Roll of Grain-boundary Free Energy & Surface Free Energy for Tin Whisker
GrowthKiyotaka Tsuji
Ishihara Chemical Co., Ltd.
5-26, Nishi-yanagihara-cho, Hyogo-ku, Kobe 652-0806, Japan

Abstract
Generally the grains of electroplated tin layers are considerably small in their sizes. Therefore the grain-boundary
free energy is expected to act as a driving force for recrystallization and possibly subsequent tin whisker growth. On
the other hand, if the dimension of a structure is sufficiently small like whiskers, then the morphology of thestructure will be determined principally by the tendency to lower its surface free energy. Hence it is supposed thatthe surface free energy may be closely related to the whisker growth process.
In this work, the grain-boundary free energy of tin electrodeposits has been estimated by determining the grain sizes
as a function of the thickness. Then the morphology of various whiskers and so-called nodules, which are theirregularly shaped mass-extrusion from the tin surface, has been studied in order to reveal the roll of surface freeenergy in the whisker growth process. While the relation between the nodule growth and the grain growth, which isa stage of recrystallization, has been also studied by electron backscatter diffraction pattern method.
It has been shown that the grain-boundary free energy is sufficient for spontaneous occurrence of grain growth of thedeposits and the grain growth lead to the nodule growth. While many whiskers were found to grow on nodules. Thismeans the whisker growth is closely related to the nodule growth. From these, it has been suggested that the whisker
growth can be understood as a result of minimization of the surface free energy of nodules, which have large surface
free energy as a result of their complicated surfaces. Farther more, it has been suggested that the minimization canbe achieved by growing in some specific directions such as <001>, <100>, <101>, and <111>.
1. Introduction
It is well known that filamentary shaped whiskers growspontaneously from electrodeposited tin layers at room
temperature. The tin whiskers show unique property,
e.g. they consist of single crystals of β-tin growing inseveral specific directions 1, 2 and include an extremely
small number of dislocations 3. Their cross sectional
shapes are constant throughout almost whole length.
Stored strain energy in deposits 4, 5 or compressive
stress, induced by formation of intermetallic compoundat the interface between deposits and substrates 6, 7,
were considered to be the driving force for the whisker
growth. Regarding growth mechanism of whiskers,
several models were proposed. J. D. Eshelby 8, F. C.
Franc 9, U. Lindborg 10, and B. Z. Lee et al. 7 attempted
to explain the whisker growth by dislocation movement.
While, W. C. Ellis et al. 11, and N. Furuta et al. 12
proposed the models based on recrystallization.
However we have not yet achieved complete
understanding of this peculiar phenomenon at thepresent.
Generally the grain sizes of electrodeposited tin layers
are extremely smaller than that of bulk material.
Therefore grain-boundary-free energy of the deposits is
expected to act as the driving force of the whisker
growth. In this work, to reveal the roll of thegrain-boundary-free energy on whisker growth, it has
been estimated by determining the grain sizes as afunction of the thickness. Then the morphology ofvarious whiskers, including the fractured section of
their roots, and so-called nodules, which are theirregularly shaped mass-extrusion from the tin surface,has been studied to understand whisker growth process
in detail. If the dimension of a structure is sufficiently
small like whiskers, the tendency to lower its
surface-free energy is the principal motivation forchanges in structures 13. From this perspective, we havediscussed whisker growth mechanism.
2. Experimental Procedure
2.1 Electrodeposition of Specimen
Three different types of plating solution, shown in
Table 1, were used for preparing specimens forobservation of whiskers and nodules. Matt Sn, bright
Sn and matt Sn-0.5%Cu were electrodeposited on
copper substrate. (Percent in this paper means masspercent, unless otherwise stated.) Matt Sn-0.5%Cu wasused for acceleration of the whisker growth as reported
in literatures14, 15. The thickness of the deposits wasmeasured with an X-ray fluorescent coating thickness
gauge. The specimens were stored at room temperatureand 100 oC to grow whiskers and nodules for 751 days
at maximum.
2.2 Determination of Grain Size
The grain sizes were determined as a function of the
thickness of deposits. For matt Sn and matt Sn-0.5%Cu,
a line of certain length l was drawn arbitrarily on thescanning electron microscope (SEM) image of grains,
and the number n of the grains located on the line wascounted. Then l/n gives the average grain size d. For
bright Sn, specimens were etched with an aqueous
solution of 100 g/L ammonium persulfate prior to theTable 1 Composition of plating solution and plating conditions.
Items

