polymers: the size of a coil - andreas b. dahlin · 2017-09-22 1 soft matter physics 1 polymers:...

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

[email protected]

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.

mailto:[email protected]

http://www.adahlin.com/

2017-09-22

2


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.


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


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!


Video: Plant Bottle

30% sustainable production of plastics from plants (not necessarily biodegradable).

2017-09-22

4


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.


Branching

Can occur for instance in starch and cellulose.

Drastically changes the properties of the polymer!

2017-09-22

5


Copolymers

Different monomers!

The same monomer can appear in blocks or alternate regularly. Alternatively, the

sequence is random.

blocks

regular

random


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


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


Video: Acreo Display

Display made of polymers (no metals or semiconductors).

2017-09-22

7


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


Exercise 3.1

What is the polydispersity of poly(ethyleneoxide) if

10% of the molecules have N = 98,




10% of the molecules have N = 102.

Comment on the value!

– CH2 – CH2 – O –

2017-09-22

8


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


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:


Freely Jointed Chain

R

N

i

iar1

rmax = Na

r

a


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


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,


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!


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


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


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


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


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?


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


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


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


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


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


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


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


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


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!


The Size of a Coil

???

2/1

g6

1

RR


Reflections and Questions

?

2017-09-22

19


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


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


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


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.

polymers: the size of a coil - andreas b. dahlin · 2017-09-22 1 soft matter physics 1 polymers:...

Documents