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Last Lecture:• The root-mean-squared end-to-end distance, <R2>1/2, of
a freely-jointed polymer molecule is N1/2a, when there are N repeat units, each of length a.
• Polymer coiling is favoured by entropy.• The elastic free energy of a polymer coil is given as
• Copolymers can be random, statistical, alternating or block.
• Thinner lamellar layers in a diblock copolymer will increase the interfacial energy and are not favourable. Thicker layers require chain stretch and likewise are not favourable! A compromise must be reached.
.++=)( constTNa
kRRF 2
2
2
3
Free Energy Minimisation
dNa
Na
dkTFtot
3
2
2 +
2
3
220d
Na
Na
dkT
dddFtot
=)(
2
3
22d
Na
Na
dkT
=
32315
2//)(= N
kTa
d Chains are NOT fully stretched -
but nor are they randomly coiled!
kTaN
d2
523
=
Two different dependencies on d!
Poly(styrene) and poly(methyl
methacrylate) copolymer
The thickness, d, of lamellae created by diblock copolymers is proportional to N2/3. Thus, the molecules are not fully-stretched (d ~ N1) but nor are they randomly coiled (d ~ N1/2).
Polymers in Solvent and Rubber Elasticity
3SCMP
16 March, 2006
Lecture 9
See Jones’ Soft Condensed Matter, Chapt. 5 and 9
Radius of Gyration of a Polymer Coil
R
For a hard, solid sphere of radius, R, the radius of gyration, Rg, is:
RRRRg 6320510
52
.===
21212
661
Na
RRg ==
R
A polymer coil is less dense than a hard, solid sphere. Thus, its Rg is significantly less than the rms-R:
The Self-Avoiding Walk
In describing the polymer coil as a random walk, it was tacitly assumed that the chain could “cross itself”.
In reality, of course, it cannot. Consequently, when polymers are dissolved in solvents, they are often expanded to sizes greater than a random coil.
Such expanded conformations are described instead by a “self-avoiding walk” in which <R2>1/2 is given by aN (instead of N1/2 as for a random coil).
What is the value of ?
Excluded Volume
Paul Flory developed an argument in which a polymer in a solvent is described as N repeat units confined to a volume of R3.
From the Boltzmann equation, we know that entropy, S, can be calculated from the number of ways of arranging a system, : S = k ln .
Each repeat unit prevents other units from occupying the same volume. The entropy associated with the chain conformation (“coil disorder”) is decreased by the presence of the other units. There is an excluded volume!
In an ideal polymer coil with no excluded volume, is inversely related to the density of units,:
NcR
RN
cc 3
3
~~~
where c is a constant
Entropy with Excluded Volume Hence, the entropy for each repeat unit in an ideal polymer coil is
)ln(=ln=N
cRkkSideal
3
In the non-ideal case, however, each segment is excluded from the volume occupied by the other N segments, each with a volume, b:
)ln(=))(
ln(= cbN
cRk
NNbRc
kSni
33
)]ln(+)[ln(=)](ln[= 3
3
3
3
11R
bNN
cRk
R
bNN
cRkSni
)]ln[(+= 31R
bNkSS idealni
But if x is small, then ln(1-x) -x, so:3R
kbNSS idealni
RNth unit
Unit vol. = b
Excluded Volume Contribution to F
For each unit, the entropy decrease from the excluded volume will lead to an increase in the free energy, as F = U - TS:
3R
bNkTFF idealni +=
Of course, a polymer molecule consists of N repeat units, and so the increase in F for a molecule, as a result of the excluded volume, is
3
2
R
bNkTRFexc =)(
Larger R values reduce the free energy. Hence, expansion is favoured by excluded volume effects.
In last week’s lecture, however, we saw that the coiling of molecules increased the entropy of a polymer molecule. This additional entropy contributes an elastic contribution to F:
2
2
2
3
Na
RkTRFel +=)(
Elastic Contributions to F
Coiling up of the molecules is therefore favoured by elastic contributions.
Reducing the R by coiling will decrease the free energy.
