Journal of the Physical Society of Japan
Vol. 75, No. 10, October, 2006, 104802#2006 The Physical Society of Japan

Shape Selection of Kompeitoh
Isamu S AKAI and Yoshinori HAYAKAWA
Department of Physics, Graduate School of Science, Tohoku University, Sendai 980-8578(Received May 2, 2006; accepted July 26, 2006; published October 10, 2006)

We carried out experiments on the collective crystal growth of sucrose into a form of candy known as
Kompeitoh. We examined the process for the selection of granule size and the number of spikes formedon the crystal surface. We found that the characteristic size of spikes was proportional to the size of
granules in the steady-state regime, although the proportionality coeﬃcient was dependent on conditions
such as the initial shape of the crystals. A selection process of surface structure during the crystallization
appears to have been caused by the distribution process of sucrose solution during the mixing of granules.
In addition, a phenomenological model of surface growth based on the obtained experimental results ispresented herein.
KEYWORDS: crystal growth, dendrite, sucrose crystal, granule, diffusion-limited growth
DOI: 10.1143/JPSJ.75.104802

1.

Introduction

Kompeitoh is a common type of Japanese candy made ofsugar (sucrose) crystals that has been produced for centuries.
Kompeitoh is traditionally produced by a standard method,which is described below. First, small seed granules, such as
sesame seeds, are placed in a large rotating pan that is heated
to an appropriate temperature and warm syrup is then added.
Highly viscous syrup is distributed over the granules throughthe mixing process, and the sucrose crystallizes as the water
evaporates from the surface of the syrup. Several days are
required to obtain large sucrose crystals having a diameter ofapproximately one centimeter. Remarkably, several spikelike structures emerge autonomously during crystal growth.
A number of scientiﬁc reports (mostly in Japanese) describing the characteristic shape of kompeitoh, as shown in
Fig. 1, have been published.1–7)
The growth process of kompeitoh has a few interesting
characteristics that may diﬀer from ordinary crystal growthin the liquid or vapor phase. Throughout growth, viscous
sucrose solution (syrup) is supplied from outside so that thesurface of the sucrose crystal is covered by a wetting layer.
To consider the boundary conditions for the growinginterface, we therefore may need to account for the factthat the ﬂuid motion aﬀects both heat and material transportson the crystal surface.
As another feature of this system, the mixing of granules
with avalanches constantly occurs, while there also exists anapproximately closest packed region beneath the ﬂowing
region. Therefore, we may not be able to assume theuniformity and steadiness of the environment surrounding
the crystals, and the interactions among granules may play
an important role.
From the viewpoint of the physics of pattern formation,some authors have suggested that the similarity between
kompeitoh growth and diﬀusion-limited aggregation(DLA)8,9) originates from the symmetry-breaking mechanism during crystallization.3,5) However, as described above,
simple diﬀusive transport and aggregation processes may
not be appropriate to describe kompeitoh growth. For
instance, in a pioneering study on kompeitoh, Fukushima2)E-mail: hida@phys.tohoku.ac.jp

pointed out the existence of regular crystal shapes with
deﬁnite numbers of spikes, typically 27 or 28, which could
not be explained by a DLA-like growth model.
In dendritic crystal growth, the capillary length for thesolid-liquid interface is typically on the order of nanometers,
and resulting crystals correspondingly exhibit ﬁne micrometer structures. In contrast, the characteristic spacing of
spikes observed in kompeitoh becomes much larger than thislength scale, and is typically on the order of millimeters. The
simpliﬁed boundary conditions assumed in the study ofordinary dendritic crystals with the Gibbs–Thomson eﬀect
may not be suﬃcient to explain the morphology selection
for kompeitoh. To explain the large-length-scale instability
seen in icicle growth, a theory that takes into account for theheat transport through laminar water ﬂow on an ice crystal
surface has recently been proposed.10–12) These studies
suggest that the hydrodynamic eﬀects of a thin liquid layer
may change pattern selection markedly, although the growth
process seems to be very diﬀerent from that of the present
subject.
In this paper, in an attempt to clarify the relevant
physical process in this seemingly complex system, wereport the experimental results for shape selection during the
growth processes of kompeitoh. In the following section, we
brieﬂy describe the experimental procedures and importantparameters. In §3, we summarize the results of theseexperiments. In §4, we present an interpretation of thegrowth process of kompeitoh on the basis of our experiments, followed by a phenomenological dynamical model of
the wet interface.
2.