Matt Sn

Bright Sn

Matt Sn-0.5%Cu

2+

20

20

59.3

Sn (g/L)*2+0.75

Cu (g/L)**Free Acid (g/L)***

100

100

140

Additives commercially provided were used.

Additive
Current density (A/dm2)oTemperature ( C)

2

2

2

25

25
Copper plates (25x25x0.3mmt)
Copper plates (5.0x0.4x0.25mmt)

45

Substrate material*: Sn(II) methanesulfonate**: Cu(II) methanesulfonate***: Methanesulfonic aciddetermination of grain sizes because the grains were
not recognized apparently.
2.3 Observation of Whiskers and Nodules
Whiskers and nodules were observed by SEM. To
examine the root of whiskers and nodules, specimens
were bent to 180 degree to produce cracks of thedeposits as illustrated in Fig. 1. The fractured sections
at the cracks where whiskers or nodules were expected
to exist were observed by SEM.
Fig. 2 Preparation of the specimen for EBSP.
The polished surface of the specimen was analyzed.
3. Results and Discussion

Fig. 1 Preparation of specimens for observation of
the fractured section of the deposits.

2.4 Determination of Crystal Orientation and Grain
Size at the Root of Nodules
The specimens were polished with colloidal silica
whose particle size was 0.04µm to smoothen thesurface and to reveal the roots of nodules as shown in
Fig. 2. The crystal orientation and the grains at theroots of nodules were examined by electron backscatter
diffraction pattern method (EBSP) with a fieldemission scanning electron microscope (FESEM).
In this paper, “lateral direction” means the direction
parallel to the substrate and “z-axis direction” means
the direction perpendicular to the substrate as
illustrated in Fig. 2.

3.1 Estimation of Grain-boundary Free Energy
Fig. 3 shows the grains of matt Sn of various
thicknesses. Apparently the grain sizes increase withincreasing thickness. Matt Sn-0.5%Cu shows the sametendency as matt Sn. Here note that extremely small
grains appear in the thin deposits.
In Fig. 4, the average grain sizes d of matt Sn, bright
Sn and matt Sn-0.5%Cu are plotted against square rootof the thickness t. As seen in this figure, matt Sn andmatt Sn-0.5%Cu showed good linearity. Then, for mattdeposits this relation can be written byd = k t(Eq. 1)
where k is a constant depending on plating conditions.

The obtained values of k are 1.57 µm1/2 for matt Sn and1.06 µm1/2 for matt Sn-0.5%Cu. Regarding matt Sn, kvaried from 1.5 to 2.0 µm1/2 depending on the Sn2+
concentration and the current density.

(a) 1.0 µm(b) 10 µm(c) 30 µm

Fig. 3 SEM images of grains of matt Sn of various thickness t.
Conversely, the average grain size of bright Sn is foundto be independent of the thickness. Then, for brightdeposits, d can be written by(Eq. 2)d = k′
where k’ is a constant. For bright Sn, k’ was found tobe 0.56 µm.
The difference in these tendencies between matt andbright deposits may be due to the difference in growthmechanism of the deposits. According to A. N.
Kolmogorov 16, Eq. 1 holds in the case ofpolycrystalline growth, which is known as“geometrical selection” as schematically illustrated inFig. 5. In the figure, arrows mean the preferential
growth directions of each grain. Here let θ be the angle
between the preferential growth direction and the z-axis
direction. If θ of a grain is large, then the growth of thegrain by electrodeposition will be obstructed by aneighboring grain whose θ is small. Therefore thegrains having small θ can survive and continue to growdominantly. Hence the number of survived grains
gradually decreases with increasing thickness,consequently those grain sizes become larger.

