polymers: the size of a coil - andreas b. dahlin · 2017-09-22 1 soft matter physics 1 polymers:...
Post on 10-Aug-2018
215 Views
Preview:
TRANSCRIPT
2017-09-22
1
1Soft Matter Physics
Polymers: The Size of a Coil
2017-09-22
Andreas DahlinLecture 3/6
Jones: 5.1-5.3 Hamley: 2.1-2.4, 2.14
adahlin@chalmers.se
http://www.adahlin.com/
2017-09-22 Soft Matter Physics 2
Outline
We will have four lectures related to polymer physics but also touching on other topics:
• Introduction (this lecture) to polymers and basic models for determination of coil size.
• The influence from the solvent and attachment of polymers to surfaces.
• Polymer melts, mixtures, crystals and gels.
• Mechanical properties, rubbers and viscoelasticity of polymers.
2017-09-22
2
2017-09-22 Soft Matter Physics 3
What is a Polymer?
• Chain-like molecules where chemical groups are (at least to some extent) repeated.
• Covalent bonds between the units.
polyethylene poly(ethylene oxide)
poly(dimethyl siloxane)
poly(N-isopropyl acrylamide)
Jones: ”physicists should note with due
humility the tremendous intellectual and
practical achievements of polymer chemists”
But now we focus on the physics!
We are interested in finding generic models
that apply to all these molecules!
We are often interested in knowing how a
property scales with the size of the polymer.
2017-09-22 Soft Matter Physics 4
Natural and Synthetic Polymers
• Plastics, simple organic compounds as monomers.
• Cellulose and starch, sugars as monomers.
• DNA.
• Rubber.
• Nylon, polyesters (clothes).
Wikipedia: Rubber
Polarn o Pyret
http://www.polarnopyret.se/
2017-09-22
3
2017-09-22 Soft Matter Physics 5
Plastics
Plastics are everywhere in our modern world. They usually consist of synthetic polymers
originating from the petroleum industry.
But oil is a finite resource and plastic recycling is not so simple…
Possible solution: Plastics made of
polymers produced biologically and that
are biodegradable (e.g. corn starch).
The polymer physics will still be valid!
2017-09-22 Soft Matter Physics 6
Video: Plant Bottle
30% sustainable production of plastics from plants (not necessarily biodegradable).
2017-09-22
4
2017-09-22 Soft Matter Physics 7
Synthesis
Polymers are generally synthesized by letting a solution of monomers attach to
each other and form chains.
• The most common mechanism is free radical polymerization, where highly
reactive unpaired electrons form covalent bonds.
• The reactions may require a catalyst to run efficiently.
• Some sort of initiation is done to start the reaction.
• There is also a termination step, or one waits until all monomers are gone.
2017-09-22 Soft Matter Physics 8
Branching
Can occur for instance in starch and cellulose.
Drastically changes the properties of the polymer!
2017-09-22
5
2017-09-22 Soft Matter Physics 9
Copolymers
Different monomers!
The same monomer can appear in blocks or alternate regularly. Alternatively, the
sequence is random.
blocks
regular
random
2017-09-22 Soft Matter Physics 10
Stereochemistry
Different directions (usually two) are possible for the sidegroups sticking out
from the main chain. They can be facing the same direction (isotactic),
alternating (syndiotactic) or random (atactic).
Wikipedia: Tacticity
isotactic polypropylene
syndiotactic polypropylene
2017-09-22
6
2017-09-22 Soft Matter Physics 11
Organic Electronics
Electronic components like light-emitting diodes and transistors can now be made of
polymers!
Polymers containing aromatic rings with delocalized (π-conjugated) electrons are central
because they conduct electricity.
Since the polymers are soft they are excellent for flexible electronic devices!
polypyrrole
2017-09-22 Soft Matter Physics 12
Video: Acreo Display
Display made of polymers (no metals or semiconductors).
2017-09-22
7
2017-09-22 Soft Matter Physics 13
Polydispersity
There is almost always a distribution in molecular weight M because the degree of
polymerization N, the number of monomers, is a random variable.
Number average Mn based on number of molecules n:
Weight average Mw based on weight fraction w:
The polydispersity index (PDI) is given by Mw/Mn.
