Rate Determining Processes of Growth of Ice Crystals from the Vapour Phase. Part II: Investigation of Surface Kinetic Process

Transcription

1 June 1984 T. Kuroda and T. Gonda 563 Rate Determining Processes of Growth of Ice Crystals from the Vapour Phase Part II: Investigation of Surface Kinetic Process By Toshio Kuroda The Institute of Low Temperature Science, Hokkaido University, Sapporo 060, Japan and Takehiko Gonda Faculty of Science and Technology, Science University of Tokyo, Chiba 278, Japan (Manuscript received 11 January 1984, in revised form 9 April 1984) Abstract Growth rates of basal and prism faces of ice crystals from the vapour phase have been measured as functions of the supersaturation at infinity from the crystal surface at -30* and at air pressures 250 and 0.3 Torr*. Contributions of the volume diffusion process of water molecules, the surface kinetic process and the conduction process of the sublimation heat to the measured growth rates have been quantitatively discussed on the basis of theoretical consideration in the companion paper (Kuroda, 1984). The dependence on supersaturation at the crystal surface and air pressure of the kinetic coefficient or the condensation coefficient has been obtained by combining the theory of the companion paper with the experiments. It should be noted that an increase in air pressure makes not only the diffusion process slower, but also the surface kinetic process inactive. 1. Introduction It is well known that growth forms and growth rates of snow crystals, i. e, ice crystals grown from the vapour phase, sensitively depend on growth conditions, e. g, temperature, supersaturation, pressure of foreign gases and so on. According to the results of experimental studies of the relation between growth forms of artificial snow crystals and growth conditions (Nakaya, 1951, 1954; Aufm Kampe et al., 1951; Mason, 1953; Kobayashi, 1957, 1958, 1961; Hallett and Mason, 1958), the remarkable variation of growth forms has following features. Firstly, three changes occur in the basic habit (plate or column) with decreasing temperature. Kuroda and Lacmann (1982) interpreted the habit change by considering the dependence on temperature and crystallographic orientation of the surface structure as well as growth kinetics of ice crystals. Secondly, increasing supersaturation causes the preferred growth of corners and edges of crystals producing dendrites, hollow prisms, needles and so on. This morphological instability is attributable to the non-uniformity in supersaturation at the crystal surface and preferred two-dimensional nucleation at the corners (Chernov, 1974; Frank, 1974, 1982; Kuroda et al., 1977; Kuroda, 1982; Irisawa et al., 1983). Experimental studies of this problem have been done by Gonda et al. (1970, 1971, 1976, 1977, 1980, 1982a, 1982b, 1983), Nenow and Stoyanova (1977, 1984) and Keller and Hallett (1982).

