Stimuli-responsive Polymer Gels

Fernando P. Duda

Programa de Engenharia MecânicaPrograma de Engenharia da Nanotecnologia

Instituto Luiz Alberto Coimbra de Pós-Graduação e Pesquisa - COPPEUniversidade Federal do Rio de Janeiro

School on Biological Soft Matter: from molecularinteractions to engineered materials,

March 13 – 24, 2017São Paulo, Brazil

ICTP-SAIFR/IFT-UNESP

Bibliography

Doi, Masao. Soft matter physics. Oxford University Press,2013.Flory, Paul J. Principles of polymer chemistry. CornellUniversity Press, 1953.Treloar, Leslie Ronald George. The physics of rubberelasticity. Oxford University Press, USA, 1975.Gurtin, Morton E., Eliot Fried, and Lallit Anand. Themechanics and thermodynamics of continua. CambridgeUniversity Press, 2010.

What is a polymer gel?

Expanding Water Beads Laughing Water Brewing Co.

What is a polymer gel?

A polymer gel is a material composed of two basic ingredients:An elastic cross-linked network of polymer chainsSolvent filling the interstitial spaces of the network

(Horkay and McKenna, 2007)

Chemical gels (molecules joined by chemical bonds)Physical gels (molecules joined by physical bonds)Hydrogels (solvent is water)Ionic gels (polymer chains contain ionazible groups)

What is a polymer gel?

A polymer gel is a rubber-like solid and hence can sustain:Large deformations (changes of shape)

But unlike rubber, it can undergo:Large volume changes by swelling (taking in solvent) ordeswelling (expelling solvent)Solvent-induced changes in physical properties

(Imran & Takeoka, 2015)

The aforementioned changes are reversible under appropriateconditions

What is a polymer gel?

When immersed in a compatible solvent, it will expand, or swell,by imbibing (a large amount) of solvent

(Popular Water Grow Toys. AliExpress.com)The amount of (inhomogeneous) swelling will depend on:

Compatibility of the solvent with the polymer (χ)Cross-link density of the network (N)

Stimuli-responsive polymer gels

Stimuli-responsive polymer gels have the ability to swelland deswell in response to changes in the environmentalconditions(mechanical forces, solvent composition, temperature, pH,and supply of electric field, light)Since the discovery of “volume phase transition” by Tanakain 1978, many stimuli-responsive polymer gels, thatchange volume abruptly in response environmetal changeshave been developed.

Polymer gel

A polymer gel has a solid-fluid nature:

It has a definite shape provided by the networkIt allows solvent flow through the interstices of the network

Because of this unique feature:Living organisms are largely made of polymer gels. Thisfacilitates the transport of ions and molecules within theorganism while keeping its solidity (shape)Polymer gels find practical and potential applications infields as diverse as robotics, drug delivery and tissueengineeringPolymer gels are among the most attractive and versatilesoft materials. They are used in our daily lives for contactlenses, superabsorbent polymers, etc., and more high-techapplications are in development

Related applications

Lauga, Eric. "Life at high Deborah number." EPL(Europhysics Letters) 86.6 (2009): 64001.

“ In order to reach the uterus of the female and continuetheir journey towards the ovum, mammalian spermatozoacells have to progress through the cervical mucus, a highlyviscous and highly elastic cross-linked polymeric gel.”Gel electrophoresis

“...uses a gel as an anticonvective medium and/or sievingmedium during electrophoresis... Gels suppress thethermal convection caused by application of the electricfield, and can also act as a sieving medium, retarding thepassage of molecules...’bigskip https://en.wikipedia.org/wiki/Gel_electrophoresis

Polymer gel

Stage interdependent phenomena:large deformation & solvent permeationStimuli-responsive polymer gels

Degree of swelling affected by mechanical stresses,temperature, PH, light, electromagnetic field, etcDisplays many unusual and interesting behaviors

Volume phase transtionMechanical instabilities induced by inhomogeneousswellling

Mechanical instability

Gel allowed to swell in water. An extremely fine pattern appearson the free surface of the disk gel, and evolves with time(Tanaka et al, 1987)

Why do fingers wrinkle in the bath?