Experimental Procedure

The crystal growth is performed in a cylindrical copper
drum having a radius of 12 cm and a width of 12 cm. The
axis along which the container rotates is tilted fromapproximately 0 to 5 deg from the horizontal plane to avoid
the segregation of granules into diﬀerent sizes. The container is heated from the bottom using an electrical heating
device, and the temperature of the container can be changed
using a current controller attached to the heater. A liquid
sucrose solution is delivered from a reservoir into thecontainer through a thin tube, and drops of the solution fallfrom the outlet of the tube located near the axle of rotation

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I. S AKAI and Y. HAYAKAWA

a

c

b

Fig. 1. Samples of kompeitoh. (a) The growth times are 4, 10, and 18 h from left to right, respectively. (b) Cross section of kompeitoh. Sucrose solutions
with/without dye were added alternately for 2 h. The white area at the center is the seed particle, which consists of tapioca starch. (c) Single larger seedparticle (left) added to granules of smaller size (right). Samples are taken 4 h after the start of growth.

down to the spherical crystals. The ﬂow rate can becontrolled within the range from 10 to 500 cm3 /h.
The controllable parameters in our experiments are theﬂow rate of the sucrose aqueous solution Q (cm3 /h), theweight concentration of the solution C (g/cm3 ), the temperature at the copper side wall of the container T ( C), and therotation speed of the container ! (rpm). The experiments arecarried out at room temperature, 20  3  C.
The typical single time course of an experiment takes
about 18 hours as described in the following:(1) Place the seed granules in the rotating container andpreheat them for approximately 30 min. In most cases,
spheres of tapioca starch of approximately 3 mmdiameter were used as the seed granules. The numberof initial granules was approximately 1,500, whichcorresponds to a bulk volume of 40 cm3 . In addition,
we tested various types and sizes of seed granules.
Inside the rotating container, we observe a periodic
avalanche of granules on the surface of the granularpile that occurs at intervals of approximately 1 s (for! ¼ 3 rpm), which become shorter as ! increases.
(2) Start supplying the sucrose solution. At the very
beginning, some of the seed granules stick together
104802-2

to form clusters. To avoid continued adhesion of theseparticles, we stir the particles with a rod during the ﬁrst
stage of growth. Thus, the resulting sucrose crystals
contain only one seed granule.
Observations revealed that the surfaces of thesucrose crystals were covered by a layer of sucrose
solution. Although the thickness of the layer could not
be measured with the present experimental setup, thecharacteristic thickness was apparently less than submillimeters from visual observation. If the surface ofthe granules is dry owing to insuﬃcient sucrose
solution supply, the spherical shape remains stable
and spike-like undulations are not observed. Sincedrops of syrup are supplied from a ﬁxed position inside
the container, only a very small number of granules
near the top layer of the pile are subject to wetting at
the same time.
To maintain constant the amount of sucrose solution
supplied to the unit surface area on the granules, we
increased the ﬂow rate of the solution as a function oftime t (min) according to the change in the total surface
area. Otherwise, the surface conditions of the crystal
may have changed drastically during the experiment.

J. Phys. Soc. Jpn., Vol. 75, No. 10

I. S AKAI and Y. HAYAKAWA

3.

