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# Indentation of polydimethylsiloxane submerged in organic solvents

This paper uses a method based on indentation to characterize a polydimethylsiloxane (PDMS) elastomer submerged in an organic solvent (decane, heptane, pentane, or cyclohexane). An indenter is pressed into a disk of a swollen elastomer to a fixed depth, and the force on the indenter is recorded as a function of time. By examining how the relaxation time scales with the radius of contact, one can differentiate the poroelastic behavior from the viscoelastic behavior. By matching the relaxation curve measured experimentally to that derived from the theory of poroelasticity, one can identify elastic constants and permeability. The measured elastic constants are interpreted within the Flory-Huggins theory. The measured permeability indicate that the solvent migrates in PDMS by diffusion, rather than by convection. This work confirms that indentation is a reliable and convenient method to characterize swollen elastomers.

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## Comments

## The compressibility of swollen gels

Hi Yuhang，

Thank you for sharing your interesting paper with us.

I have noted in your paper the individual polymer chains and solvent molecules are both assumed to be incompressible, the Poisson’s ratio of the swollen gel should be 0.5 if no solvents flow out. When the indenter is pressed into the swollen gel, the measured Poisson’s ratio is below 0.5, is this compressibility only caused by the solvent migrating out of the gel?

Thanks

Lianhua

## Hi Lianhua, Thank you for

Hi Lianhua,

Thank you for raising this good question.

You are right that the measured Poisson's ratio is below 0.5 is only due to solvent migrating out of the gel, because the individual polymer network and solvent molecules have been assumed to be incompressible in this model.

Actually the Poisson's ratio for poroelastic material is defined in equilibrium state. You can think about it in the following way. Taking tensile test for instance, you pull the material in one direction to a certain length and hold it there for long enough until the migration of solvent reaches equilibrium, then you count how much contraction you get in other directions. This is how Poisson's ratio is defined for poroelastic material.

## Thanks,

Thanks for your prompt response, Yuhang. Under your assumption, the Poisson’s ratio may be measured also by the volume change of external solvent.

It is interesting that it seems that there is difference between your defined elastic parameters and those in traditional Biot’s poroelasticity theory. In your work, these two parameters represent elastic properties of the saturated swollen gel (containing fluid) , whereas in the traditional poroelasticity theory, the modulus and Poisson’s ratio characterize the properties of drained solid skeleton (there is no fluid). Anyway, the formulation is really similar to that of Biot’s poroelasticity theory, despite some items have different physical meaning.

Another question,the modulus and Poisson’s ratio are defined in equilibrium state as you mentioned above, if we perform a transient analysis, can the elastic parameter be regarded as a constant? if it is so, maybe we need another assumption of quasistatic process.

## Hi Lianhua, The Poisson's

Hi Lianhua,

The Poisson's ratio in our famula is exactly the same as in Biot's definition. To my understanding, the word "drained" raised by Dr. Rice refers to equilibrium state (constant pore pressure but conaining fluid).

For small deformation, shear modulus does not change during solvent migration. So the two parameters (shear modulus, Poisson's ratio) are constant and enough to discribe the whole time dependent behavior as in Biot's.

## Hi Yuhang, Many thanks for your clarification.

Hi Yuhang, Many thanks for your clarification. Perhaps my understanding is incorrect. The porous medium is a material containing open pores. Let’s give two states of the material, state (a) is the case where the pores have no fluid, in such a case, the material is just a solid skeleton containing some voids; another one is the state (b) in which the pores are filled with a fluid at a constant pressure.

If we apply an external force on the two states, do you agree that they have the same mechanical behaviors (having same elastic modulus and same Poisson’s ratio) ?

Thanks

Lianhua

## The incompressibility of swollen gels

I am wondering if there is the case where no solvents flow out even if we compress the swollen gel?

In other words, under what circumstances could we assume the membrane is not permeable to solvent molecules?

## In my perspective, the

In my perspective, the material itself is permeable because there are pores in it. Whether the solvents flow or not is controled by chemical potential. Both stress and concentration influence the chemical potential gradient. So under the circumstance of no chemical potential gradient, there is no fluid flowing out or in.

