Supercooled vapors

The Kelvin equation helps explain an assortment of supersaturation phenomena. All of these —supercooled vapors, supersaturated solutions, supercooled melts —involve the onset of phase separation. In each case the difficulty is the nucleation of the new phase Chemists are familiar with the use of seed crystals and the effectiveness of foreign nuclei to initiate the formation of the second phase. 

M. von Smoluchowski (1903) ° studied this problem of "irregularity from several angles in connection with the following problem Consider the deviations from the "most probable spacial distribution which the molecules of a gas in thermal equilibrium show at various times. What is the effect of these deviations on the equation of state and the F-value Smoluchowski in this paper referred to the relationship of this problem to the stability limit for a superheated fluid and supercooled vapor, which he investigated more closely in a later paper (1907).s ... 

PRIMARY NUCLEATION. In scientific usage, nucleation refers to the birth of very small bodies of a new phase within a supersaturated homogenous existing phase. Basically, the phenomenon of nucleation is the same for crystallization from solution, crystallization from a melt, condensation of fog drops in a supercooled vapor, and generation of bubbles in a superheated liquid. In all instances, nucleation is a consequence of rapid local fluctuations on a molecular scale in a homogenous phase that is in a state of metastable equilibrium. The basic phenomenon is called homogeneous nucleation, which is further restricted to the formation of new particles within a phase iminfluenced in any way by solids of... 

Figure 2.2. The van der Waals-type phase diagram for one mole of argon. C = 150 K is the critical point, TP is the triple point line of the coexistence between solid, liquid and vapor. The upper shaded area represents the liquid-vapor coexistence, while the lower that of solid-vapor. Vapor exists on the right hand side of the shaded areas, while liquid and solids on their left. The supercritical region is that above the critical point, where (at high pressures) the vapor density is comparable to that of liquid. Supercooled vapor is also indicated by a series of solid points. After [Flowers and Mendoza, 1970].

He also says that metastable states have a good deal in common with constrained states, but the only example he gives is of a supercooled vapor phase which requires no physical constraints, but rather some good luck and/or a sense of humor to be treated as an equilibrium state. He does not distinguish between real states and thermodynamic states. In addition, his statements about the applicability of the fundamental equation do not take into account all possibilities. 

Note that the rate expression can be written as the product of three terms. The first one [/(n )] is the product of p times the surface area of the critical nucleus, and represents the frequency of arrival of single molecules to the critical nucleus. The third term, N(n ), is the equilibrium concentration of critical nuclei. Therefore, the second term, Z, can be interpreted as a factor that corrects for the fact that the concentration of critical nuclei differs from the equilibrium value. This term is frequently referred to as the Zeldovich non-equilibrium factor [9]. For incompressible embryos (e.g., droplets in a supercooled vapor) use of (11)-(15) in (21) yields... 

If we neglect the pressure dependence of the liquid chemical potential, and assume the vapor phase to behave ideally, we obtain for the homogeneous nucleation rate in a supercooled vapor. 

Progress in the development of improved diffusion (e.g., [37]), and expansion chambers [38-40] during the last twenty years has made it possible to measure rates of nucleation in supercooled vapors directly (see [41,42] for excellent reviews). Two important conclusions can be drawn from the experiments ... 

The scaling properties of the critical nucleus have recently been derived by McGraw and Laaksonen [54]. Their analysis predicts that the size (number of molecules) of the critical nucleus is well predicted by the classical theory that the difference between the true reversible work of formation of the critical nucleus and that (xedicted by classical nucleation theory depends on temperature, but not on supersaturation and that the classical theory is recovered exactly at coexistence (saturation), with departures from the classical prediction W /(n Aji) = 1/2 (see eqs. 13, 15) being quadratic in Ap, the chemical potential difference between the supercooled vapor and the stable liquid at the bulk temperature and pressure. 

For droplet condensation in supercooled vapors or bubble formation in superheated liquids, density functional theory predicts that the free energy barrier to nucleation vanishes at the spinodal curve. This is an important improvement on classical nucleation theory, which predicts finite barriers irrespective of the depth of penetration into the two-phase region. Density functional theory is an extremely powerful technique for the rigorous calculation of free energies barriers to nucleation. Examples of calculations in non-ideal systems include bubble nucleation in the superheated Yukawa and Lennard-Jones liquids [55, 57] liquid nucleation in dipolar vapors [61] binary nucleation of liquids from vapors [58] and of bubbles from liquids [62] and crystal nucleation [59]. 

The microscopic theory of the physically consistent cluster due to Reiss and co-workers [25, 63-68] addresses the rigorous calculation of the energetics of embryo formation from statistical mechanics. This approach is only applicable to nucleation in supercooled vapors. The key result of the theory is an expression for the free energy of embryo formation. 

The work of Reiss and co-workers puts the question of the equilibrium distribution of liquid embryos in dilute supercooled vapors on sound conceptual ground. However, having to calculate embryo free energies by simulation rules out the use of such an approach in practical applications. To overcome this limitation, Weakliem and Reiss [67] developed a modified liquid drop theory that combines elements of the physically consistent cluster with the conventional capillarity approximation. These same authors have also developed a rate theory which allows the calculation of nucleation rates in supercooled vapors [68]. The dependence of the predicted rates on supersaturation agree with classical nucleation theory, but the temperature dependence shows systematic deviations, in accordance with scaling arguments [54]. 

McGraw, R. (1981) A corresponding states correlation of the homogeneous nucleation thresholds of supercooled vapors, /. Chem. Phys. 75, 5514. 

In addition, we should find that the vapor is slightly superheated and the liquid slightly supercooled. The opposite situation (supercooled vapor and superheated liquid) is, normally, unlikely to occur. In any case, though, whatever the situation, the equilibrium calculation will give a temperature (which we shall not use) and the composition of the two phases present. 

In accurately adjusted experiments, metastable states characterized by segments MO and NL (Figure. 3.24) can be obtained. These states are supercooled vapor (line MO) and a superheated liquid (line NL). The supercooled vapor is in a state such that, according to the parameters it should be a liquid, but its properties continue to foUow the behavior of a gaseous state-it aspires to extend, for example, as the volume increase. On the other hand, a superheated liquid is in a state such that, according to the parameters, it should be a vapor, but its properties remain liquid. Both these states are metastable at small external influences they pass into a two-phase state. A line OL corresponds to a negative factor of compression it is unstable and cannot be realized at all. 

---