Unformatted text preview: Lecture 13
Networks and Gels
Rubber Elasticity and Polymer Melt Rheology
references for this lecture:
Rubenstein and Colby, Chapter 7
Painter and Coleman, Essentials Chapter 13
(Painter and Coleman Fundamentals Chapter 11)
Strobl CHapter 9 1
Monday, November 15, 2010 Polymers and Entropy Driven Elasticity
•
•
•
• The chain folded structure and/or coiled up chains explain the ability of
polymers to be extensively deformed, compared to metals and ceramics
Once they are deformed, what is the driving force to recover their original
shape?
Polymer (rubber) elasticity is entropy driven
For a polymer exhibit elasticity, there needs to be some sort of network on a
time scale relevant to the deformation 2
Monday, November 15, 2010 Elastomers are entropy “springs” uncoiling
coiling
relaxed structure(s)
many energy states large Ω
high entropy Δ S < 0 stretched structure
few energy states
small Ω
low entropy Δ S = k ln [Ωstretched / Ωrelaxed] < 0 3
Monday, November 15, 2010 Entropy driven elasticity
•
•
•
• Polymer (rubber) networks exhibit predominately entropydriven
elasticity
Uniaxially stretched rubber, under constant load, contracts reversibly
upon heating (WHY?)
Rubber gives out heat reversibly when stretched
The entropy decreases, due to the orientation of the chain in the
direction of deformation 4
Monday, November 15, 2010 But wait… •
• How do the chains get tied together?
What do we really mean by network formation? 5
Monday, November 15, 2010 Networks •Polymer chain networks interpenetrate
• If you stretch a chain or a network from its equilibrium end to end
distance, there is a retractive force to return the molecule to its equilibrium
coil dimension, this is the origin of rubber elasticity
6
Monday, November 15, 2010 Polymer networks
•
• Polymer networks are very important soft solids, used in a range of
biomedical and tissue engineering applications
In contrast to plastic materials, which can be molded and shaped
(sometimes repeatedly) but lack reversible elasticity, in crosslinked
polymers deformation is reversible 7
Monday, November 15, 2010 Elastomers
• •
•
•
• An elastomer is a polymer which exhibits rubber elastic properties, for
example, a material that can be stretched to several times its original
length without breaking, and, upon release of the stress, immediately
returns to its original length (reversible deformation, not limited to
stretching)
Crosslinking of unsaturated bonds can occur using sulfur (vulcanization),
peroxides or other radicals (other mechanism possible too)
Natural rubber is a crosslinked, elastomeric network
Different modes of crosslinking?
How will different modes affect elastic properties? 8
Monday, November 15, 2010 Crosslinked Networks
• • crosslinks
– chemical or physical
– covalent; ionic; Hbonding; microphase segregation,
entanglements?
crosslinking
– increased molecular weight
– swell in solvents
• organogel
• hydrogel 9
Monday, November 15, 2010 Chemical Crosslinking
• Forming chemical bonds between polymer chains • Adding crosslinks increases elasticity, toughness • If enough crosslinking, the material can lose crystallinity and never have a Tm • Can result in a network structure • As extent of crosslinking increases (or crosslinking density), the material
becomes less deformable • Can crosslink across unsaturated bonds (in polybutadiene, for example), or
by adding a crosslinking agent (during or postpolymerization) (vulcanization) 10
Monday, November 15, 2010 Physical Crosslinking
• Physical crosslinks: crosslinking as a result of noncovalent interactions,
especially microphase segregation •
• Segregation as a result of secondary interactions (hydrogen bonding in
