rheology gels and networks

Structural PropertiesElastic Properties

Viscoelastic Properties

Rheology Gels and Networks

Tyler Shendruk

March 31, 2011

Tyler Shendruk Rheology Gels and Networks

Outline1 Structural Properties2 Elastic Properties

Thermodynamics3 Viscoelastic Properties

Crosslinks

Gel Solid formed by linking molecular strands intonetwork

Crosslink Bonds can be

Physical linking

Strong effectively permanentMicrocrystalsGlassy clusters

Weak Reversible and temporaryassociations

Hydrogen bondsHydrophobic associationIonic interactions

Chemical linking

Covalent bonds.

Crosslinks

Physical linking

Chemical linking

Covalent bonds.

Crosslinks

Physical linking

Chemical linking

Covalent bonds.

Crosslinks

Physical linking

Chemical linking

Covalent bonds.

Chemical Gelation

Chemical Gelation Mechanisms

Condensation of monomers in a melt or solution

Vulcanization cross-linking of long chains

Chemical Gelation

Chemical Gelation Mechanisms

Condensation of monomers in a melt or solution

Vulcanization cross-linking of long chains

Formation

Linking

Sol polydisperse mixture of branched polymers in asolvent

Gel “Infinite Polymer”

Incipient Gel One structure perculates the entire system

Sol-gel Transition The moment the incipient gel forms

Formation

Linking

Formation

Linking

Gel Point

Mean-Field Theory

Linking does not necessarily lead to gelation

There’s some condition on the probablity of making a link andthe number of links

That sounds amenable to a mean-field theory

Coordination number?

Gel Point

Mean-Field Theory

Gel Point

Mean-Field Theory

Gel Point

Mean-Field Theory

Gel Point

Bethe Lattice

Monomers have some functionality f

Each has some probablity p to form a bond (independent ofother bonds)

Gel Point

Bethe Lattice

Gel Point

Bethe Lattice

Gel Point

Conclusion

The transition occurswhen

f − 1

Bond Percolation Model (Mean-FieldModel of Gelation)

Consider some parent site (who has adefinite grandparent)

There are (f − 1) potential neighbours

Thus, the average number of bonds is

p (f − 1)

If p (f − 1) < 1 then the next generationhas a smaller population than the lastand the sol must be finite

Else if p (f − 1) > 1 then each newgeneration has a larger population thanthe last and sol branches to infinitybecoming an incipient gel

Gel Point

Conclusion

f − 1

p (f − 1)

Gel Point

Conclusion

f − 1

p (f − 1)

Gel Point

Conclusion

f − 1

p (f − 1)

Gel Point

Conclusion

f − 1

p (f − 1)

Assessment

Comments

The probablity p is often called the extent of reaction andcan be used to specify the sol fraction and the gel fraction,the moments of the size distribution.

Mean-field percolation only holds on a Bethe Lattice and so in2 and 3D the number of functional groups and the degree ofpolymerization are not easily related (researchers resort tonumerical studies).

But Mean-field percolation works exceptionally well forvulcanization (where f is now the the number of crosslinkablemonomers).

Viscoelastic PropertiesThermodynamics

Entropy

Helmholtz Free Energy

The fascinating elastic properties arise from the entropy as follows:

F = U − TS

dF = −SdT − pdV + fdL

∣∣∣V ,L

dT +∂F

∣∣∣T ,L

dV +∂F

∣∣∣T ,V

Therefore,

f =∂F

∣∣∣T ,V

S = − ∂F

∣∣∣V ,L

Maxwell Relation

−∂S

∣∣∣T ,V

∣∣∣V ,L

Entropy

Force to Deform Network

f =∂F

∣∣∣T ,V

∣∣∣V ,L

Stress-Elongation

Shear Modulus

σ = Gλ

Free Energy of Deformation

Li = λiL0,i

N, ~R)

= −3

Nb2+ S0

Free Energy

∆F = −T ∆S =n

(λ2x + λ2y + λ2z

)IncompressibilityUniaxial Deformation

Shear Modulus

G =ρRT

Li = λiL0,i

N, ~R)

= −3

Nb2+ S0

Free Energy

∆F = −T ∆S =n

(λ2x + λ2y + λ2z

IncompressibilityUniaxial Deformation

Shear Modulus

G =ρRT

Li = λiL0,i

N, ~R)

= −3

Nb2+ S0

Free Energy

∆F = −T ∆S =n

(λ2x + λ2y + λ2z

Shear Modulus

G =ρRT

Li = λiL0,i

N, ~R)

= −3

Nb2+ S0

Free Energy

∆F = −T ∆S =n

(λ2x + λ2y + λ2z

Shear Modulus

G =ρRT

Phantom Network

Fluctuating Crosslinks

This does not take into account fluctuations which should lowerfree nergy by reducing cummulative stretch.

Effective Free Chains

The fluctuations of some monomer in an ideal chain with ficedends is identical to the fluctuations of the end monomer with aneffective length

Shear Modulus

G =ρRT

(f − 2

Phantom Network

Fluctuating Crosslinks

This does not take into account fluctuations which should lowerfree nergy by reducing cummulative stretch.

Effective Free Chains

The fluctuations of some monomer in an ideal chain with ficedends is identical to the fluctuations of the end monomer with aneffective length

Shear Modulus

G =ρRT

(f − 2

)Tyler Shendruk Rheology Gels and Networks

Entanglement

Can entanglement give us the softening?

Entanglement

The tube diameter is

a ≈ bN2e

So a can be interpreted as the end-to-end distance betweenentanglement points. If an entanglement is equivalent to acrosslink and the shear modulus is kBT per crosslinked strand then

Ge =ρRT

where Me = Nem.

Which is independent of N but has notchanged the deformation dependence.

Entanglement

Can entanglement give us the softening?

Entanglement

The tube diameter is

a ≈ bN2e

So a can be interpreted as the end-to-end distance betweenentanglement points. If an entanglement is equivalent to acrosslink and the shear modulus is kBT per crosslinked strand then

Ge =ρRT

where Me = Nem. Which is independent of N but has notchanged the deformation dependence.

Subtle Tube

However, the tube also limits fluctuations.

When the network is deformed by some λi , we expect thetube diameter to change. Instead of

a ≈ bN1/2e

we expect N ′e∼= NeNeλ (since L ∝ N between

entanglements). The effective tube diameter is only

a′ ≈ bN′1/2e

= bN1/2e λ1/2

= aλ1/2

Subtle Tube

Shear Modulus

The effective entanglement molar mass goes like the effectiveentanglement number so the shear modulus becomes:

G = Gx + Ge

= Gx +Ge

Wrong-ish

Mooney-Rivlin Equation:

G = C1 +C2

Non-affin Tube Model:

G = Gx +Ge

λ− λ1/2 + 1

Subtle Tube

Shear Modulus

G = Gx + Ge

= Gx +Ge

Wrong-ish

G = C1 +C2

G = Gx +Ge

λ− λ1/2 + 1

Subtle Tube

Shear Modulus

G = Gx + Ge

= Gx +Ge

Wrong-ish

G = C1 +C2

G = Gx +Ge

λ− λ1/2 + 1

Maxwell Fluid

Network

rheology gels and networks

Documents