Fig. 4 Relation between average grain size d andsquare root of thickness t.
In the case of matt deposits, it is supposed thatgeometrical selection may take place in their growth
process because Eq. 1 holds. Meanwhile, for brightdeposits, the grain growth in electrodeposition isstrongly prohibited by adsorption of additives.
Therefore the deposits grow by continuous nucleation
predominantly, and then the grain sizes remain small
and constant throughout the whole thickness of thedeposits.

Fig. 5 Schematic explanation of geometricalselection. Arrows mean the preferential growth
directions of each grain. Let θ be the angle between
the preferential growth direction and the z-axis
direction. If θ of a grain is large, then the growth of
the grain will be obstructed by a neighboring grain
whose θ is small. Therefore the grains having small
θ can survive and continue to grow dominantly.
Here, in order to estimate the grain-boundary-freeenergy per unit volume, Gb, the grains are assumed tobe regular hexagons, which can tessellate the wholesurface of a plane, as shown in Fig. 6 (a), and let thelength of their side be a. Consider the grain boundaries
on a plane P whose area is A, and the plane is parallelto the substrate and located at the distance z from thesubstrate, as shown in Fig. 6 (b). Then the area of agrain is 3 3 ⋅ a 2 / 2 . Two neighboring grains share aside of the hexagons. Therefore the circumference per
grain is 3a, and the number of grains on the plane P is
2 A / 3 3 ⋅ a 2 . Hence the total length of the grainboundaries on the plane becomes 2 A / 3 ⋅ a .

(

)

(

)

Here let us consider the small segment between theplane P and a plane Q shown in Fig. 6 (b). The plane Q

is located at a small distance dz from the plane P. The
area of the grain boundaries involved in the small
segment is given by (2 A / 3 ⋅ a )⋅ dz . Then the total
grain-boundary-free energy involved in this segment
can be expressed by 2 A σ b / 3 ⋅ a ⋅ dz , where σb isthe grain-boundary-free energy per unit area.

()

Fig. 6 Regular hexagon model for a grain, (a), andcross section of a deposit, (b).
Hence the grain-boundary-free energy per unit volume
for matt deposits, G bM , can be formulated by
G bM =

2σ b
3 ⋅a

.

(Eq. 3)

G bB =(4 +)3 ⋅σ b3 ⋅k′=

3 . 31 σ b .
k′(Eq.6)In Eq. 5, if z approaches to 0, then G bM becomes
infinity. However, at the initial stage ofelectrodeposition, it may begin with nucleation even in
the case of matt deposits. After the nuclei havetessellated the whole surface of the substrate,
subsequent grain growth in electrodeposition can takeplace. Therefore the thickness less than the size of a
nucleus is needless to consider.
As we discuss later, it is substantially important forwhisker and nodule growth whether recrystallization
can take place in the deposits. Next, we discuss on thepossibility of the occurrence of recrystallization.
According to K. T. Aust et al. 17, thegrain-boundary-free energy of tin was determined to beapproximately 0.1 J/m2 at 220 oC. We can use thisvalue at room temperature assuming the temperaturedependence of the grain-boundary-free energy is
negligible 18 and also applicable for Sn-0.5%Cu. Fig. 7
shows Gb distribution along z-axis direction of deposits
calculated by Eqs. 5 and 6. In the case of matt deposits,
Gb increases steeply when the distance z is less than 2µm. On the other hand, bright deposits show constant
Gb independent of the distance z. It is said that grain
growth as a stage of recrystallization requires 104 to
6x105 J/m3 for driving force 19. Furthermore it is said
that recrystallization can take place at the temperature
higher than approximately Tm/2, where Tm is themelting point in absolute temperature. In the case of tin,
Tm/2 is 252.5 K (-20.5 oC). Therefore not only brightdeposits but also matt deposits have sufficient freeenergy to initiate recrystallization at room temperature.
If the grain sizes are sufficiently small even in the caseof matt deposits, then recrystallization can take place
without existence of strain energy. Of course, if there
exists strain energy in the deposits, it can also leads torecrystallization.