polyurethanes)
Example: a thermoplastic elastomer has two blocks  a short, rigid block of
polyurethane, and a longer, flexible chain. The polyurethane blocks
segregate into small domains. This prevents the chains from slipping during
deformation, and also acts as small particulate filler to toughen the sample. • Segregation as a result of Tg differences • Example: polystyrene and 1,3diene copolymers at RT, polystyrene is below
its Tg, so the polystyrene blocks are glassy, while the polydiene is
amorphous. The polystyrene blocks phase segregate into small domains
that act to crosslink the material. 11
Monday, November 15, 2010 What is happening? 12
Monday, November 15, 2010 Entropy Driven Elasticity
•
• • •
•
• In a network at its equilibrium position, the chains are not fully stretched
When they become stretched, their entropy decreases, so they release
the heat that was being “used” by the chain to rotate about different
conformations
When we allow the network to relax to its equilibrium position (for a bulk
network, this will be the balance between the chains wanting to be in the
theta state, and the shrinkage stress due to the volume change that
occurs upon crosslinking); the chains will coil up again, and this requires
energy (heat) to allow the many different conformations
Because of high segmental mobility (we are above the Tg) of rubbers,
instantaneous deformations are possible
Reversible deformation is a consequence of the fact that rubbers are
lightly crosslinked materials, typically
The segments between the crosslinks can move freely and easily rotate
around their sigma bonds 13
Monday, November 15, 2010 Some thermo…
• Consider the free energy of a system at constant temperature and volume
(Helmholtz free energy) F = E − TS
• If we stretch a sample to length L, the free energy changes; this change
can be related to the applied force, f: ⎛ ∂F ⎞
⎛ ∂E ⎞
⎛ ∂S ⎞
f = ⎜ ⎟ = ⎜ ⎟ − T⎜ ⎟
⎝ ∂L ⎠V ,T ⎝ ∂L ⎠V ,T
⎝ ∂L ⎠V ,T
• In an ideal crystalline material, the change in internal energy is enthalpic in
origin (stretching bonds):
⎛ ∂E ⎞
f ≈⎜ ⎟
⎝ ∂L ⎠V ,T • In an ideal rubber, the change in internal energy is entropic in origin
(stretching chains):
⎛ ∂S ⎞
f ≈ −T⎜ ⎟
⎝ ∂L ⎠V ,T
14 Monday, November 15, 2010 Some more thermo…
•
•
• The change in entropy, resulting from the change in the distribution of
conformations, is responsible for generating the restoring force
This is true of ideal elastomers, but real systems are more complex
For now, we’ll approximate the system as ideal ⎛ ∂F ⎞
⎛ ∂E ⎞
⎛ ∂S ⎞
f = ⎜ ⎟ = ⎜ ⎟ − T⎜ ⎟
⎝ ∂L ⎠V ,T ⎝ ∂L ⎠V ,T
⎝ ∂L ⎠V ,T
This term is zero for
ideal elastomer 15
Monday, November 15, 2010 Entropy driven elasticity
•
•
• Flory constructed a simple method to separate energetic from entropic
contribution to the elastic force
Consider a typical temperature dependence of a retractive force, f, for a
network of constant volume V at constant elongation L.
The slope of the curve at T is: ⎛ ∂f ⎞
slope = ⎜ ⎟
⎝ ∂T ⎠V ,L
• The change in the ordinate from the point on
the curve to the intercept of the tangent with
the faxis is the entropic contribution to the
force ⎛ ∂E ⎞
fE = ⎜ ⎟
⎝ ∂L ⎠T ,V (in ideal networks, this is
approximately zero) ⎛ ∂f ⎞
⎛ ∂S ⎞
f S = T⎜ ⎟ = −T⎜ ⎟
⎝ ∂T ⎠V ,L
⎝ ∂L ⎠T ,V
Monday, November 15, 2010 (in crystalline materials, this is
low or approximately zero) 16 Molecular Models of Rubber Elasticity
• •
•
•
• • In the affine model, chain segments deform independently; the relative
deformation of each network strand is the same as the relative
macroscopic deformation imposed on the whole network
In the affine model, the crosslinks are assume fixed in space, and the
