Spiral Cracks in Drying Precipitates

Abstract

We investigate the formation of spiral crack patterns during the desiccation of thin layers of precipitates in contact with a substrate. This symmetry-breaking fracturing mode is found to arise naturally not from torsion forces but from a propagating stress front induced by the foldup of the fragments. We model their formation mechanism using a coarse-grain model for fragmentation and successfully reproduce the spiral cracks. Fittings of experimental and simulation data show that the spirals are logarithmic. Theoretical aspects of the logarithmic spirals are discussed. In particular we show that this occurs generally when the crack speed is proportional to the propagating speed of stress front.

Full text

arXiv:cond-mat/0201272v1 [cond-mat.soft] 16 Jan 2002
Spiral cracks in drying precipitates
Z. N´edaa, K.-t. Leungb, L. J´ozsacand M. Ravasza
aBabe¸s-Bolyai University, Dept. of Physics, RO-3400, Cluj, Romania
bInstitute of Physics, Academia Sinica, Taipei, Taiwan 11529, R.O.C.
cBabe¸s-Bolyai University, Dept. of Chemistry, RO-3400, Cluj, Romania
(Last revised February 1, 2008)
We investigate the formation of spiral crack patterns during the desiccation of thin layers of pre-
cipitates in contact with a substrate. This symmetry-breaking fracturing mode is found to arise
naturally not from torsion forces, but from a propagating stress front induced by the fold-up of the
fragments. We model their formation mechanism using a coarse-grain model for fragmentation and
successfully reproduce the spiral cracks. Fittings of experimental and simulation data show that the
spirals are logarithmic, corresponding to constant deviation from a circular crack path. Theoretical
aspects of the logarithmic spirals are discussed. In particular we show that this occurs generally
when the crack speed is proportional to the propagating speed of stress front.
PACS numbers: 61.43.Bn, 45.70.Qj, 46.50.+a
Fracture of solids produces a large variety of fascinat-
ing patterns. Straight and wiggling cracks in fragmented
dried out fields, rocks, tectonic plates and paintings, and
the self-affine fractured surfaces are just a few well stud-
ied examples. Understanding fracture and fragmenta-
tion phenomena is of increasing interest in physics and
engineering [1]. Many recent studies have analyzed the
morphology of fractured surfaces and fracture lines [2],
most of which showing a cellular and hierarchical pat-
tern. Concomitantly, successful models have been pro-
posed to describe crack propagation and to reproduce
the observed structures [3].
Apart from the cellular type, spiral, helical and in gen-
eral smoothly curving fracturing modes are known in ma-
terial science [4]. Most of the known spiral cracks are ei-
ther due to imposed torsion (twist) as in the spiral frac-
ture of the tibia [5], or due to geometric constraints as in
the fracture of pipes [6]. Spiral cracks can, however, also
arise in situations where no obvious twisting is applied,
so that the symmetry is spontaneously broken. An ex-
ample of this was recently given by Hull [7] in the study
of the shrinkage of silica based sol-gel. Similar structures
were reported by us for the post-fragmentation process
of a thin layer of drying precipitate [8]. The formation of
spiral cracks under specific conditions was also recently
considered by Xia and Hutchinson [9]. In this letter, the
mechanism leading to this special cracking mode shall be
investigated and modeled. We shall discuss some math-
ematical properties in the shape of the spiral and repro-
duce them by computer simulations.
The experimental conditions leading to such struc-
tures are simple, one can do that without a labora-
tory. The first step is to produce by chemical re-
actions a fine suspension of precipitate. The spi-
ral cracks are not restricted to one peculiar mate-
rial, as we obtain them with different compounds, in-
cluding nickel phosphate (Ni3(P O4)2, from the reac-
tion between NiSO4and N a3P O4), ferric ferrocyanide
(F e4[F e(CN )6]3, from K4[F e(CN )6] and F eCl3) and
ferric hydroxide (F e(OH)3from F e Cl3and NaOH ).
The reacting salts are diluted in distiled water to concen-
trations between 0.3−10% for all reactions. Mixing the
two reacting solutions produces the desired compound.
The solution is then left to sediment and the dissolved
ions (Na+,K+,Cl−,SO−
4,...) are removed by rinsing
with distiled water. This gives us an aqueous suspension,
which is then poured into a Petri-dish and let dry. Dur-
ing drying a thin solidified layer is formed which is then
fragmented into isolated parts (Fig. 1a).
a
c
b
d
FIG. 1. Fracture of nickel phosphate precipitate at differ-
ent length scales. (a) Typical fragmentation pattern inside a
Petri-dish, (b) spiral and circular structures inside the frag-
ments, (c) close-up of a spiral crack and (d) traces of the spiral
cracks left on the glass surface.
1

