# Tuning and predicting the onset of desiccation cracks in air-dried aqueous
clay suspensions

Vaibhav Raj Singh Parmar  Ranjini Bandyopadhyay 111Corresponding author email:
<EMAIL_ADDRESS>
###### Abstract

Hypothesis: Stacks of clay platelets in aqueous suspension swell and break up
gradually, with the exfoliated clay particles eventually self-assembling to
form transient, fragile, gel-like networks. As a consequence of this physical
aging phenomenon, clay suspensions transform spontaneously from viscoelastic
liquids to viscoelastic solid-like phases. When subjected to desiccation,
these colloidal systems accumulate drying-induced stresses and are prone to
cracking. For a fixed evaporation rate and sample-boundary adhesion, it should
be possible to control the formation of cracks by tuning the sample aging
dynamics.
Experiments: Clay suspensions layers were dried in controlled environments.
The location and time of crack onset were recorded in imaging experiments. The
elasticity of each partially dried sample was extracted by performing
microindentation experiments in an atomic force microscope.
Findings: Clay aging dynamics and desiccation crack onset were controlled by
tuning clay and salt concentrations in the aqueous medium. Our experiments
reveal that the crack onset time, $t_{c}$, varies with sample elasticity, $E$,
according to $t_{c}\propto 1/\sqrt{E}$. We explain this correlation by
proposing a simple scaling argument that incorporates the Griffith’s fracture
criterion in a poroelastic model. Fits to our model reveal the high ductility
of aging clay suspensions. Besides allowing us to control the cracking process
by tuning sample elasticity, our results are potentially useful in formulating
crack-resistant coatings and paints.

###### keywords:

Desiccation cracks , Clay suspension , Microindentation , Poroelasticity ,
Fracture energy , Griffith theory of fracture

[inst1]organization=Soft Condensed Matter Group, Raman Research
Institute,addressline=C. V. Raman Avenue, Sadashivanagar, city=Bangalore,
postcode=560 080, country=India

##  1 Introduction

Evaporating films, coatings and surface layers constituted by aqueous
colloidal suspensions tend to shrink and crack [1, 2, 3, 4, 5, 6]. As water
evaporates from colloidal samples, the air-water interface at the free
evaporation surface develops a meniscus, resulting in enhanced internal pore
pressure. Tensile stresses that build up in the sample due to desiccation-
induced shrinkage are sensitively dependent on the adhesion of the sample with
the frictional substrate [3, 7]. As the tensile stresses exceed the tensile
strength of the material, the sudden release of stresses at the surface of the
sample results in the formation of fractures/cracks. Once a crack is
nucleated, its growth kinetics follow the Griffith’s criterion [8], with the
released strain energy exceeding the energy required to create new fracture
surfaces [6]. Drying-induced cracks form successively, such that a previously
formed crack determines the new stress field governing the formation of a new
crack [7]. In such hierarchical crack patterns, a new crack meets an existing
one at 90∘. In contrast, for very thin drying layers [9] and samples
undergoing repeated drying and wetting cycles [10], crack joint angles of 120∘
have been reported.

Crack formation is a complex phenomenon observed over a range of length scales
and in various systems such as monolayers of microbeads [11], old paintings
[12], craquelures in pottery glaze [1], industrial coatings [4], colloidal
crystals [13], crocodile head scales [14], columnar jointing in cooling lava
[15] and dried lakes [16]. Though undesirable in many applications,
understanding drying-induced cracks is useful in the investigation of ancient
geophysical processes [15], identifying health problems by drying drops of
blood [17], as a forensic tool for monitoring drying blood pools in crime
scenes [18] and while detecting the authenticity of old paintings [12, 19].
Morphological changes in crack patterns also reveal memory formation in drying
pastes [20]. The morphologies and growth kinetics of desiccation cracks in
drying colloidal samples depend on the evaporation rate [21], initial volume
fraction [22], rheology of the material [23, 24], shapes [25], sizes [26] and
surface charges of the constituent colloidal particles [27], sample thickness
[9, 28, 29], substrate-sample adherence [9, 30], substrate wettability,
applied fields [26, 31], substrate elasticity [32] and shape of the confining
boundary [33].

Desiccation cracks in soils are particularly interesting due to their
geophysical relevance [6, 34, 35]. Clay, a major ingredient of natural soil,
exhibits very interesting rheology when mixed in water, with the initially
liquid-like clay suspension losing its ability to flow and evolving gradually
into a viscoelastic solid via a physical aging mechanism [36, 37, 38, 39].
Physical aging is a common phenomenon that appears in various systems such as
colloidal gels [37, 40, 41], molecular glasses [42] and amorphous polymers
[43]. It has been established that aging a gel before drying increases gel
network strength and reduces the risk of fracture formation [44, 45, 46, 47].
While cracking because of solvent loss has been studied in great detail, the
effect of physical aging of the sample during desiccation has received less
attention. Here, we show _via_ experiments and theory that the risk of
fracturing of an isotropic gel-like aging clay sample can be evaluated from
its mechanical properties in the fully-developed state.

