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Journal Club Theme of February 2013: Surface energy and mechanical instabilities in soft materials

Cai Shengqiang's picture

When a material is soft or the size of the material is small, the effect of surface energy on its deformation can be significant.The importance of surface energy on the deformation of a structure could be evaluated by the magnitude of a dimensionless number, called elastocapillary number: γ/μL, where γ is surface energy density, μ is shear modulus and L is the characteristic length of the structure.  Many intriguing phenomena of surface energy induced deformation of even instabilities have been observed in different experiments.  In this journal club, I want to initiate a discussion on how surface energy may affect mechanical instabilites in soft materials. In the following, I would like to use our recent work as exmaples. Any thoughts and comments on this topic are welcome.

Recently, we have studied the influence of surface energy on the creasing instability of an elastomer under uniaxial compression. In experiments we found that creases form by nucleation at preexisting defects and grow by channeling across the surface of the film.  Surface energy provides a nucleation barrier and also resists channeling for finite values of the elastocapillary number. While the heterogeneous nucleation makes it difficult to characterize the critical strain for nucleation, the condition for channeling is well characterized and depends on the elastocapillary number.  We further show that adhesion, rather than plastic deformation, is responsible for the dramatic hysteresis between the first and subsequent cycles of compression.

Our paper can be found in the following link. Some experimental videos have been put in the supplemental materials.

Surface energy as a barrier to creasing of elastomer films: An elastic analogy to classical nucleation

Another paper of us illustrates the influence of surface tension and streching limit of polymers on the snap-through instabilities of a cavity inside an elastomer. The link is given in the following.

Snap-through expansion of a gas bubble in an elastomer


Zhigang Suo's picture

Dear Shengqiang:  Thank you so much for this timely post.  A number of us are still talking about your work.  Here are some quick notes.

The ratio (surface energy)/(elastic modulus) defines a length, called the elastocapillary length.  

For a hard material, such as a metal or a ceramic, the surface energy is ~1 N/m and the elastic modulus is ~ 100 GPa, so that the elastocapillary length is 10^-11 m.  Consequently, the elastocapillary phenomena in hard materials involve actions that weaken the effective of the large modulus.  

One example is the Griffith theory of fracture, where strain is small, say 10^-2, and the surface energy competes with the elastic energy.  The latter scales with the strain squared.   

Another example is a flexible structure, such as adhesion of whiskers or thin foils.

In a soft material, however, the elastocapillary length is large.  For example, taking surface energy 0.1 N/m and modulus 1 kPa, we estimate the elastocapillary length to be 0.1 mm.  Thus, in soft materials, the surface energy can compete with finite deformation, with strain of order 1.

You gave two examples in your initial post.  Here are a few more recent papers:

To make the surface energy competitive with elasticity, you have several options:

  • Make strain small.
  • Use a flexible structure
  • Use a soft material

I taught a course on evolving small structures, in which I descibe phenomena involving surface energy and solids.

Incidentally, I'm teaching ES 241 Advanced elasticity again this semester.  I plan to add a few lectures on elastocapillarity.  This time, the focus will be on elastocapillarity in soft materials.  I have been drafting notes on elastocapillarity, and will post them when I'm done. 

    Anand Jagota's picture

    This is an interesting topic to me.  In addition to the references put up by Zhigang, I would like to point to some others that might contribute to the discussion.  


    One geometrical idea to consider is what shapes amplify (beams/plates), attenuate (ripples, ridges), or leave unchanged (rounding of a corner) the displacements resulting from the elasto-capillarity length.  That is, is the characteristic displacement roughly the same as elasto-capillarity or not?


    Another idea that has been much explored for stiff solids is whether surface stress (tension) is numerically equal to surface energy.  For the kinds of soft materials we generally discuss, the intuitive expecation is that this should be the case, but there isn't enough data to establish it fully.


    It might be useful to distinguish between elastocapillarity phenomena driven by liquid surface tension/energy and those driven by solid surface energy/stress. The latter are less understood.


    Here are some papers: 

    1) "Jagota, A., D. Paretkar, and A. Ghatak, Surface-tension-induced flattening of a nearly plane elastic solid.
    Physical Review E, 2012. 85: p. 051602.

     This paper looks at flattening of an undulating surface by surface stress.

    2) "Hui, C.-Y., et al., Constraints
    on Micro-Contact Printing Imposed by Stamp Deformation”, .
    Langmuir, 2002.
    18: p. 1394-1404.

    This paper shows that the rounding of a sharp corner gives a radius of about the elastocapillarity length.


    3) "Jerison, E.R., et al., Deformation
    of an Elastic Substrate by a Three-Phase Contact Line.
    Physical Review
    Letters, 2011. 106(18): p. 186103.
    "  Previous paper from Eric Dufresne's group on contact line deformation.  


    4)  "Carre,
    A., Gastel, J. -., & Shanahan, M. E. R. (1996). Viscoelastic effets in the
    spreading of liquids.
    Nature, 379, 432-434.