For fine and thin precipitate (grain-size smaller than a
few hundreds of nm and layer thickness between 0.2 to
0.5 mm), regular spirals as well as other smoothly curv-
ing cracks (such as circles, ellipses, and intersecting arcs)
finally show up inside the fragments (Fig. 1b). Depend-
ing on the grain size and layer thickness the size of these
fascinating structures varies widely, ranging from several
hundred microns to about 5mm. Easily overlooked by
naked eye, they are revealed under a microscope (Fig. 1c).
Moreover, after removal of the layer, they leave beautiful
marks on the glass surface (Fig. 1d).
By digitally analyzing their shape we found that all
those spirals are approximately logarithmic, i.e., de-
scribed by the functional form
r(θ) = r0ekθ ,(1)
where r0and kare constants, with kdetermining the
overall tightness of the spiral. rand θare the polar
coordinates in plane (in our convention θcan take any
real value, not restricted to [0,2π].) A characteristic fit
is presented in Fig. 2, showing their logarithmic nature.
0 10 20 30
θ(rad)
2
3
4
ln(r)
FIG. 2. A spiral crack in ferric ferrocyanide precipitate and
a fit of its trajectory in polar coordinates. The linear fit has
a slope of 0.072.
Read from the slope, the values of kfrom 14 instances
of spirals are summarized below, for two different com-
pounds and five different layer thickness. For ferric fer-
rocyanide we got: 0.075 (sample I.), 0.067 (sample I.),
0.080 (sample I.), 0.082 (sample I.), 0.086 (sample I.),
0.059 (sample II.), 0.072 (sample II.); for nickel phos-
phate: 0.078 (sample III.), 0.075 (sample III.), 0.063
(sample III.), 0.061 (sample III.), 0.061 (sample IV.),
0.078 (sample V.), 0.058 (Sample V.) Some observations
can be made at this point:
(i) The logarithmic spiral, also known as the equian-
gular spirals, has been studied extensively since the 17th
century by Descartes, Torricelli and Jacques Bernoulli.
In addition to the well-known example in nautilus shells
[10], it also describes the shapes of the arms of some spi-
ral galaxies such as NGC 6946 [11], and the flight path
of a peregrine falcon to its prey [12].
(ii) The apparent length-scale r0from equation (1)
can be absorbed in the phase: r(θ) = ek(θ−θ0), with
θ0=−ln(r0)/k. This demonstrates that the logarith-
mic spirals are scale-free.
(iii) Surprisingly, the majority of experimentally mea-
sured values of kfall between 0.06 and 0.08. The fluc-
tuation among different spirals of the same sample is
of the same magnitude as that among different samples
with different compounds and thickness. This indicates
a mechanism independent of layer thickness and precipi-
tate type for the formation of the observed spiral cracks.
(iv) There are pronounced oscillations accompanying
the linear trend in all our fittings, as exemplified in Fig. 2.
These oscillations are almost periodical, and increases
with r(note the logarithmic scale.)
substrate
detachment
front
(
b)
(
a)
(
c)
FIG. 3. The proposed mechanism for generating spiral
cracks: (a) After primary fragmentation, a fragment folds
up due to desiccation with a shrinking detachment front, (b)
crack nucleation along the front and (c) the advancement of
the front from time t1to t2leads to an inward propagating
spiral crack.
In-situ observations under microscope during the des-
iccation process suggests the following mechanism. The
spirals and circle-shaped structures are formed only after
the primary fragmentation process is over. Due to the hu-
midity gradient across the thickness, the fragments grad-
ually fold up and detach from the substrate (Fig. 3a-3b),
generating large tensile stress in the radial direction, at
and normal to the front of detachment. The extend of the
attached area shrinks as the ring-shaped front advances
inward due to ongoing desiccation. When the stress at
the front exceeds the local material strength, a crack is
nucleated (Fig. 3c). Since the nucleation is seldom sym-
metrical with respect to the boundaries, the crack tends
to propagate along the front in one preferred direction
2