In this paper, we report experiments on desiccation-induced crack formation in
thin aqueous layers of the synthetic clay
Laponite${}^{\text{\textregistered}}$. In powder form, monodisperse disk-
shaped Laponite clay nanoparticles of 25 nm diameter and 1 nm thickness are
arranged in one-dimensional stacks or tactoids. When this powder is mixed with
water, the soluble Na+ ions that reside in the interlayer gallery spaces [36]
migrate towards the bulk aqueous medium due to osmotic pressure gradients. To
maintain overall charge neutrality, the faces of the clay particles acquire
negative charges. Screened electrostatic double layer repulsions between the
faces of like-charged clay particles within the tactoids cause tactoid
swelling and exfoliation [40, 48], as illustrated in Supplementary Information
Fig. S1(a). Diffuse electric double layers of Na+ ions that form around the
negatively charged clay particles faces evolve with time as tactoid
exfoliation progresses gradually [49, 50]. If the medium pH $<$ 11, the clay
particle edges acquire slight positive charges due to the hydration of
magnesia groups [51, 52]. Individual disk-shaped clay particles in aqueous
suspensions therefore have negative faces and positive rims, and spontaneously
self-assemble into overlapping coin and house of cards configurations, as
shown in Supplementary Information Fig. S1(b,c). The gradual formation and
restructuring of system-spanning fragile clay gel networks result in a
continuous increase in suspension elasticity [40, 48, 52, 53].

As shown schematically in Figs. 1(a,b), the addition of common salt leads to a
reduction in double-layer repulsion and a strong temporal increase in the
mechanical properties of the clay suspension due to accelerated structural
buildup. When tetrasodium pyrophosphate is introduced to the medium, the
attachment of P2O74- groups to clay particle rims enhances interparticle
repulsions, ruptures suspension microstructures and fluidises the sample [54,
55]. Moreover, the aging dynamics of clay suspensions can be accelerated by
increasing clay concentration [49]. Systematic studies of the evolution of the
mechanical response of aging clay suspensions [37, 38, 39] due to the
incorporation of additives [49, 56, 57], application of electric fields [58,
59, 60] and by tuning the solvent temperature and pH [51, 57] are available in
the literature.

Since clay is a major ingredient of soil, there have been several attempts to
understand the formation of desiccation cracks in thin contracting clay
layers. In one such study, it was reported that the cumulative crack area
decreased monotonically with increase in the ratio of the minimum crack width
and clay layer thickness [35]. Interestingly, a combined experimental and
numerical study demonstrated that the addition of polymers to clay suspensions
can substantially modify desiccation crack morphologies [61]. It was reported
that crack onset could be accelerated by increasing the strength of a static
electric field [62]. Enhanced cracking was also reported when alternating
voltages of progressively larger amplitudes and frequencies were applied [63].
A transition from multi-mode to single-mode periodic crack features was noted
when the amplitude of the alternating voltage was increased. These results
were modelled successfully using a spring chain model.

We performed direct visualisation experiments and digital image correlation
analyses to monitor the time and location of crack onset in desiccating clay
layers. We recorded the time, $t_{c}$, at which the sample first cracks, and
noted that the first crack always appears near the vertical boundary of the
sample cell. We demonstrate here that crack formation can be promoted or
inhibited by respectively adding common salt or tetrasodium pyrophosphate to
the aqueous suspension medium. We measured the elasticity, $E$, of each
partially dried sample by performing microindentation experiments in an atomic
force microscope and observed that crack onset time $t_{c}\propto 1/\sqrt{E}$.
Furthermore, by controlling the rate of water evaporation, $V_{E}$, from the
drying sample by tuning the chamber temperature, we noted that $t_{c}\propto
1/{V_{E}}$. We explain our results by incorporating the Griffith’s criterion
for crack formation in a poroelastic formalism. Besides describing our
experimental data successfully, our model allows us to estimate a fracture
energy of $\approx$ 103 J/m2 for the drying clay suspension layers studied
here. These samples are therefore highly ductile and deform significantly
before crack onset. Our study establishes that the formation of desiccation
cracks can be controlled by tuning the sample aging dynamics. Since clay is a
widely used rheological modifier, our results are useful in the formulation of
crack-resistant films, coatings [64] and paints.

