Example: Uniaxial Deformation
y
z
With Axi-symmetric sample cross-section
→
Deform along x
αx = αx, α y = αz =
1
l0 → lx , α x =
x
lx
dl
, dα x = x
l0
l0
Poisson contraction in lateral directions
αx
since αxαyαz = 1
Rewriting ΔS (α i ) explicitly in terms of α x
⎞
k⎛ 2 1
1
⎜
ΔS( αi ) = − ⎜ α x +
+
− 3 ⎟⎟
2⎝
αx αx
⎠
⎞
∂
kT ∂ ⎛ 2 2
Fx = −T
(ΔS(α x )) T ,P =
⎜α x + − 3 ⎟
αx ⎠
∂lx
2 ∂lx ⎝
Uniaxial Deformation cont’d
kT ⎛⎜
2 ⎞⎟ kT ⎛⎜
1 ⎞⎟
Fx =
αx − 2
2α x − 2 =
⎜
⎟
⎜
2l0 ⎝
α x ⎠ l0 ⎝
α x ⎟⎠
At small extensions, the stress behavior is Hookean (Fx ~ const αx)
Stress-Extension Relationship
σ xx
Fx (α x ) zkT ⎛
1 ⎞
⎜
= total
=
α x − 2 ⎟⎟ for z subchains in volume V
⎜
A0
A0l0 ⎝
αx ⎠
z
= # of subchains per unit volume = N xlinks
A0l0 = V
V
Usually the entropic stress of an elastomeric network is written in terms of Mx
where Mx = avg. molecular weight of subchain between x-links
mass /vol
=
molesofcrosslinks/volume
Mx =
ρ
N x /N A
1 ⎞
ρ N A kT ⎛
⎟
⎜
σ xx ( α x ) =
α
−
x
2
M x ⎜⎝
α x ⎟⎠
uniaxial
Young’s Modulus of a Rubber
dσ xx ρRT ⎛
2 ⎞
⎜⎜1 + 3 ⎟⎟
=
E ≡ lim
α x →1 dα
M x ⎝ αx ⎠
x
E = 3ρRT/Mx
Notice that Young’s Modulus of a rubber is :
1. Directly proportional to temperature
2. Indirectly proportional to Mx
• Can measure modulus of crosslinked rubber to derive Mx
• In an analogous fashion, the entanglement network of a melt, gives
rise to entropic restoring elastic force provided the time scale of the
measurement is sufficiently short so the chains do not slip out of
their entanglements. In this case, we can measure modulus of
non-crosslinked rubber melt to derive Me !
Stress-Uniaxial Extension Ratio Behavior of Elastomers
Uniaxial Stretching
tensile
5
?
σ(α)
4
Please see Fig. 5 in Treloar, L. R. G. “StressStrain Data for Vulcanised Rubber Under
Various Types of Deformation.” Transactions
of the Faraday Society 40 (1944): 59-70
compressive
Stress / MN m-2
Image removed due to copyright restrictions.
3
Theoretical
2
1
0
1
2
3
4
5
6
7
8
Small deformations
Figure by MIT OCW.
High tensile elongations
Rubber Demos
Tg for Happy Ball and Unhappy Ball
Deformation of Rubber to High Strains
Heating a Stretched Rubber Band
How stretchy is a gel?
Gel – A Highly Swollen Network
Several ways to form a gel:
• Start with a polymer melt and produce a 3D network
(via chemical or physical crosslinks) and then swell it with
a solvent
Example:
styrene
divinyl benzene
• Start with a concentrated polymer solution and induce
network formation, for example, crosslink the polymer by:
• Radiation- (UV, electron beam) (covalent bonds)
• Chemical- (e.g. divinyl benzene & PS) (covalent bonds)
• Physical Associations - (noncovalent bonds induced by
lowering the temperature or adding a nonsolvent)
Q: describe some types of physical (ie noncovalent) crosslinks
Flory-Rehner Treatment of Gels
• Assumes elastic effect and mixing effect are linearly additive
ΔG = ΔGmix + ΔGelastic
Polymer-solvent Thermodynamics, ΔGmix
Flory Huggins Theory (χ, x1, x2, φ1, φ2)
mix
solvent
Rubber Elasticity Theory ΔGelastic
deform
Flory-Rehner Theory
ΔG = ΔGmix + ΔGelastic
At equilibrium,
1.