chains between the crosslinks have conformational freedom
In the phantom network model, a certain free motion of the crosslinks
about their average affine deformation positions is also allowed
These two models can be used to predict the modulus of the network
These two are limiting cases  that is the affine network theory gives the
upper modulus boundary and the phantom network gives the lower
boundary
Assumes the dimensions of these chains are unperturbed by excluded
volume effects 17
Monday, November 15, 2010 Rubber Elasticity What this picture doesn’t capture well (due to being 2D) is that we
are talking about unentangled networks
18
Monday, November 15, 2010 Affine Model
ASSUMPTIONS
• The chain segments between crosslinks can be represented by
Gaussian statistics of unperturbed (phantom) chains
• Networks consist of Nchains per unit volume
• The entropy of the network is the sum of the entropies of the individual
chains
• All different conformation states have the same energy
• The deformation on the molecular level is the same as that on a
macroscopic level (affine)
• The unstressed network is isotropic
• The volume remains constant during deformation 19
Monday, November 15, 2010 Affine Model
2 2 2 3 R2
3 Rx + Ry + Rz
S N , R = − k 2 + S ( N , 0) = − k
+ S ( N , 0)
2 Nb
2
Nb 2 () 2 2 2
2 2 2 3 Rx + Ry + Rz 3 Rx 0 + Ry 0 + Rz0
ΔS = S N , R − S N , R0 = − k
+k
2
Nb 2
2
Nb 2 ()( ) (sum over every strand in the network)
Rx = λx Rx 0
Ry = λy Ry 0 ΔSnet = − l
λ=
l0 Rz = λz Rz0 nk 2
(λx + λ2y + λ2z − 3)
2 If there’s only an entropic contribution to the free energy:
ΔF = −TΔSnet = nkT 2
(λx + λ2y + λ2z − 3)
2 Typically, dry networks are incompressible, so volume does not change upon deformation:
V = Lx 0 Ly 0 Lz0 = Lx Ly Lz = λx Lx 0 λy Ly 0 λz Lz0 = λx λy λzV For uniaxial deformation:
Monday, November 15, 2010 λx λ y λ z = 1 ΔFnet = −TΔSnet = λx = λ
λy = λ z = 1
λ ⎞
nkT ⎛ 2 2
⎜ λ + − 3⎟
2⎝
λ⎠
20 Affine Model
ΔFnet = −TΔSnet = f= ∂ΔFnet ∂ΔFnet
1 ∂ΔFnet nkT ⎛
1⎞
=
=
=
⎜λ − 2 ⎟
∂Lx
Lx 0 ⎝
λ⎠
∂ ( λLx 0 ) Lx 0 ∂λ σ= f
f
nkT ⎛
1 ⎞ nkT ⎛ 2 1 ⎞
= x=
⎜λ − 2 ⎟ =
⎜λ − ⎟
A Ly Lz Lx 0 Ly Lz ⎝
λ⎠ V ⎝
λ⎠ G= • ⎞
nkT ⎛ 2 2
⎜ λ + − 3⎟
2⎝
λ⎠ nkT
ρRT
= vkT =
V
Ms • This equation relates the true stress and the extension ratio
The modulus ρRT/Ms is proportional to T • The modulus is also inversely proportional to Ms (Mc) •
• Rubbers with high crosslink density (low Mc) are stiff
With constant stress, the modulus increases with increasing
temperature Monday, November 15, 2010 21 Phantom Model
• In the phantom model, crosslinks are not fixed in space, but can
fluctuate • Effective chains are used to represent the way elasticity is transmitted
from macroscopic scales down to the individual chains • The phantom network model (with fluctuating junctions but fixed
endpoints) is equivalent to an affine network model with a combined
effective chain instead of just the chain between two nonfluctuating
crosslinks
The expression for free energy extension includes a term, f, which
expresses functionality at the crosslink site: • ⎛ 2 ⎞ nkT ⎛ 2 2
⎞
ΔFnet = ⎜1 − ⎟
λ + − 3⎟
⎜
f⎠ 2 ⎝
λ⎠
⎝ • The stress becomes:
⎛ 2 ⎞ nkT ⎛ 2 1 ⎞
⎛ 2 1⎞
σ = ⎜1 − ⎟
⎜ λ − ⎟ = G⎜ λ − ⎟
⎝
f⎠ V ⎝
λ⎠
λ⎠
⎝
22 Monday, November 15, 2010 Phantom Chain Model
• The stress becomes:
⎛ 2 ⎞ nkT ⎛ 2 1 ⎞
⎛ 2 1⎞
σ = ⎜1 − ⎟
⎜ λ − ⎟ = G⎜ λ − ⎟
⎝
f⎠ V ⎝
λ⎠
λ⎠
⎝ • If f=4, the phantom model predicts a modulus half that given by the
affine model G= G= nkT
ρRT
= vkT =
V
Ms
affine ⎛ 2 ⎞ ρRT ⎛ 2 ⎞
nkT ⎛ 2 ⎞
⎜1 − ⎟ = vkT⎜1 − ⎟ =
⎜1 − ⎟ phantom
V⎝
f⎠
f ⎠ Ms ⎝
f⎠
⎝ G= ρRT
Mx Here, Mx is greater than Ms, and is the
combined strand length in the
phantom network 23
Monday, November 15, 2010 Comparison: Rubber Elasticity Models
