Important Points from 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 h decreases.
• van der Waals’ energy acting 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 the molecular level.
Forces are found by differentiating the interaction energy with
respect to the distance of separation.
• The Hamaker constant, A, contains information about molecular
density (r) and the strength of intermolecular interactions (via
the van der Waals’ constant, C): A = p2r2C
PH3-SM (PHY3032)
Soft Matter
Lecture 8
Introductions to Polymers and
Semi-Crystalline Polymers
29 November, 2011
See Jones’ Soft Condensed Matter, Chapt. 5 & 8
Definition of Polymers
Polymers 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.
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”.
Synthetic polymers never have the same value of N for all of its
constituent molecules, but there is a Gaussian distribution of N.
The average N (or M) has a huge influence on mechanical
properties of polymers.
Examples of Repeat Units
Molecular Weight Distributions
Fraction of
molecules
M
M
In both cases: the number average
molecular weight, Mn = 10,000
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:
MN
n i M i
=
ni
= Total mass divided by the total
number of molecules
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:
ni Mi 2
MW = w i Mi =
ni M i
The polydispersity index describes the width of the distribution. In all
cases:
MW/MN > 1
Types of Copolymer Molecules
Within a single molecule, there can be “permanent
order/disorder” in copolymers consisting of two or more
different repeat units.
Diblock
Random or
Statistical
Alternating
Can also be multi (>2) block.
Semi-Crystalline Polymers
• It is nearly impossible for a polymer to be 100%
crystalline.
• Typically, the level of crystallinity is in the range from 20
to 60%.
• The chains surrounding polymer crystals can be in the
glassy state, e.g. in poly(ethylene terephthalate) or PET
• The chains can be at a temperature above their glass
transition temperature and be “rubbery”, e.g. in
poly(ethylene) or PE
• The density of a polymer crystal is greater than the
density of a polymer glass.
Examples of Polymer Crystals
Poly(ethylene) crystal
15 mm x 15 mm
Crystals of poly(ethylene oxide)
5 mm x 5 mm
Polymer crystals can grow up to millimeters in size.
Crystal Lattice Structure
•The unit cell is repeated
in three directions in
space.
•Polyethylene’s unit cell
contains two ethylene
repeat groups (C2H4).
Polyethylene
•Chains are aligned along
the c-axis of the unit cell.
From G. Strobl, The Physics of Polymers (1997) Springer, p. 155
Structure at Different Length Scales
• Chains weave back and forth to create crystalline sheets,
called lamella.
• A chain is not usually entirely contained within a lamella:
portions of it can be in the amorphous phase or bridging two
(or more) lamella.
• The lamella thickness, L, is typically about 10 nm.
Lamella stacks
L
From R.A.L. Jones, Soft Condensed Matter, O.U.P. (2004) p. 130
Structure at Different Length Scales
• Lamella usually form at a nucleation site and grow outwards.
• To fill all available space, the lamella branch or increase in
number at greater distances from the centre.
• The resulting structures are called spherulites.
• Can be up to hundreds of micrometers in size.
From G. Strobl, The Physics of Polymers (1997) Springer, p. 148
Hierarchical Structures of Chains in a
Polymer Crystal
• Chains are aligned in the
lamella in a direction that is
perpendicular to the
direction of the spherulite
arm growth.
From I.W. Hamley, Introduction to
Soft Matter, p. 103
• Optical properties are
anisotropic.
Crossed Polarisers Block Light
Transmission
• An anisotropic
polymer layer between
crossed polarisers will
“twist” the polarisation
and allow some light to
pass.
• The pattern is called a
“Maltese cross”.
Crossed polarisers:
Parallel polarisers:
No light can pass!
All light can pass
http://www.kth.se/fakulteter/TFY/kmf/lcd/lcd~1.htm
Observing Polymer Crystals
Under Crossed Polarisers
Light is only transmitted when anisotropic optical properties
“twist” the polarisation of the light.
Free Energy of Phase Transitions
• The state with the lowest free energy is the stable one.
• Below the equilibrium melting temperature, Tm(), the
crystalline state is stable.
Free energy, G
•The thermodynamic driving force for crystallisation, DG,
DG
increases when cooling below the equilibrium Tm ().
Crystalline state
Liquid (melt) state
Undercooling, DT, is
defined as Tm – T.
Tm()
Temperature, T
Thermodynamics of the Phase Transition
Heat flows in
DHm
•Enthalpy of melting, DHm:
heat is absorbed when going
from the crystal to the melt.
•Enthalpy of crystallisation:
heat is given off when a molten
polymer forms a crystal.
Heat flows out
•The melting temperature, Tm,
is always greater than the
crystallisation temperature.
•The phase transitions are
broad: they happen over a
relatively wide range of
temperatures.
From G. Strobl, The Physics of Polymers (1997) Springer
Crystal Growth Mechanisms
Crystals from small
molecules grow one
molecule at a time.
Entire chains must join
the polymer crystal at
one time.
L
Xu et al., Nature Materials (2009) 8, 348.
Thermodynamics of the
Crystallisation/Melting Phase Transition
Melt to crystal: Increase in Gibbs’ free energy from the creation of an
interface between the crystal and amorphous region. When a single
chain joins a crystal:
2
a2
DG  2 f a
 f is an interfacial energy
L
Melt to crystal (below Tm): Decrease in Gibbs’ free energy
because of the enthalpy differences between the states
(Enthalpy change per volume, H m) x (volume) x (fractional undercooling)
DT
DT
2
DG  DH mV
 DH m La
Tm ()
Tm ()
At equilibrium: energy contributions are balanced and DG = 0.
Thermodynamics of the
Crystallisation/Melting Phase Transition
From DG = 0:
DH m DT
La 2  2a 2 f
Tm ()
Re-arranging and
writing undercooling in
terms of Tm(L):
2 f
Tm ()  Tm ( L)