For bright deposits, because the grain sizes remains
constant throughout the whole thickness, the brightdeposits can be regarded as a layered structure whose
each layer is assumed to be tessellated with hexagonal
prisms. Therefore the grain boundaries parallel to thesubstrate should be taken into account. Assuming theheight of the hexagonal prism to be 2a, thegrain-boundary-free energy of such boundaries per unit
volume is given by σb/2a. Then the total
grain-boundary-free energy per unit volume for brightdeposits, GbB , can be expressed by
G bB =

3) ⋅σ b .

(4 +2 3 ⋅a(Eq. 4)

Meanwhile a equals to d/2 approximately. Then from
Eqs. 1 and 3, GbM at the distance z from the substrate
can be given by
G bM =

4σ b
3 ⋅k

z

=

2 . 31 σ b .
k(Eq. 5)z

From Eqs. 2 and 4, G bB for bright deposits can be
written by

Fig. 7 Grain-boundary free energy per unit volume,
Gb , distribution along z-axis direction of deposits.
It is reported that reflow or annealing of tin deposits
can effectively prevent the whisker growth 20. It can beunderstood that such treatment make the grain sizes
sufficiently large, and then the grain-boundary-freeenergy will be reduced significantly. However the graingrowth by annealing is recrystallization process itself.
If recrystallization is directly related to the whisker
growth, it seems to be a contradiction because
annealing prevents the whisker growth. As we discuss
this later, whiskers cannot be formed at elevatedtemperature even though recrystallization develops
more rapidly as compared with room temperature.
Fig. 8 shows the crystal orientation of each grain formatt Sn-0.5%Cu, stored at room temperature for 314
days, analyzed by EBSP. Original thickness of thedeposits was 4.0 µm, and EBSP analysis was(a) SEM image

conducted after polishing 0.5 µm from its original
surface. The sites of nodules designated as A to E on
the SEM image, (a), correspond to the sites designatedas the same symbols on the inverse pole figure map, (b).
In Fig. 8 (b), black spots are intermetallic compound
between tin and copper, which was verified by energy
dispersion X-ray (EDX) analysis, and othercolor-corded segments are all tin grains. As seen in thisfigure, recrystallized grains are developing underneaththe nodules encroaching surrounding small grains. Thissuggests that a nodule itself may be a recrystallized
single crystal of tin.

(b) Inverse pole figure map
by EBSP(c) Color code of tin
crystal orientation

Fig. 8 Crystal orientation of each grain of matt Sn-0.5%Cu analyzed by EBSP. The electrodeposited specimen
was stored at room temperature for 314 days. Original thickness of the deposit was 4.0 µm, and EBSP
analysis was conducted after polishing 0.5 µm from its original surface.
The nodules B and E show approximately the sameorientation as that of the matrix. On the contrary, thenodules A, C, and D show different orientation fromthose. From these crystal orientations of nodules, itdoes not seem to exist specific growth direction in
nodule formation.
Recrystallization itself cannot be the cause of
mass-extrusion, i.e. the whisker or the nodule growth,because it can develop only in the lateral direction
encroaching the surrounding grains. To extrude massfrom the surface of deposits, tin atoms should flowfrom the surrounding regions to the roots of nodules bydiffusion due to mass-balance. The chemical potentialof tin atoms in the recrystallized grain is expected to beconsiderably smaller than that of un-recrystallized
grains because grain-boundary-free energy and also
possible strain energy are decreased in therecrystallized grain. Therefore this diffusion process