##  2 Materials and methods

Figure 1: Physical aging of aqueous Laponite clay suspensions, oscillatory
time sweep rheology, experimental setup, and AFM microindentation
measurements. (a) Schematic illustration of the self-assembly of clay
particles in suspensions with and without additives. (b) Temporal evolution of
complex viscosity,
$\eta^{*}=\sqrt{G^{\prime^{2}}+G^{\prime\prime^{2}}}/\omega$, of aqueous
Laponite suspensions with no additive (navy blue), 4mM NaCl (olive green) and
4 mM TSPP (fuscia) at 25∘C. Here, $G^{\prime}$ and $G^{\prime\prime}$ are,
respectively, the elastic and viscous moduli of the suspension and were
measured in rheology experiments at a small strain $\gamma$ = 0.1% and angular
frequency $\omega$ = 6 rad/sec. (c) Schematic diagrams of the experimental
setup for imaging the onset and growth of desiccation cracks. (d) Schematic
diagrams of a partially dried sample and a microindentation measurement
performed to estimate sample elasticity. A thin layer of silicon oil was
applied to the sample surface to reduce capillary attraction between the AFM
cantilever and the sample.

###  2.1 Sample preparation

Hygroscopic Laponite${}^{\text{\textregistered}}$ XLG powder was procured from
BYK Additives Inc. and baked at 120∘C for $18-24$ hours to remove moisture.
Aqueous solutions were prepared by adding a specific amount of sodium
chloride, NaCl (common salt, LABORT Fine Chem Pvt. Ltd), or tetrasodium
pyrophosphate, TSPP (E. Merck (India) Ltd), to deionized and distilled water
(Millipore Corp., resistivity 18.2 M$\Omega$-cm). A predetermined amount of
dried Laponite powder was slowly added to 50 ml Milli-Q water to prepare clay
sample without additive, or to an aqueous solution prepared with an additive
(NaCl or TSPP) under vigorous stirring conditions. Homogeneous suspensions
with clay concentrations 2.0, 2.5, 3.0 and 3.5% w/v were prepared using this
method. After 40 minutes of vigorous mixing of the transparent clay
suspension, 200 $\mu$l of 2.09 mM/L Rhodamine B dye was added to enhance
contrast during imaging. Finally, the dyed clay suspensions were stirred for
an additional 5 minutes.

###  2.2 Monitoring the physical aging phenomenon in clay suspensions via
rheological measurements

To understand the evolution of microstructures in aging clay suspensions,
oscillatory rheological measurements at 0.1% strain were performed at 25∘C in
an Anton Paar MCR702 rheometer while using a double gap geometry (DG26.7).
Figure 1(b) shows the distinct time-evolution behaviours of complex viscosity
$\eta^{*}=\sqrt{G^{\prime^{2}}+G^{\prime\prime^{2}}}/\omega$ for suspensions
with and without additives. In the presence of NaCl, the rapid evolution of
$\eta^{*}$, plotted in Fig.1(b), indicates the enhanced formation of
suspension microstructures [49, 57]. The presence of externally added Na+ and
Cl- ions shrinks the Debye layer surrounding each clay particle (Fig 1(a)),
thereby reducing interparticle repulsion and promoting particle self-assembly
in overlapping coin and house of cards configurations that eventually span the
entire system [52]. In contrast, the low and approximately constant $\eta^{*}$
values in the presence of TSPP arise due to the absence of system-spanning
microstructures in these repulsive suspensions.

###  2.3 The desiccation experiment

12 ml of the freshly prepared suspension was poured into a transparent
polymethylpentene (TPX) Petri dish with an inner diameter of 4.65 cm. The
initial clay layer thickness was kept constant at 7 mm in all the desiccation
experiments reported here. The Petri dish was placed immediately in a
temperature and humidity-controlled chamber, a schematic illustration of which
is presented in Fig. 1(c). The temperature in the chamber was maintained at
35, 40, 45 and 50 ∘C in separate experimental runs using Arduino-based relay
switching of a heating element. Humidity was maintained at 27 $\pm$ 4 % by
placing a saturated salt solution of magnesium chloride in the chamber [65].
The variations of temperature and relative humidity inside the chamber during
the experiment, shown in Supplementary Information Fig. S2, were seen to be
minimal. The cracking of the drying suspension layer was recorded using a
Nikon D5600 camera, with consecutive images taken at 3-minute intervals.
Depending on the chamber temperature, the total drying duration of all the
samples studied here varied between 22 and 46 hours.

####  2.3.1 Imaging crack onset and propagation

A representative video of the temporal evolution of a desiccating sample with
clay concentration of 3.0% w/v and NaCl concentration of 4mM is presented in
Supplementary Video 1. The acquired images were converted to grayscale format
and analysed using the MATLAB@2021 image processing toolbox. These images were
binarized after setting a threshold. We recorded the time of crack onset,
$t_{c}$, by determining the instant at which the first crack appeared. The
procedure for determining $t_{c}$ is schematically illustrated in
Supplementary Information Fig. S3.