## thanks

Thanks,Yuhang,

When the dry gel is submerged in a solvent for a long time, the gel swells and the chemical potential in the gel equilibrates with that of external solvent. On this basis we have the swollen gel bounded by an impermeable membrane, and we further apply compression load on the enveloping gel, in such case no fluid flows out, how to describe the chemical potential in the gel? Is it equal to that of external solvents?

## In this case, on the

In this case, on the boundary the chemical potential gradient equals to zero. But within the gel, in transient state the chemical potential is an inhomogeneous field which will drive fluid flow and in equilibrium state the chemical potential becomes a homogeneous field but not equal to that of external solvents.

## little clarifications

Hi Lianhua,the question you raised really confused lots of people. I 'd like to add some more comments for possible clarification. In biot's theory, he treated solid part and fluid part separately and define the possion's ratio for solid part. Now his model is named multi-phasic model by most of people. However, we treat the porous material(eg. gel) as a whole and define a number named poison's ratio since when gel equlibriate with external solvent, this number indeed relates the deformation in one direction to the deformation in its perpendicular direction. The number can be measured by doing tension test mentioned by Yuhang.

If the gel is wrapped around by a impermeable membrane, no solvent can come out. However, the solvent can still migrate inside the gel to make the chemical potential in the gel homogenous finally. In this situation, since the solvent inside the gel does not equlibrate with the external solvent, the number named poison's ratio does not directly relate the deformation in two directions anymore.

## Hi Yuhang and Shengqing

Hi Yuhang and Shengqiang,

It’s nice to discuss the basic issue of Biot’s theory with all of you.

I agree with your opinion about the elastic parameters defined in your work. As you said, the elastic parameters can be measured at equilibrium state by the tension of the swollen gel (constant pore pressure but containing fluid). In Biot’s theory, the modulus and Poisson’s ratio are defined by the solid skeleton, it is worth noting that they are defined by the solid skeleton containing voids, not the solid part (solid polymer in your work).

To my understanding, the two ways of the definition of elastic parameters are identical. I quite agree with Shengqiang’s comment that Biot treated solid part and fluid part separately, and your work treated the porous material as a whole. Although the definitions of elastic parameters are different, I think the numbers of the defined parameters in your work is identical to those in Biot’s framework. For the two states of a porous medium given above (please see the fifth comment), in my perspective, the materials at state (a) and state (b) have the same mechanical behaviors (having same elastic modulus and same Poisson’s ratio), because the fluid at state (b) can not resist any deformations under external loads, and it has no contribution to the effective mechanical properties of the solid skeleton.

Cheers

Lianhua

## Hi Lianhua, In

Hi Lianhua,

In equilibrium state and in the state that chemical potential equals to chemical potential of the external solvent everywhere (in other words, pore pressure equals to zero), they behave the same.

## Adapt Biot's theory of poroelasticity to elastomeric gels

Here is the original paper of Biot:

M.A. Biot, General theory of three-dimensional consolidation, Journal of Applied Physics 12, 155-164 (1941)

The theory was developed for soil saturated with water. The theory is phenomenological, and does

nottreat water and solid separately. Biot's approach differs from the biphasic approach. His theory has been adapted recently for elastomeric gels. The following paper contains a description of this adaptation, along with an interpretation of Poisson's ratio: http://www.seas.harvard.edu/suo/papers/232.pdf## thanks

Thank you, Prof. Suo, for the helpful clarification. In Biot’s theory, the total stress acting on the soil consists of two parts: one is the effective stress (the stress may be understood as contact stress among solid particles) caused by the solid skeleton deformation; the other is the pore stress (the hydrostatic stress of the fluid). From here, I usually misunderstood that Biot treated solid and fluid separately. In fact, in Biot’s theoretical framework, the governing field equations were derived as a whole. If we want to treat every phase separately, we should correspondingly give the field equations for every phase, which is also one of the ideas of multiphasic theory. Despite Biot’s model differs from the multiphasic theory, we can recover the Biot’s formulation from the biphasic approach. Biot’s theory is the special version of multiphasic model.