will be driven by this chemical potential difference oftin atoms. The diffusion path may be the interface
between the deposits and the substrate.
Furthermore, in the case of thin deposits, the extrusionof the recrystallized grain will be more advantageous insurface-free energy than the lateral grain growth
without mass-extrusion because the surface area of thegrain in the former case will be less than that in thelatter case. On the contrary, in the case of substantiallythick deposits, the grain growth by recrystallization
will take place within the inside of the deposits withoutmass-extrusion. It is reported that the whisker growth
can be reduced when the deposits are thick 7, 20. Thismay be related to the lower possibility ofmass-extrusion in the case of thick deposits.
The grain growth as a stage of recrystallization is arelaxation process of the grain-boundary-free energy.

However another relaxation process should beconsidered, i.e. the formation of intermetallic
compound (IMC) at the interface. It is reported that thewhisker growth tends to be strongly affected by thekind of substrate materials 20, 21. If thegrain-boundary-freeenergycanberelaxedpredominantly by IMC formation, whisker growth will
be reduced significantly especially in the case of mattdeposits.
As shown in Fig. 8 (b), the grain growth in the lateral
direction seems to be interfered by surrounding IMC
particles although several IMC particles are included in
the grains of nodules. As mentioned previously, copper
accelerates the whisker growth. We found larger
number of whiskers and also nodules on mattSn-0.5%Cu as compared with matt Sn. If the IMC
particles interfere the grain growth in the lateral
direction on the way of growth, the grain will befacilitated to extrude from the surface of the deposits.
Therefore some impurities, such as copper, accelerate
the formation of nodules and subsequently whiskers.
W. C. Ellis et al. 11 and N. Furuta et al. 12 proposed thatwhiskers were formed directly by recrystallization
introducing the immobile and stable grain boundary ofa grain, which acts as the nucleus of a whisker.
However, from the above facts and discussion,recrystallization process can only lead to the formationof nodules. To form whiskers, another stage of progressshould be taken into consideration.
3.2 Relation between Nodules and Whiskers
Fig. 9 shows an example of a whisker that grew on a
nodule formed on bright Sn. Regarding bright Sn,
almost all whiskers were found to grow on nodules.
Generally the shapes of the nodules formed on brightdeposits are extremely complicated. While, in the caseof matt Sn and Sn-0.5%Cu, it was found that whiskers
were growing not only on nodules, Fig. 10, but also
directly on the deposits, Fig. 11. Several nodules
designated as A to D can be seen in Fig. 10 (a), andthey seem to represent early stages of the whisker
growth on nodules. Among them, the nodules A and B
indicate the earliest forms of mass-extrusion followedby the nodules C and D. It can be seen that theseFig. 9 An example of a whisker growing on a nodule.
The specimen was plated with 4.34 µm of bright Sn,
and stored at room temperature for 680 days.