Thermodynamically good solvent swells rubber
favorable χ interaction
favorable ΔSmix
•
•
2.
ΔGmix = −ΔGelastic
BUT Entropy elasticity of network (ΔSelastic) exerts retractive force to
to oppose swelling
Swollen Rubber
Crosslinked Rubber
Swell the network,
add n1 moles of solvent
x’
x
x
x’
x
volume, Vo
subchain, r02
x’
1
2
subchain
volume, V
2
subchain, r
1
2
subchain
Thermo of Swollen Network cont’d
x'
αx = linear swelling ratio = x
V0
φ2 = volume fraction polymer = V
N = # of subchains in the network =
ρV 0 N A
Mx
Isotropic swelling α x = α y = α z = α s
V0α = V
3
s
chemical potential,
μ1 − μ10 =
⇒
φ2 =
V0
V
so α s3 =
1
φ2
⎛ ∂ΔG tot ⎞
⎟⎟
μ i = ⎜⎜
n
∂
i
⎠ T,P,n j
⎝
chemical potential difference for solvent in gel vs pure solvent
⎛ ∂ΔG elastic ⎞⎛ ∂α s ⎞
⎛ ∂ΔG mix ⎞
⎟⎟⎜⎜
⎟⎟
⎟⎟
+ ⎜⎜
μ1 − μ10 = ⎜⎜
⎝ ∂n 1 ⎠ T,P,n 2 ⎝ ∂α s ⎠⎝ ∂n 1 ⎠ T,P,n 2
ΔGmix and Flory Huggins
• n1 moles of solvent
• n2 moles of polymer
⎡
⎤
ΔGm
φ
φ
= kT ⎢ χφ1φ2 + 1 ln φ1 + 2 ln φ2 ⎥
N0
x1
x2
⎣
⎦
recall
equivalently
some math
[n1 + n2x2]NA = N0
ΔG mix = RT [n1 ln φ1 + n2 ln φ 2 + n1 χ 12φ 2 ]
⎛ ∂ΔG mix ⎞ = RT[ln φ + φ (1− 1 ) + χ φ 2 ]
⎜⎜
⎟⎟
1
2
12 2
x2
⎝ ∂n1 ⎠
Since
x2 → ∞ for a network (one megamolecule)
⎛ ∂ΔG mix ⎞
⎜⎜
⎟⎟ = RT[ln φ + φ + χ φ 2 ]
1
2
12 2
⎝ ∂n1 ⎠
ΔGelastic and Rubber Elasticity
ΔG elastic = −TΔS elastic
ΔSelastic
new term
k⎛
V⎞
2
2
2
= − ⎜ [ (α x + α y + α z ) − 3 ] − ln ⎟
2⎝
V0 ⎠
• new term is an additional entropy increase per subchain, due to increased
volume available on swelling, of
k ⎛V ⎞
ln ⎜⎜ ⎟⎟
2 ⎝ V0 ⎠
• for N subchains
ΔSelastic
k ⎛ 2
V⎞
= − N ⎜ 3α s − 3 − ln ⎟
2 ⎝
V0 ⎠
ΔGelastic cont’d
⎛ ∂ΔG elastic ⎞ ⎛ ∂ΔG elastic ⎞⎛ ∂α s ⎞
⎟⎟⎜⎜
⎟⎟
⎜⎜
⎟⎟ = ⎜⎜
⎝ ∂n1 ⎠ ⎝ ∂α s ⎠⎝ ∂n1 ⎠
⎛ ∂ΔG elastic ⎞ NkT ∂ ⎛ 2
α s3V0 ⎞
⎜⎜ 3α s − 3 − ln
⎟⎟
⎜⎜
⎟⎟ =
2 ∂α s ⎝
V0 ⎠
⎝ ∂α s ⎠
⎞ NkT ⎛
NkT ⎛
1
3⎞
2
=
⎜ 6α s − 3 (3α s )⎟ =
⎜6α s − ⎟
αs
αs ⎠
2 ⎝
2 ⎝
⎠
To calculate
∂α s
∂n1
α s3 =
V V0 + n1v1
=
V0
V0
⎛ n1v1 ⎞
α s = ⎜1+
⎟
V0 ⎠
⎝
1/ 3
or
where v1 is the molar volume of the solvent
−2 / 3
∂α s 1 ⎛ n1v1 ⎞
v1
= ⎜1+
⎟
∂n1 3 ⎝
V0 ⎠ V0
=
1 v1
2
3α s Vo
ΔGelastic cont’d
collecting terms
⎛ ∂ΔG elastic ⎞ ⎛ ∂ΔG elastic ⎞⎛ ∂α s ⎞ NkT ⎛
3 ⎞ 1 v1
⎜
⎟=⎜
⎟⎜
⎟=
⎜6α s − ⎟ 2
∂n
∂
α
∂n
α s ⎠ 3α s V0
2
⎝
⎠ ⎝
⎠⎝ 1 ⎠
⎝
1
s
Now express
in terms of
φ 2 and M x
⎛ ∂ΔGelastic⎞
RTρv1 ⎛ 13 φ2 ⎞
v1 ⎛ 13 φ2 ⎞
⎜φ2 − ⎟
⎜
⎟ = NkT ⎜φ2 − ⎟ =
V0 ⎝
2⎠
Mx ⎝
2⎠
⎝ ∂n1 ⎠
Collecting the terms we’ve worked out,
RT ρ v1 ⎛ 1 3 φ 2 ⎞
μ1 − μ = RT[ln φ1 + φ 2 + χ φ ] +
⎜φ 2 − ⎟
Mx ⎝
2⎠
0
1
2
12 2
At equilibrium, μ1 = μ1
0
ln(1− φ 2 ) + φ 2 + χ φ = −
2
12 2
ρ v1 ⎛
1
φ2 ⎞
⎜φ2 − ⎟
Mx ⎝
2⎠
3
Determination of χ12, Mx from Swelling Experiments
I.
Generally ρ, v1 are known
φ2 measured from equilibrium swelling ratio
V
V0
If Mx is known from elastic modulus of dry rubber
then, χ12 available from single swelling experiment
II.
ρ, v1, φ1, φ2 known as above
π
Assume, χ12 known from c measurements for polymerz
solvent solution. Then, Mx available from single swelling
