T}_{s}}}{\partial x}-{{h}_{\delta }}({{T}_{\infty }}-{{T}_{\delta }})={{\rho }_{s}}{{h}_{sv}}\frac{d\delta }{dt} \qquad \qquad(1)$

where hδ is the convective heat transfer coefficient at the solid-vapor interface, hsv is the latent heat of sublimation, and δ is the thickness of the sublimable or deposited material. The interfacial velocity dδ / dt in eq. (1) can be either positive or negative, depending on the direction of the overall heat flux at the interface. While a negative interfacial velocity signifies sublimation, a positive interfacial velocity signifies deposition. When the vapor phase is superheated, as shown in Fig. 2(a), the solid-vapor interface is usually smooth and stable.

Figure 1: Phase diagram for solid-liquid and solid-vapor phase change.

Figure 2: Temperature distribution in sublimation and deposition.

In another possible scenario, as shown in Fig. 2 (b), the solid temperature is above the interfacial temperature and the vapor phase is supercooled. The interfacial energy balance for this case can still be described by eq. (1). Depending on the degrees of superheat in the solid phase and supercooling in the vapor phase [the relative magnitude of the first and second terms in eq. (1)], both sublimation and deposition are possible. During sublimation, a smooth and stable interface can be obtained. During deposition, on the other hand, the interface is dendritic and not stable, because supercooled vapor is not stable. The solid formed by deposition of supercooled vapor has a porous structure. During sublimation or deposition, the latent heat of sublimation can be supplied from or absorbed by either the solid phase or the vapor phase, depending on the temperature distributions in both phases.

Naphthalene sublimation is also a technique whereby a heat transfer coefficient can be obtained through the measurement of a mass transfer coefficient and the analogy between heat and mass transfer (Eckert and Goldstein, 1976). The significant advantages of this method include its high accuracy and the simplicity of the experimental apparatus. In addition, the local heat transfer coefficient can be obtained by measuring the local sublimed depth of the specimen.

Vapor deposition, which finds applications in coating and thermal manufacturing processes, is classified into two broad categories (Seshau, 2001): Physical Vapor Deposition (PVD) and Chemical Vapor Deposition (CVD). PVD operates at a very low pressure and transports the species generated by one of two means: (1) evaporation, or (2) bombarding the target materials to the substrate through free molecular flow or transition flow. CVD, on the other hand, is a process in which material is formed on a substrate by chemical reaction of gaseous precursors using activation energy. The deposited film thickness can range from a few nanometers, as applied to optical coating, to tens of microns, as applied to wear-resistance coating (Jenson et al., 1991). Conventional CVD has been extensively investigated by many researchers and a detailed literature review is given by Mahajan (1996). In a pyrolytic CVD process, the entire substrate is heated and vapor deposition occurs over the whole substrate. When a laser beam is used to heat the substrate, only a very small spot on the substrate is heated by the laser beam and vapor deposition occurs only on the heated spot. In this case the activation energy is provided by the laser beam and it is therefore referred to as Laser Chemical Vapor Deposition (LCVD; Kwok and Chiu, 2003). LCVD can also be based on chemical reactions initiated photolytically, which involves tuning the laser to an electrical or vibrational level of the gas (Bauerle, 1996). The irradiated material decomposes, and the products deposit on the cooler substrate to form the solid film (Mazumder and Kar, 1995).

Section 7.2 presents analytical solutions of sublimation over a flat plate in parallel flow and inside a tube; the problems are treated as a conjugated heat and mass transfer problem. Section 7.2 also includes a detailed analysis of a sublimation process with chemical reaction. Section 7.3 presents an in-depth discussion of CVD, including various CVD configurations, governing equations, transport properties and several selected applications.