Experimental Results

The characteristic spike-like shape of kompeitoh is seen to
be robust against the change in experimental parameters.
Within the ranges of ﬂow rates corresponding to 0:2 <  <2:4 cm3 /h3 in eq. (1), container temperatures of 55 < T <120  C, and sucrose concentrations of 30 < C < 70 wt %,we found that the initially rounded surface became unstable,
allowing the formation of spikes. As a standard condition,
we set the rotation speed of the drum to be ! ¼ 3 rpm.
If ! is small or zero, particles form clusters that consist of
several seed particles. Consequently, without suﬃcient
mixing due to the rotation, separated granules would not
be formed. For ! > 3, aggregation of crystal particles was
not seen except in the very early growth stage. The rotation
speed appears to aﬀect the stability of the crystal surface as
well. For ! ¼ 3 rpm, initial spikes were observed approximately one hour after the sugar solution supply was started.
When ! was doubled (6 rpm), the ﬁrst instability occurred
after approximately 15 minutes. For larger !, a portion of
the granules remained spherical without spikes, and weeventually obtained inhomogeneous granular crystals having
large deviations in size and shape. For ! > 4:5, crystals
segregated into diﬀerent sizes in the container, and we founda broad distribution in diameter.
Figure 1(a) shows the typical shape of crystals at diﬀerent
growth stages. As discussed in the following, there is atendency for the number of spikes to be reduced as the
crystal grows. Figure 1(b) shows the cross section of acrystal formed over two hours that was alternatively coloredwith dye.
Figure 2 shows the distance between nearest spikes at the
point when tiny spikes ﬁrst appear. As seed particles, weused several types of approximately spherical objects,
including beans. Note that the diameter of the granules
plotted in the ﬁgure does not indicate the initial size of theseeds, but rather the diameter at which the instability is ﬁrst
seen. There is an apparent proportionality between the

λ = 0.2d − 0.1

λ : Distance (mm)2.5Adzuki beans

2

Soybeans

1.5

Tapioca (large)

Tapioca (small)

1
Glassbeads

0.5
0Sugar particle

0

2

4

6

8

10

12

14

d : Diameter (mm)Fig. 2. Spacing of initially observed spikes for diﬀerent sizes of seedparticles.

diameter d and the selected interspike distance , and fromﬁtting the data, we found that  ’ 0:20d. If we assume thatthe spherical surface is divided into equitriangles having
vertices at the positions of the spikes, the number oftriangles Nt required to cover a sphere of diameter d isestimated as Nt ’ 7:25ðd=Þ2 . When each vertex is sharedby an average of six triangles, the number of spikes Ns isgiven by Ns ¼ 3Nt =6. Using this formula, our estimation of
the initial number of spikes is approximately 90, which ismuch larger than that observed in the later stage of growth.
Figure 3 shows the distribution function of Ns for the
randomly sampled crystals at diﬀerent times. The number of
spikes constantly decreases to a certain number, and the
distribution becomes sharper with growth. This is thegeneral tendency for spherical seeds other than sucrose
single-crystal of cuboidal shape. As shown in Fig. 4, the size
distribution also narrows at the same time.
Although sharpening of the distribution is clear, thenumber of spikes that eventually form may change depending on the experimental conditions. In addition to 24 spikes,
we also observed the 20-spike case, as shown in the inset of
Fig. 3. Furthermore, for the case of starting from a sucrose
single crystal, which is common for commercially available

Frequency (%)

The desirable condition is, therefore, QðtÞ=SðtÞ ¼
const., where SðtÞ is the total surface area of crystals.
Since the total volume VðtÞ of crystals increases as
dV=dt  QðtÞ, assuming a similar growth process V 
S3=2 , QðtÞ needs to satisfy Q1=2 dQ=dt  Q. As thesolution of this equation, we used the following
quadratic form as the sucrose supply:2QðtÞ ¼  t þ  ;ð1Þ
R0
where  is determined such that  QðtÞ dt is equal to
the seed volume equivalent to the sucrose crystal
volume, assuming perfect spherical growth. The typical growth speed of the surface was approximately
1 mm/h.
(3) During crystallization, periodic avalanches of particles
occurred so that the particles collided and changed
mutual positions. Depending on the number of seedparticles and the growth rate, the total volume of thepile of sucrose crystals may exceed the limit of thecontainer. In such cases, we remove some of thecrystals to prevent the container from overﬂowing. In
addition, we remove a small portion of the crystals foruse as samples for measurement.

