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the interaction of soft substrates with fluid interfaces is common to many biological , medical , and industrial processes . for this reason , the field of elastocapillarity @xcite ,
in which one studies how surface tension forces couple to the deformations of elastic substrates , has been the subject of much recent attention .
even the most basic characterization of the wetting forces at the three - phase contact - line remains in dispute , especially for soft viscoelastic materials @xcite that have both liquid and solid properties .
for such materials , the young - dupr law for a liquid wetting a hard solid , neumann s law for a liquid wetting another liquid , or some hybrid thereof may apply @xcite .
theoretical efforts have been made to bridge the gap between the hard solid and liquid regime , such as introducing the concept of solid surface tension @xcite .
alternative methods employ computational approaches such as density functional theory ( dft ) @xcite and molecular dynamic ( md ) simulations @xcite to gain a more thorough understanding of the wetting forces acting at the contact - line .    in general , the elastic resistance must be comparable in magnitude to the surface tension forces applied to the elastic medium for many elastocapillary phenomenon .
the elastocapillary number @xmath1 is a typical measure of the relative importance of capillarity to elasticity . here
@xmath2 is the surface tension , @xmath3 the elastic modulus and @xmath4 a characteristic length scale . in experiments ,
it is typically easiest to adjust @xmath4 , as seen in the wrinkling of elastic sheets @xcite , capillary origami @xcite and buckling of elastic fibers @xcite .
however , it is also possible to use a gel as the solid phase @xcite , which permits @xmath3 to be tuned over several orders of magnitude .    [ cols="^,^ " , ]     we end this section with a comment on the failure criteria .
the results presented in this section have assumed that the substrate is always weak enough to generate deformations that exceed the failure threshold for agar gel . as such
, we have focused our study on identifying a critical wavenumber and not the failure threshold .
additional experiments would be needed to extract such information .
the model developed above captures the central feature required to generate a starburst with a particular number of arms : a critical wavenumber @xmath5 for which the tangential displacement is maximal . in experiments
, we observe that for a given set of experimental conditions , a narrow range of wavenumbers are selected ; this is not surprising given that values of @xmath6 can have a maximum displacement similar to @xmath5 .
however , the failure criteria does not provide a failure mechanism . in physical gels such as agar or gelatin ,
the stiffness of the material is provided by a network of entangled polymers @xcite . under strain
, these polymers are stretched further away from equilibrium , although not necessarily to the point of failure .
because polymers are subject to thermal fluctuations , even a sub - critical strain can be triggered by a sufficiently large thermal fluctuation .
this is the mechanism behind delayed fracture , whereby there is an exponential distribution of waiting times ( indicating a poisson process ) before which a strained material fractures . this has been directly observed in both stiffer gelatin rods ( @xmath7  kpa ) @xcite and in vibration - controlled experiments in this system .
( color online ) definition sketch of a fluid - filled crack of length @xmath4 driven by a pressure head @xmath8 .
, scaledwidth=37.0% ]    once a fracture is initiated , fluid begins to flow into the crack tip and the crack propagates outward ( away from the central droplet ) until the reservoir of fluid is exhausted .
we develop a two - dimensional model for the asymptotic propagation of a mode i fracture driven by a pressure head @xmath8 . a schematic
is shown in figure  [ fig : fracture ] .
to begin , we consider an incipient crack of length @xmath9 , filled with an inviscid fluid of density @xmath10 , propagating into an elastic solid.in this section , we model the solid as either i ) a linear elastic material or ii ) an incompressible neo - hookean material , as is typically used for polymeric gels , and contrast the computed propagation rates .
we assume the capillary pressure head @xmath11 at the fluid reservoir @xmath12 is greater than the elastic stress in the solid in front of that reservoir , so as to drive fluid into the crack tip .
the flow @xmath13 of this inviscid fluid is governed by bernoulli s law , @xmath14 and , hence , driven by the gradient in fluid pressure @xmath15 , which is related to the elastic stress induced by deformations of the crack surface .
here we assume there is no ambient stress field in the elastic solid , or the substrate is not pre - stressed
. the localized stress ( pressure ) field near the crack tip @xmath16 of a linear elastic solid has the well - known asymptotic form @xcite , @xmath17    for an incompressible neo - hookean solid , the asymptotic stress field is given by @xcite , @xmath18 here @xmath19 and @xmath20 are stress intensity factors for a linear elastic and incompressible neo - hookean solid , respectively .
in general , @xmath19 is a function of the crack geometry and far - field loading conditions , but we may simplify our model by utilizing several experimental observations .
recent experiments have demonstrated self - healing behavior of fluid - filled cracks in soft substrates @xcite .
this observation is characteristic of stable , or transport - limited , crack growth .
that is , the speed of crack growth is limited by the extent to which the fluid can be supplied ( transported ) to the crack tip . for reference , unstable crack growth
would be controlled by the speed of an elastic wave in the solid , a characteristic which is not observed experimentally .
hereafter , we assume the crack propagates in the marginal state @xmath21 , where @xmath22 is the fracture toughness , consistent with stable crack growth .