• Affine gives upper limit modulus:
σ true = • nkT ⎛ 2 1 ⎞
⎜λ − ⎟
V⎝
λ⎠ Phantom gives lower limit modulus:
⎛ 2 ⎞ nkT ⎛ 2 1 ⎞
⎛ 2 1⎞
σ = ⎜1 − ⎟
⎜ λ − ⎟ = G⎜ λ − ⎟
⎝
f⎠ V ⎝
λ⎠
λ⎠
⎝ • Affine and Phantom fail to predict the stress strain behavior at large
strain
affine nkT ρRT
G=
=
V
Ms G= ρRT ⎛ 2 ⎞
⎜1 − ⎟ phantom
Ms ⎝
f⎠ 24
Monday, November 15, 2010 Deviations from Theory
• In classical theories
– Chains are assumed to be
endless, though loose chain
ends transfer stress less
efficiently than other chain
parts
– Other types of defects exist in
networks 25
Monday, November 15, 2010 Deviations from Theory
• Deficiencies in the statistical mechanics:
– The segment vectors do not follow Gaussian statistics when the
rubber is highly stretched  experimental results do not follow
theoretical values (the number of states in high deformations is
smaller than predicted by theories)
– Gaussian theory does not describe highly crosslinked materials very
well 26
Monday, November 15, 2010 MooneyRivlin Model
•
• Alternatively, we can use a phenomenological model (doesn’t include
molecular interpretations)
The MooneyRivlin model uses strain invariants to derive an expression
for stress; in classical models, 2C1=G and C2=0
⎛
⎛
1⎞
1⎞
σ = 2C1⎜ λ − 2 ⎟ + 2C2 ⎜1 − 3 ⎟
⎝
⎝ λ⎠
λ⎠ A MooneyRivlin plot, the stress divided by the
classical model prediction is on the y axis, and
the inverse of the deformation is on the xaxis; C2
shows deviation from classical theories 27
Monday, November 15, 2010 Comparison of the Models 28
Monday, November 15, 2010 Entangled Networks
• •
•
•
• The phantom model should be the lower modulus limit, and the affine
model the upper modulus limit, but the modulus of many real systems is
considerably larger than either model predicts
Remember, both the phantom and affine models assume that the chains
do not entangle, that chain motion is not restricted by other chains
In reality, network chains impose topological constraints on one another
(entanglements)
Imagine a single chain in a polymer network, where many chains form
loops around it (entanglements)
These loops or entanglements limit the movement of that single polymer
chain to an effective tube 29
Monday, November 15, 2010 Plateau Modulus for Polymer Melts
• Recall that the elastic shear modulus of a network depends on molecular
weight between crosslinks, Mc. In a polymer melt, G therefore depends on
Me .
• Just as for a polymer network: ρRT
Ge =
Me 30
Monday, November 15, 2010 Entanglement Parameters ρRT
Ge =
Me 31
Monday, November 15, 2010 Entangled Networks
Entanglements confine the movement of a polymer chain
The molar mass of the entanglement strand is Me=NeM0, and this strand
replaces the network strand in our determination of the modulus for networks
made from long strands
Although the entanglements are not permanent, the time scale over which they
disentangle is not instantaneous; think of it like “temporary” crosslinks
Ge = ρRT
Me ⎛1
1⎞
G ≅ Gx + Ge = ρRT⎜
+
⎟
Mx Me ⎠
⎝ 32
Monday, November 15, 2010 Entangled Networks
This graph shows computer simulations of network modulus for networks with
three different strand lengths that do not entangle (open) and that do entangle
(filled) This discrepancy
is due to
entanglement 33
Monday, November 15, 2010 Now that we’ve covered polymer elasticity…
• It’s time to talk about the viscous nature of polymer melts (polymer melt
rheology) 34
Monday, November 15, 2010 Boston Molasses Disaster 35
Monday, November 15, 2010 Shear
Shear:
stress (τ) : applied force per unit area
strain (γ) : deformation (as a result of stress)
θ τ= Fs
A0 δ
γ = = tan θ
z A perfectly elastic
undergoing nondestructive shear will
deform almost instantly
proportionally to G
!