2 564 Journal of the Meteorological Society of Japan Vol. 62, No. 3 Isono et al. (1957) studied the growth forms of ice crystals falling in the gases, such as H2, CO2 and air and pointed out the importance of the vapour diffusivity and thermal diffusivity for their growth forms. Then Gonda and Komabayashi (1970, 1971) and Gonda (1976, 1977, 1980) investigated the effect of the vapour diffusivity and thermal diffusivity on the growth of ice crystals falling in heliumargon mixtures. Beckmann and Lacmann recently investigated the growth and evaporation kinetics of basal and prism faces of ice crystals on substrates in the temperature range -7 to-15* under pure water vapour conditions (Beckmann, 1982a, Beckmann and Lacmann, 1982) and under partial pressures of nitrogen (Beckmann, 1982b). Under pure water vapour conditions, the growth rates at low supersaturations (excess vapour pressure *p <2Pa) were mostly controlled by two-dimensional nucleation mechanism at the surface covered with a quasi liquid layer, i. e, the so-called V-QL-Smechanism by Kuroda and Lacmann (1982). Those at higher supersaturations were linearly proportional to *p with the condensation coefficient *0.14, which seems to be due to entropy constraints. In the whole range of under-saturations (-*p<1pa), the evaporation rate was linear in ap with evaporation coefficient av*0.14. Beckmann (1982b) distinguished the influence of the surface kinetic process and of the volume diffusion process on the measured growth rates on the assumption of a linear law of the growth rate with p and modelling the crystals with * half spheres of equivalent diameter. Then it was shown that not only the resistance of volume diffusion, but also that of surface kinetics increased with increasing partial pressure of nitrogen. Kuroda (1984) (Paper I) showed how the surface supersaturation and growth rates are determined by taking account of relevant three processes to the growth, i. e. 1) volume diffusion process of water molecules towards a crystal surface, 2) surface kinetic process for incorporating the water molecules into crystal lattice, and 3) transport process of latent heat of sublimation by heat conduction. Then the rate determining role of each process was discussed. As pointed out in Paper I, growth mechanism or the surface kinetic process is characterized by the dependence on surface supersaturation *s of the kinetic coefficient (*s) defined as where Vk is the growth rate determined by the surface kinetic process. For an example, in the case of lateral growth of smooth surfaee on a molecular level, *(*s) increases with *s until it reaches a maximum value *max. Nonlinearity of the growth rate as a function of the supersaturation is expected, if the kinetic coefficient is not constant but depends on *s, and if the resistances of diffusion process and heat conduction process are much smaller than that of the surface kinetic process. We are still in need of experimental and theoretical investigation of the surface kinetic process of ice crystals at various supersaturations, temperatures, pressures of foreign gases and so on. The purposes of this paper are firstly to measure the supersaturation dependence of the growth rates of basal and prism faces of ice crystals at -30* and at two air pressures, i. e. 250 Torr and 0.3 Torr (Sections 2 and 3), secondly to estimate the dependence of the kinetic coefficient *(*s ; Pa) of the both faces on surface supersaturation and on air pressure Pa by *s applying the theory of Paper I to the experiments (Sections 4 and 5), and then to discuss the rate determining processes of the growth of ice crystals from the vapour phase (Section 5). 2. Experimental procedures Fig. 1 shows a cold chamber for in situ observation of ice crystals growing at -30*. The cold chamber is designed to cool independently an ice plate (B) for supplying water vapour and a substrate (C) for the growth of ice crystals. The cooling of the chamber is carried out by an electric current to the thermoelectric cooling panels (A) set at the top and the bottom of the chamber, keeping the ice plate (B) at a temperature lower than that of the growth substrate (C). The accuracy of the temperature on the ice plate and on *

3 June 1984 T. Kuroda and T. Gonda 565 Fig. 1 Cold chamber for in situ observations of ice crystals growing at -30*. the growth substrate is *0.05*. The temperature control is carried out by automatically turning on and off the electric current flowing to the thermolectric cooling panels at the top and the bottom of the chamber. Thereafter, the air in the chamber is evacuated using a vacuum pump until the pressure of air reaches about 3*10-1 Torr. When a sufficiently diluted silver iodide smoke of about 3cm3 is inserted into the chamber, keeping the ice plate (B) at a temperature, slightly higher than that on the growth substrate (C), minute ice crystals are nucleated in air and fall on the growth substrate. The nucleation by this seeding is so controlled as to grow only an ice crystal on the substrate within the field of view of a microscope. The supersaturation is held constant by controlling the temperature difference between the ice plate (B) and the growth substrate (C). Then, the residual air in the chamber is evacuated, and photomicrographs of ice crystals are taken.. Ice crystals were grown at -30*and under various constant supersaturations. 3. Experimental results Fig. 2 shows the lengths along the <0001> and <1010> directions of ice crystals grown in air at (a) 0.3 and (b) 250 Torr at -30* and an ice-supersaturation of 3* plotted as a function of growth time. As shown in Fig. 2(a), the length along the <0001> and <1010> directions of the ice crystals growing in air at 0.3 Torr increases linearly with time at the size below about 160*m, that is, the growth rate of ice crystals below about 160*m is independent of the crystal size. On the other hand, as shown in Fig. 2(b), the length along the (0001> and <1010> derections of ice crystals growing in air at 250 Torr increases nonlinearly with time at the size above about 10 Fig. 2 Lengths along the <0001> and <1010> directions of ice crystals grown in air at (a) 0.3 and (b) 250 Torr at -30* and an ice-supersaturation of 3* versus growth time. (a) in air at 0.3 Torr (b) in air at 250 Torr