we apply bernoulli s law ( [ bernoulli ] ) using the crack tip ( state 1 ) and a point upstream ( state 2 ) to derive an equation relating the crack - tip velocity @xmath23 to the pressure gradient in the linear elastic solid ( [ fstress ] ) :    @xmath24    whose solution gives @xmath25 this predicts a characteristic exponent of @xmath26 for the growth of the arm .
a similar relationship is derived for an incompressible neo - hookean solid by applying the respective stress field ( [ nhstress ] ) instead : @xmath27 for this system , the characteristic exponent is instead @xmath28 : @xmath29    note that both constitutive models predict power law behavior , but the neo - hookean model is able to reproduce the universal exponent @xmath30 observed experimentally @xcite .
we attribute this result to the observation that the gel substrate is incompressible and that the large scale deformation of the soft solid are more aptly described by the nonlinear neo - hookean constitutive law .
( color online ) histogram for the crack length exponent @xmath31 with mean @xmath32 .
, scaledwidth=40.0% ]    because the gels are spatially - heterogeneous , cracks nucleate with different widths and propagate at varying rates .
we fit the length of each fracture arm to the form @xmath33 and examine the distribution of @xmath31 values observed .
no systematic variation with material parameters was found for the values of @xmath31 , but the width of the crack determines the pre - factor @xmath34 @xcite .
for this same data set of @xmath35 individual arms , we find that @xmath36 ( standard error ) .
the full probability distribution is show in figure  [ fig : alpha ] , which allows for comparison to the @xmath37 ( neo - hookean ) and @xmath38 ( linear elastic ) constitutive models .
we have analyzed the deformations of a soft elastic substrate induced by the liquid / solid interactions with a liquid droplet , as well as the propagation of a fluid - filled crack in a soft elastic medium .
the deformations associated with a corrugated contact - line yield a critical disturbance with wavenumber @xmath5 that generates the largest elastic response within the substrate .
we quantify the response with the tangential displacement , a failure criteria which we correlate with the initiation of a starburst fracture ( see figure  [ fig : intro ] ) .
computations reveal that the location of the unbalanced contact - line force is the most important parameter in wavenumber selection .
our theoretical predictions compare favorably to i ) previously - reported experiments with fixed volume droplets @xcite and ii ) experiments where the contact - angle @xmath39 is fixed and the droplet volume @xmath40 is varied , thereby confirming our interpretation of the initiation process .
our model also predicts that the substrate thickness @xmath41 is also an important parameter in wavenumber selection , an observation which we see in experiments . for the crack propagation problem
, we develop a model by considering the transport of an inviscid fluid into the crack tip of either a linear elastic or incompressible neo - hookean solid .
while both elastic models yield power law growth , the neo - hookean model predicts the crack length grows with universal exponent @xmath0 and does not scale with any material parameters , consistent with experimental observations @xcite .
studies on fracture necessarily take place on materials which are strong enough to resist fracture until the point when the measurement is made . however , many biological materials are soft enough that conventional fracture measurements are not possible because the material is too weak to support its own weight .
the present study suggests that by depositing a droplet of known size and wettability , it could be possible to use starburst fractures to measure the fracture toughness of very soft materials so long as they remain fully supported by the dish . furthermore , because the growth rate of fractures is sensitive to whether or not the material is linear - elastic , this could also provide a new method for materials characterization .
the authors are grateful for support from the national science foundation under grant number dms-0968258 , as well as nc state s undergraduate research office , to michael shearer for valuable discussions , and to mark schillaci for preliminary experiments useful in formulating the model .
tadashi kajiya , adrian daerr , tetsuharu narita , laurent royon , francois lequeux , and laurent limat .
advancing liquid contact line on visco - elastic gel substrates : stick - slip vs. continuous motions . , 9:454461 , 2013 .
a liquid droplet resting on a soft gel substrate can deform that substrate to the point of material failure , whereby fractures develop on the gel surface that propagate outwards from the contact - line in a starburst pattern . in this paper , we characterize i ) the initiation process in which the number of arms in the starburst is controlled by the ratio of surface tension contrast to the gel s elastic modulus and ii ) the propagation dynamics showing that once fractures are initiated they propagate with a universal power law @xmath0 . 
 we develop a model for crack initiation by treating the gel as a linear elastic solid and computing the deformations within the substrate from the liquid / solid wetting forces . 
 the elastic solution shows that both the location and magnitude of the wetting forces are critical in providing a quantitative prediction for the number of fractures and , hence , an interpretation of the initiation of _ capillary fractures_. this solution also reveals that the depth of the gel is an important factor in the fracture process , as it can help mitigate large surface tractions ; this finding is confirmed with experiments . 
 we then develop a model for crack propagation by considering the transport of an inviscid fluid into the fracture tip of an incompressible material , and find that a simple energy - conservation argument can explain the observed material - independent power law . 
 we compare predictions for both linear elastic and neo - hookean solids finding that the latter better explains the observed exponent .