“The epidermis, or outer layer of the skin, is made up of cellscalled keratinocytes, which form a very strong intracellularskeleton made up of a protein called keratin....When hands aresoaked in water, the keratin absorbs it and swells. The inside ofthe fingers, however, does not swell. As a result, there isrelatively too much stratum corneum and it wrinkles”

https://www.scientificamerican.com/article/why-do-fingers-wrinkle-in/(still the subject of some debate)

Mechanical instability

Wu et ., Three-dimensional shape transformations of hydrogelsheets induced by small-scale modulation of internal stresses,Nat. Commun., 2013, 4, 1586

Volume phase transition (VPT)

(Hirotsu, 1994)

Harnessing mechanical instabilities

Environmetal changes regulate permeation

Non-homogeneous swelling/deswelling

Stress generation

Bifurcation instabilities

Final shape

Goals

1) Formulation of a coupled theory for large elastic deformationand solvent permeation in polymer gels ∗

Framework of modern continuum mechanics

2) Applications

Mechanical instabilitiesVolume phase transitionOthers

∗Duda, Souza & Fried, JMPS, 2010; Duda, Souza & Fried, JMPS, 2011

Coupling elastic deformation and solvent permeation

Equilibrium of elastic solids capable of absorbing fluids(Gibbs (1878))Fluid chemical potential in the solid definableChemical potential as the driving force for diffusionPillar of many subsequent developments in metallurgy,polymer science, geology ...

Equilibria & dynamics of solid solutions(Li, Oriani & Darken, 1966; Larché & Cahn, 1985 )Swelling equilibria of polymer networks and gels(Flory, 1953; Treloar, 1975)

Alternative formulations

mixture theoryporoelasticity

Some features of the proposed formulation

Basic balancesbalance laws of mechanics, solvent-content balance

Dissipation inequalityenergy inflow = external power (mechanics andpermeation) (solvent chemical potential introduced here)

Constitutive theorymultiplicative decomposition: mechanical andsolvent-induced deformations

Solvent permeation and boundary conditionsprescription of the fluid flux : naturalprescription of the fluid chemical potential: essential

The continuum viewpoint

Any material object of interest is idealized via the notion of“body":

A body B is a collection of elements called particles ormaterial points†

The material points comprised by B can be placed in aone-to-one correspondence with the points of some regionof the Euclidean point space‡

The one-to-one correspondence between material pointsand varies with time as the body moves and deformsThe material point retains its identity as it moves

†A particle in the continuum is different to what we refer to as a particle inclassical mechanics.‡Since a “region" is an open connected set, a discrete set of particles

does not constitute a body.

The continuum viewpoint

Thus, given a body B, there is necessarily a one-to-onemapping χ that takes material points p ∈ B into their geometriclocations y ∈ R in three-dimensional Euclidean space:

y = χ(p), where p ∈ B, y ∈ Rχ is called a configuration of the body B;y is the position occupied by p in the configuration χ;R is the region occupied, or the shape assumed, by the body inthe configuration χ.

(B is a mathematical abstraction whereas R is a spatial region)

Motion of a body

A motion of a body is a one-parameter family of configurationsparametrized by the time. However, to describe geometricalchanges experienced by B, it is convenient to describe themotion of B relative to a fixed reference shape by the motionmap χ:

x = χ(X, t), x ∈ B, X ∈ B, t0 ≤ t ≤ t1

Bt : region occupied by the body at time t ; B: reference shape

Kinematical quantities

Velocity of the material point p at time t :

v(x, t) = χ(X, t), x = χ(X, t)

Acceleration of the material point p at time t :

v(x, t) = χ(X, t), x = χ(X, t)

Deformation gradient tensor

F(X, t) = ∇χ(X, t), det F > 0

It is the linear approximation of χ in a neighborhood of X:

χ(Y, t) = χ(X, t) + F(X, t)(Y− X) + o(Y− X)

Deformation of lines, areas and volumes

The deformation of small objects is fully described by F

In particular, the motion χ deforms the line, area and volumeelements

dL, dA, dV

at (X, t) intod l, da, dv

at x = χ(X, t) where:

d l = FdL, da = (det F)F−>dA, dv = det FdV

A pause for reflection

The description of distortion and flow of a continuum body astime passes relies on the notion of material points. But

What is a material point? Does it consist of the tribe ofmolecules belonging, at a certain instant t , to theimmediate neighborhood of a place x?