20 10 hours

4 hours20

18 hours

10

15

0

38

48

24
30
36
42
:
Numberoftips
perparticle
NS

48

18

28
NS

10
5
0

18

Fig. 3. Distribution of number of spikes during diﬀerent growth stages (4,10, and 18 h). The inset shows the plot for the same experimental
conditions at 18 hours, which has a diﬀerent peak position (20). Morethan 100 samples were used to make these plots.

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I. S AKAI and Y. HAYAKAWA

d2 =d1 , is maintained constant at approximately 0:8.2)
We also conﬁrmed that this ratio becomes constant during
the later stage of growth. However, the measured valueof the ratio in the present experiments was approximately
0:6, which appears to be signiﬁcantly smaller. This discrepancy may have arisen owing to insuﬃcient growth ofthe crystals used in ref. 2 to discuss a ‘‘self-similar’’ regionin which the shape and number of spikes are selected.

30
4 hours
10 hours
18 hours

Frequency (%)

25
20
15
10

4.
5
0

4

6

8

10

12

14

16

d : Diameter (mm)Fig. 4. Distribution of tip-to-tip diameter of samples for data shown in
Fig. 3.

3.5

λ : Distance (mm)32.5

α = 0.2
α = 0.5
α = 1.5

2
1.5
1
0.5
0

0

1

2

3
4
5
d : Diameter (mm)67Fig. 5. Tip-to-tip distances  of spikes with respect to crystal diameter dduring growth starting from seeds of 3 mm diameter. Data for diﬀerent
ﬂow rates of sucrose solution are plotted in the same plane. The solid line
represents the –d relation for the case in which each spike is located atthe vertex of the dodecahedron.

kompeitoh candy, the selected number becomes smaller than
20.6) These numbers are signiﬁcantly smaller than thosereported in ref. 2.
In Fig. 5, we plot the interspike distance during growth fordiﬀerent ﬂow rates of sucrose solution. In the plot, the data
for the very early growth stage are excluded. The ﬂow rate Q
has only a minor eﬀect on the shape selection process. Since
the ﬂow rate exhibits an asymptotically linear relationshipwith diameter, from the overall slope of the plots, we canestimate that an average of 20 spikes will be selected bycomparison with the intervertex distance versus diameter
relation of dodecahedron with 20 vertices (solid line in
Fig. 5).
To measure the crystal size and irregularity, it isconvenient to introduce two crystal diameters: the ﬁrst
diameter is that of a circumscribed sphere of the crystal, d1 ,and the second diameter is that of the ‘‘inner’’ basesphere upon which spikes are formed, d2 . On the basis of
the measurement of commercially available samples,
Fukushima claimed that there is a steady-state growth
regime in which the ratio of the radii of the two spheres,