Guidelines: PDI < 1.1 good, PDI < 1.5 OK, PDI > 2 bad.
i
i
i
ii
n
Mn
M n
i
ii
i
iii
i
iiMn
MMn
MwM w
2017-09-22 Soft Matter Physics 14
Exercise 3.1
What is the polydispersity of poly(ethyleneoxide) if
10% of the molecules have N = 98,
20% of the molecules have N = 99,
40% of the molecules have N = 100,
20% of the molecules have N = 101,
10% of the molecules have N = 102.
Comment on the value!
– CH2 – CH2 – O –
2017-09-22
8
2017-09-22 Soft Matter Physics 15
Exercise 3.1
The monomer has a weight of 2×12 (carbon) + 4×1 (hydrogen) + 1×16 (oxygen) = 44
gmol-1. The average based on number of molecules is then:
This is actually obvious since the distribution is symmetric around 100 monomers. The
weight fraction average is:
PDI = Mw/Mn = 1.00012, very monodisperse, probably not possible to achieve in reality.
In reality the distribution is usually not symmetric around the mean value.
1-
n gmol 44001.02.04.02.01.0
441021.0441012.0441004.044992.044981.0
M
122222
22222
w
gmol 528.4400100
1021.01012.01004.0992.0981.044
441021.0441012.0441004.044992.044981.0
441021.0441012.0441004.044992.044981.0
M
2017-09-22 Soft Matter Physics 16
Experiment: Chain Configuration
End to end distance of cables!
2017-09-22
9
Polymers normally form random coils due to the entropy gain.
The number of microstates is much higher for a random coil configuration compared to a
stretched chain.
How can we estimate the size of the coil? What is the expected distance (R) between the
end points if r is a random variable?
Freely jointed chain segment model: At each “connection point” a new direction is
acquired by random! If each segment is equal to the chemical monomer in size the
”walk” along the chain consists of N vectors that each point in a random direction:
2017-09-22 Soft Matter Physics 17
Freely Jointed Chain
R
N
i
iar1
rmax = Na
r
a
2017-09-22 Soft Matter Physics 18
The Size of a Coil
We can write the magnitude of the vector r as scalar products:
Consider the average or expected value of these sums. When i ≠ j the expected value of
any of these scalar products must be zero. Only the self products are important:
If we skip vector denotation from now on, we can write the “random walk” distance as:
2/1aNR
N
i
j
N
ij
i
N
i
j
i
j
ii
N
i
i
N
i
j
N
j
i
N
j
j
N
i
i aaaaaaaaaarrr1 11
1
111 111
2
2
1
20 aNaar i
N
i
i
Note that R is the expected value of the
random variable r.
The configuration of the coils can be seen
(in 2D) by atomic force microscopy!Wikipedia: Polymer
2017-09-22
10
2017-09-22 Soft Matter Physics 19
End to End Distance Distribution
The probability distribution for r is approximately Gaussian for large N (Jones appendix):
2
22/3
2 2
3exp
π2
3,
Na
r
NaNrp
Note that the dimension is inverse
volume! The probability is obtained
by integration in 3D space:
dP = 4πr2p(r)dr
Plot shows dP vs normalized
extension for different N.
Not Gaussian!
relative extension (r/[aN])
pro
ba
bili
ty d
ensity
(a.u
.)
N = 10
N = 100
N = 1000
Na
rkrNrWk
Na
rkNak
rNrWkNa
rNak
rNrWkNrpkrNrWNrpkNrWkNrS
2
2
B
0
B2
2
B
2
B
0
B2
22/3
2
B
0
BB
0
BB
2
3constantd,log
2
3
3
π2log
2
3
d,log2
3exp
3
π2log
d,log,logd,,log,log,
2017-09-22 Soft Matter Physics 20
Conformational Entropy
We want the number of configurations W for a given r. It must be true that:
Now Boltzmann’s formula can be used:
does not contain r
0
d,
,,
rNrW
NrWNrp
contains r but does not
really depend on it
2017-09-22
11
When the entropy is known as a function of chain elongation, we can get the free energy
as a function of r for the random walk model:
The force required to stretch the chain is:
The ”spring constant” is:
Here all is based on conformational entropy, no interaction energies accounted for!
2017-09-22 Soft Matter Physics 21
Polymers as Entropic Springs
constant2
32
2
B Na
TrkrTSrG
2
B3
Na
Trk
r
GrF
2
B
2
2 3
Na
Tk
r
F
r
G
Wikipedia: Spring
easier to pull long
polymers
2017-09-22 Soft Matter Physics 22
Kuhn Length
Problem: We know that a chain of covalently bound atoms is not free to rotate at certain
”joint points” and fully stiff in between those. The chemistry is more complex than that.
