a membrane theory for swelling polymer gels...a membrane theory for swelling polymer gels!!...

A membrane theory for swelling polymer gels !

Alessandro Lucantonio !!

Joint work with Luciano Teresi and Antonio DeSimone

Complex Materials Workshop - Oslo, 10-12 June 2015

AdG “MicroMotility”

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Physics and applications of polymer gels

A polymer gel is a soft elastic material consisting of a polymer network swollen with a fluid (solvent).

In thermo-responsive gels, the solvent-polymer chemical affinity varies with temperature and affects the swelling degree.

Actuators for microfluidics Drug delivery systems

Contact lenses

Plants and biological tissues

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Modeling of swelling-induced deformations

‣ Non-linear 3D models of coupled elasticity and solvent transport.!

‣ Structural (reduced) models of polymer gels: beams, plates, shells.!

‣ Shape morphing and instabilities.

Swelling-induced wrinkling of a stretched elastomeric membraneTransient free swelling of a cubic gel

a

A.L. et al., Transient analysis of swelling-induced large deformations in polymer gels, JMPS 61, 2013.!A.L. et al., Buckling dynamics of a solvent-stimulated stretched elastomeric sheet, Soft Matter 10, 2014.!

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Background: wrinkling of free-standing films

(from Cerda & Mahadevan, 2002)

‣ The wavelength follows a scaling law with applied strain!‣ For certain aspect ratios, wrinkling does not occur swelling-

induced wrinkling!‣ How can we combine swelling and stretching for shape control?

�!

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Swelling-induced wrinkling of an elastomeric membrane

A.L. et al., Buckling dynamics of a solvent-stimulated stretched elastomeric sheet, Soft Matter 10, 2014.!

‣ A homogeneous free-standing elastomeric sheet is stretched between clamps!‣ The stretched sheet is swollen with oil contained in a dish below the elastomer!‣ Upon reaching a critical solvent uptake, wrinkling occurs

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time (s)

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0 20 40 60

u z (m

m)

s (mm)

g

clam

p

clamp

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Stretch-controlled wrinkling of an elastomeric membrane

A.L. et al., Buckling dynamics of a solvent-stimulated stretched elastomeric sheet, Soft Matter 10, 2014.!

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mx /

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my /

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p/G

z/h0

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t

t

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itude

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¡ ¡

‣ The membrane swells laterally against the confinement due to the clamps and the Poisson effect; as a result, the transverse stress becomes negative, thus triggering the instability

‣ Wavelength and amplitude of the wrinkles at equilibrium decrease with an increasing applied strain

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Modeling of smart membranes - Balance equations

Z

Pb

(Ss ·rsevs + �ne⌘) =Z

@Pb

ts · evs +

Z

Pb

(fs · evs + cne⌘)

Principle of Virtual Power Pb ⇢ S, 8 evs, e⌘

Ss : surface stress

ts : contact force�n : thickness stress

fs : distributed surface load

Kinematics: membrane with thickness microstructure (swelling material surface)

cn : distributed thickness loadA.L., L. Teresi, A. DeSimone, Continuum theory of swelling material surfaces with applications to thermo-responsive gel membranes and surface mass transport, under review.!

: surface velocity : thickness stretch ratevs = us ⌘ = �bF = PT +rsus , Fs = Pt

bFP,Pt

: surface deformation gradients: projectors to the reference and current tangent space of the surface

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Z

Pb

(Ss ·rsevs + �ne⌘) =Z

@Pb

ts · evs +

Z

Pb

(fs · evs + cne⌘)

Principle of Virtual Power Pb ⇢ S, 8 evs, e⌘

Ss : surface stress

ts : contact force�n : thickness stress

fs : distributed surface load

Kinematics: membrane with thickness microstructure (swelling material surface)

cn : distributed thickness load

divsSs + fs = 0 , on S ⇥ I ,

ts = Ssms , on @S ⇥ I ,

�n = cn , on S ⇥ I .

A.L., L. Teresi, A. DeSimone, Continuum theory of swelling material surfaces with applications to thermo-responsive gel membranes and surface mass transport, under review.!

: surface velocity : thickness stretch ratevs = us ⌘ = �bF = PT +rsus , Fs = Pt

bFP,Pt

: surface deformation gradients: projectors to the reference and current tangent space of the surface

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Balance of solvent mass

: solvent flux: surface solvent concentration

: solvent mass source

d

dt

Z

Pb

cs = �Z

@Pb

hs ·ms +

Z

Pb

qs

cshs

qs


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Balance of solvent mass

: solvent flux: surface solvent concentration

: solvent mass source

d

dt

Z

Pb

cs = �Z

@Pb

hs ·ms +

Z

Pb

qs

cshs

qs

cs = �divshs + qs , on S ⇥ I ,

� hs ·ms = q on @S ⇥ I .