Discussion

4.1 Growth process
In our experiments, we could not ﬁnd a ‘‘magic number’’of spikes to be ﬁnally selected. In contrast, the selected
number of spikes appears to be aﬀected by subtle changes inthe experimental conditions. However, there is an evident
tendency in the selection process of the number of spikes, aslong as the conditions are such that near monodispersity of
the granules is maintained (Figs. 3 and 4). That is, the initialbroad distribution is focused into a single peak throughgrowth. Therefore, we expect that the interaction of granules
during crystal growth is crucial to spike formation and thatthe number and spacing of spikes are determined collectively through the growth.
In support of this theory, we observed that a similarcorrugated surface of sucrose crystals with spikes was
also deposited on the cylindrical wall of the container when
the sucrose drops happened to condense on the hot surface.
The spacing of the spikes became identical to that of
the granular crystals inside.
Furthermore, we intentionally added single particles oflarger diameter to the granules of standard size. Spikes
formed on the larger seeds in a manner such that the
spacing of the spikes was ‘‘duplicated’’ as a result of smaller
granules surrounding the larger seeds [see the photograph inFig. 1(c)], although the spacing of the spikes was proportional to the particle diameter if the process was started frommonodispersed granules, as shown in Fig. 2.
From the above observations, the kompeitoh growth
process is described as follows:(1) At the very early stage, instability ﬁrst occurs due tothe crystallization of the sucrose solution that isdelivered onto the surface by contact processes. Since
the distance between initial spikes is proportional to thediameter of the granules (Fig. 2), the contact process
and/or packing of granules, rather than intrinsic
instability mechanisms of crystal growth, such as the
Mullins–Sekerka instability, is inﬂuential on theunstable wavelength.
(2) The protruding surface has a greater chance ofacquiring sucrose solution from other granules so thatthe growth of such parts is accelerated. It is thoughtthat the instability is also promoted by a higher
evaporation rate of water at the convex superior
region. On the other hand, the tips of the spikes also
have a greater chance to provide solution to othergranules. It is very likely that the exchange of ﬂuid
takes place simultaneously at more than one contact
point because the granules are densely packed. As
a consequence, spatial correlation of the crystal
surface is also transferred during the contact process,which will bring about an autocatalytic eﬀect, exhib-

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I. S AKAI and Y. HAYAKAWA

iting the focusing eﬀect mentioned in the previous
section.
(3) During crystal growth accompanied by mixing, thedistance between spikes, as well as the diameter ofparticles, is regulated. There is a tendency for thenumber of the spikes to decrease gradually. After thetransient growth stage, the system enters a self-similar
growth regime, where the shape of crystals remains thesame and the number of spikes has a sharp single
distribution.
In a diﬀusion-limited growth process with some stabilizing eﬀects, the stability of the moving interface is expected
to depend on the growth speed.13,14) On the other hand, on
the basis of Fig. 5, the characteristic length scale to be
selected is independent of the average growth rate of theinterface, which is proportional to the ﬂow rate Q. We
therefore conclude that, at least, for the selection of the spikepattern, the transport of heat and vapor is not the governing
process.
Kametani and Yamauchi reported the appearance of
crystals similar to kompeitoh during the electrodepositionof nickel in a vibrating bed cell using a suspension
electrolysis technique.15) Despite the diﬀerences in thematerial properties and growth mechanism, they obtained
very similar metal objects, typically having 28 to 30 spikes.
This may not be a coincidence if the interaction of granules
is crucial to pattern formation.
4.2 Phenomenological model of wet surface growth
Next, we consider the phenomenological model for thesurface growth in terms of the interface equation of motion
based on the above discussions. Upon contact a wet granule,a portion of viscous liquid is delivered near the contact
point. Since the concentration of the solution is chosen to benear saturation, crystallization can occur quickly, upon water
evaporation from the interface. From conservation of
sucrose, if the wetting layer is suﬃciently thin, the underlying solid–liquid interface will follow the liquid–vapor
interface.
During the condensation, we can expect the ﬂuid motion
to be driven by surface tension. Imagine that the surface ofthe sucrose crystal is covered by an added wetting layer ofcharacteristic thickness w, the initial shape of which is given
by an adhesion and peering process of liquid during mixing.
For the case in which w is small and changes very slowly inspace, the local pressure P is determined by the Laplace lawasP  P0 ¼ H;

ð2Þ

where H is the mean curvature of the liquid–vapor interface,
P0 is the constant atmospheric pressure, and  is the surface
tension of the interface.
On the solid–liquid interface, the nonslip boundary
condition is appropriate, whereas the liquid–vapor boundaryis stress-free. Therefore, if the change in w is suﬃciently
small, by solving the Navier–Stokes equation for the highviscousity limit, the velocity, u, for uniform ﬂow along theinterface becomes
 2


z
u ¼ rH wz ;
ð3Þ2

where  is the viscosity. Taking the average in the zdirection, the ﬂuid motion along the interface is described by
Z
1 w
w2 
uðzÞ dz ¼ 
rH:
ð4Þ
u ¼
w 0
3
For simplicity, let us consider the growth of the wet planar
surface with small amplitude undulation. From the conservation of ﬂuid, the height h of the liquid-vapor interface
obeys @h=@t þ r  ðwuÞ ¼ 0. Using the above expression ofu,@hw3  2
¼
r H;
@t
3