The notion of material point has been motivated quite differentlyfor solids and fluids. It corresponds to a small cluster of

neighboring molecules (solids)hypothetical immersed objects whose motion is induced byfluid molecules (fluids)

In the case of a polymer gel:the notion of a material particle corresponds to a smallcluster of polymer molecules in the network solid

Physical laws involved

The fundamental laws of physics appropriate to the kind ofcontinua considered here are:

Law of mass conservation for the polymer network, andbalance laws of linear momentum, angular momentum,solvent contentFirst and second principles of thermodynamics

Under isothermal conditions, the principles of thermodynamicscan replaced by the

Free-energy imbalance

Network mass conservation

The mass conservation law for the polymer network states that

ddt

M(Pt ) =ddt

∫Pt

ρdv = 0

for all material region Pt , where ρ is the network mass density§

Using the Reynolds transport theorem and the locationtheorem, one arrives the local form of the network massconservation holding for points in the interior of Bt :

ρ+ ρ(divv) = 0 (continuity equation)

where ρ is the material time derivative¶ of ρ and divv thedivergence of the velocity vector field v.§ρ(x, t) represents the network mass density at the spatial point x in the

deformed body Bt .¶Time derivative following a material particle

Linear momentum balance

The linear momentum balance states that

ddt

l(Pt ) = fc(Pt ) + fd (Pt )

where l(Pt ) is linear momentum of Pt , fc(Pt ) the net contactforce on Pt , and fd (Pt ) the net distance force. These quantitiescan be represented by

l(Pt ) =

∫Pt

p dv , fc(Pt ) =

∫∂Pt

t(n,x, t),da, fd (Pt ) =

∫Pt

b dv

p is linear momentum density; t(n,x, t) - with n the outward unitnormal to ∂Pt at x - gives the surface force, per unit area,exerted on the body at x by contact with its exterior; b isexternal body force density

Linear momentum balance

The linear momentum balance now reads:

ddt

∫Pt

p dv =

∫Pt

(p + (divv)p) dv =

∫∂Pt

t(n,x, t) +

∫Pt

b dv

for every material region Pt . Notice that the Reynolds transporttheorem has been used and that p is the material timederivative of pA fundamental result to all of continuum mechanics is theCauchy’s theorem

As consequence of the linear momentum balance, thereexists a spatial tensor field T, called the Cauchy stress,such that

t(n,x, t)) = T(x, t)n

Linear momentum balance

On using Caucy’s theorem and the divergence theorem to write∫Pt

divT dv =

∫∂Pt

Tn da,

where divT is the divergence of the stress tensor field T, onegets that ∫

Pt

(p + (divv)p) dv =

∫∂Pt

divT dv +

∫Pt

b dv

for every material region Pt , from which one the Cauchymomentum balance:

p + (divv)p = divT + b

or, after assuming that p = ρv:

ρv = divT + b

Angular momentum balance

The angular momentum balance states that

ddt

a(Pt ) = mc(Pt ) + md (Pt )

for every material region Pt . Here: a(Pt ) is the angularmomentum of Pt ; mc(Pt ) the net contact moment on Pt , andmd (Pt ) the net distance moment on Pt . For the standardCauchy continua, the following expressions hold:

a(Pt ) =

∫Pt

r×p dv , mc(Pt ) =

∫∂Pt

r×Tn da, md (Pt ) =

∫Pt

r×b dv

where r = x− x0 the position vector relative to a fixed point.Under these conditions, the following important result holds:

The Cauchy stress tensor is symmetric:

T = T>

Solvent content balance

Following Cauchy’s procedure, the solvent content balance canbe written as

ddt

∫Pt

c dv = −∫∂Pt

j · n da +

∫Pt

h dv

for every material region Pt . Here: c is solvent content density,j, the counterpart of the Cauchy stress tensor, the solvent fluxvector relative to the network, and h the solvent supply rate.The local sovent content is given by:

c + c(divv) = −div j + h

Free-energy imbalance

The free energy of an arbitrary material part Pt of Bt can notincrease at a rate faster that the power expended on Pt :

ddt

∫Pt

ψ dV ≤ Wm(Pt ) +Wc(Pt ),

where ψ is the free energy density and

Wm(Pt ) =

∫Pt

b∗ · v dv +

∫∂Pt

Tn · v dv (mechanical power)

and

Wc(Pt ) =

∫Pt

µh dv −∫∂Pt

µj · n dv (chemical power)

The scalar field µ is the chemical potential of the solvent in thebody. After a lengthy calculation one arrives at

ψ + ψ(divv)− µ(c + c(divv))− T · grad v + j · gradµ ≤ 0 ,

Summary of the equations

Field equations (for points in Bt )

ρ+ρ(divv), ρv = divT+b, T = T>, c+c(divv) = −div j+h

Dissipation inequality (for points in Bt )

ψ + ψ(divv)− µ(c + c(divv))− T · grad v + j · gradµ ≤ 0,

Remark:The field equations must be solved the unknown domain Bt(moving boundary problem). This problem can becircumvented by formulating all the laws using densities inthe reference shape.