Total Free Energy of an Expanded Coil
.++=)( constTNa
kRT
R
kbNRFtot 2
2
3
2
2
3
The total free energy is obtained from the sum of the two contributions: Fexc + Fel
At equilibrium, the polymer coil will adopt an R that minimises Ftot. At the minimum, dFtot/dR = 0:
Fel
Ftot
RFexc
Ftot
24
2 3+
3=0=
Na
kRT
R
kbTNdR
dFtot
Characterising the Self-Avoiding Walk
24
2 33
Na
kRT
R
kbTN=
325 bNaR =So,
53 /= aNaNR
The volume of a repeat unit, b, can be approximated as a3.
355 NaR
This result agrees with a more exact value of obtained via a computational method: 0.588
Measurements of polymer coil sizes in solvent also support the theoretical (scaling) result.
Re-arranging:
Visualisation of the Self-Avoiding Walk2-D Random walks
21212 //= aNR
2-D Self-avoiding walks
53212 //= aNR
Polymer/Solvent Interaction EnergySo far, we have neglected the interaction energies between the components of a polymer solution.
Units in a polymer molecule have an interaction energy with other nearby (non-bonded) units: wpp
There is similarly an interaction energy between the solvent molecules (wss), and when the polymer is dissolved in the solvent, a new interaction energy between the polymer units and solvent (wps) is introduced.
wss
wps
Polymer/Solvent -ParameterWhen a polymer is dissolved in solvent, new polymer-solvent contacts are made, while contacts between like molecules are lost.
Following arguments similar to our approach for liquid miscibility, we can derive a -parameter for polymer units in solvent:
( )SSPPPS wwwkTz
= 22
where z is the number of neighbour contacts per unit or solvent molecule.
Observe that smaller coils reduce the number of P-S contacts because more P-P contacts are created. Uint is more negative and F is reduced.
We note that N/R3 represents the concentration of the repeat units in the “occupied volume”, and the volume of the polymer molecule is Nb.
When a polymer is added to a solvent, the change in potential energy, (from the change in w) will cause a change in internal energy, U:
3int )(=).)(2(=R
NNbkTunitsnowwwU SSPPPS 2
Significance of the -ParameterWe recall that excluded volume effects favour coil swelling:
3
2
R
bNkTRFexc =)(
Opposing the swelling, will be the polymer/solvent interactions, as described by Uint. (But also - elastic effects, in which Fel ~ R2, are also still active!)
3
2
21R
NkTbUFexc )(=+ int
As the form of the expressions for Fexc and Uint are the same, they can be combined into a single equation:
The value of then tells us whether the excluded volume effects are significant or whether they are counter-acted by polymer/solvent interactions.
Types of Solvent
• When = 1/2, the two effects cancel: Fexc + Uint = 0.
The coil size is determined by elastic (entropic) effects only, so it adopts a random-coil conformation.
2121
2 aNR =
3
2
21R
NkTbUFexc )(=+ int
• When < 1/2, then the excluded volume effects contribute to determining the coil size: Fexc + Uint > 0.
The solvent is called a “theta-solvent”.
5321
2 aNR =
The molecule is swollen in a “good solvent”.
Types of Solvent
3
2
21R
NkTbUFexc )(=+ int
When > 1/2, then the polymer/solvent interactions dominate in determining the coil size. Fexc + Uint < 0.
The molecule forms a globule in a “bad solvent”.
Energy is reduced by coiling up the molecule (i.e. by reducing its R).
Elastic (entropic) contributions likewise favour coiling.
TNa
kR
R
NFFRF eltot 2
2
3
2
int 2
3+~+=)(
Determination of Polymer Conformation
Good solvent: I q1/(3/5)
Scattering Intensity, I q-1/ or I-1 q1/
Theta solvent: I q1/(1/2)
Applications of Polymer Coiling
Nano-valves
Bad solvent: “Valve open”
Good solvent: “Valve closed”
Switching of colloidal stability
Good solvent: Sterically stabilisedBad solvent: Unstabilised
A Nano-Motor?
• The transition from an expanded coil to a globule can be initiated by changing .
A possible “nano-motor”!
> 1/2 < 1/2
Changes in temperature or pH can be used to make the polymer coil expand and contract.
Rubber ElasticityA rubber (or elastomer) can be created by linking together linear polymer molecules into a 3-D network.