The Petri dish was placed on an Arduino-controlled weighing scale (Fig 1(c))
with a transparent acrylic measuring plate. The mass of the drying sample was
monitored at intervals of 10 minutes. Supplementary Information Fig. S4 shows
a monotonic decrease in the mass of the aqueous sample till the completion of
evaporation. The evaporation rate, $V_{E}$, was determined from the change in
sample mass due to loss of solvent, using
$V_{E}=\frac{dm/dt}{\pi\rho_{w}r^{2}}$ [66]. Here, $dm/dt$ is the mass loss
per unit time, $\rho_{w}$ = 1 g/cm3 is the density of water and $\pi r^{2}$ is
the area occupied by the evaporating clay surface in the Petri dish of radius
$r$ = 2.325 cm. The temporal variation of the evaporation rate, $V_{E}$, is
shown in Supplementary Information Fig. S4. The sample, prepared at room
temperature, initially equilibrates to 50∘C, after which we note distinct
evaporation zones. After maintaining a constant rate for a few hours, $V_{E}$
decreases rapidly to zero as the sample dries up. These distinct evaporation
regimes were earlier also reported for drying nanopore gels [30] and wet
granular media [67]. As in earlier observations [34, 68], we note that the
first crack emerges during the constant evaporation rate regime in all our
desiccating samples.

####  2.3.2 Strain field analysis using digital image correlation

A random texture was created by sprinkling fine black pepper powder over the
surface of the drying clay samples. The deformation of the texture due to
desiccation-induced strains was imaged and analysed using an open-source 2D
digital image correlation (DIC) Matlab software ‘Ncorr’ [69, 70]. Ncorr
determines one-to-one correspondences between a reference and subsequent
images to estimate the deformation fields in the drying sample using linear
combinations of translation, shear and stretch transformations. This protocol
is used to evaluate the diagonal components of the two-dimensional strain
tensor and estimate the extents of elongative and compressive stresses on the
sample surface. Highly elastic Laponite suspensions of particle concentration
3.0% w/v and NaCl concentration 4 mM were studied to ensure only deformation
and no flow during DIC analysis. Supplementary Video 2 shows the temporal
evolution of drying clay suspensions for different sample-boundary adhesion
conditions. Snapshots of these samples at various stages of drying are shown
in Supplementary Information Fig. S5.

###  2.4 Determining sample elasticity by microindentation using atomic force
microscopy

The elasticity of partially dried clay samples was determined using a
microindentation technique [71] in an atomic force microscope (Asylum
Research, Oxford Instruments). The clay samples were allowed to dry for $t$ =
14h or 15h to ensure fully-developed gel structures when elasticity
measurements were performed [49, 57]. Since clay samples contain water,
microindentation can measure force with changing indentation depth only when
capillary attraction between the cantilever (Tap190-G, stiffness = 28 - 75 N/m
(Budget Sensors)) and sample is minimal. As illustrated in Fig. 1(d), a thin
layer of silicon oil (Sigma-Aldrich) of viscosity $\eta=5$ cSt was applied on
the surface of the partially dried sample to reduce capillary attraction
between the cantilever and sample. The protocol adopted for measuring force-
indentation depth curves is presented in section ST1 and Fig. S6 of the
Supplementary Information. The force $F(\delta)$ at an indentation depth
$\delta$ was estimated from raw AFM data. A representative force-indentation
depth curve corresponding to a clay sample of particle concentration 2.5% w/v
and NaCl concentration 4mM for $\delta\geq 0$ is presented in Supplementary
Information Fig. S6(e). Indentation experiments could not be performed for
clay samples with added TSPP due to immeasurably low elasticity values.

Even though the tip of the cantilever used in these experiments is pyramid-
shaped, the oil on the cantilever is expected to assume a spherical shape due
to surface tension. This is illustrated in Fig. 1(d). The force-indentation
depth curve was fitted with the Hertz contact model for a spherical tip,
$F(\delta)=\frac{4E\sqrt{R}\delta^{3/2}}{3(1-\nu^{2})}$, to extract sample
elasticity, $E$, from the approaching cycle of the force-indentation depth
curve. Here, $\nu=0.5$ is the Poisson’s ratio and $R=10$ nm is the cantilever-
tip radius. Force-indentation depth data for a standard polymer sample (PDMS),
acquired using microindentation and fitted to the Hertz model, yielded a
reasonable elasticity value of 1.62 MPa, as shown in Supplementary Information
Fig. S7.