4 566 Journal of the Meteorological Society of Japan Vol. 62, No. 3 Fig. 3 The normal growth rates of ice crystals growing in air at -30* as a function of supersaturation *. (a) in air at 0.3 Torr (100*m in lengths along the <0001> and <1010> directions) (b) in air at 250 Torr (40*m in length along the <0001> direction) * m. The rates of ice crystal growth in air at 0.3 Torr were obtained at 100*m in length along the <0001> and <1010> directions, and those at 250 Torr were obtained at 40*m in length along the <0001> direction, at which polyhedral crystals can grow in morphologically stable way. Fig. 3 shows the ice-supersaturation dependence of the normal growth rates of the basal {0001} and prism {1010} faces of ice crystals in air at (a) 0.3 Torr and (b) 250 Torr at -30*. As shown in Fig. 3(a), the growth rates of the both faces are nearly the same in air at 0.3 Torr, although, strictly speaking, the growth rate of the {1010} faces is slightly larger than that of the {0001} faces at ice supersaturations above 5*. On the contrary, as shown in Fig. 3(b), the growth rate of {1010} faces is fairly smaller than that of {0001} faces in the whole range of measured ice-super-saturations. That is, an introduction of the air into the chamber makes the growth rate of {1010} faces smaller than that of {0001} faces. 4. Method of investigating the behaviour of surface kinetic process According to Paper I, the growth rate of an ice crystal surface from the vapour phase is expressed as The notations used are as follows :

5 June 1984 T. Kuroda and T. Gonda 567 where ps and TS are the vapour pressure and the temperature at the surface, respectively (*s) : the kinetic coefficient * (see Section 1 and Eq. (1)) d : the thickness of * the diffusion boundary layer which is defined by (*p/*x)x=0*d*p*-ps, where (*p/ x)x=, is the normal derivative of the pressure of the water vapour at the surface r : the thickness * of the thermal boundary layer defined by (*T/*x)x-0*t D : the diffusion coefficient of water molecules in air * and (c) the heat conduction process As mentioned in Paper I, we can estimate the value of Rd and Rh among these three resistances. Therefore, we may evaluate the resistance Rk depending on *(*s) as a function of the supersaturation * at infinity from the surface by substituting the measured dependence on * of growth rates Vexp(*) on the left-hand side of Eq. (2). We actually obtain a relation for estimating Rk by the procedure mentioned above, i. e. Eq. (2) is formal expression for the growth rate, since it includes the supersaturation *s which is defined by two unknown parameters, i. e, the vapour pressure ps and the temperature Ts at the surface. These parameters are solved from the heat and mass flow continuity conditions at the surface. It was shown that is the supersaturation satisfying the following *s equation : The resistance R,, (the left-hand side of Eq. (8)) is evaluated as a function of * by substracting the resistances Rd and Rh (the second and third term of the right-hand side) from the measured total resistance (the first term of the right-hand side of Eq. (8)). Furthermore, by substituting Eq. (8) in Eq. (4), we obtain an expression for estimating the supersaturation *s(*) at the surface as a function of *: We should notice that the numerator of Eq. (2) corresponds to a macroscopic driving force for growth of snow crystals and the denominator represents a sum R=R k+rd+ Rh of the resistance of each relevant process of the growth, i. e. (a) the surface kinetic process (b) the volume diffusion process By eliminating * between Eqs. (8) and (9) using the measured relation Vexp(*) at a given temperature T* and a given air pressure Pa, we can investigate the dependence of the kinetic coefficient *(*s; T*, pa) on the surface supersaturation *s at the given T* and pa, to which the surface kinetic process for incorporating the molecules into crystal lattice is essentially reflected. 5. Resistances of growth process and the kinetic coefficient 5.1 Pa=250 Torr First of all we shall estimate the resistance