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Modeling of smart membranes - Thermodynamics


d

dt

Z

Pb

s Z

Pb

fs · us +

Z

@Pb

ts · us +

Z

Pb

cn� �Z

@Pb

µshs ·ms +

Z

Pb

µsqs

Free energy inequality

µs : solvent chemical potential

Coleman-Noll procedure

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d

dt

Z

Pb

s Z

Pb

fs · us +

Z

@Pb

ts · us +

Z

Pb

cn� �Z

@Pb

µshs ·ms +

Z

Pb

µsqs



Helmholtz free energy (Flory-Rehner like)


s(Fs, cs) =1

2Gsh(Fs · Fs + �2 � 3) + g(cs;T )� ps(Js� � 1� ⌦scs)

Elastic energy (Neo-Hooke)

hGs : network shear modulus

: membrane ref thickness

Fs : surface def gradient

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d

dt

Z

Pb

s Z

Pb

fs · us +

Z

@Pb

ts · us +

Z

Pb

cn� �Z

@Pb

µshs ·ms +

Z

Pb

µsqs





Mixing energy: temperature

s(Fs, cs) =1


T




Fs : surface def gradient

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d

dt

Z

Pb

s Z

Pb

fs · us +

Z

@Pb

ts · us +

Z

Pb

cn� �Z

@Pb

µshs ·ms +

Z

Pb

µsqs



Swelling constraintJs� = 1 + ⌦scs

ps : Lagrange multiplier (pressure)



Js = detFs : Area change


s(Fs, cs) =1


T




Fs : surface def gradient ⌦s = ⌦/h⌦ : solvent molar volume

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d

dt

Z

Pb

s Z

Pb

fs · us +

Z

@Pb

ts · us +

Z

Pb

cn� �Z

@Pb

µshs ·ms +

Z

Pb

µsqs






s(Fs, cs) =1


T

An increase in temperature leads to a reduction in chemical affinity between solvent and polymer, which hampers swelling.

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d

dt

Z

Pb

s Z

Pb

fs · us +

Z

@Pb

ts · us +

Z

Pb

cn� �Z

@Pb

µshs ·ms +

Z

Pb

µsqs





s(Fs, cs) =1


Constitutive equations

Ss =@ s

@bF� ps�P

Tt F

?s , �n =

@ s

@�� psJs ,

µs =@ s

@cs+ ps⌦s , hs(bF, cs,rsµs) ·rsµs 0.

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3D theory of swelling - Governing equations

A.L. et al., Transient analysis of swelling-induced large deformations in polymer gels, JMPS 61, 2013.!

divS+ f = 0 , on B ⇥ I ,

t = Sm , on @B ⇥ I .

c = �divh , on B ⇥ I ,

� h ·m = q on @B ⇥ I .

J = detF = 1 + ⌦c

S =@

@F� pF? ,

µ =@

@c+ p⌦ ,

h(F, c,rµ) ·rµ 0 .

‣ Balance of forces

‣ Balance of solvent

‣ Swelling constraint

‣ Constitutive equations

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3D theory of swelling - Governing equations

A.L. et al., Transient analysis of swelling-induced large deformations in polymer gels, JMPS 61, 2013.!

divS+ f = 0 , on B ⇥ I ,

t = Sm , on @B ⇥ I .

c = �divh , on B ⇥ I ,

� h ·m = q on @B ⇥ I .

J = detF = 1 + ⌦c

S =@

@F� pF? ,

µ =@

@c+ p⌦ ,

h(F, c,rµ) ·rµ 0 .

‣ Balance of forces

‣ Balance of solvent

‣ Swelling constraint

‣ Constitutive equations fs = fe � t

qs = qe � q

Couplings with the boundary membrane

µ = µs

S ⇢ @B

fe, qe : External “loads”

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Smart membranes: applications (1/3)

t = 12 h

t = 24 h (continuous) t = 24 h (on-off)

t = 48 h (on-off)

cdr0 0.5 1

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0 6 12 18 24 30 36 42 48D

rug

rele

ase

perc

ent

Time [h]

Smart drug delivery system

A hydrogel tablet coated with a thermo-responsive gel membrane. Drug loaded in the tablet is released by diffusion. When temperature is increased beyond the transition temperature, the coating gel shrinks, causing a reduction of the permeability of the surface, which hampers further drug release.!

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x/L

(T-To)/(T

m-To)

t/τ

xp/L

xa/L

∆/L

Temperature-activated crawler

A crawler advancing on a directional substrate by exploiting dry friction. When temperature is increased beyond the transition temperature, the crawler bends and the posterior edge in contact with the surface moves over it, while the anterior edge stays fixed.!

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Spreading and absorption of a liquid on the surface of a gel layer

A gel layer coated with a surface having a different solvent permeability. A solvent flux is prescribed for 1 s in a circular region at the center of the layer. The solvent partially spreads over the surface and is also absorbed by the layer. Plot of the swelling ratio J at time t = 1 s over 1/4 of the gel layer for (left) Ds = 1×10-10 m2/s = D and for (right) Ds = 1 × 10−7 m2/s.

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1.15

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1

1.01

J J

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Ongoing work: modeling of swelling shells

Wrinkling of a corona gel

A corona gel is clamped on its inner edge and swollen with a solvent. Compressive stresses arise due to the constraint and a wrinkling pattern emerges. Consistent with experiments, the wavenumber of the wrinkles increases with the ratio Ri/H between the inner radius of the annulus and the thickness of the membrane (the ratio Ro/H between the outer radius and the thickness is fixed).!

Ri/H = 12Ri/H = 8

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Modeling of swelling shells for shape morphing

Morphing of 3D shapes starting from non-homogeneous swelling polymer sheets

(from Kim et al. 2012)

(from Wu et al. 2013)

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Conclusions

‣ Instability patterns in soft materials are common in both natural and synthetic systems applications harnessing instabilities!

‣ Shape morphing applications involve 3D transformations of soft/smart/swelling membranes!

‣ Controlled wrinkling of a single-layer swelling membrane may be achieved by tuning the pre-stretch!

‣ A swelling polymer gel membrane theory with thickness microstructure can describe boundary spreading/absorption and temperature-activated bending in polymer gels

�!

a membrane theory for swelling polymer gels...a membrane theory for swelling polymer gels!!...

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