##  3 Results and discussions

Figure 2: Onset of desiccation cracks. Time evolution of desiccation cracks in
aging Laponite clay suspensions of particle concentration $C_{L}$ = 2.5% w/v
as the solvent evaporates in the presence of (a1-a5) externally added TSPP of
concentration 4 mM, (b1-b5) no additives and (c1-c5) externally added NaCl of
concentration 4 mM. Here, $t$ is the time elapsed since suspension loading and
$t_{f}$ is the total time duration for completion of the desiccation
experiment. The scale bar at the bottom right of (c5) is 1 cm. (d) The crack
onset time $t_{c}$, defined as the time elapsed when the first crack forms,
decreases with increasing NaCl and clay concentrations. The error bars
represent standard errors calculated from 3-4 experiments.

###  3.1 Accelerated aging in clay suspensions leads to earlier onset of
cracks

In our first set of desiccation experiments, the evaporation rate of the
sample was fixed by maintaining a constant temperature of 50∘C and relative
humidity of 27% in the experimental chamber. The aging dynamics, and therefore
the rate of microstructural buildup of the sample, was tuned by controlling
clay and additive concentrations. Figures 2(a-c) display the temporal
evolution of desiccation cracks in drying clay suspensions, prepared at a
concentration 2.5% w/v, both with and without additives. The horizontal axis
shows the normalised elapsed time $t/t_{f}$, where $t$ is the elapsed time and
$t_{f}$ is the total time required for completion of the desiccation process.
Addition of TSPP (Figs. 2(a1-a5)) resulted in the lateral shrinkage of the
drying sample, which detached from the Petri dish wall without the formation
of desiccation cracks. For the clay suspension without additive (Figs.
2(b1-b5)), cracks formed before the occurrence of lateral shrinkage. In the
presence of NaCl (Figs. 2(c1-c5)), the first crack appeared even earlier and
cracking increased substantially.

Clearly, solvent loss due to evaporation and the accelerated buildup of
suspension microstructures in the aging sample both contribute to
consolidating the drying clay sample and accelerating the onset of desiccation
cracks. Crack onset time vs. added NaCl concentration is plotted in Fig. 2(d)
for different clay concentrations. We observe that cracks form earlier with
increase in salt and clay concentrations. Although the first crack always
forms in the constant evaporation rate regime, there is a large standard
deviation in the crack onset times, presumably due to local strain field
heterogeneities in the drying samples. The final crack patterns, imaged after
completion of the desiccation process, are shown in Supplementary Information
Fig. S8(a). The surface crack ratio, which is a measure of the extent of crack
formation, is estimated by dividing the total crack area by the total surface
area of the drying sample. As expected, the surface crack ratio, plotted in
Fig. S8(b), is higher for samples with larger NaCl content.

Figure 3: Mapping strain fields during desiccation. Temporal evolution of
strain fields along the x and y directions, $\epsilon_{xx}$ and
$\epsilon_{yy}$, are displayed vs. elapsed time for a drying Laponite clay
suspension of concentration 3.0% w/v with 4 mM externally added NaCl. The
colour bars at the bottom of each row show the magnitudes of $\epsilon_{xx}$
and $\epsilon_{yy}$, with positive and negative strain values indicating
elongative and compressive strains respectively. The time stamps at the bottom
right of each sub-figure indicate the elapsed time. The scale bar is 1 cm.

###  3.2 Strain field maps reveal that the first crack nucleates at the
vertical wall of the Petri dish

Figure 3 displays the strain fields on the surface of a drying clay sample of
concentration 3.0% w/v with 4 mM added NaCl, analysed from the captured images
using digital image correlation. Figures 3(a1-a4) and 3(a5-a8) show the
evolutions of the diagonal components, $\epsilon_{xx}$ and $\epsilon_{yy}$, of
the strain tensor till the formation of the first crack. Estimates of
elongative (+ve values) and compressive (-ve values) strains along the x and y
directions, with the first image frame as reference, are represented by colour
bars at the bottom of each row of figures. It is seen from Figs. 3(a1-a8) that
while strong compressive strains develop near the vertical boundary as drying
progresses, elongative strains are comparatively more uniformly distributed
across the sample surface. In all the samples studied, the first crack forms
near the vertical wall of the Petri dish, where the buildup of adhesion-
induced strains is strongest due to constrained lateral shrinkage.

To further study the role of the boundary on crack formation, we reduced
sample-boundary adhesion by coating the Petri dish with vacuum grease. These
experiments, presented in Supplementary Information Fig. S9 and discussed in
section ST2, show that reduction in sample-boundary adhesion promotes
isotropic shrinkage and results in delayed cracking. For our subsequent
studies on crack onset in drying clay suspension layers, we ensured maximum
sample-boundary adhesion by loading samples in uncoated Petri dishes.