experiment
The Most Relaxed Gel: Swell polymer in a θ solvent so that the chains
overlap and then crosslink. In this case, swollen chains have their
unperturbed dimensions. Mx will be large since there are fewer
subchains per unit volume in the dilute and swollen state (c2 ~ c2* in
order to form network). The maximum extension of this type of gel
can be HUGE!!
Practice Problems to Try
(1) Calculate Mx from known swelling ratio and
V
=4
V0
= Š 0.2
=1.0g/cm3 v1 = 40 cm3/mole
ANS:400 g/mole
(2) Suppose Mx=5,000 g/mole and
=1.0g/cm3, v1 = 40 cm 3 /mole
= 0.5,
What is the % solvent in the network for
equilibrium swelling?
End of material that will be covered by the 1st exam.
Self Organization
Order
Homogeneous
Disorder
state
Structurally
ordered state
Order
Structurally
ordered state
• Competing interactions: Enthalpy (H) vs. Entropy (S)
• Free energy landscape: entropic frustration, multiple pathways
• Order forming processes
- (Macro)Phase separation
- Microphase separation
- Mesophase formation
- Adsorption/complexation
- Crystallization
• Selection of symmetries and characteristic lengths
- Chemical affinities (long range correlations)
- Interfacial tension
Muthukumar, M., Ober, C.K. and Thomas, E.L., "Competing Interactions
and Levels of Ordering in Self-Organizing Materials," Science, 277, 1225-1237 (1997).
Competing Interactions and Levels of Ordering in
Self-Organizing (Soft) Materials
Materials
• liquid crystals
• block copolymers
• hydrogen bonded complexes
• nanocrystals
Structural order over many length scales
• atomic
• molecular
increasing size scale
• mesogens
• domains
• grains
Outcome:
Precise shapes, structures and functions
Strategic Design for Materials with Multiple Length Scales
•
Synthetic design strategy
- Intramolecular shapes and interaction sites (molecular docking, etc)
- Control multistep processing to achieve long range order
•
Interactions
- sequential
- simultaneous
- synergistic
- antagonistic
•
Reduction of disorder (S )
Strengthening of intra- and (H )
inter-molecular interactions
Structural design strategy
- organize starting from initially homogeneous state
- organize from largest to smallest length scale
(induce a global pattern, followed by sequential development of finer details)
•
Selection of growth directions
- applied bias field(s)
- substrate patterning
•
Prior-formed structures impose boundary conditions
- commensuration of emergent and prior length scales
- compatibility of structures across interfaces
Principles of Self Organization:
Microphase Separation Block Copolymers
The min - max principle:
• Minimize interfacial area
• Maximize chain conformational entropy
Result:
• Morphology highly coupled to molecular characteristics
• Morphology serves as a sort of molecular probe
Homogeneous
Disordered State
Gas of junctions
Ordered State
Junctions on
Surfaces
T < TODT
IMDS
Figure by MIT OCW.
