Last Lecture:• The Peclet number, Pe, describes the competition

between particle disordering because of Brownian diffusion and particle ordering under a shear stress.

• At high Pe (high shear strain rate), the particles are more ordered; shear thinning behaviour occurs and decreases.

• van der Waals’ energy between a colloidal particle and a semi- slab (or another particle) can be calculated by summing up the intermolecular energy between the constituent molecules.

• Macroscopic interactions can be related to molecular.• The Hamaker constant, A, contains information about

molecular density () and the strength of intermolecular interactions (via the London constant, C): A = 22C

3SCMP

Polymer Structure and Molecular Size

6 March, 2007

Lecture 8

See Jones’ Soft Condensed Matter, Chapt. 4, 5 and 9

Definition of PolymersPolymers are giant molecules that consist of many repeating units. The molar mass (molecular weight) of a molecule, M, equals moN, where mo is the the molar mass of a repeat unit and N is the number of units.

Synthetic polymers never have the same value of N for all of its constituent molecules, but there is a Gaussian distribution of N.

Polymers can be synthetic (such as poly(styrene) or poly(ethylene)) or natural (such as starch (repeat units of amylose) or proteins (repeat unit of amino acids)).

Synthetic polymers are created through chemical reactions between smaller molecules, called “monomers”.

The average N (or M) has a huge influence on mechanical properties of polymers.

Examples of Repeat Units

Molecular Weight Distributions

In both cases: the number average molecular weight, Mn = 10,000

M M

Fraction of molecules

Molecular Weight of Polymers

The molecular weight can be defined by a number average that depends on the number of molecules, ni, having a mass of Mi:

The polydispersity index describes the width of the distribution. In all cases:

MW/MN > 1

The molecular weight can also be defined by a weight average that depends on the weight fraction, wi, of each type of molecule with a mass of Mi:

ii

iiii Mn

MnMw

2

==MW

MN i

ii

nMn

== Total mass divided by number of molecules

Polymer Architecture

Linear

Star-branched

Branched

Side-branched

Types of Copolymer Molecules

Within a single molecule, there can be “permanent disorder” in copolymers consisting of two or more different repeat units.

Diblock

Alternating

Random orStatistical

Can also be multi (>2) block.

Polymer Structures

Glassy Polymers: molecules in a “random coil” conformation

Crystalline Polymers: molecules show some degree of ordering

Lamellar growth direction

Lamella thickness

Polymer Crystals

AFM image of a crystal of high density poly(ethylene) - viewed while “looking down” at the lamella.

15 m x 15 m Lamella grows outwards

Polymer Crystals

Several crystals of poly(ethylene oxide)

5 m x 5 m

Polymers are usually polycrystalline - not monocrystalline. They are usually never completely crystalline but have some glassy regions and “packing defects”.

Thermodynamics of Glass Transitions

V

T

Crystalline solid

Tm

Liquid

Glass

Tg

Crystals can grow from the liquid phase (below Tm) or from the glassy phase (below Tg).

Temperature Dependence of Crystal Growth Rate, u

From Ross and Frolen, Methods of Exptl. Phys., Vol. 16B (1985) p. 363.

T-Tm (K)T-Tm (K) T-Tm (K)

Tm = crystal melting temperature

Why is crystal growth rate maximum between Tg and Tm?

As T decreases towards Tg, molecular motion slows down.

Viscosity varies according to V-F equation:

Temperature Dependence of Crystal Growth Rate, u

)exp(=oTT

Bo

Growth rate, u, is inversely related to viscosity, so

u ~ 1/ ~ exp (- B/(T-To))

Hence, u decreases as T decreases toward To, because of a slowing down of configurational re-arrangements.

• Above Tm, the crystal will melt. The liquid is the most favourable state according to thermodynamics.

• Crystallisation becomes more favourable with greater “undercooling” (i.e. as T decreases below Tm) because the free energy difference between the crystal and glass increases. There is a greater “driving force”.

• Hence u increases exponentially with the amount of undercooling (defined as Tm - T) such that:

Temperature Dependence of Crystal Growth Rate, u

( )TTT

mu exp~

• Considering the previous argument, there is an intermediate T where u is maximum.

Data in Support of Crystallisation Rate Equation

J.D. Hoffman et al., Journ. Res. Nat. Bur. Stand., vol. 79A, (1975), p. 671.

( )TTT

TTB

mou exp)exp(~

V-F contribution: describes molecular slowing down with decreasing T Undercooling

contribution: considers greater driving force for crystal growth with decreasing T

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 be valid for polymer glasses and melts.

( ) ( )ji aaRR •=•

i=1 j=1

N N

ji aaR

•= 2

22 aaaa ijiji =cos=• 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,

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.

If the molecule is straightened out, then its length will 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 Distances

In 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:

)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)

Entropic Effects

Recall the Boltzmann equation for calculating the entropy, S, of a system by considering the number of microstates, , for a given macrostate:

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

.+=)( 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 (be less disordered).

Concept of an “Entropic Spring”

Decreasing entropy

Fewer configurations

Helmholtz free energy: F = U - TS

Internal 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 Coil

In 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 =

Interfacial Width, w, between Immiscible Polymers

A B

w

loop

loopNaw ~

• Consider the interface between two immiscible polymers (A and B), such as in a phase-separated blend or in a diblock copolymer.

• The molecules at the interface want to maximise their entropy by maintaining their random coil shape.

• Part of the chain - a “loop” – from A will extend into B over a distance comparable to the interfacial width, w. Our statistical analysis predicts the size of the loop is ~ a(Nloop)1/2

loopNaw ~

Simple Scaling Argument for Polymer Interfacial Width, w

1~1~ looploop NN In which case:

a

w ~Substituting in for Nloop:

NkTU ~int

But every unit of the “A” molecule that enters the “B” phase has an unfavourable interaction energy. The total interaction energy is:

kTU ~int

At equilibrium, this unfavourable interaction energy will be comparable to the thermal energy:

Example of Copolymer MorphologiesPolymers that are immiscible can be “tied together” within the same molecules. They therefore cannot phase separate on large length scales.

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

copolymers

2m x 2m

Self-Assembly of Di-Block Copolymers

Diblock copolymers are very effective “building blocks” of materials at the nanometer length scale.

They can form “lamellae” in thin films, in which the spacings are 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

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.

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

Determination of Lamellar Spacing

• Free energy increase caused by chain stretching:

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: Fstr + Fint

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

dNa

F3

int

Free Energy Minimisation

Chains are NOT fully stretched (N1) - but nor are they randomly coiled (N1/2)!

Two different dependencies on d!d

Na

Na

dkTFtot

3

2

2 +

kTaN

d2

523

=2

3

22d

Na

Na

dkT

=

32315

2//)(= N

kTa

d

2

3

220d

Na

Na

dkT

dddFtot

=)(

Find minimum:

Poly(styrene) and poly(methyl

methacrylate) copolymer

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).

Experimental Study of Polymer Lamellae

Small-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 Copolymers

When diblock copolymers are asymmetric, lamellar structures are not favoured.

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

Density within phases is maintained close to bulk value.

Interfacial “energy cost”: (4r2)

Reduced stretching energy for shorter block

Copolymer Micelles

Diblock copolymer of poly(styrene) and poly(viny pyrrolidone): poly(PS-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

Nanolithography to make electronic structures

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

Creation of “photonic band gap” materials Images from website of Prof. Ned Thomas, MIT

Nanolithography

From Scientific American, March 2004, p. 44

Used to make nano-sized “flash memories”