Figure 4:  Elasticity values. The inset displays representative force-
indentation depth curves for partially dried Laponite clay suspensions of
concentration 2.5 % w/v with different NaCl contents at $t=14$ h. The sample
elasticity values, $E$, in the main figure are extracted from the fits of the
force-indentation depth data, shown by black lines in the inset, to the Hertz
model. Standard errors in $E$ are calculated from 3-8 measurements.

###  3.3 AFM microindentation measurements reveal an inverse correlation
between crack onset time and sample elasticity

The onset of cracking in layers or sessile drops of colloidal suspensions
depends on the elastic properties of the material [23] and the evaporation
rate [72]. We performed microindentation experiments in an atomic force
microscope to correlate the crack onset time, $t_{c}$, with the elasticity,
$E$, of the drying clay suspension in a fully-developed gel state. For
measuring meaningful force-indentation depth curves, we ensured that the
samples tested were sufficiently elastic ($E\geq$ 0.5 MPa). Since clay samples
with added TSPP do not have measurable elasticity and do not crack, these
samples were excluded from further analysis. The inset of Fig. 4 shows
representative force-indentation depth curves at elapsed time $t=14$ h for
partially dried clay suspensions with different NaCl contents. The elasticity
of each sample was extracted by fitting the data to the Hertz model, shown as
solid black lines in the inset, using a protocol outlined in Supplementary
Information Section ST1. Elasticity values thus estimated at $t=14$ h are
plotted in Fig. 4. $t_{c}$ and $E$ values for all the samples are tabulated in
Supplementary Information Table S10. Sample elasticity measurements
corresponding to another elapsed time in the constant evaporation rate regime
are presented in Supplementary Information Fig. S11(a).

Crack onset times, $t_{c}$ are plotted vs. sample elasticity, $E$, in Fig.
5(a), for data acquired at elapsed time $t=14$ h. The error bars represent
standard errors estimated from at least 3 independent measurements. The
observed inverse correlation between $t_{c}$ and $E$ verifies that highly
elastic samples crack earlier.

Figure 5: Inverse correlation between crack onset time and sample elasticity.
(a) Crack onset time, $t_{c}$, vs. sample elasticity, $E$. (b) Crack onset
time, $t_{c}$, vs. inverse square root of sample elasticity,
$\frac{1}{\sqrt{E}}$. Apart from three data points (in red) that are expected
to be driven into the nematic gel phase upon desiccation, all other points lie
on a (blue) straight line. Horizontal and vertical error bars represent
standard errors calculated from at least 3 measurements. (Inset) The variation
of $t_{c}$ with evaporation rate, $V_{E}$, of the drying sample. The blue line
is a straight-line fit to this data.

###  3.4 Modelling the inverse correlation between crack onset time, $t_{c}$,
and sample elasticity, $E$

We model the observed variation of crack onset time, $t_{c}$, with sample
elasticity, $E$, displayed in Fig. 5(a), by incorporating the Griffith’s
criterion for crack formation [8] in a one-dimensional poroelastic model [73].
In the context of our experiments wherein water diffuses through the
microscopic pores of the deformable clay gel during desiccation, the pressure
distribution in the pores should satisfy the following diffusion equation [23,
47, 74, 75]:

$\frac{\partial p}{\partial t}=\frac{\kappa
E}{\eta}\frac{\partial^{2}p}{\partial{z^{2}}}$ (1)

In this equation, $z=0$ and $z=h$ correspond to the bottom and top surfaces of
the drying clay suspension respectively, with free evaporation allowed only at
$z=h$ as illustrated in Fig. 1(d). Furthermore, $\frac{\kappa E}{\eta}$ is the
consolidation coefficient, where $\eta$ is the viscosity of water, $E$ is the
elasticity and $\kappa$ is the permeability of the porous gel. As discussed in
Supplementary Information section ST4, the permeability of our samples is
estimated to be $\approx 10^{-19}m^{2}$ using the Carman–Kozeny relationship
[76]. Assuming $\eta=10^{-3}$ Pa-s and $E=10^{6}$ Pa, the consolidation
coefficient $\approx 10^{-10}$ m2/s. The low value of the consolidation
coefficient indicates an extremely slow rate of water diffusion through the
pores of the gel and inadequate compensation for solvent loss at the free
evaporation surface. In addition, capillary pressure at the surface is
expected to remain high due to meniscus formation. These factors [2], along
with constrained lateral shrinkage of the clay sample due to its adhesion with
the Petri dish wall, result in the observed cracking.