To observe “stretchiness”, the temperature should be > Tg for the polymer.
Chemical bonds between polymer molecules are called “crosslinks”. Sulphur can crosslink natural rubber.
Affine Deformation
With an affine deformation, the macroscopic change in dimension is mirrored at the molecular level.
We define an extension ratio, , as the dimension after a deformation divided by the initial dimension:
o
=
oo ll
==
o
Bulk:
l
Strand:
lo
y x
z
x
z
y yy
zz
xx
z
y
xR2 = x2+y2+z2
Transformation with Affine Deformation
z
y
x
Bulk:
Ro
Single Strand
Ro = xo+ yo+ zo
R
R = xxo + yyo + zzo
Entropy Change in Deforming a Strand
We recall our expression for the entropy of a polymer coil with end-to-end distance, R:
The entropy change when a single strand is deformed, S, can be calculated from the difference between the entropy of the deformed coil and the unperturbed coil:
S = S(R) - S(Ro) = S(xxo, yyo, zzo) - S(xo, yo, zo)
)++(~.+=)( 22222222
2
2
3
2
3ozoyox zyx
Na
kconst
Na
kRRS
])(+)(+)[(~)()( 2222222 111
2
3ozoyoxo zyx
Na
kRSRS
Finding S:
)++(~)( 22222
3oooo zyx
Na
kRSInitially:
Entropy Change in Polymer Deformation
])(+)(+)[(~ 2222222 111
2
3ozoyox zyx
Na
kS
But, if the conformation of the coil is initially random, then <xo
2>=<yo2>=<zo
2>, so:
)](+)(+)[(~ 1112
3 2222
2
zyxo
Na
kxS
For a random coil, <R2>=Na2, and also R2 = x2+y2+z2 = 3x2, so we see:
3
22 Na
xo >=<
)++)((~ 332
3 2222
2 zyxNa
Na
kS Substituting:
)3++(2
~ 222zyx
kS
This simplifies to:
)++(~ 32
222zyxbulk
nkS
F for Bulk Deformation
If the rubber is incompressible (volume is constant), then xyz=1.For a one-dimensional stretch in the x-direction, we can say that x = . Incompressibility then implies
1== zy
)+(~ 32
22
nkSbulk
Thus, for a one-dimensional deformation of x = :
The corresponding change in free energy will be
)+(+~ 32
22
nkTFbulk
If there are n strands per unit volume, then S per unit volume for bulk deformation:
Force for Rubber DeformationIf the initial length is Lo, then = L/Lo.
)+)((+~ 32
22
LL
LLnkT
F o
obulk
Substituting: ))+(
+)+((+~ 31
21
22
nkTFbulk
Realising that Fbulk is an energy of deformation (per unit volume), then dF/d is the force (per unit area) for the deformation, i.e. the tensile stress, T.
])1+(
2)1+(2[
2== 2
nkTddF
T
In lecture 3, we saw that T = Y. The strain, , for a 1-D tensile deformation is
1===oo
o
o LL
LLL
LL
Young’s and Shear Modulus for Rubber
])1+(
1)1+[(= 2
nkTT
In the limit of small strain, T 3nkT, and the Young’s modulus is thus Y = 3nkT.
The Young’s modulus can be related to the shear modulus, G, to find a very simple result: G = nkT
This result tells us something quite fundamental. The elasticity of a rubber does not depend on the chemical make-up of the polymer nor on how it is crosslinked.
G does depend on the crosslink density. To make a higher modulus, more crosslinks should be added so that the lengths of the segments become shorter.
Experiments on Rubber Elasticity
])1+(
1)1+[(= 2
nkTT
Treloar, Physics of Rubber Elasticity (1975)
Rubbers are elastic over a large range of !
Alternative Equation for a Rubber’s G
We have shown that G = nkT, where n is the number of strands per unit volume.
xMRT
=
x
A
MN
n
=
For a rubber with a known density, , in which the average molecular mass of a strand is Mx (m.m. between crosslinks), we can write:
)(
)#)((=
#
moleg
molestrands
mg
m
strands 3
3
Looking at the units makes this equation easier to understand:
kTMN
nkTGx
A==Substituting for n:
strand