ð5Þ

where H ’ r2 h in the leading order. This is similar to thesurface diﬀusion for microscopic dynamics on solids at ﬁnite
temperature.
It is plausible to assume that extruding a portion of thesurface into vapor provides a larger growth rate. One reason
for this is an excluded volume eﬀect caused by ﬁnite-sizedsucrose crystals. When the growing interface is surround by
spherical crystals of diameter a, one can virtually draw aneﬀective surface extended by the thickness a. If the granules
distribute uniformly on the eﬀective surface, the probabilityof occurrence of the contact process between a perimeter site
and the crystals increases by a factor of ½ðR1 þ aÞðR2 þ aÞ=
ðR1 R2 Þ ’ 1 þ aðR1 þ R2 Þ=ðR1 R2 Þ ¼ 1 þ 2aH, where Rx is
the radius of curvature in the principal direction at theperimeter, as long as a
Rx . Thus, if crystallization islimited owing to the supplement of solution throughcollisions, the growth velocity of the curved interface
having mean curvature H is modiﬁed by the factor 2v0 aH,
where v0 is the velocity of the interface. Even in the case of
a ’ Rx , this tendency remains, and the lowest relevant term
has the same form as c2 H, where c2 is a constant that is
proportional to v0 . In general, we can assume c2 ¼ v0  using
a characteristic length , which depends on the origin of theinstability.
Another possible reason for the instability of the curved
front is the ﬂux nonuniformity of the water evaporation
process on an irregular surface, which causes ‘‘Laplacian
instability’’. For small-amplitude surface ﬂuctuations, thiscan be also represented in the form c2 H, although thederivation of the explicit form of c2 may not be straightforward.
Summarizing the above arguments with the ﬁrst nonlinear
correction term for an inclined surface with average growthvelocity, the height h of the growing surface in the average
growth direction will obey the following:


@h
1
2
4
2
¼ c2 r h  c4 r h þ g 1 þ ðrhÞ þ F; ð6Þ@t2
where c2 ¼ v0 , c4 ¼ w3 =3, g ¼ v0 , and F represent theexchange of sucrose solution, which is dependent on space
and time. This equation is simply the Kuramoto–Sivashinsky(KS) equation16) with an external forcing term.
In the study of the microscopic surface growth, Cuerno
et al.17) introduced a KS model with a random noise term
and found a crossover from a periodic surface structure to an
irregular surface structure. For the case of kompeitoh, Fshould have correlation in space according to the shape andpacking structure of the granules that deliver the liquid. We

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I. S AKAI and Y. HAYAKAWA

need to derive a self-consistent formula that includes theliquid exchange process in conjunction with eq. (6). This is a
topic for future study. Since the KS equation has a broaddispersion relation for long wavelengths, collective forcing
around a wavelength may cause reinforcement and focusing,which leads to the selection of spikes other than those at themost unstable intrinsic mode. As a preliminary study, in the
one-dimensional case, we carried out computer simulationsof eq. (6) with a sinusoidal modulation Fðx; tÞ ¼  sinðk0 x þðtÞÞ, where ðtÞ is a temporally independent noise in thephase and  is a small number. We found that k0 , rather than
the intrinsic unstable mode k ¼ ðc2 =2c4 Þ1=2 , is ﬁnallyselected after a transient time. We expect that this softly
unstable nature of the wetting interface is intrinsicallyimportant for shape selection through an adhesion andpeeling cycle of sucrose solution accompanied by deposition.
Acknowledgment
The authors would like to thank Dr. T. Kawakatsu, Dr. T.
Hondou, and Dr. N. Uchida for helpful discussions andcomments. This study was supported by a Grant-in-Aid for

Scientiﬁc Research (#15654050) from the Ministry of
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