The pressure $p$ in the sample along the $z$-axis satisfies the following
conditions [23]: (i) $p(z,t=0)=p_{i}$, the atmospheric pressure at the start
of desiccation at $t$ = 0, (ii) $\frac{\partial p}{\partial z}=0$ for $t>$ 0
at $z$ = 0, and (iii) $\frac{\partial p}{\partial z}=-\frac{\eta
V_{E}}{\kappa}$ at $z=h$ due to solvent loss for an evaporation rate, $V_{E}$.
The solution of Eqn. 1 under these conditions yields the pressure at the
evaporating surface of the sample: $p(z=h,t)\approx\frac{EV_{E}}{h}t+p_{i}$
[23, 77, 78]. The desiccation-induced stress generated at the free evaporating
surface, $z=h$, is given by [23, 77]:

$\sigma(h,t)=p(h,t)-p_{i}\approx\frac{EV_{E}}{h}t$ (2)

According to Griffith’s criterion for equilibrium crack propagation [8], the
first crack develops at a critical cracking stress $\sigma_{c}$ [6]:

$\sigma_{c}=\sqrt{\frac{\mathcal{G}_{c}E}{\pi a}}$ (3)

Here, $\mathcal{G}_{c}$, the fracture energy of the sample, is the total
energy per unit area required for crack propagation and $a$ is the crack
length. Since for crack onset at $t=t_{c}$, the stress at the surface of the
sample $\sigma(h,t=t_{c})$ must be equal to the critical cracking stress
$\sigma_{c}$, we combine Eqns. 2 and 3 to obtain:

$t_{c}\approx\left[\sqrt{\frac{\mathcal{G}_{c}}{\pi
a}}\frac{h}{V_{E}}\right]\sqrt{\frac{1}{E}}$ (4)

Our simple model therefore predicts that crack onset time scales inversely
with the square root of sample elasticity: $t_{c}\propto\frac{1}{\sqrt{E}}$.
To check the validity of our model, we have re-plotted the data in Fig. 5(a)
in the form $t_{c}$ vs. $1/\sqrt{E}$ in Fig. 5(b). Apart from three data
points (marked in red) that correspond to samples with the highest clay and
lowest salt concentrations, all other data points (marked in black) lie on a
straight line as predicted by our model. In Supplementary Information Fig.
S12, we have reproduced a phase diagram for aqueous Laponite suspensions in
the clay concentration - NaCl concentration plane [40]. The samples
corresponding to the red data points in Fig. 5(b) lie in close proximity to
the phase boundary at $t=0$ and are expected to evolve into directed nematic
gel phases as the aging samples consolidate due to desiccation-induced solvent
loss. The poroelastic model, which describes the deformation of saturated
isotropic gels, is therefore not applicable to these samples.

We now turn our attention to the black data points in Fig. 5(b) that agree
with our model prediction, $t_{c}\propto\frac{1}{\sqrt{E}}$. According to Eqn.
4, the slope of the blue straight line in Fig. 5(b) is
$\sqrt{\frac{\mathcal{G}_{c}}{\pi a}}\frac{h}{V_{E}}$. Assuming $h\approx
a\approx 10^{-3}$ m and evaporation rate $V_{E}=10^{-7}$ m/sec at 50∘C, the
fracture energy $\mathcal{G}_{c}\approx$ 103 J/m2, which indicates that drying
clay samples are highly ductile and deform significantly before the appearance
of the first crack.

The predicted scaling between the crack onset time and sample elasticity was
also verified by performing microindentation measurements at an elapsed time
$t$ = 15 h. A linear fit to the data, displayed in Supplementary Information
Fig. S11(b), yields $\mathcal{G}_{c}\approx$ 103 J/m2 and agrees very well
with our previous estimate. Next, the crack onset time for drying suspensions
of 3% w/v clay with 4 mM NaCl was recorded at different chamber temperatures
and therefore at different evaporation rates, $V_{E}$. The data, presented in
the inset of Fig. 5(b), satisfies $t_{c}\propto 1/V_{E}$ as predicted by Eqn.
4, with $\mathcal{G}_{c}\approx$ 103 J/m2. The computation of $V_{E}$ at
different temperatures is displayed in Supplementary Information Fig. S13.

Our simplified one-dimensional theory can therefore be successfully employed
to explain the complex cracking phenomenon in heterogeneous viscoelastic aging
clay suspensions. Finally, Eqn. 4 is fitted to data extracted from Dauphiné
and Pauchard’s study [23] on crack onset during the drying of silica particle
suspensions and presented in Supplementary Information Fig. S14. The observed
linear relationship between $t_{c}$ and $1/\sqrt{E}$ indicates that our
prediction of an inverse scaling relation between crack onset time and square
root of sample elasticity can be successfully extended to desiccation crack
onset in other gel-forming colloidal suspensions.