Microdomain Morphologies and Symmetries
- Diblock Copolymers
IMDS
Spheres
Cylinders
0-21%
21-33%
Double
Gyroid
Double
Diamond
33-37%
Lamellae
37-50%
"A" block
Increasing volume fraction of minority phase polymer
"B" block
Junction point
Figure by MIT OCW.
Hierarchical Structure & Length Scales
[ CH
2
CH CH
CH2]n
Polybutadiene
[ CH2 CH]n
Polystyrene
-0.5 nm
loop
bridge
-20 nm
-40 nm
Figure by MIT OCW.
Computing the characteristic length scale:
Equilibrium Domain Spacing
Min-Max Principle
∑, γ
Sharp Interface
G = Free Energy per Chain
N = # of segments = NA + NB
segmental
aA ~ aB
a = Step Size
volume ~ a 3
λ = Domain Periodicity
Σ = Interfacial Area/Chain
kT χ AB
γAB = Interfacial Energy = 2
6
a
χ AB = Segment - Segment Interaction Parameter =
AB
λ
2
z ⎡
1
⎤
[
]
ε
−
ε
+
ε
AB
AA
BB ⎥
kT ⎢⎣
2
⎦
Strong Segregation Limit → Nχ very large (high MW and positive χ),
=> pure A domains & pure B microdomains
Characteristic Period (Lamellae)
State 1
State 2
Melt
IMDS
ΔG = ΔH − TΔS
= γ AB ∑
2
⎤
3 ⎡ (λ /2)
− N χ AB φ A φ B kT +
kT⎢
−1⎥
2
2 ⎢⎣ Na
⎥⎦
Enthalpic Term
ΔG( λ) =
Microdomains
kT
a2
Note: Na =
3
Entropic Spring Term
2
⎡
⎤
λ
/2
3
χ AB Na
(
)
− N χ AB φ A φ B kT + kT⎢
−1⎥
2
2 ⎢⎣ Na
6 (λ /2)
⎥⎦
3
α
ΔG( λ) = − const1 + βλ2 − const2
λ
∂ΔG
=0
∂λ
0 =
−α
+ 2βλ
2
λ
λ
2
Σ
Free Energy of Lamellae con’t
Thus, the optimum period of the lamellae repeat unit is :
λopt = 3
α
≅ aN 2 / 3 χ 1/ 6
2β
Important Result: Domain dimensions scale as λ ~ N 2 / 3
Chains in microdomains are therefore stretched
compared to the homogeneous melt state
ΔG(λopt ) = 1.2kT N χ AB
13
13
3
− kT
2
Order-Disorder Transition (ODT)
Estimating the Order-Disorder Transition:
GLAM ≅ GDisordered
1.2kTN 1/ 3 χ 1/ 3 ≈ Nχ AB φ A φ B kT
For a 50/50 volume fraction,
since both terms >>
φA φB = 1 / 4
3
kT
2
so
1.2 N1 / 3χ1 / 3 = Nχ / 4
The critical Nχ is just ( Nχ) c = ( 4.8) 3 / 2 ~ 10.5
Nχ < 10.5
Nχ > 10.5
Homogeneous, Mixed Melt
Lamellar Microdomains
Original Order-Disorder Diblock Phase Diagram computed by L. Leibler ,
Macromolecules, 1980
Diblock Copolymer Morphology Diagram
B
B
B
B
Spheres
Cylinders
Double Gyroid
Lamellae
Im3m
p6mm
Ia3d
pm
strong
segregation
limit
Nχ >> 100
80
χN
Figure by MIT OCW.
40
S`
S
L
C
60
C'
G
20
CPS
CPS`
10.5
Disordered
0
0
0.2
0.4
0.6
0.8
weak
segregation
limit
Nχ ~ 10
1.0
fA
Figure by MIT OCW.