However, the correlation in direction when moving along the chain must disappear
eventually since there is always some freedom to rotate!
In other words there must be a certain distance b, the Kuhn length, along the polymer
that corresponds to a segment in a freely jointed chain model but it is not necessarily
equal to the monomer length a.
One can estimate b by experiment or theory (the chemical nature of the covalent bonds
such as rotational freedom).
θmax
θ
one monomer
2017-09-22
12
2017-09-22 Soft Matter Physics 23
Flory’s Characteristic Ratio
Unfortunately, much literature uses N to denote number of monomers also in terms of
Kuhn segments! Here we will stick to N in its original definition, which is number of
chemical monomers.
There is a difference between actual end to end distance and that predicted by the
random walk with steps based on monomer length (Flory’s characteristic ratio).
In principle b could be either bigger or smaller than a, but it is always bigger in reality.
Rubinstein & Colby
Polymer Physics Oxford 2003
[b/a]2
2017-09-22 Soft Matter Physics 24
Rescaling
Very important: In this and following lectures we will derive many expressions
containing a and N based on the random walk principle. However, since the real
effective step length b is larger than the monomer step a we must often rescale:
• Replace the monomer length a with the Kuhn length b.
• Replace N with the number of Kuhn lengths, which is rmax/b = aN/b.
Note that sometimes rescaling is not necessary:
• The task at hand is just theoretical, so we can pretend the monomer is like a Kuhn step.
• The physics of the situation only relates to contour length or physical monomer size.
2017-09-22
13
2017-09-22 Soft Matter Physics 25
Exercise 3.2
In water, poly(ethylene oxide) has a Kuhn length of 0.72 nm and a monomer length of
0.28 nm. What is the expected end to end distance of a 20 kgmol-1 coil in a random walk
configuration?
2017-09-22 Soft Matter Physics 26
Exercise 3.2
The monomer weight is 44 gmol-1 (check previous exercise). This give N = 20000/44.
Rescaling the random walk model by replacing a with b and N with aN/b:
Sanity check: If a = b we recover the previous expression. So the answer is R = 9.6 nm.
nm ...57.944
2000072.028.0
2/1
2/1
2/1
abN
b
NabR
2017-09-22
14
2017-09-22 Soft Matter Physics 27
Worm-Like Chain Model
One commonly used alternate model is a continuously bending worm-like chain. The
correlation in direction decays exponentially:
Here lp is the persistence length, which clearly contains the same information as the
Kuhn length. For rmax = Na >> lp we recover the random walk:
Random walk means R = [abN]1/2, so b = 2lp!
The worm-like chain is a useful model for stiff polymers (e.g. Kevlar or double stranded
DNA) that bend little and continuously over long distances.
However, we still have one big problem to deal with: The chain has to avoid itself!
2/1
p
max
max
p
maxpwlc exp112
l
r
r
lrlR
Rubinstein & Colby
Polymer Physics Oxford 2003 2/1
p
2/1
maxp 22pmax
aNlrlRlr
2017-09-22 Soft Matter Physics 28
Self Avoiding Chains
Recall relation between volume and entropy for a free particle:
Assume we have our N chain segments, each with volume v, in a total volume V. The
excluded volume occupied by the coil is Nv.
We can now do a mean field approximation: We assume that the segment density is
homogenous throughout the volume that the coil occupies.
The entropy loss per segment is then:
We thus treat the polymer as a “gas” of monomers. Remember that the molecule is
assumed to be long and flexible. (It wobbles around very fast so the assumption is not
totally crazy.)
V
Nvk
V
Nvk
V
NvVk
V
VkS B
BB
i
fBvol 1logloglog
for small polymer
volume fractions
i
fB log
V
VkS
2017-09-22
15
2017-09-22 Soft Matter Physics 29
Excluded Volume Entropy
We can assume that V ≈ r3 and that v ≈ a3 so that:
We get the total free energy increase for the polymer due to the presence of itself by
multiplying with N (all segments) and temperature:
The chain will want to expand to minimize ΔGvol, but we must not forget the free energy
cost of stretching the chain:
constant2
33
32
B
2
2
Btot
r
aTNk
Na
TrkrG
a3 r3
3
32
Bvolvol
r
aTNkSNTrG
3
3
Bvol
r
TNakrS
2017-09-22 Soft Matter Physics 30
Example of the Excluded Volume Effect
As an example we can plot the free
energies as a function of r for:
N = 1000
a = 1 nm
For small r the excluded volume
effect dominates entirely, but
disappears very fast with (r-3).