where more stresses can be released. In the absence of
further nucleation event, by the time the crack growth
completes a cycle the front has already advanced, forcing
the crack to turn further inward, resulting eventually in
a spiral crack (Fig. 3d). Since the stresses on the top
of the layer are mostly relieved by folding, they are con-
centrated at the layer-substrate interface. Therefore, the
spiral runs like a tunnel, with 20 −60% penetration into
the layer thickness. The patterns being largely spiral
means that crack propagation is favored over nucleation,
for otherwise we would have observed more cylindrical
concentric structures.
We now investigate in more detail the conditions un-
der which the logarithmic spiral cracks can be produced.
Denote the speed of the inward propagating drying front
by uand the speed of the crack tip by v. We choose the
origin as the center of the spiral. The trajectory of the
crack is parameterized by the linear distance s, the radial
distance r, and the polar angle θ, as shown in Fig. 4.
ydr
dθ
dl ds
Φ
x
r
θ
crackpath
FIG. 4. Parametrization of the crack path.
For infinitesimal dθ, we have dl =rdθ, (ds)2= (dr)2+
(dl)2, and tan Φ = dr/dl. Then:
u2=dr
dt 2
=dr
dθ 2dθ
dt 2
,(2)
v2=ds
dt 2
="dr
dθ 2
+r2#dθ
dt 2
.(3)
Eliminating dθ/dt from (2) and (3) we obtain:
dr
dθ =ur
√v2−u2=kr, (4)
where
k≡u
√v2−u2= tan Φ.(5)
For constant k, we get (1), i.e., a logarithmic spiral.
Physically, since k(u, v) is a function of u/v alone, this
result implies that the general condition for a logarith-
mic spiral to occur is to have a constant ratio between
the two velocities, despite the possibility that the actual
dynamics may be very complicated and neither speed is
constant. In other words, the logarithmic spiral structure
is surprisingly robust to dynamical details. On the other
hand since it is reasonable to expect a quasi-static crack
to follow the direction of maximal stress relief, the geo-
metric result of a constant angle Φ suggests that the ori-
entation of the maximal stress component is constantly
bending away from the instantaneous direction of prop-
agation. The fact that all fitted values of kfall within
a narrow range means that the degree of such bending
is insensitive to details like the thickness or the chemi-
cal composition. More theoretically, one could ask if the
Cotterell and Rice criterion [13] applies here: whether the
stress intensity factor, KII , vanishes along a constant in-
clination as the crack propagates under the influence of a
radially symmetric stress field. If so, a logarithmic spiral
is naturally expected.
Now we turn our attention to the oscillations that dec-
orate the logarithmic behavior, as displayed in Fig. 2.
Such oscillations indicate the departure from radial sym-
metry in the underlying stress field. It can be explained
on the basis of boundary effects. Clearly the shape of
a typical fragment is generally polygonal, not circular,
and so its free boundaries will modify the shape of the
shrinking stress front, more so the closer the front is to
the boundaries. This leads to periodic variations in Φ,
and hence oscillation in k. For rectangular fragments we
expect to see two cycles of oscillation per revolution of
the spiral. This is indeed roughly the case in our fits.
Moreover, the observed diminishing amplitude of oscilla-
tion at smaller ris also in agreement.
To confirm and test the robustness of our proposed
mechanism, we implement it in a mesoscopic spring-block
model [14] which describes fracture of an overlayer on
a frictional substrate. In this model, the grains in the
layer are represented by blocks which form a triangular
array of linear size Lwith interconnecting bundles each
of which has Hbonds (Hookean springs). The bond has
a breaking strength Fcand a relaxed length l. While
Hplays the role of thickness, the initial block spacing a
prescribes the residual strain s= (a−l)/a. Each block
has the same slipping threshold Fssuch that whenever
the magnitude of the resultant force ~
Fon a block ex-
ceeds Fs, the block slips to an equilibrium position de-
fined by ~
F= 0. The temporally increasing stress in the
layer during desiccation can be modeled by increasing the
stiffness of the bonds. This is however equivalent to the
case of constant stiffness but decreasing Fsand Fc, with
fixed κ≡Fc/Fs. In this way, the competition between
stick-and-slip and bond breaking, quantified by the set
of parameters Γ = {s, κ, H, L}, was shown to give rise to
realistic fragmentation and select the emerged scale [14].
Here we focus instead on what happens after the frag-
mentation process has settled. The simulated system is
thus assumed to represent a fragment stable against the
primary fragmentation (ensured by choosing Γ properly),
but susceptible to secondary curved crackings induced by
3

an advancing stress front. Therefore, an inhomogeneous
stress field is imposed, with a profile constant everywhere
at σ0except on an annular region of radius R, where there
is a hump of height ∆σand width w(see upper inset of
Fig. 5). By decreasing Rat a constant speed u, we model
the advancing front of stress field caused by detachment.
In a rather narrow parameter region (mainly small u,
large κ, ∆σ∼σ0and w≈a), the desired spiral cracks
are successfully reproduced (lower inset of Fig. 5). Con-
sistent with experiment, the simulated spirals also follow
a logarithmic form as illustrated in Fig. 5. The value of k
depends on the parameters of the model. In particular,
imposing smaller penetration results in tighter-binding
spirals and hence smaller k. This is consistent with the
screening effects of the existing crack on the stress field
that influences further propagation of the crack.
−20 −15 −10 −5 0 5
θ(rad)
1
2
3
4
5
ln r
R
−R 0
σ
∆σ
w0
σ
x
FIG. 5. A simulated spiral crack (lower inset) and a fit
of its trajectory. The simulation parameters are: L= 300,
s= 0.3, κ= 2, H= 4, ∆σ/σ0= 1.4, w=a, speed
u= 0.005a/step, and full penetration. The linear behav-
ior indicates a logarithmic spiral, with a slope≈0.184. Upper
inset: schematic plot of the stress profile imposed on the frag-
ment.
In conclusion, spiral crack, an astonishing member of
the family of fascinating patterns produced by fracture, is
realized in a surprisingly simple setup of desiccating pre-
cipitate. Their formation is argued to be driven by an
unusual stress relaxation process governed by the fold-up
of the fragments. A discrete spring-block model incor-
porating such a driving appears to capture successfully
the observed phenomena. The spirals have interesting
properties, of which several quantitative aspects can be
understood theoretically.
The work of K.-t.Leung is supported by the National
Science Council of R.O.C. and the work of Z. Neda is
sponsored by the Bergen Computational Physics Labo-
ratory. We are grateful for the professional advices and
comments from Professor Derek Hull.
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