Basic laws in the reference shape

All quantities below are now functions of (X, t).Field equations

ρR = 0, Div S+bR = ρRχ, SF> = FS>, −Div J+hR = cR,

Dissipation inequality

ψR − µcR − S · F + J · ∇µ ≤ 0

where

ρR = ρdet F, bR = (det F)b, cR = c det F, hR = h det F

ψR = ψ det F, S = TF∗, J = (F∗)> j

S: Piola stress tensor (force per unit reference area)J: referential solvent flux (flux per unit reference area)cR: solvent content per unit reference volume

Basic laws in the reference shape

Field equations

Div S + bR = ρRχ, SF> = FS>, −Div J + hR = cR,

Dissipation inequality

ψR − µcR − S · F + J · ∇µ ≤ 0

Problem: Given bR, hR and ρR , find χ and cUnderdetermined system (more unknows than equations)Constitutive theory (thermodynamically consist)Relate (S,J, ψR) with (χ, cR, µ)

(the subscript R will be omitted from now on)

Constitutive theory

Consider a reference shape B that is dry (solvent free)

Multiplicative decomposition:

F = FeFe

Fe: mechanical strain; Fg: solvent-induced (swelling) strainVolume changes are solely due to swelling which by itsturn is isotropic:

det Fe = 1, Fg = (1 + υc)13 I.

Thus,

det F = 1 + υc =1φ

where υ is the volume occupied by a solvent molecule and φ isthe polymer volume fraction

Constitutive theory

In view of the constraint det F = 1 + υc:

S = SA − qF∗, F∗ = (det F)F>

SA: active part of S; q: Langrange multiplierThe local form of the dissipation inequality can be rewritten as

−ψ + SA · F + (µ− υq)c − J · ∇µ ≥ 0

Guided by this inequality we consider constitutive equations(chemoelasticity with diffusion)

ψ = ψ(F, c,∇µ), SA = S(F, c,∇µ),

µ− υq = γ(F, c,∇µ), J = J(F, c,∇µ),

Dissipation principle

The dissipation inequality must hold in every admissibleprocesses.

1

ψ(F, c,∇µ) = ψ(F, c)

2

S =∂ψ

∂F, γ =

∂ψ

∂c,

3 Reduced dissipation inequality

J · ∇µ ≤ 0

Solvent flow relative to the solid is the only dissipationprocess involved

J = −M(F, c,∇µ)∇µ,

M : positive semi-definite tensor (mobility responses)

Constitutive responses

The constitutive theory is fully specified by the followingconstitutive responses:

Free energy ψ: the so-called Frenkel–Flory–Rehnerhypothesis is invoked:

ψ(F, c) = µ0c + ψe(F) + ψm(c)

where µ0 is a reference chemical potential for the solvent,ψe the elastic contribution of the network and ψm themixing contribution.Solvent mobility in the solid M: a simple choice is given by

M = − cDkBT

(F>F

)−1

where D is the solvent diffusion coefficient in the solid

Elastic and mixing contributions

Within the context of Treloar’s theory of swelling:Elastic energy (Gaussian network theory)

ψe(F) =NkBT

2(F · F− 3),

N: number of cross-linked units per unit reference volumePolymer-fluid (Flory-Huggins) mixing energy

ψm(c) =kBTυ

(ln(1− φ) + χφ)c

χ: Flory interaction parameterφ = 1/(1 + υc): polymer volume fraction

Remark:The elastic and mixing energies relative to a swollenreference state, with polymer volume fraction φ0, can beobtained by multiplying the corresponding expressionsprovided above by φ1/3

0 and φ0, respectively.

Consequences

Within the context of Treloar’s theory of swelling, the followingrelatios hold for the Piola stress S and the solvent chemicalpotential µ:

S = NkBT F− qF∗, µ = µ0 +KBTυ

(ln(1− φ) + φ+ χφ2)

The latter implies that

q = p + π, p :=µ− µ0

υ, π = −kBT

υ(ln(1− φ) + φ+ χφ2)

andπ = −kBT

υ(ln(1− φ) + φ+ χφ2)

The scalar p is called fluid pressure and π the osmoticpressure. See next slide.