The conformational entropy loss
increases steadily (r2).
Minimum for some value of r!
0 200 400 600 8000
1
2
3
4
5
6
7x 10
-18
end to end distance (r, nm)
free
en
ergy (
Gto
t, a
.u.)
excluded
volume entropy
conformational
entropy
2017-09-22
16
2017-09-22 Soft Matter Physics 31
The Flory Radius
We can minimize Gtot with respect to r by taking the derivative:
Setting the derivative to zero will give the free energy minimum and thus the expected
value of r (in other words R) from:
This gives us:
So we arrive at an exponent of 3/5 instead of 1/2. We can denote this as the Flory radius:
(We cheated a little because the excluded volume should actually not be rescaled.)
4
32
B
2
Btot 33
r
aTNk
Na
Trk
r
G
4
32
B
2
B 33
R
aTNk
Na
TRk
535 aNR
5/3
F aNR
2017-09-22 Soft Matter Physics 32
Validity?
The “accurate” value of the exponent, based on the math of self-avoiding random walks,
is 0.588… But experiments cannot discriminate this value from 3/5 (though from 1/2).
radius of gyration of polystyrene
chains in a theta-solvent
(cyclohexane at 34.5°C) and in a
good solvent (benzene at 25°C)
Fetters et al. J. Phys. Chem. Ref. Data 1994
2017-09-22
17
2017-09-22 Soft Matter Physics 33
Who Cares About 1/2 or 3/5?
The value of the exponent is very important because N is a large number!
Equal to 1.58 for N = 100 and 2.00 for N = 1000.
10/1
2/1
5/3
F NaN
aN
R
R
2017-09-22 Soft Matter Physics 34
Nobel Prize in Chemistry 1974
For predicting the ”spatial configuration of macromolecular chains”.
Paul Flory
2017-09-22
18
So after all this work, what is now the size of a polymer molecule?
The physical size of a coil is often described by the radius of gyration Rg, which is “the
mean squared distance of each point on the object from its center of gravity”.
For a random walk, one can relate Rg to R by:
Note that Rg is indeed a radius, so the diameter is 2Rg.
Experimental data (like light scattering) will tend to give the hydrodynamic radius,
which is not the same thing as Rg but similar in magnitude.
Whatever parameter we use it will be proportional to the end to end distance, so R (or
RF) is a characteristic length that represents the size of the polymer!
2017-09-22 Soft Matter Physics 35
The Size of a Coil
???
2/1
g6
1
RR
2017-09-22 Soft Matter Physics 36
Reflections and Questions
?
2017-09-22
19
2017-09-22 Soft Matter Physics 37
Exercise 3.3
What is the force required to maintain a coil following a random walk model at r = 2R,
i.e. twice its expected distance between endpoints, as a function of R and temperature?
→
R
TkF B6
2017-09-22 Soft Matter Physics 38
Exercise 3.4
Consider a poly(ethylene oxide) coil with N = 100. Calculate its size by the random walk
model, the worm-like chain model and the Flory radius (use monomer length 0.28 nm
and Kuhn length 0.72 nm). Explain any differences in size estimates physically.
→
R = 4.5 nm by a random walk. With two digits precision Rwlc is the same (but would be
smaller for lower N). The polymer is flexible since rmax is much larger than the
persistence length so the worm-like chain model is “not needed”. RF is 6.5 nm, i.e.
significantly larger. This is because the excluded volume effect is taken into account.
2017-09-22
20
2017-09-22 Soft Matter Physics 39
Exercise 3.5
Repeat problem 4.3 with the excluded volume entropy taken into account! (The force is
now a function of a and N instead of R.)
→
5/2
B
16
93
aN
TkF
2017-09-22 Soft Matter Physics 40
Exercise 3.6
Compare the end to end distance of 100 kgmol-1 polyethylene (monomer CH2) by the
random walk model and the Flory excluded volume model! The monomer is 0.2 nm and
the Kuhn length 1.4 nm.
→
Random walk R = 45 nm, excluded volume RF = 89 nm.
top related