6 568 Journal of the Meteorological Society of Japan Vol. 62, No. 3 Rk(*) of the surface kinetic process as a function of *. in the case of the growth at -30* and 250 Torr of air pressure by substituting the measured relation Vexp(*) (Fig. 3(b)) and estimated values of Rd and Rh in Eq. (8). In this case, the ratio lc/la of the length along the c-axis and the length la lc along the a-axis is about two to three in the investigated range of * because of anisotropy of the growth rates, V*exp of basal (0001) face and Vpxpe P of prism (1010) face. To obtain the strict supersaturation distribution surrounding such an anisotropic polyhedral crystal, which determines Rd, we must solve the Laplace's equation subject to the following boundary condition : at infinite distance from the hexagonal columnar crystal, and the normal gradient (*/*n) of supersaturation at each surface is constant all over the surface, i. e. since diffusion flow of water molecules towards each surface should be constant all over the surface to preserve the polyhedral form (Irisawa et al., 1983). For estimating Rd, however, we shall tentatively use the thickness of diffusion boundary layer *d of the (0001) face and *pd of the (1010) face obtained on the following assumptions : (ii) Mean extent of the diffusion boundary layer due to the sink action of a growing crystal on the substrate is equal to the radius *c of a half sphere with the same volume as the crystal, and of the assump- We shall discuss the validity tions later. For an example, *bd=11*m and *pd=24*m Fig. 4 The estimated resistances of the surface kinetic process Rk of the basal {0001} and the prism {1010} faces and of the volume diffusion process Rd as functions of supersaturation at infinity from the surface. * See text for details. are obtained from Eqs. (12) and (13) for a crystal of lc=40*m growing at *=2.5* (Fig. 3(b)). Since D=0.528cm2/s in air at -30* and 250 Torr, Rd=1.89*d. Thus the diffusion resistance for (0001) face may be 2.1*10-3s/cm and that for (1010) face may be 4.5*10-3s/cm. Rd of each surface is shown in Fig. 4 with dot-dashed line. It should be noted that Rd increases with time, since *d increases with crystal size (Eqs. (2) and (4) in Paper I). As shown in Paper I, Rh=4*10-1 *t=4* 10-4s/cm was taken in the case of the growth of ice crystals in air using the assigned values =2.2*103erg/cm s, *l =8.4*10-13 erg * and t=10*m. On the other hand, the value * of Rh in our experiment described in Sections 2 and 3 may be smaller than the above value by a factor of 10-2, since an ice crystal grows on the substrate with larger thermal conductivitiy than that of air by a factor of 102. Therefore, we can neglect Rh in Eqs. (8) and (9). The estimated resistances Rk(*) of basal and prism faces at pa=250 Torr and lc=40 m are shown in Fig. 4 with solid * lines as functions of *. The resistances of both faces increase with decreasing *. This results are interpreted as follows : The surface super-

7 June 1984 T. Kuroda and T. Gonda 569 saturation *s decreases with decreasing * (Fig. 5), and the rate of generation of steps necessary for lateral growth of smooth surface decreases with decreasing *s. Therefore, with decreasing *, the kinetic coefficient *(*s) decreases and thus the resistance Rk increases. The Rk of the fast growing basal face is of course smaller than that of the slow growing prism face (Fig. 4). The estimated supersaturation *s at the basal and prism faces from Eq. (9) at 250 Torr is represented in Fig. 5. with a solid line. The value of *s may increase with in- creasing diffusion coefficient D because of easier supply of molecules by diffusion to the surface. Hence, *s may be regarded as equal to * (dashed line in Fig. 5. ), if the resistance Rd is completely negligible. The growth at 0.3 Torr is nearly the case as shown later. Let us estimate the dependence on cr of the kinetic coefficient *(*s) at 250 Torr by eliminating * between the relations Rk= Pe(T*)*/kT*(*s) (Fig. 4) and *s(*) (Fig. 5). The kinetic coefficient of the basal and prism faces at 250 Torr are represented in Fig. 6 with solid lines. They increase with *s, since the rate of step generation increases with *s. The kinetic coefficient of the fast growing basal face is larger than that of the slow growing prism face. The curves in Fig. 6 correspond to the curves in Fig. 3. To indicate the amount of scatter in *(*s) caused by deviation of growth rates from the curve in Fig. 3, two values of Fig. 5 The estimated supersaturations *s at the {0001} and {1010} faces as a function of * in air at 250 Torr (solid line) and at 0.3 Torr (broken line). Fig. 6 The estimated kinetic coefficients *(*s) of the {0001} and {1010} faces as a function of the surface supersaturation *s in air at 250 Torr (solid line) and at 0.3 Torr (broken line). The value of the condensation coefficient * (*s) corresponding to * (*s) is shown on the right ordinate. where F* is the fundamental solution for (*/*n)*=1, (*/*n)p=0, la=1, given lc/la and *=0, and Fp is that for (*/*n)*=0, (*/*n)p=1, 1a=1, a given lc/la, and *=0.