    186103."  Older paper on contact line deformation due to liquid capillary force.


    5)  "Roman, B. and J. Bico, Elasto-capillarity:
    deforming an elastic structure with a liquid droplet.
    Journal of Physics:
    Condensed Matter, 2010. 22(49)

     a very nice review of liquid capillarity driven elasto-capillarity. 


    6) "Mora, S., et al., Capillarity
    Driven Instability of a Soft Solid.
    Physical Review Letters, 2010. 105 p.
    "  Rayleigh-Plateau instability of a soft solid driven by solid surface stress.

    7) "Gurtin, M.E. and A.I. Murdoch, Surface Stress in Solids. International Journal of Solids and
    Structures, 1978. 14: p. 431-440. 
    "   Gurtin & Murdoch formulation of surface stress driven deformation.  It postulates surface stress as a basic quantity.


    8) "Jagota, A., C. Argento, and S. Mazur, Growth of adhesive contacts for Maxwell viscoelastic spheres.
    Journal of Applied Physics, 1998. 83(1): p. 250-259.
    "  An example of where action of surface energy and surface stress can be distinguished in the same phenomenon.  Contact growth as in the JKR theory is by material points coming together and surface annihilation.  Contact growth by sintering, as in the Frenkel mode, is by contact stretching. 

    Cai Shengqiang's picture

    Dear Anand,

    Thanks for pointing out the differences between surface energy and solid surface stress.

    I will  read all the papers you included with great interest.

    Lihua Jin's picture

    Thanks for the interesting topic!

    I have one more example to show that for soft enough materials
    with small size, surface energy can be comparable to elastic energy, and
    surface tension can deform or affect the deformation of soft materials. In the
    following paper, we encapsulate two microgels in a water drop flowing in oil.
    The water-oil surface tension squeezes the drop and deforms the microgels. By a
    simple model and image analysis, we are able to measure the modulus of those
    microgels. The Young's modulus of the microgels E is around 10kPa, the radius r
    is around 40um, and surface tension \gama is around 4mN/m. We can calculate the
    dimensinless number, as Shengqiang defined,

    \gama/Er=0.01, which means surface energy can play a role.


    So we are convinced that surface energy is important for soft materials in small scales. However, there is almost no commercial way to add surface tension into commercial finite element softwares. People have made a lot of effects to write their own code to simulate surface energy based on methods such as level set method.

    There is also few work to include surface tension into commercial FEM softwares by user subroutine.

    However, in general, a deformation problem including surface tension, and espacially when it becomes a moving boundary value problem, is hard to be solved in commercial FEM softwares. More computational efforts are needed. After that, more interesting phenomena driven by surface tension can be simulated.

    Cai Shengqiang's picture

    Dear Lihua,

    Thanks for providing additional examples.

    Computational techniques being capable of sovling elasticity with including surface tension effect are certainly important.

    Zhigang Suo's picture

    I have just posted a draft of notes on elastocapillarity.  I'll teach this topic later in the semester.  Let me know if you have suggestions.

    Cai Shengqiang's picture

    Dear Zhigang,

    Thanks for the comments. I am so glad to know that you will include elastocapillarity into your lectures. I always can clear up some confusions in my mind through reading your notes.

    Here is another example showing capillarity driven instability of a soft solid

    Zhigang Suo's picture

    Dear Shengqiang:  Thank you so much for point out the paper on Rayleigh instability in soft elastic solid.  In writing the class notes, I was unsure if this problem had been studied, and could only write vaguely about it.  I'll use the paper you pointed out to update my notes.

    It appears that one can look for an interesting elastocapillary phenomenon in soft elastic materials by

    • adding elasticity to a phenomenon known in liquids
    • reducing elastic modulus in a phenomenon in hard solids

    In the case of the Rayleigh instability, once elasticity is added, a length scale--the elastocapillary length--emerges.  the undulated shape can reach a state of equilibrium.  By contrast, the Rayleigh instability in a liquid jet will lead to an array of droplets.  

    Thank you for getting the discussion started. 

    Alfred Crosby's picture

    Dear Shengqiang:  Thank you for initiating this discussion.  This is a topic that is at the heart of many projects in our group--and we are definitely interested in finding new ways to think about it and take advantage of it. 

    One thing that we have been recently considering is the scaling of the observed geometric form with the ratio of g/E. In many problems (see review by Roman and Bico for examples), the observed geometric length scales with the square root (or weaker power) of g/E.  Can anyone show us examples in the literature of higher scaling powers?

     In terms of some interesting papers that you might enjoy, here are a few based on the method that we call "cavitation rheology":

    Jessica Zimberlin,
    Naomi Sanabria-DeLong, Gregory Tew,
    Alfred J. Crosby. “Cavitation
    Rheology for Soft Materials.”
    Soft Matter,
    3, 763-767. 

    Jessica A. Zimberlin,
    Jennifer J. McManus,
    Alfred J. Crosby “Cavitation Rheology of the
    Vitreous: Mechanical Properties of Biological Tissue
    in vivo.” Soft Matter,  2010, 6, 3632-3635.

    Jessica Zimberlin, Alfred
    J. Crosby
    “Water Cavitation of Hydrogrels.” Journal of Polymer Science: Part B: Polymer Physics, 2010, 48,
    13, 1423-1427.

    Jun Cui, Cheol Hee
    Lee, Aline Delbos, Jennifer J. McManus, and
    Alfred J. Crosby.
    “Cavitation rheology of the eye lens.”
    , 2011, 7, 7827-7831. 

    The one on Water Cavitation may be particularly interesting since we changed the surface tension contribution and observed the same elastic contribution, in the context of bubble formation.

     Beyond cavitation, here are two other papers that might be interesting to consider in this topic:

    Miquelard-Garnier, Andrew B. Croll, Chelsea S. Davis, and
    Alfred J. Crosby
    “Contact-line mechanics for pattern control.”
    Soft Matter, 2010, 6,
    , 5789-5794. 

    This paper describes an interesting method for using elasto-capillarity to wrinkle a surface.  We have used this method extensively in recent papers.  It creates many interesting questions and powerful advantages.

    Also, the following two papers present a method that we used to quantify the "residual stress" of a living cell sheet.  These types of cellular responses are often referred to as "surface tension" for the cells.   Although we don't have a current project on this, we are excited about hopefully continuing these in the near future.  Any ideas or ways to collaborate on these would be great! 

    Jessica Zimberlin, Patricia Wadsworth, Alfred J Crosby,
    “Living Microlens Arrays.” Cell Motility
    and the Cytoskeleton,
     2008, 65, 762-767.

    Miquelard-Garnier, Jessica A. Zimberlin, Christian B. Sikora, Patricia
    Wadsworth and
    Alfred J. Crosby. “Polymer microlenses for quantifying
    cell sheet mechanics.”
    Soft Matter,
    6, 398-403.


    Thanks again! 

    Cai Shengqiang's picture

    Dear Al, thanks for the comments and providing several more examples.

    I also have great interest in working on soft biological tissues. Recenlty, I have read an excellent review article about different models for developing tissues and tumors.

    Surprisingly, at least to me, many phenomenon observed in soft tissues can be well explained by liquid model with considering surface tension effects.

    I am wondering if the predictions or explainations can be improved somehow by modelling soft biological tissues by soft elastomers with including surfac energy.

    The review paper can be found in the following link:

    Soft matters model for developing tissues and tumors

    RobStyle's picture

    Thanks for kicking off the interesting discussion! I work in Eric Dufresne's group, and a lot of this is very relevant to what we do. There are several papers here that we hadn't come across, which have been very interesting to read.

    One of our experimental papers was just published today which might be of interest. We place droplets on soft solids where droplet size ~ elastocapillary length. We found that Young's law breaks down, and that you can measure surface stresses by looking locally at the contact line.

    There's also been some nice work done on this problem by the Bonaccurso group in Germany:

    Pericet-Camara R., Best A., Butt H.-J., Bonaccurso E., Effect of capillary pressure and surface tension on the deformation of elastic surfaces by sessile liquid microdrops: an experimental investigation, Langmuir, 24, 10565-10568 (2008)

    Pericet-Camara R., Auernhammer G., Koynov K., Lorenzoni S., Raiteri R., Bonaccurso E., Solid supported thin elastomer films deformed by microdrops, Soft Matter, 5, 3611 (2009)

    Sokuler M., Auernhammer G.K., Roth M., Liu C.J., Bonaccurso E., Butt H.-J., The softer the better: fast condensation on soft surfaces, Langmuir, 26, 1544 (2010)

    As Anand mentioned, there's an interesting distinction between surface stress and surface energy for solids. If anyone knows of any further papers that look at this for squishy materials, we'd be very interested.

    All the best

    this is goodCool

    mohamedlamine's picture


    Is Surface Energy different From Strain Energy ?

    Why you do not use the definition of Strain Energy : U=0.5*integral({σ}T{ε}dV ) with corresponding Assumptions to your case and dV=1*dA ?



    Zhigang Suo's picture

    Atoms on the surface of a body have a different bonding environment from atoms in the interior of the body.  

    When the body is stretched, the atoms on the surface change energy by an amount different from the atoms inside the body.

    This difference can be represnted in continuum mechanics.  See my lecture notes on elastocapillarity.  The notes just sketch out the basic ideas.  I'll improve them and post a new version later in the semester. 

    mohamedlamine's picture

    If it is Distinct it can be a Heat Energy (and/or Sound Energy) by a Convection Process. The Strain Energy is the Effect of Deformations Generated by Internal Forces.

    Anand Jagota's picture

    My colleague, Manoj Chaudhury, showed me a nice paper that covers previous work on drops on soft surfaces.  It might be of interest:


     Ying-Song Yu, "Substrate elastic deformation due to vertical component of liquid-vapor interfacial tension", Appl. Math. Mech. -Engl. Ed v33(9) 1095-1114 (2012)

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