##  4 Conclusions

Desiccation cracks [5] in drying soil or clay structures are driven by the
development of evaporation-induced differential strains [6, 34, 35]. As the
underlying clay suspension microstructures consolidate due to the combined
effects of aging and desiccation, the ability of the sample to support large
stresses and store strain energy improves considerably. While aging a gel
prior to drying has been shown to reduce the risk of fracturing by
strengthening the sample microstructures [44, 45, 46, 47], a systematic
investigation of the combined effects of aging dynamics and solvent loss on
the formation of desiccation cracks in drying samples has never been explored
in the literature. In the present work, we experimentally studied the onset of
cracking in desiccating clay suspension layers by first investigating the role
of substrate adhesion. Our digital image correlation analyses [69, 70]
revealed that the generation of substantial differential strains in the
presence of sample-boundary adhesion results in faster crack onset.

It was previously reported that static electric fields can rapidly accelerate
the aging dynamics of aqueous Laponite clay suspensions [58], with cracks
forming earlier when the static electric field strength was increased [63]. In
our present study, we accelerated [49, 63] and inhibited [54] sample aging,
and therefore sample solidification due to microstructural build-up [37, 38,
39], by incorporating sodium chloride (NaCl) and tetrasodium pyrophosphate
(TSPP) respectively in the suspension medium. For maximal sample-boundary
adhesion, direct imaging studies demonstrated that cracks formed earlier in
drying clay suspensions with added NaCl, while the presence of TSPP fluidised
the suspension and delayed crack formation. Furthermore, we showed via
microindentation measurements [71] that clay suspensions with higher NaCl
content are characterised by higher elasticity values due to the rapid
formation of elastic clay gels.

Combining our microindentation results with our imaging data, we demonstrated
that the crack onset time, $t_{c}$, is inversely correlated with the square
root of sample elasticity, $E$, of the partially dried sample, i.e.
$t_{c}\propto 1/\sqrt{E}$. We successfully modelled this correlation by
incorporating the Griffith’s criterion in a one-dimensional poroelasticity
theory. We experimentally validated another model prediction,
$t_{c}\propto\frac{1}{V_{E}}$, where $V_{E}$ is the evaporation rate, by
performing additional experiments at different environmental chamber
temperatures. Fits of our model to our experimental datasets yielded fracture
energy values of the order of 103 J/m2 [6], indicating that desiccating clay
suspensions undergo ductile fracturing and can sustain large deformations due
to avalanches of localised plastic events before yielding or cracking [79].
This is reasonable, given the fragile nature [80] of clay gels. Finally, we
analysed previously published data on the cracking of drying silica gel
droplets [23] and verified an inverse correlation between crack onset time and
the square root of sample elasticity. Our findings can therefore be extended
to other desiccating systems.

One of the major shortcomings of our model is its applicability to only
isotropic gel-forming materials. Deriving a correlation between crack onset
time and mechanical modulus of an anisotropic gel, such as those expected to
form at higher clay concentrations [40], would be an interesting future
direction. We must also note that incorporating the temporal evolution of
sample elasticity and film height in the poroelastic model would ensure a more
accurate prediction of crack onset time. However, despite its limitations, it
is indeed encouraging that our toy model can successfully capture several
features of the complex desiccation process.

A deeper understanding of the cracking phenomenon is important in heritage
science, for example, in detecting the authenticity of old paintings or in
creating craquelure art. Since clays are biodegradable [81] and ubiquitous in
various industrial products [82], our results are useful in the formulation of
crack-resistant films, coatings [64], rheological modifiers, etc. The scaling
relations between crack onset time, sample elasticity and evaporation rate, as
unearthed by us in this study, can be exploited to predict and prevent
desiccation-induced cracking in natural clayey soils, clay pottery and other
clay-based structures. Indeed, improved predictions of time and location of
desiccation crack onset may be achieved by employing machine learning
algorithms [83] for analysing strain fields on the surface of the desiccating
sample. The location and risk of fracture could then be predicted with
improved accuracy, thereby preventing catastrophic failure events.

##  5 CRediT authorship contribution statement

VRS Parmar: Methodology, Software, Writing - Original Draft, Validation,
Formal analysis, Investigation, Writing - Review & Editing, Visualization. RB:
Conceptualization, Supervision, Project administration, Writing - Original
Draft, Visualization, Writing - Review & Editing, Funding acquisition,
Resources.

##  6 Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have influenced the work reported in this
paper.

##  7 Acknowledgement

The authors thank Yatheendran K. M. for help with AFM microindentation
measurements, Shivprasad Patil for their experities in microindentation
technique, Lucas Goehring for their experties on desiccation cracks and
Abhishek gadhai for useful discussions related to digital image correlation
experiments.

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