8 570 Journal of the Meteorological Society of Japan Vol. 62, No. 3 Using a relaxation method, we have obtained numerical values of Fb and Fp for lc/la=2, e, g. at the centre of the basal face, F*(2)= and Fp(2)= at the edge of the cylinder, and F*(3)= and Fp(3)= at the middle of the side face. Thus, the value of *s(*) at the corresponding position * in the case of the growth at *= 2.5* (Fig. 3(b)) is as follows : a,(1)=1.9*, *s(2)=2.0* and *s(3)=2.1*. On the other hand, *s evaluated from Eqs. (9), (12) and (13) is 2* at a=2.5* (Fig. 5). Therefore, the way estimating *bd and *pd is fairly good. 5.2 p=0.3 Torr The diffusion coefficient in air at 0.3 Torr and at -30* is D-447cm2/s. Thus the resistance is estimated to be Rd = 2.2*10-3 *d =2.2*10-5s/cm for *d=100*m, while the total resistance (pe*/ kt*)(*/v exp(* )) is of the order of 10-4s/cm in the whole range of supersaturations of the experiments. Therefore, the total resistance almost consists of the resistance Rk of surfacc kinetic process alone. That is, we can ignore the second and third terms on the right-hand side of Eq. (8). If the resistances of volume diffusion and heat conduction are negligibly small, the growth rate is expected to be independent of crystal size, because the size dependence results from d in Rd and *t in Rh. Actually the crystal * size was linearly proportional to time from 10 to 160*m (Fig. 2(a)). The resistances of kinetic process of both faces at 0.3 Torr are represented in Fig. 4 with a dashed line. In this case, * s, is regarded as equal to *, since the second term and the third term in Eq. (9) are much smaller than unity (see dashed line in Fig. 5). In other words, as can be kept at * because of quick supply of molecules through the gaseous phase with very large D. The dependence on *s of *(*s) at 0.3 Torr are shown in Fig. 6 with a dashed line. With increasing *s, *(*s) increases and reaches the maximum value 9.7*10-4cm/s. The right ordinnate of the Fig. 6 corresponds to the value of condensation coefficient *(*s), which is defined by From eqs. (1) and (10) we obtain between * and * : and therefor a relation Then the maximum value of * corresponding to *max at 0.3 Torr is 0.2 (Fig. 6). This value is good agreement in the result by Beckmann and Lacmann (1982) at temperatures -7 to -15* under pure water vapour conditions. It should be noticed that the kinetic coefficient in air of large pa is smaller than that in air of small pa for a given a s and T : where the notation klmn corresponds to basal (0001) and prism (1010) faces. In other words, an increase in pressure of air makes not only diffusion process slower, but also surface kinetic process inactive. Beckmann (1982b) have already reported the same influence of partial pressure of nitrogen at higher supersaturations where the growth rate is linearly proportional to *, i. e. * reaches a maximum value. In this paper, we have found the influence of air pressure on the surface kinetic process for wide range of supersaturation where the non-linear dependence on a. of growth rate was measured. It should be also noticed that the influence of air pressure to the kinetic process is larger for prism face than for basal face (Fig. 6). The surface kinetics of ice crystals growing under various pressures of foreign gases is a worthwhile subject to investigate theoretically and experimentally in future. Concluding remarks By combining the theory of Paper I with

9 June 1984 T. Kuroda and T. Gonda 571 the measured growth rates Vexp(*) of basal and prism faces of ice crystals at -30* as a function of the supersaturation * at infinity, we have estimated the supersaturation s at the surface as a function * of *and the dependence on * of the kinetic coefficient N(*s ; T, pa) at given temperature T and air pressure pa. We can draw the following concluding remkars : (1) The kinetic coefficient * representing the surface activity for incorporating the water molecules into crystal lattice increases with It is attributable to an increase in *s the rate of generation of the steps necessary for the lateral growth with increasing *s. (2) With increasing *s or *, * reaches a maximum value *max=9.7*10-4cm/s corresponding to the maximum value *max=0.2 of the condensation coefficient in the case of growth at 0.3 Torr of air pressure and at -30*. This value of *max is in good agreement with the results obtained by Beckmann and Lacmann (1982) in the case of growth under pure water vapour conditions at temperatures -7 to -15*. (3) An increase in air pressure makes not only the diffusion process slower, but also the surface kinetic process at a given *s inactive. This result is consistent with the experiments by Beckmann (1982b) under various pressures of nitrogen at -11 and -15* and at such high supersaturations that * reaches *max. (4) The influence of air pressure to the surface kinetic process is larger for the prism faces than for the basal faces. Acknowledgement The authors wish their cordial thanks to Prof. T. Kobayashi for critical reading of the manuscript. The authors are indebted to Mr. T. Irisawa for valuable discussions concerning the supersaturation at the surface of a growing cylinder. References Aufm Kampe, H. J., H. K. Weickmann and J. J. Kelly, 1951: The influence of temperature on the shape of ice crystals growing at water saturation. J. Meteor., 8, Beckmann, W., 1982a: Interface kinetics of the growth and evaporation of ice single crystals from the vapour phase I. Experimental techniques. J. Crystal Growth, 58, b: III. Measurements -, under partial pressures of nitrogen. J. Crystal Growth, 58, , and R. Lacmann, 1982: II. Measurements in a pure water vapour environment. J. Crystal Growth, 58, Chernov, A. A., 1974: Stability of faceted shapes. J. Crystal Growth, 24/25, Frank, F. C., 1974: Japanese work on snow crystals. J. Crystal Growth, 24/25, : Snow crystals. -, Contemp. Phys., 23, Gonda, T. and M.. Komabayashi, 1970: Growth of ice crystals in the atmosphere of helium-argon mixture. J. Meteor. Soc. Japan, 48, : Skeletal and dendritic structures,- of ice as a function of thermal conductivity and vapour diffusivity. J. Meteor. Soc. Japan, 49, , 1976: The growth of small ice crystals in gases of high and low pressures. J. Meteor. Soc. Japan, 54, : The growth -, of small ice crystals in gases of high and low pressures at -30* and -44*. J. Meteor. Soc. Japan, 55, Gonda, T., 1980: The influence of the diffusion of vapour and heat on the morphology of ice crystals grown from the vapour. J. Crystal Growth, 49, , and T. Koike, 1982a: Growth rates and growth forms of ice crystals grown from the vapour phase. J. Crystal Growth, 56, and T. Yamazaki, 1982b: Morphological -, stability of polyhedral crystals growing from the vapor phase. J. Crystal Growth, 60, and T. Koike, 1983: Growth mechanism -, of single ice crystals growing at low temperature and their morphological stability. J. Crystal Growth, 65, Hallett, J. and B. J. Mason, 1958: The influence of temperature and supersaturation on the habit of ice crystals grown from the vapour. Proc. Roy. Soc., A247, Irisawa, T., T. Kuroda and A. Ookawa, 1983: Growth of hexagonal crystal from vapor and its morphological stability. Presented at 7th Intern. Conf. Crystal Growth in Stuttgart, September; to be published in J. Crystal Growth. Isono, K., M. Komabayashi and A. Ono, 1957: On the habit of ice crystals grown in the atmospheres of hydrogen and carbon dioxide. J. Meteor. Soc. Japan, 35, Keller, V. W. and J. Hallett, 1982: Influence of air velocity on the habit of ice crystal growth from the vapor. J. Crystal Growth, 60, Kobayashi, T., 1957: Experimental researches on

10 572 Journal of the Meteorological Society of Japan Vol. 62, No. 3 the snow crystal habit and growth by means of diffusion cloud chamber. J. Meteor. Soc. Japan, 75th Ann. Vol., Kobayashi, T., 1958: On the habit of snow crystals artificially produced at low pressures. J. Meteor. Soc. Japan, 36, : The growth -, of snow crystals at low supersaturations. Phil. Mag., 6, Kuroda, T., T. Irisawa and A. Ookawa, 1977: Growth of polyhedral crystal from solution and its morphological stability. J. Crystal Growth, 42, , 1982: Growth kinetics of ice single crystal from the vapour phase and variation of its growth forms. J. Meteor. Soc. Japan, 60, , and R. Lacmann, 1982: Growth kinetics of ice from vapour phase and its growth forms. J. Crystal Growth, 56, : Rate determining -, processes of growth of ice crystals from the vapour phase I. Theoretical consideration. J. Meteor. Soc. Japan, 62, Mason, B. J., 1953: The growth of ice crystals in a supercooled water cloud. Quart. J. Roy. Meteor. Soc., 79, Nakaya, U., 1951: The formation of ice crystals. Compendium Meteor. Soc., Boston, : Snow crystals-natural and artificial. -, Harvard Univ. Press, Nenow, D. and V. Stoyanova, 1977: On the formation of ice dendrites from the vapour phase. J. Crystal Growth, 41, and N. -,- Genediev, 1984: Morphologicalinstability of vapour-grown ice crystals, to be published in J. Crystal Growth.