nodules have whisker-like filamentary structures(whisker-embryos) on the tops of them, which may beformed at the earliest stage of mass-extrusion. These
whisker-embryos are expected to grow up to be longwhiskers. In Fig. 10 (b), substantially planar surfaces
can be seen at the whisker-embryo. These planar
surfaces may be low index planes, so-called facets interms of crystallography. It is important to understandthe meanings why these facets are produced, as we
discuss this later.
In the case of directly growing whiskers on the deposits
as seen in Fig. 11 (a), at the tip of the whisker, agranular object can be seen. It seems to be a grain,
which was located at the original surface of the deposit.
Therefore this type of whiskers may be initiated to
grow from the inside of the deposits, probably from thevicinity of interface because there exists considerable
amount of grain-boundary-free energy as shown in Fig.
7. It is supposed that this type of whisker growth may
be related to the recrystallization process hidden in thedeposits. Hence it is suggested that the whisker growthis closely related to the nodule growth, to be precise, tothe grain growth stage of recrystallization.
There exists another important feature of whiskers,
typically shown in Figs. 11 (a) and (b). When thewhisker is thinner, (a), its lateral surface is simple andconsists of only a few facets. On the contrary, when itis thicker, (b), its lateral surface becomes complicatedand consists of many facets. If the diameter of awhisker becomes thinner, it will be required to be asimpler form to reduce its surface-free energy further
more because its surface area becomes larger ascompared with its volume. This means the morphologyof the microstructure, such as whiskers, depends on thetendency to lower the surface-free energy.
Fig. 12 shows the fractured sections of a nodule, (a),and whiskers, (b) and (c). From Fig. 12 (a), the nodule
seems to grow from the interface between the deposit
and the substrate. This seems to support that the
nodules were formed by the grain growth accompaniedwith the interfacial diffusion of tin atoms as discussed
previously. As seen in Figs. 12 (b) and (c), nodules arefound at the roots of the whiskers. These nodules also
seem to grow from the vicinity of interface. Any
whiskers growing without nodules were not found in
this investigation. These facts support the close relation
between nodules and whiskers.
Whiskers and nodules can be distinguished by theirmorphology and surface condition. The shapes ofnodules are very irregular and complicated and theirsurfaces are substantially rough and consist of generalsurface (not specific surface such as facets). While
typical whiskers are surrounded by regular and smooth
facets on their lateral surfaces. From this, it issuggested that whiskers may grow on nodules as aresult of the tendency of reducing surface-free energy
of nodules. In other words, the driving force for theformation of whiskers from nodules is supposed to be
the surface-free energy difference between whiskers
and nodules. In this case, strain energy cannot be thedriving force because nodules are already recrystallized
structures.

(a) Matt Sn

another nodule, such as segment E. However the
residual segment, from B to D, may be formed bydifferent growth mechanism. At the present, a grainboundary exists at D, which separates the nodule andthe whisker (N/W boundary). It is supposed that thisN/W boundary might have been migrating from B to D
in the past continuously creating new configurations oftin atoms at the whisker side of the grain boundary.
Therefore the shapes of the N/W boundary in the past
are recorded on the lateral surface of this whisker. It
can be seen that the lateral surface becomes graduallysmoother to reduce the surface-free energy as thediameter of this whisker becomes smaller. In this case,the force reducing the diameter of the whisker may be
the grain boundary tension, equivalent tograin-boundary-free energy, at the N/W boundary, andthe driving force for the migration of the N/Wboundary is supposed to be the surface-free energy
difference between the nodule and the whisker.

(b) Matt Sn-0.5%Cu(a)(c) Matt Sn-0.5%Cu
Fig. 10 Examples of whiskers growing on nodules
formed on (a) 1.85 µm of matt Sn, which was stored
at room temperature for 751 days, (b) 2.39 µm ofmatt Sn-0.5%Cu stored at room temperature for732 days, (c) 3.96µm of matt Sn-0.5%Cu stored atroom temperature for 732 days.
3.3 Whisker Formation Process
Fig. 13 shows a whisker growing on a nodule formedon bright Sn. To be precise, it is not a typical whisker
because its diameter is not constant. This type ofwhiskers can be often seen in bright deposits. From A
to D, designated in the figure, this whisker may consistof one or two single crystals because there may exists a
possible grain boundary at C. The segment, from A to
B, may be formed by general nodule growth process. It
seems that this segment was originally located on(b)Fig. 11 Examples of whiskers growing directly on
6.03 µm of matt Sn-0.5%Cu stored at room
temperature for 66 days.

phenomenon is known as “faceting” 13, 23. The facets,
which can be seen in Fig. 10 (b) on a whisker-embryo,
may be formed by such faceting.
Generally facets have cusped minima in surface-freeenergy 13, 22. Fig. 14 explains schematically suchcusped minima, e.g. {100} planes, in the case of tin.
The facets are stable in terms of stiffness due to suchnature of the surface-free energy. It means that thewhiskers whose lateral surfaces consist of facets cangrow only in the direction maintaining the planes of the
facets themselves. This can explain the reason why thecross sectional shapes of whiskers remains unchanged
throughout almost whole length.
(a) Fractured section of a nodule.