Network and fluid stresses

Taking into account that q = p + π:

Piola stress S

S = SN + SF, SN = NkBT F− πF∗, SF = −pF∗

Cauchy stress T = S(F∗)−1

T = TN + TF, TN = NkBTF(F∗)−1 − πI, TF = −pI

ThusThe stress in a polymer gel can be viewed as the sum oftwo contributions, one from the network and other from theinterstital fluid. The network stress by its turn is the sum ofelastic and osmotic contributions.

Jain et. al 2010, Delivering nanomedicine to solidtumors, Nature Reviews

Summary of the governing equations

Unknowns: motion χ, solvent content per unit reference volumec, fluid pressure p (or chemical potential µ)(inertia, body forces and solvent supply neglected)

Incompressibility constraint‖

det F = 1 + υc

Mechanics

Div S = 0, S = NkBTF−(π+p)F∗, π = −kBTυ

(ln(1−φ)+φ+χφ2)

Solvent permeation

c = −Div J, J = − υDkBT

(F>F

)−1∇p

‖Change of volume due to fluid permeation only.

Boundary conditions

MechanicsPrescription of either y or SnFluid permeationPrescription of either µ or J · n

Impermeability

J · n = 0

Local equilibrium at the solid-environment interface

µ = µa = µ0 + υpa

µa: ambient chemical potential(pure and incompressible fluid bath)µ0: reference chemical potentialpa: ambient pressure

Free sweling

The amount of solvent imbibed by the gel?Chemical equilibrium: p is constantChemical boundary condition: p = pa

Free swelling: F = λeI, λe constant∗∗

Incompressibility constraint: λ3e = 1 + υce = 1/φe

Piola Stress: S = (NkBTλe − π(φe)− pa)IDiv S = 0 trivially satisfiedMechanical Equilibrium: NkBTλe − π(φe)− pa = −pa

Equation to be solved for φe:

f (φe) :=Nυφ3

e+ (ln(1− φe) + φe + χφ2

e) = 0

(Flory & Rehner, 1943)∗∗F = ∇χ = αI iff alpha is constant

Small departures from a swollen state

Suppose the reference shape is a swollen equilibrium state withpolymer volume fraction φe. If one is interested in smalldeparture from this reference shape, an approximate theorycan be constructed based on a suitable linearization of thegeneral equations presented before:

Mechanics

Div S = 0, S = KαI + 2G(

E− 12αI),

where K = π′(φe), G = NkBTφ1/3e , and

E =12

(∇u + (∇u)>

)α = Div u (volume change); K : osmotic modulus; G: shearmoduusPermeation

α = Dc4α

Net free-energy potential

For a conservative loading environment:

˙L[y] = −∫B

b · y dV−∫∂sB

s · y dA

and the following permeation boundary conditions:

µ = µa in ∂Be, J · n = 0 in ∂B\∂Be

Net free-energy potential:

F [y, c] =

∫B

(ψ(F, c)− µac ) dV + L[y].

(F [y] =

∫B

(Ψ(F)− ((µa − µ0)/υ) det F ) dV + L[y])

Equilibrium states:

δF = 0 det F = 1 + υc (δF = 0)

1D example

Cylindrical body B of lentgh L and radius A

B := {(X ,R,Θ) | 0 ≤ X ≤ L, 0 ≤ R ≤ A, 0 ≤ Θ < 2π},

Boundary conditions:

Surface X = 0 fixed and in contact with a pure andincompressible fluid bath at pressure p0:Surface X = L subject to a normal traction (−pL + σ) andin contact with a pure and incompressible fluid bath atpressure pLσ: extra Piola tractionLateral surface fixed at radial direction, free of tangentialstress and impermeable

1D example

Ansatz:x = f (X , t), r = R, θ = Θ,

Governing equations

λ = 1 + υc =

1φ

(incompressibility constraint)

∂SX

∂X= 0 (force balance)

c = − ∂J∂X

(fluid content balance)

whereλ =

∂f∂X

, SX = cSkBTλ− qλ

J = −m(c)∂µ

∂X, µ = µ0 + kBT (ln(1− φ) + φ+ χφ2) + υqφ

1D example

Taking into account the incompressibility constraint to write λ as1/φ it follows that

Governing equationsqφ =

cSkBTφ

+ K (force balance)

φ = υφ2∂2g(φ)