E δ
z $ K G " #
36 Monday, November 15, 2010 Viscosity
Unlike an elastic solid, a fluid continues to deform under the action of
shear stress, dissipating the energy as flow
This time, the top surface moves a distance δ in t sec τ= . Fs
A0 . θ . δv
γ= = 0
zz δ
z dv d ⎛ dx ⎞
γ xy =
=⎜⎟
dz dt ⎝ dz ⎠ 37
Monday, November 15, 2010 Viscosity
Viscosity: resistance of a fluid to deformation under shear stress
stress (τ) : applied force per unit area
.
Strain rate (γ) : change in strain over time
Newtonian fluid: linear response to stress
nonNewtonian fluid: nonlinear response to stress
.
τ = ηγ n
n < 1 pseudoplastic (shear thinning)
n = 1 Newtonian
n > 1 dilatent (shear thickening) pseudoplastic
newtonian fluid ! η
G
dilatent ."
thixotropic  fluid decreases viscosity with time under shear
rheopectic  fluid increases viscosity with time under shear
38
Monday, November 15, 2010 Viscosity 39
Monday, November 15, 2010 Viscosity ηm = K L ( DP )1.0
w
ηm = K H ( DP ) 3.4
w
Monday, November 15, 2010 40 Entanglements and Polymer Chain Dynamics 41
Monday, November 15, 2010 Development of Reptation Scaling Theory
Pierre de Gennes (Paris) developed the concept of polymer reptation and
derived scaling relationships. Sir Sam Edwards (Cambridge) devised tube models and predictions of the
shear relaxation modulus.
In 1991, de Gennes was awarded the Nobel Prize for Physics. 42
Monday, November 15, 2010 Concept of Chain Entanglements
If the molecules are sufficiently long (N >100  corresponding to the entanglement
mol. wt., Me), they will entangle with each other.
Each molecule is confined within a dynamic tube Tube Monday, November 15, 2010 G.Strobl, The Physics of
Polymers, p. 283 Network of Entanglements There is a direct analogy between chemical crosslinks in rubbers and “physical”
crosslinks that are created by the entanglements. The physical entanglements can support stress (for short periods up to a time
τT), creating a “transient” network. 44
Monday, November 15, 2010 Snakes! There are obvious similarities between a collection of snakes and the
entangled polymer chains in a melt.
The source of continual motion on the molecular level is thermal energy, of
course.
45
Monday, November 15, 2010 Reptation Theory • Polymer molecules “disentangle” after a time τT. • Chain entanglements create restraints to other chains, defining a “tube”
through which they must travel.
The process by which a polymer chain moves through its tube formed by
entanglements is called reptation
Reptation (from the Latin reptare: to crawl) is a snakelike diffusive motion that
is driven by thermal motion. •
•
•
•
• Models of reptation consider each repeat unit of the chain as diffusing
through a tube with a drag coefficient, ξ
The tube is considered to be a viscous medium surrounding each segment.
For a polymer consisting of N units: ξpol = Nξ 46
Monday, November 15, 2010 Polymer Reptation
• The distance traveled, l, by a particle diffusing in a medium in a time t
goes by as t1/2 (related to random walk)
l 2 Dt • We can use these equations to determine the time it takes a polymer
chain to diffuse out of its tube
kT
Dtube =
Nξ
lc2
τd =
Dtube • Since l ~N, we get: • The relaxation time is related to ratio of η to G at the transition between
elastic and viscous behavior τ~η/Ge η ~ τd ~ N3 47
Monday, November 15, 2010 Relaxation Modulus for Polymer Melts
Elastic Viscous flow t Gedde, Polymer Physics, p. 103
Monday, November 15, 2010 Experimental Evidence for Reptation
Fluorescentlystained DNA molecule manipulated with optical tweezers
After a brisk tug, the DNA chain relaxes back along the path of its reptation
tube Initial state Stretched Chain follows the path of the front 49
Chu et al., Science (1994) 264, p. 819.
Monday, November 15, 2010 Next time… • Combining the viscous and elastic behavior of polymer networks to get
viscoelasticity 50
Monday, November 15, 2010 ...
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