PH3-SM (PHY3032)

Soft MatterLecture 9

Glassy Polymers, Copolymer Self-Assembly, and Polymers in Solutions

6 December, 2011

See Jones’ Soft Condensed Matter, Chapt. 5 & 9

Polymer Conformation in Glass

Describe as a “random walk” with N repeat units (i.e. steps), each with a size of a:

12

3

N

aR

iN aaaaaR

=...+++= 321i=1

N

The average R for an ensemble of polymers is 0.

But what is the mean-squared end-to-end distance, ?2R

In a “freely-jointed” chain, each repeat unit can assume any orientation in space.

Shown to hold true for polymer glasses and melts.

( ) ( )ji aaRR •=•

i=1 j=1

N N

ji aaR

•= 2

22 cos aaaa iiiji Those terms in which i=j can be

simplified as:

ijaNaR cos+= 222 2

ij

N N

0=cos ijThe angle can assume any value between 0 and 2 and is uncorrelated. Therefore:

ijjiji aaaa cos=•

By definition:

Random Walk Statistics

22 NaR =

Finally, Compare to random walk Compare to random walk statistics for colloids!statistics for colloids!

a1

a2

a4

a3

22 NaR =

Defining the Size of Polymer Molecules

aNR 21

21

2 =

We see that and

Often, we want to consider the size of isolated polymer molecules.

In a simple approach, “freely-jointed molecules” can be described as spheres with a characteristic size of 2

12R

Typically, “a” has a value of 0.6 nm or so. Hence, a very large molecule with 104 repeat units will have a r.m.s. end-to-end distance of 60 nm.

On the other hand, the contour length of the same molecule will be much greater: aN = 6x103 nm or 6 m!

(Root-mean squared end-to-end distance)

21

21

2 ~ NR

Scaling Relations of Polymer Size

Observe that the rms end-to-end distance is proportional to the square root of N (for a polymer glass).

Hence, if N becomes 9 times as big, the “size” of the molecule is only three times as big.

However, if the molecule was straightened out, then its length would instead be proportional to N.

Concept of Space Filling

Molecules are in a random coil in a polymer glass, but that does not mean that it contains a lot of “open space”.

Instead, there is extensive overlap between molecules.

Thus, instead of open space within a molecule, there are other molecules, which ensure “space filling”.

Distribution of End-to-End DistancesIn an ensemble of polymers, the molecules each have a different end-to-end distance, R.

In the limit of large N, there is a Gaussian distribution of end-to-end distances, described by a probability function (number/volume):

)2

3exp()]2/(3[)(

2

22/32

Na

RNaRP

Larger coils are less probable, and the most likely place for a chain end is at the starting point of the random coil.

Just as when we described the structure of glasses, we can construct a radial distribution function, g(r), by multiplying P(R) by the surface area of a sphere with radius, R:

)2

3exp()]/2/(3[4)(

2

22/322

Na

RNaRRg

From U. Gedde, Polymer Physics

aNR =2

g(R)P(R)

Radius of Gyration of a Polymer Coil

RFor 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 radius of gyration is the root-mean square distance of an objects' parts from its centre of gravity.

Entropic EffectsRecall the Boltzmann equation for calculating the entropy, S, of a system by considering the number of microstates, , for a given macro-state:

S = k lnIn the case of arranging a polymer’s repeat units in a coil shape, we see that = P(R), so that:

.+=)( constNa

kRRS 2

2

2

3

If a molecule is stretched, and its R increases, S(R) will decrease (become more negative).

Intuitively, this makes sense, as an uncoiled molecule will have more order (i.e. be less disordered).

Concept of an “Entropic Spring”

Decreasing entropy

Fewer configurations

Helmholtz free energy: F = U - TSInternal energy, U, does not change significantly with stretching.

2

3

Na

kTR

dR

dFf .++=)( constT

Na

kRRF 2

2

2

3 Restoring force, f

R

R

ff

Spring Polymer

x

S change is large; it provides the restoring

force, f.

Entropy (S) change is negligible, but U is large, providing the

restoring force, f.

22

1 xkU s)(=

Difference between a Spring and a Polymer CoilIn experiments, f

for single molecules can be

measured using an AFM tip!

Molecules that are Not Freely-JointedIn reality, most molecules are not “freely-jointed” (not really like a pearl necklace), but their conformation can still be described using random walk statistics.

Why? (1) Covalent bonds have preferred bond angles.

(2) Bond rotation is often hindered.

In such cases, g monomer repeat units can be treated as a “statistical step length”, s (in place of the length, a).

A polymer with N monomer repeat units, will have N/g statistical step units.

The mean-squared end-to-end distance then becomes:

22 sgN

R =

Example of Copolymer MorphologiesImmiscible polymers can be “tied together” within the same diblock copolymer molecules. Phase separation cannot then occur on large length scales.

Poly(styrene) and poly(methyl methacrylate) diblock copolymer Poly(ethylene) diblock copolymers

2m x 2m

Self-Assembly of Di-Block CopolymersDiblock copolymers are very effective “building blocks” of materials at the nanometer length scale.

They can form “lamellae” in thin films, in which the spacing is a function of the sizes of the two blocks.

At equilibrium, the block with the lowest surface energy, , segregates at the surface!

The system will become “frustrated” when one block prefers the air interface because of its lower , but the alternation of the blocks requires the other block to be at that interface. Ordering can then be disrupted.

Thin Film Lamellae: Competing Effects

There is thermodynamic competition between polymer chain stretching and coiling to determine the lamellar thickness, d.

d

The addition of each layer creates an interface with an energy, . Increasing the lamellar thickness reduces the free energy per unit volume and is therefore favoured by .

Increasing the lamellar thickness, on the other hand, imposes a free energy cost, because it perturbs the random coil conformation.

The value of d is determined by the minimisation of the free energy.

Poly(styrene) and poly(methyl methacrylate)

copolymer

Interfacial Area/Volume

e

e

3= eV

Area of each interface: A = e2

Interfacial Area/Volume:

dee

eV

A 1=

3=

3= 3

2

d=e/3Lamella thickness: d

In general, d = e divided by an integer value.

e

Determination of Lamellar Spacing

• Free energy increase caused by chain stretching (per molecule):

2

2

Na

dkTFstr

Ratio of (lamellar spacing)2 to (random coil size)2

• The interfacial area per unit volume of polymer is 1/d, and hence the interfacial energy per unit volume is /d.

• The volume of a molecule is approximated as Na3, and so there are 1/(Na3) molecules per unit volume.

Total free energy change: Ftotal = Fstr + Fint

• Free energy increase (per polymer molecule) caused by the presence of interfaces:

dNa

F3

int

Free Energy Minimisation

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!

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).

dNa

Na

dkTFtot

3

2

2 +

2

3

220dNa

Nad

kTd

Ftot Finding the

minimum, where slope is 0:

d

Ftot

FstrFint

F

Experimental Study of Polymer LamellaeSmall-angle X-ray Scattering (SAXS)

Transmission Electron Microscopy

(°)T. Hashimoto et al., Macromolecules (1980) 13, p. 1237.

Poly(styrene)-b-poly(isoprene)

Support of Scaling Argument

2/3

T. Hashimoto et al., Macromolecules (1980) 13, p. 1237.

Micellar Structure of Diblock CopolymersWhen diblock copolymers are asymmetric, lamellar structures are not favoured – as too much interface would form!

Instead the shorter block segregates into small spherical phases known as “micelles”.

Density within phases is maintained close to the bulk value.

Interfacial “energy cost”: (4r2)

Reduced stretching energy when the shorter block is in the micelle.

Copolymer Micelles

Diblock copolymer of poly(styrene) and poly(vinyl pyrrolidone): poly(PS-b-PVP)

5 m x 5 m

AFM image

Diblock Copolymer Morphologies

Lamellar Cylindrical Spherical micelle

Gyroid DiamondPierced Lamellar

TRI-block

“Bow-Tie”

Gyroid

Copolymer Phase Diagram

N

~10 From I.W. Hamley, Intro. to Soft Matter, p. 120.

Applications of Self-Assembly Creation of “photonic band gap” materials

Images from website of Prof. Ned Thomas, MIT

In photovoltaics for solar cells, excitons decay into electrons and holes. Controlled phase separation of p-type/n-type diblock copolymers could allow a large contact area between the two phase.

http://crg.postech.ac.kr/korean/viewforum.php?f=90

Nanolithography

From Scientific American, March 2004, p. 44

Nanolithography to make electronic structures, such as “flash

memories”

Thin layer of poly(methyl methacrylate)/ poly(styrene) diblock copolymer. Image from IBM (taken from BBC website)

The Self-Avoiding Walk In describing the polymer coil as a random walk, it was tacitly assumed that the chain could “cross itself”.

But, when polymers are dissolved in solvents (e.g. water or acetone), they are often expanded to sizes greater than a random coil.

Such expanded conformations are described by a “self-avoiding walk” in which <R2>1/2 is given by aN (instead of aN1/2 as for a coil described by a random walk).

What is the value of ?

The conformation of polymer molecules in a polymer glass and in a melted polymer can be adequately described by random walk statistics.

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 microstates, , for a macrostate: 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 number 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 unit is excluded from the volume occupied by the other N units, 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 FFor 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.

Earlier in the lecture (slide 18), however, we saw that the coiling of polymer molecules increased the entropy. 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 (entropic) 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 change 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:

But when are excluded volume effects important?

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 (polymer + solvent).

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). Finally, 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 (ps) contacts are made, while contacts between like molecules (pp + ss) are lost.

Following arguments similar to our approach for liquid miscibility, we can write out 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. For a +ve , 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:

).)(--2(int unitsnowwwU SSPPPS 3

2

3 2)(2-R

bNkT

RN

NbkT

Significance of the -ParameterWe recall (slide 31) that excluded volume effects favour coil swelling:

3

2

R

bNkTRFexc =)(

Also, depending on the value of , the swelling will be opposed by 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, the term is positive, and the excluded volume/energetic effects contribute to determining the coil size: Fexc + Uint > 0.

The solvent is called a “theta-solvent”.

5321

2 aNR =

as shown previously (considering the balance with the elastic energy). The molecule is said to be swollen in a “good solvent”.

Types of Solvent

3

2

21R

NkTbUFexc )(=+ int

• When > 1/2, the term goes negative, and the polymer/solvent interactions dominate in determining the coil size. Fexc + Uint < 0.

Both terms lower F (which is favourable) as R decreases. 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+~+=)(

sin4

=2

=d

q

Determining Structure: Scattering Experiments

d= characteristic

spacing

Scattered intensity is measured as a function of the wave vector, q:

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 CoilingNano-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.

Polymer Particles Adsorbed on a Positively-Charged Surface

Particles can contain small molecules such as a drug or a flavouring agent. Thus, they are a “nano-capsule”.

1 m 100 nm

Comparison of Particle Response in Solution and at an Interface

Light scattering from solution

Ellipsometry of adsorbed particles

Good solvent: particle is open

Bad solvent: particle is closed

V. Nerapusri, et al. , Langmuir (2006) 22, 5036.