THE MECHANISM OF ICE CRYSTAL GROWTH AND HABIT FORMATION

The formation of multiple layers of adsorbed water molecules on the basal and prism faces of ice may be reponsible for the remarkable temperature dependence of all growth variables:

Linear growth rate Step velocity Mean migration distance This effect results from an increased residence time of molecules in the adsorbed state

as the melting point is approached.

A quantitative treatment based on the Brunauer, Emmett and Teller model of multi-layer adsorption exemplifies these concepts and appears to explain the measured trends with temperature.

When the theoretical treatment is used in conjunction with a growth model based on the propagation of spiral steps, reasonable values for the condensation coefficient emerge.

The alternation of the primary habit of ice crystals with temperature is explained when the theoretical treatment is applied to the basal and prism faces, respectively.

Nakaya (1954) showed that the habits of ice grown in air are a complicated function of supersaturation and temperature

However, Kobayashi (1961) found that these variations in shape may be subdivided into two broad classifications:

Primary habits, which depend mainly on temperature

Secondary features, which are strongly dependent on supersaturation (saturation????). The secondary features appear superimposed upon the basic primary habit in the presence of external diffusion fields containing strong gradients of moisture or temperature.

A peculiar characteristic of ice is the tendency for the primary habit to alternate between plates and columns as the temperature is lowered from the melting point; the transition temperatures are roughly -4, -9 and -200 C.

This alternation of the primary habit is a manifestation of the strange dependence on temperature of the linear growth rates of the individual faces.

It should be realized that for understanding the origins of the primary habits ( dependent on temperature), we should only use the relative growth rates.

A convenient measure for the relative rates is the condensation coefficient α defined by

Gmax is the maximum growth rate possible at the net impingement flux δF, Ω is the volume of a water molecule in the lattice. The net impingement flux is related to the local supersturation s by:

Where Fs(T) is the molecular flux hitting the surface at equilibrium. If δP is the corresponding excess vapor pressure, then from the kinetic theory of gases:

In which k is Boltzmann’s constant, kT is defined as the kinetic theory factor, m is the mass of a water molecule, and T the temperature.

From these equations the condensation coefficient can be calculated from linear growth rate data at known values of temperature and excess pressure (supersaturation).

The condensation coefficient may be interpreted as the fraction of impinging vapor molecules which are successful at actually being incorporated into the ice lattice.

As pointed out by Lamb and Hobbs and discussed in detail by Lamb and Scott, there exists a basic similarity in the shape of the trend to the linear rates (condensation coefficients) and the measured velocities of step propagation across the basal face (Hallett, 1961; Kobayashi, 1967).

This, it was suggested, implies an indirect correlation between the step velocity ν and the frequency f at which steps are generated on the face since the step velocity ν itself is not directly coupled with the rate G at which the face advances parallel to itself

The spiral step mechanism of Frank (1949) was invoked as being the simplest mechanism which allows a natural physical relation between G, f and ν. Then, if the steps gather adsorbed molecules from within a limited distance from the step, all of the growth parameters may be qualitatively explained.

This catchment distance is usually taken to be the mean migration distance on the surface, χs (Mason et al, 1963). The purpose of this paper to reconsider this basic parameter and at the same time develop a total picture of the growth process, both qualitatively and quantitatively

PREVIOUS ATTEMPTS TO UNDERSTAND χS

The crucial parameter χs is a function of the mean time τ that a typical molecule resides in the adsorbed state:

Ds is the surface diffusion coefficient and U the activation energy. The residence time

The residence time is the inverse of the desorption constant used by Hobbs and Scott (1965), where W is the energy of attachment to the surface.

In a simplified way, the trends with temperature of all of the growth variables (linear growth rates, step velocity, and migration distance) may be thought of as a superposition of a strong average trend upward with increasing temperature together with a nearly discontinuos change between each maximum and associated minimum.