Tm ()
DH m L
2 f
2 f 

Tm ( L)

 Tm ( L)  Tm () 1 
Solving for Tm(L): 1 

DH m L Tm ()
D
H
L
m 

Conclusion: We see that a chain-folded crystal (short L) will melt at
a lower temperature than an extended chain crystal (very large L).
Lamellar Crystal Growth
Lamella
thickness, L
Lamellar growth direction
a
From U.W. Gedde, Polymer Physics (1995)
Chapman & Hall, p. 145
L
Crystal growth is from the edge of the lamella.
The lamella grows a distance a when each chain is added.
From U.W. Gedde, Polymer Physics
(1995) Chapman & Hall, p. 161
The Entropy Barrier for a Polymer Chain
to Join a Crystal
Free energy
Crystalline state
TDS
Melted
state
DG
Re-drawn from R.A.L. Jones, Soft Condensed Matter, O.U.P. (2004) p. 132
The Rate of Crystal Growth, u
Melt to crystal: the rate of crystal growth is equal to the product of the
frequency (t 1) of “attempts” and the probability of going over the
energy barrier (TDS):
 TDS 
 DS 
1
umc  t 1 exp 

t
exp



 kT 
 k 
Crystal to melt: the rate of crystal melting is equal to the product of the
frequency (t1) of “attempts” and the probability of going over the
energy barrier (TDS + DG):
ucm
 (TDS  DG ) 
 DS 
 DG 
1
 t exp 
 t exp 
exp 


kT


 k 
 kT 
1
Net growth rate, u: the net rate of crystal growth, u, is equal to the
difference between the two rates:
DG 
 DS  1
 DS 
 DG 
 DS  
1
umc  ucm  t 1 exp 

t
exp

exp


t
exp

1

exp(

)








kT 
 k 
 k 
 kT 
 k 
Lamellar Crystal Growth
Lamella
thickness, L
Lamellar growth direction
a
From U.W. Gedde, Polymer Physics (1995)
Chapman & Hall, p. 145
L
Crystal growth is from the edge of the lamella.
The lamella grows a distance a when each chain is added.
From U.W. Gedde, Polymer Physics
(1995) Chapman & Hall, p. 161
The Velocity of Crystal Growth, n
The velocity of crystal growth can be calculated from the product of the rate
of growth (u, a frequency) and the distance added by each chain, a. Also, as
DG/kT << 1, exp(-DG/kT)  1  DG/kT:
DG 
 DS  
 DS   DG 
1
v  ua  t 1a exp 
1

exp(

)

t
a
exp





kT 
 k 
 k   kT 
But from before (slide 19) DG is a function of L:
DT
DG  DH m La
 2 a 2 f
Tm ()
The entropy loss in straightening out a
chain is proportional to the number of
units of size a in a chain of length L:
Finally, we find:
2
DS ~
L
a
1
 cL 
2 DT
v~
exp   (DH m La
 2a 2 f )
kT
Tm ()
 a 
We see that the crystal growth velocity is a function of lamellar thickness, L.
The Fastest Growing Lamellar Thickness, L*
 cL 
2 DT
v ~ exp   (DH m La
 2a 2 f )
Tm ()
 a 
L dependence
To find the maximum n, set the differential = 0, and solve for L = L*.
 DH m DT 2    cL   c 
dn
  cL   DH m DT 2   2
 0  exp 
a    2a  f 
La exp 



dL
Tm ()
 a  Tm ()
 
  a  a 
DH m DT
DH m DT
Lc 
a  2c f
Tm ()
Tm ()
Solve for L = L*:
2 f Tm ()
a
L*  
c DH m (Tm ()  T )
n
L*
L
Lamellar Thickness is Inversely Related to Undercooling
2 f Tm ()
a
1
L*  
~ const .  m
c DH m (Tm ()  T )
DT
L* (nm)
L* (nm)
Experimental
data for
polyethylene.
DT  Tm ()  T
Jones, Soft Condensed
Matter, p. 134
Original data from Barham et al.
J. Mater. Sci. (1985) 20, p.1625
Tm()-T
Chains Can Re-organise to Reduce the Number of Folds
Xu et al., Nature Materials (2009) 8, 348.
Temperature Dependence of Crystal Growth
Velocity, n
1
The rate at which a chain attempts to join a growing crystal, t , is expect to have the
same temperature-dependence as the viscosity of the polymer melt (h ~ Gt):

B 

 T  T0 
t  t 0 exp 
This temperature-dependence will contribute to the crystal growth velocity:
 B 
  cL  DG
  cL  DG
1



a
t
exp
exp



0


a
kT
T

T
a



 kT
0 

Recall that DG depends on T and on L as:
n  at 1 exp 
DT
DG  DH m La
 2 a 2 f
Tm ()
2
Finally, recall that the fastest-growing lamellar size, L = L*, also depends on
temperature as:
a 2 T ()
L* 
We see that DG(L*) becomes:
c

f
m
DH m DT
DHDTa 3
DG ( L*) 
Tm c
Temperature Dependence of Crystal Growth
Velocity, n
We can evaluate n when L = L* and when DG = DG(L*):
n  at 0
1
 B 
 cL *  DG( L*)

 exp 
exp 

 a  kT
 T  T0 
We finally find that:
1
 B 
 2c f Tm () 
at 0 a 3 DH m DT
 exp  

n
exp 
ekT c Tm ()
 T  T0 
 aDH m DT 
Describes molecular
slowing-down as T
decreases towards T0
Describes how the driving force for
crystal growth is smaller with a lower
amount of undercooling, DT.
Recall that T0 is approximately 50 K less than the glass
transition temperature.
n (cm s-1)
Experimental Data on the Temperature
Dependence of Crystal Growth Velocity
T-Tm() (K)
T-Tm() (K)
T-Tm() (K)
Tm() = crystal melting temperature
From Ross and Frolen, Methods of Exptl. Phys., Vol. 16B (1985) p. 363.
Data in Support of Crystallisation Rate Equation
describes molecular
slowing down with
decreasing T
Undercooling
contribution: considers
greater driving force for
crystal growth with
decreasing T
n exp (B/(T-T0))
V-F contribution:
[cm s-1]
 B 


1
 exp 

n ~ exp 
 T  T0 
 Tm ()  T 
1/(T(Tm()-T)) [10-4 K-2]
J.D. Hoffman et al., Journ. Res. Nat. Bur. Stand., vol. 79A, (1975), p. 671.
Why Are Polymer Single Crystals (Extended
Chains) Nearly Impossible to Achieve?
• Crystal with extended chains are favourable at very low levels of
undercooling, as L* ~ 1/DT
n (cm s-1)
• But as temperatures approach Tm(), the crystal growth velocity is
exceedingly slow!
T-Tm()
(K)
T-Tm()
(K)
T-Tm()
(K)
Factors that Inhibit Polymer
Crystallisation
1. Slow chain motion (associated with high
viscosity) creates a kinetic barrier
2. “Built-in” chain disorder, e.g. tacticity
3. Chain branching
Tacticity Builds in Disorder
Isotactic: identical repeat units
Easiest to crystallise
Syndiotactic: alternating repeat
units
Atactic: No pattern in repeat units
Usually do not crystallise
R.A.L. Jones, Soft Condensed Matter (2004) O.U.P., p. 75
Polymer Architecture
Linear
Branched
Side-branched
Star-branched
Effects on Branching on Crystallinity
Linear Poly(ethylene)
Branched Poly(ethylene)
Lamella are packed less tightly together when the chains are branched.
There is a greater amorphous fraction and a lower overall density.
From U.W. Gedde, Polymer Physics (1995) Chapman & Hall, p. 148
Determining Whether a Polymer Is (Semi)-Crystalline
Raman Spectra
“Fully” crystalline
Amorphous
Partially crystalline
From G. Strobl, The Physics of Polymers (1997) Springer, p. 154
Crystal Nucleation from “Seeds”
Original
crystal
Re-crystallised
from “seed
crystals”
Xu et al., Nature Materials (2009) 8, 348.
Summary
• Polymer crystals have a hierarchical structure: aligned chains, lamella,
spherulites.
• Melting point is inversely related to the crystal’s lamellar thickness.
• Lamellar thickness is inversely related to the amount of undercooling.
• The maximum crystal growth rate usually occurs at temperatures below the
melting temperature (Tm) but above the glass transition temperature, Tg.
•Tacticity and chain branching prevents or interrupts polymer crystal growth.
Further Reading
1. Gert Strobl (1997) The Physics of Polymers, Springer
2. Richard A.L. Jones (2004) Soft Condensed Matter, Oxford University Press
3. Ulf W. Gedde (1995) Polymer Physics, Chapman & Hall
Problem Set 5
This table lists experimental values of the initial lamellar thickness for
polyethylene crystallised at various temperatures. The equilibrium melting
temperature was independently found to be 417.8 K.
Temperature, T (K)
358.95
368.95
385.75
396.15
397.55
399.15
400.85
401.65
403.05
404.15
405.55
Lamellar thickness, L (nm)
8.9
9.9
12.0
14.1
16.1
15.9
17.3
18.2
17.9
20.1
22.2
(a) Are the data broadly consistent with the predictions of theory?
(b) Predict the melting temperature of crystals grown at a temperature of
400 K.