∂X 2 (fluid content balance)

where K is a constant to be determined and g is defined by

g′(φ) = φm(φ)µ′(φ)

with

µ(φ) = µ0 + kBT (ln(1− φ) + φ+ χφ2) + υ

(cSkBTφ

+ K)

1D example

Boundary conditions:Mechanical:f (0, t) = 0 at X = 0; K = pL − σ at X = LPermeation

X = 0: σ(φ(0, t)) = (p0 − pL) + σ

X = L: σ(φ(L, t)) = σ

where

σ(φ) =cskBTφ

+kBTυ

(ln(1− φ) + φ+ χφ2)

RemarksThe value of φ at X = 0 depends on the applied traction atX = L as well as of the applied pressure at X = 0p0 = pL implies that φ(0, t) = φ(L, t)The behavior of σ plays a crucial role. It depends on theparamenters cs, υ and χ

1D example: uniform equilibrium states

Suppose that p0 = pL. In this case, φ = φ(0, t) = φ(L, t) isconstant. To obtain the values of φ at equilibrium for a givenload σ, the following equation must be solved:

Fσ ′(φ) = 0⇔ σ = σ(φ).

A solution of this equation is stable (local minimum) providedthat

Fσ ′′(φ) > 0⇔ σ′(φ) < 0

1D example: uniform equilibrium states

Consider a cylindrical (polymer gel) bar fixed at one end andwith an external (extra) force P. The bar is impermeable exceptat its free end which is in contact with a pure a incompressiblesolvent bath. In We want to study the role played by cS and χon the stress-stetch behavior of this bar.It can be shown that

σ = σ(φe), φe = 1/λe

admits a solution for any value of σ = P/A.Stability

a solution is stable if σ′(φe) ≤ 0 and unstable otherwiseMultiphase equilibria (nonuniqueness)

the possibility of multiple solutions, a prerequisite for phasecoexistence, is indicated by the condition σ′ < 0 over aportion of (0,1), i.e., by a nonmonotic behavior of σ

Necessary condition for stress-induced phasetransition: χ > χc

Behavior of σ via χ− φ diagram:

0 1 2 30

0.2

0.4

0.6

0.8

1

χ

φ

σ′ = 0

σ′ > 0 :Unstable

σ′ < 0 :Stable

χc

χ < χc : σ is monotonicχ > χc : possibility of multi-phase equilibriaχc > 1/2

Necessary condition for stress-induced phasetransition: χ > χc

0 2 4 6 80

2

4

6

8

λ = φ− 13

σ/ν

(λ2, σ2)

(λ1, σ1)

σ < σ1 or σ > σ2: unique and stable solution;σ = σ1 or σ = σ2: two solutions, both of which stable;σ1 < σ < σ2: admits three solutions, two stable and oneunstable

Pressure-induced diffusion in swollen membranes

(Paul & Ebra-Lima, 1970)

Pressure-induced diffusion in swollen membranes

Permanent regime with σ = 0 and 4p := pL − p0 > 0fluid flow is constant and given by

J =g(φ0)− g(φL)

L

Polymer-volume fractions at X = 0 and X = L

cskBTφ0

+kBTυ

(ln(1− φ0) + φ0 + χφ20) = −4p

cskBTφL

+kBTυ

(ln(1− φL) + φL + χφ2L) = 0

Polymer-volume fraction at an arbitrary X

JX = g(φ0)− g(φ(X )),

Pressure-induced diffusion in swollen membranes

Diffusion of toluene through a rubber membrane induced bypressure difference(m(c) = Mcn)

L 0.0265 cmcS 1.06× 10−4 mole/ccυ 106 cc/moleχ 0.425T 300 Cn 3M/(kBTυn−1) 3.46× 10−5 cm2/s

Table: Parameters used in diffusion problem.

Comparison with experimental results of Paul &Ebra-Lima

Volumetric flux versus pressure difference & polymer volumefraction distribution

Final remarks

A continuum framework for coupling fluid permeation andlarge deformation in polymer gels has been presented.The interplay between fluid permeation and largedeformation in polymer gels gives rises to a rich variety ofphenomena of practical and scientific interest.

I envisage plenty of opportunitiesMultiscale and multiphysics analysisMaterial design and shape controlDesign a polymer gel to achieve a given shape undercertain conditionsMaterial and drug deliveryDesign a polymer gel to meet drug release requirementsBiomedicine in generalFracture...

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