The discontinuity, the origin of the maximum and minimum, is often considered to be an effect of a surface reorientation of Faraday’s so-called “quasi-liquid” layer

THE EFFECTS OF MULTI-LAYER ADSORPTION

A. THE ADSORPTION MODEL

In order to understand growth characteristics of ice crystals more quantitatively, consider the model of multi-layer adsorption developed by Brunauer, Emmett and Teller (1938).

The effects they are trying to illustrate would arise from any model in which the concentration n [cm-2 ] of adsorbed moleculed increases with increasing pressure sufficiently strongly near the liquid condensation pressure.

The approach to the liquid condensation pressure is usually specified by the relative pressure:

Where P is the vapor pressure in dynamic equilibrium with the adsorbed layer and PL (T) a reference pressure which is usually considered to be in equilibrium with the supercooled liquid state. Generally, when the solid is growing

Where PS(T) is the equilibrium vapor pressure of the solid and δP the experimentally determined excess pressure. The relative pressure X is thus a well-defined, monotonic function of temperature and, for small supersaturations, approaches unity as the melting point of the solid is approached

In BET model the total surface concentration of adsorbed molecules is given by:

Where nm is the number of adsorption sites on the solid [cm-2 ], and

C is an adsorption parameter which characterizes the system. In (11) E1 is the energy binding the first adlayer1 to the solid, and EL the biding energy between all subsequent layers.

From (10), n∞ when X=1, it means that when P approaches PL (T) the surface coverage has many layers

??? Why do they mention the multi-absorpsion layer? How the multi-absorption layer effect on ice crystal growth.

??? Finding the values of P (vapor pressure in dynamic equilibrium with the adsorbed layer); PL (T) (a reference pressure which is usually considered to be in equilibrium with the supercooled liquid state).

Relationship between R and k

21

210 ab

baC

ANRk / Where: NA : Avogadro ‘s constant (6.0221415.1023 /mol)

From the nature of the constants a1 , b1 , a2 and b2 it is evident that a1b2/b1a2 will not differ much from unity (1). They are independent of the amount of water adsorbed in the first and second layer, respectively.

B. THE RESIDENCE TIME τ In order to estimate the length of time that vapor molecules reside in the adsorbed layer,

consider the equilibrium state and small deviations from it.

At equilibrium, the desorption flux leaving the adlayer must just balance the adsorption flux entering the layer to maintain a given surface concentration.

Let the partial pressure of water vapor exceed the equilibrium value by an amount δP so as to increase the flux of molecules entering the layer by βδF , where β is a sticking or adsorption coefficient to account for impinging molecules which are simply reflected.

Since each of the additional vapor molecules which enters the adsorbed layer will, on the average, reside in the adsorbed state for a time τ before it desorbs, the surface concentration will increase by an amount δn= τ βδF. So τ may be calculated from the isotherm (Eq.10) using

From Eq. (13): τ ∞ when the term (1-X) in the denominator approaches 0 as the temperature approaches the melting point . This means that, near that point, the residence time and the number of adsorbed molecules increase greatly.

To compare Eq.(13) with the expression for τ used by others, note that if X is small,

When the vapor pressure is written in the approximate exponential form

Thus, for low surface coverages, the temperature dependence of τ is determined primarily by the energy E1 binding the first layer to the substrate and τ is determined by Eq.(7)

C. THE MEAN MIGRATION DISTANCE χs

From Eq.(5), we have the final expression for the mean migration distance on the adsorbed multi-layer:

4. SURFACE STEPS AND THE GROWTH OF ICE

A. STEP VELOCITY

In Eq.(18), the functional form of the mean migration distance χs was derived without regard to the existence or behavior of steps on the surface χs will be used as an input parameter to treat the interactions of steps with one another and to determine their ontribution to the linear growth rate of a face.

Considering the mass balance of adsorbed and migrating molecules on the surface:

1. There exists a finite equilibrium concentration ne of adsorbed molecules which serve no growth function.

2. Two parallel neighboring steps are allowed to interact by competing for the available excess of adsorbed molecules.

The geometry for this development is that used by Stickland-Constable and is shown in Fig.6

The concentration of excess admolecules δn is determined as a function of surface position x from the equation of molecular continuity: