Title

Crystallographic Orientation of Frozen Droplets on Ice Surfaces

Author(s)HIGUCHI, Keiji; YOSIDA, Tatekatu

Citation

Physics of Snow and Ice : proceedings, 1(1), 79-93

Issue Date1967

Doc URL

http://hdl.handle.net/2115/20287

Typebulletin (article)

Note

International Conference on Low Temperature Science. I. Conference on Physics of Snow and Ice, II. Conference on
Cryobiology. (August, 14-19, 1966, Sapporo, Japan)

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Hokkaido University Collection of Scholarly and Academic Papers : HUSCAPCrystallographic Orientation of Frozen Dropletson Ice Surfaces

Keiji HIGUCHI
till

1=1

1$:

=Water Research Labomtory, Nagoya University, Nagoya, Japanand

Tatekatu YOSIDA
EI3EEl ;m 1m
Sapporo Meteorological Observatory, Japan Meteorological Agency, Sapporo,' Japan

Abstract
Photogrammetric observations were made on natural snow crystals composed of an assemblageof crystallographically single crystals. It was found in the case of certain types of these crystals
that crystallographic relations exist between the base crystal and the secondary branches developed
from it.. The formation mechanism of such secondary branches is discussed on the basis of geometrical studies on lattice matching between crystal planes of ice, and on experimental work on
the crystallographic orientation of frozen droplets on ice surfaces.

1. Introduction
As is seen in the photographs of natural snow crystals of the spatial dendritic typewith a stellar base (Nakaya, 1954), the growth direction of secondary dendritic branches
is at an appreciable angle to the plane of the base crystal. This means that the crystallographic orientation of the former is different from that of the latter.
Nakaya (1954) considered that such secondary branches grew from an ice nucleus
attached to a snow crystal. In addition, Kumai (1964) proposed another theory in which
the growth of the secondary branch is generated from an ice fog particle attached toa snow crystal. It may be inferred from these two mechanisms that the crystallographic
orientation of a secondary branch should have no dependency on that of the base crystal,but should depend largely on how the ice nucleus or ice fog particle is attached to thecrystal.
Recently, Hallett (1965) proposed a new theory in which he stated that a frozen
water droplet accreted by a snow crystal can grow as a secondary branch. Now since
the crystallographic orientation of frozen water droplets should have some dependency
on that of the substrate, a crystallographic relation should exist between the secondarybranch and the base crystal. In this respect Hallett's theory differs from those of Nakayaand Kumai.
The present work was undertaken in an attempt to determine which of the threetheories provides a better explanation of the formation of secondary branches.

K. HIGUCHI AND T. YOSIDA80U.
a)

Crystallographic Orientation of Secondary Branchesof Snow Crystals

Secondary branches developed from the center of the base crystal

The simplest type of spatial dendritic crystal with a stellar base is that with one
or two branches developing from the center of the base crystal, as seen in Figs. 1 and2, reproduced from Photo. 602 and Fig. 108 in Nakaya's book. If the shape of theseIIaFig. 1. (Nakaya, 1954)

X 30.5~Itanr = 2 cos eoFig. 2.

(Nakaya, 1954)

x 16.5

Fig. 3. Projection of spatial dendriticcrystalcrystals IS as shown in Fig. 3, the geometrical relation between do and rp (Higuchi, 1957)
should betan rp = 2 cos do .
Since the values of do and rp in Figs. 1 and 2 are a solution of this equation, it canbe said for these crystals that one of the a-axes of the secondary branch is parallel tothat of the base crystal, in addition c-axis of the former inclines at a 75° angle to thatof the latter. Such a crystallographic relation could be understood if the (0001) planeof the secondary branch is combined with the (2021) plane of the base crystal or vice.
versa.
A similar shape of snow crystal to those described above might be a spatial

81

ORIENTATION OF FROZEN DROPLETS ON ICE

Fig. 4.

Fig. 5.

(Bentley, 1931)(Nakaya, 1954)

X 42.5

Fig. 6.

Projection of spatial combination of plates

combination of plates shown in Figs. 4 and 5, reproduced from p. 210 in Bentley's (1931)
book and Fig. 153 of Nakaya's book.

These seem to be projections of the same shapeof crystal as shown in Fig. 6.

cFig. 7.

(Higuchi, unpublished)

Fig. 8.

(Klinov, 1960)

X 68

Fig. 9. Projection of combinationof column and plate

82

K. HIGUCHi AND T. YosIbA
Snow crystals in Figs. 7 and 8, are examples of secondary plate branches grownfrom a columnar crystal, Fig. 8 being reproduced from a paper by Klinov (1960, p. 50).
As shown in Fig. 9, it seems to be clear that one of the a-axes of the plate is parallelto that of the column, and the c-axes of these cross each other perpendicularly. Sucha relation can be obtained if the (0001) plane of the former is combined with the (1010)
plane of the latter.
Figures 10 and 11 reproduced from Photos. 679 and 705 in Nakaya's book, are examples of snow crystals which are composed of two columns or needle crystals, their c-axes

Fig. 10. (Nakaya, 1954)

x 55.5

Fig. 11. (Nakaya, 1954)

x47.5

cross at 58.5° approximately. Such a relation can be explained by a combination of the(0001) plane of one elementary crystal with the (1211) plane of the other.

b)Secondary branches developed from main branch of base crystal
When secondary dendritic branches develop from various points of the mambranches of the base crystal, their growth direction is at an appreciable angle to thebase crystal plane. An example of the plan and side views of snch a crystal is shownin Figs. 12 and 13, after reproduction from Photos. 567 and 568 in Nakaya's book. It

Fig. 12. (Nakaya, 1954)

X 16

Fig. 13. (Nakaya, 1954)

X 16

was found in Figs. 14 and 15, that the projected direction of secondary branches IS
perpendicular to one of the a-axes of the base crystal, as indicated by Figs. 16 (A) and(B) separately. Though the combination of crystal planes is not known yet, this is an

83

ORIENTATION OF FROZEN bROpLETs ON IC£

Fig. 14. (Higuchi, unpublished)(A)
Fig. 16.

X 26.5

Fig. 15. (Higuchi, unpublished)

X 26.5(8)

Direction of secondary branches of spatial dendritic crystal

important result for discussion on the growth mechanism of secondary branches, smce
they show a systematic orientation even when the generating point of growth is different
in each case.
In summary, it can be said that some crystallographic relations exist between thesecondary branches and their base crystal.

III. Lattice Matching between Two Crystal Planes of Ice
The photogrammetric study on snow crystals described in section I indicates thatthe formation of secondary branches might be explained by ice crystal growth fromfrozen droplets attached to the crystal, as proposed by Hallett. If so, the growth of

84

K. HIGUCHI AND T. YOSIDA

secondary branches can be considered in the same way as an epitaxial growth of ice
crystals on a foreign solid surface. Therefore, the lattice matching between two crystal
planes of ice was examined as follows.
The lattice misfit between two crystal planes of ice, 0, can be given by

0=1 a:oao I 'where a is a lattice vector of one crystal plane and ao the corresponding vector of theother. When some of the crystal planes are matched with the basal plane as shownin Fig. 17, the lattice misfit in the direction of a-axis, 0lla, and that perpendicular toooooooooooooooo( 1011)

o(loTo)oo0(2021)(0001 )
Fig. 17. Basal matching with crystal planes of ice

a-axis, 0 La, can be calculated. The results are seen in Fig. 18, in which the shape ofsnow crystals expected ftom these matchings are also shown. Figure 19 shows the results
for the matching of a prism plane with others.
The calculated values of basal and prism misfit given above are of the same orderas that between ice and the most active nucleating substances such as AgI, PbI 2 andeus. It is possible, therefore, that freezing of supercooled droplets on an ice surface,
as well as ice crystal nucleation on foreign substances, would occur under the conditionsof lattice matching given above.

IV. Experimental Studies on Crystallographic Orientationof Frozen Droplets on Ice Surface
In order to study the possibility of occurrence of freezing of water droplets underthe lattice matching described in the previous section, simple experiments were carried

85

ORlENT ATION OF FROZeN 'DROPLETS ON ICE

(a)

(d)
Columna
Columns

PlatesPlatesColumna(OOOIl",-,IIOTo)(0001) ~ (1210)

Plate('010:: 0J:;110. +2.0

(b)

0,.

(e)cColumns

1 ....--"0...--<)
12..6-Columns

PlatesPlofes(0001/ ~ (lOTI)(000l) "-' (2421)~J.o • -2.5 %, SlIo-O<;.
0
Ola=fic0110 =-2.3%

(f)

(e)
ci

75.2'~cColumns
Columns
PlatesPlates(OOOtl ~ (2021)(0001 )~ ( 1211 )bJ.o. +6.7 "I. ,c0O.la::SI\ 0 • 0,(' =-4.0 %0\10.
. fi·.
t'nof
combinationofplate
anrOjec
10dcolumn
andtheirlattIce
mlS
ts
Fig. 18. P

K. HIGUCHI AND T. YOSIDA86(b)(a)c!- --.,CtO

I:fmb~c.

+ 2.0"10.--j5(loTo) '"'-' (IOTO)
li.lC •Columns

I --""7CPlate

3aColumnIIOI0)~(·1210)

SII C • -2.0 "10~(5b.l. C • -2.0 "10, ~Il c· -6.0 "10(c)72.8·cIloTO) ~(2421)

hc' -2.3 "10, ~II c"-6.0 "10

Fig. 19. Prism matching with crystal planes of ice

out in the cold chamber by using the apparatus shown schematically in Fig. 20.
Supercooled droplets were sprayed on the surface of a hoar crystal of skeleton
structure as a substrate. Following this treatment, water vapor was supplied to thehoar crystal placed in an apparatus similar to that used for the artificial production ofsnow crystals. Under suitable conditions of temperature and vapor supply, the frozen
droplets on the hoar crystal grew as ice crystals of column or plate type. The directionof crystal-axes of frozen droplets were determined from the shape of ice crystals grownfrom these droplets.

§i> ·n:ILWATERDROPLETS----r-I0o00ooSUBSTRATEHEATER

Fig. 20. Experimental apparatus

01<IENTATION OF FROZEN DROPLETS ON ICE

87

The shape of many of the ice crystals grown from the frozen droplets indicates thatthe freezing of droplets occurred under conditions of no lattice misfit, namely, the same
crystallographic orientation as that of the substrate. But, as will be described below,(a)(c)Ie)

Fig. 21.

X 34

K. HIGUCHI AND T. YOSIDA

88

some of them showed an occurrence of freezing under conditions of lattice matchingdescribed in the previous section.

a)

Ice crystals with c-axis jJCrpendicular to that of the substrate
When the air temperature is almost -SoC, ice crystals grew as hollow prisms; thiswas suitable for viewing the direction of their c-axes. Figure 21 shows a series of photographs of the hollow prisms grown from frozen droplets over an ice surface witha horizontal c-axis, taken at different focusing points from the top of the ice crystals
to the substrate. The number of sprayed droplets was so numerous in this case thatseveral of them froze as a pillar like rime growth, as shown in the sketch in Fig. 21.
It can be easily seen from these photographs that many of the ice crystals grew withthe same c-axis orientation as that of the substrate, but some of them were seen withthe c-axis perpendicular to the substrate.

(b)

(a)

Fig. 22.

X 18

(b)

(a)

Fig. 23.

X 17.5

89

ORIENTA TJON OF FROZEN DROPLETS ON ICE

Figure 22 shows another example of ice crystals similar to the above, but with a plateon the top of a hollow prism, that is to say the "Tsuzumi" type (Nakaya, 1954). It was
found from the shape of the plate-part of the ice crystals that the c-axis direction ofthis crystal is perpendicular to that of the substrate and is also in parallel to one of
the a-axes of the substrate. In addition the direction of one of the a-axes of the former
is parallel tothe c-axis of the latter.

This crystallographic relation indicatestheoccurrence of lattice matching between (0001) and (1210) in Fig. 18 d, or between (1010)
and (1010) in Fig. 19 a. Figure 23 is another example of a similar crystal.
Figures 24 and 25 are examples of ice crystal growth, different from that describedabove, when droplets were sprayed over a prismatic surface of hoar crystal. As seen inthese photographs, the direction of the c-axis of the ice crystal is perpendicular to thec-axis of the substrate; in addition one of the a-axes of the former is perpendicular tothe c-axis of the latter. This relation can be seen more clearly in Fig. 26, which shows
ice crystal growth from frozen droplets on a basal surface of hoar crystal. The hexagonal plate in this photograph indicates the direction of the a-axes of the substrate.
These crystallographic relations can be understood if the lattice matching between (0001)
and (1010) occurred as shown in Fig. 18 a.

Fig. 24.

X 64

Fig. 25.

(b)(a)

Fig. 26.

X 39

X 71.5

K. HIGUCHI A~D T. YOSIDA

90b)

Ice crystal growth with c-axis inclined to that of substrate

When ice crystals grew from frozen droplets over a prismatic surface of hoar
crystal, it was observed in some cases that their c-axes were at an inclined angle tothat of the substrate, as shown in Figs. 27",29.
Since a prismatic surface of hoar crystal with skeleton structure is composed ofbasal and prism planes of ice as shown in Fig. 30, it can be expected that freezing(bl(a)

Fig. 27.

Fig. 28.

X12Fig. 29.

X 12

X 14(0001)

7/77/777//m~_~"-'/"7//"7/'"'r/-r/-:r-/"'-'1I'(1010)~9///?

o ---7> ca
Fig. 30.

Step of a prismatic surface of hoar crystal with skeleton structure

91

ORIENT ATION OF FROZEN DROPLETS ON ICEof supercooled droplets would occur on these two crystal planes. Therefore, the lattice
matching was examined between these two planes and others than those described insection III. The prism matching was not so satisfactory, but the basal matching was
fairly good as summarized in Table 1, when the matchings are made as shown inFig. 31.

o,oooooooooooo00o( 1212100(00011Fig. 31. Basal matching with crystal planes of ice
Table 1. Basal misfit and angle between c-axes
CrystalplaneoLa(%)

eo( 0)

CrystalplaneeoOlla(%)(0)(1010)6.0

90.0(1210)- 2.0

90.0(2021)

2.5

75.2(2421)-2.3

72.8(lOll)+ 6.7

62.1(1211)-4.0

58.5(1012)

8.3

43.3(1212)+ 3.0

39.3(1013)+ 0.34+ 10.6+ 6.9

32.2(1213)

- 2.5

29.1

25.2(1214)

22.2

20.4(1215)+ 7.9+ 5.2(1014)(1015)

18.1

Figure 32 is the histogram of the measured angle between the c-axis of the subst~ate and that of ice crystals grown on one plane parallel to the a-axes of the substrate.

It can be seen in this figure that the peaks of the measured angle correspond to basal

92

K. HIGUCHI AND T. YOSIDA(J)...J25

10 Vr<)

C\lC\l NNIC\loIN

11 1 1 11~I(J)->-1;-~ 20w0
lL.

0

150::
W
CD:::E::>z15----r-5;-

f-oonir-;-60

11 90

ANGLE BETWEEN

C-AXES ( 0)fT30

Fig. 32. Histogram of angle between c-axis of ice crystals and that of substrate

matching with (1211), (1213) and (1215). However, it might be necessary for this problem
to be clarified further, since the correspondence as described above is not clear cut.

V. Conclusion
In conclusion, it may be said that crystallographic relations exist between thesecondary branches and their base crystal, as well as betw;een ice crystals grown fromfrozen droplets and their substrate. This result seems to support a formation mechanism of secondary branches in which the frozen droplet accreted with snow crystals
grew as a branch, such as suggested by Hallett. However, this paper is concerned onlywith simple crystallographic relations; the use of a polarizing microscope as attemptedby Magono and Suzuki (1964) for observing snow crystals might be used in the future
for studying more complicated relations.

References
1) BENTLEY, W. A. and HUMPHREYS, W. J. 1931 Snow Crystals, McGraw-Hill Book Co.,New York; 1962 Dover Publications, Inc., New York, 210 pp.
2) HALLETT, J. 1964 Experimental studies on the crystallization of supercooled water. J.
Atmos. Sci., 21, 671-682.
3) HIGUCHI, K. 1957 Determination of the crystal face of ice by projection method. J.
Meteorol. Soc. Japan., Ser. II. 35, 254-256.
4) KLINOV, F. lA. 1960 Voda v Atmosfere pri Nizkikh Temperaturakh, Acad. Sci. USSR,
171 pp.

ORIENTATION OF FROZEN DROPLETS ON ICE

935) KUMA!, M. 1964 A study of ice fog and ice-fog nuclei at Fairbanks, Alaska. Part 1.
CRREL Res. Rept., 150, 1-27.
6) MAGONO, C. and SUZUKI, S. 1964 The observation of snow crystals by the use of polarizing microscope. Presented to the Annual Meeting of the Japan Meteorological Society.
7) NAKAYA, U. 1954 Snow Crystal, Natural and Artificial, Harvard University Press, Cambridge, 308 pp.