(b) Fractured section of a whisker.
Fig. 13 A whisker growing on a nodule. The
specimen was plated with 4.34 µm of bright Sn
stored at room temperature for 664 days.

(c)Fractured section of a whisker.
Fig. 12 Fractured sections of a nodule, (a), andwhiskers, (b) and (c), formed on 4.45 µm of mattSn-0.5%Cu stored at room temperature for 689
days.
If this whisker continues to reduce its diameter still
further, then it will be forced to create facets on thelateral surface. In the case of Fig. 13, facets have notbeen created. However, it should be mentioned, thegrowth process of this type of whiskers suggest thepossibility that the migration of the N/W boundary
driven by surface-free energy difference may lead tocreate facets on the lateral surfaces of whiskers. Theremay be another possible way for creating facets. In acertain atmosphere and at certain temperature range,
the surface of certain metals tends to decompose to
produce facets and rather complex index planes inorder to reduce total surface-free energy, and this

It is reported that the growth direction of tin whiskers islimited to several directions, i.e. <001>, <100>, <101>and <111> 1, 2. W. C. Ellis 1 considered this fact related
to the slip directions in plastic deformation of tin single
crystal. However <100> direction is not included insuch slip directions 24. There should be different
meanings of the directions. It must be emphasized thatthese directions have zone planes, listed in Table 2,
whose surface-free energy is expected to be in cusped
minima. The zone planes are parallel to each growth
direction; therefore they can compose the lateralsurfaces of the whisker in the form of facets. It should
be noted that minimization of surface-free energy canbe achieved by growing in those specific directions.
When the temperature is elevated near the melting
point of the metal, the cusped minima in surface-freeenergy will become shallow because the entropy termof the free energy cannot be negligible at elevatedtemperature. Therefore facets cannot maintain theirstability at such temperature because of so-called“thermal roughening”. Fig. 15 shows the shapes ofwhiskers formed at 100 oC. It can be seen that thelateral surfaces of the whiskers are not planar any
longer and extinguish their stability. This experimental
result supports the above expectation. Here we canconclude that the stable filamentary shapes of whiskers
are due to the stability of the facets created on theirlateral surfaces.

important nature of whiskers, e.g. the growth directions
and the stability of the cross sectional shapes, andtemperature dependence on growth can be understood.

(a)Fig. 14 Schematic diagram of the surface-freeenergy σ(θ) of β-tin in the direction perpendicular
to [001] axis. The surface-free energy σ(θ) is afunction of angle θ. Cusped minima can be seen in<100> directions.

Table 2 Growth directions of tin whiskers and theirzone planes.
(b)Growthdirection of tin
whisker.

Zone planes.

<001>{100}, {420}, {110}<100>{001}, {101}, {301}, {100}<101>{100}, {411}, {211}, {101}<111>{110}, {321}, {211},{312}, {101}, {112}

4. Conclusion
It has been shown that the grain-boundary-free energy
is sufficient for spontaneous occurrence ofrecrystallization of the layer because of theirsignificantly small grains. While many whiskers were
found to grow on nodules, this means the whisker
growth is closely related to the nodule growth. Fromthese facts, it has been suggested that the nodule
formation can be explained by grain growth as a stageof recrystallization, and the whisker growth can beunderstood as a result of minimization of thesurface-free energy of nodules, which have large
surface-free energy as a result of their complicated andrough surfaces. Furthermore it has been suggested thatthe minimization of the surface-free energy can beachieved growing in the specific directions andcreating facets on the lateral surface. From this, some

(c)

(d)
Fig. 15 Whiskers formed at 100 oC. The specimen
was plated with 1.96 µm of matt Sn and stored at
100 oC for 48 hours.

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