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Olle Inganäs: Polymers – structure and dynamics
Polymer physics
Polymers are macromolecules formed by many identical monomers, connected through
covalent bonds, to make a linear chain of mers – a polymer. The length of the chain specifies
the weight of the macromolecule, and one and the same polymer may have chains of different
lengths and therefore weights. This was one of the reasons that a stubborn resistance was
mounted to the concept of macromolecules; they did not have a specific molecular weight and
therefore could not count as chemical species, maybe not even exist. It is only with the
success of the macromolecular hypothesis, advocated by Staudinger in the early 20th century,
that the existence of macromolecules of this kind was accepted. The growth of polymer
physics and polymer chemistry is strongly intertwined; developments of new materials have
had a strong impact on the extension of the frontier of knowledge. Still, natural polymers are
abundant, and the essential molecules of life are mainly polymers. The genetic code is
enshrined in a sequence of four nucleotides – cytosine, guanine, adenosine, thymine - in a
linear chain of DNA; translation of the genetic code is mediated through another polymer,
RNA; the genetic information is finally transformed into synthesis of proteins, linear chains
of monomers from a library of ca 20 monomers, capable of building all the functional protein
structures. These may be active in energy transduction in biological systems, as enzymes for
catalysis, they may be used for signalling, in receptors, ion channels and ion pumps. The level
of synthesis of biopolymers is so refined and advanced, compared to what we can do in the
polymer synthesis plant, that properties of synthetic polymers rarely approach what is
available through biological materials. Still, it is the properties of synthetic polymers that have
largely been influential on the formation of concepts to describe polymers, and it is mainly as
structural materials that polymers are being produced, as plastics for coatings and structural
elements. The mechanical properties of polymers have therefore been in focus during the first
centennium of polymer physics. The first and foremost physical analysis of polymers is
therefore that of polymer geometry. That is done with the help of concepts from statistical
physics and thermodynamics.
The random coil model of single polymer chains in solution
We will start our foray of polymer physics by considering a highly idealised object, a model
of macromolecules. Consider a chain of mers, a polymer, connected by covalent bonds
between mers, of length N+1 mers, and thus incorporating N bonds between mers. We may
picture this as a string of beads(point masses) connected by small links of length a. What is
the length of this chain? Clearly one answer is the total length of bonds between repeating
units L=Na, which is named the contour length. But suppose that we would like to calculate
the distance from mer 1 to mer N+1. With the help of vectors of unit length a we can
construct a vector
N
R =
r
i
1
[2.1]
Olle Inganäs: Polymers – structure and dynamics
Figure 2.1 Reducing a real polymer chain to a mathematical object
Consider now that the square of this length is calculated
R 2 = ( Error!)2
N
1
i j
= Error!ri•rj = ri2 +
N,N
ri•rj
[2.2]
Should N be a large number, and should the direction of vectors ri be completely random, the
second term in [2.2] will approach zero, and be negligible by comparison to the first term.
This is because the product ri•rj will have values between a2 and – a2, and all equally
probable, therefore leading to a sum of zero. For a random walk of this kind we note that the
distance from start to end of the chain
R = (R 2 )1/2 = a N 1/2
[2.3]
This is the dimension of the random coil, an important concept of polymer physics. This is the
geometry of the polymer chain, in a first (and second) approximation. Clearly, the
assumptions in this derivation may be less realistic, but as often in physics, brute
simplifications can be rewarding. Note that the contour length grows as N but the end-to-end
4
distance grows as N. The number N can easily reach 10 in synthetic polymers.
The size of the random coil – i.e. the distribution of mers in space- can also be evaluated by
calculating the first moment of mass distribution. This can be expressed as the radius of
gyration, and is another common measure of the size of the coil. It is related to the end-to-end
distance by a simple geometrical correction; we can therefore use these measures
interchangeably.
We noted that the product ri•rj is equal to a2 for i=j, and anything between a2 and – a2 if i is
different from j. This is an example of a stochastic variable, and we know that the central limit
theorem of mathematical statistics give a definitive form to the distribution of these if the
Olle Inganäs: Polymers – structure and dynamics
number of stochastic variables is sufficiently large. This is the case here. If we ignore
constants we find that the probability of finding an end-to-end distance of length r is
p(r) exp (- r2/Na2)
[2.4]
What does this distribution function tell us? It gives us the probability that in an ensemble of
polymer chains of length N, a fraction p(r) is characterised by having their starting and ending
mers located at a distance r. As we now attempt to calculate thermodynamic properties of the
single polymer chain, we will need to calculate the number of ways of arranging the polymer
chain, that is, how many paths can the polymer make between start and end ?
What distinguishes the polymer of length N+1 units from the collection of N+1 mers without
any covalent bonds is the number of possible arrangements of these objects in space. We see
intuitively that the number of ways of arranging a number of mers is very much larger than
that of arranging a single polymer chain, however large that number may be. As the theory of
polymer physics is basically a special case of statistical physics, we need to be able to
evaluate the partition function of polymeric chains, in order to calculate the entropy and free
energy of these objects. The random coil model is very simplified, as it does not forbid the
chain to cross itself an unlimited number of times, and as it assumes that the sequence of
vectors r has no correlation whatsoever. This is not true for real polymers, where the
chemical nature of bonding will reduce the number of possibilities of polymer geometry from
that of a random walk to something more limited.
Still, a very simple relation between the length of the polymer chain and the dimensions of the
“not-completely random” walk can be obtained, in a so called scaling relationship
R = (R2 )1/2 = a N ,
[2.5]
Where the is a coefficient between 1/2 (for ideal solutions) to 3/5 (for good solutions)
where the polymer is swollen. The interaction between chains acts to force them apart by
repulsion, leading to expansion of the random coil.
To count the number of ways of arranging a polymer chain, we may use yet another brute
approximation, in modelling the chain geometry as a random walk on a lattice of some
dimensionality d, giving the possibility to choose any of z=2d directions. Consider the
random walk illustrated in Fig. 2.2. At each node of the lattice we can let the polymer chain
continue in four different directions. There are therefore zN+1 number of chain walks on the
Fig. 2.2 A random walk on a 2D lattice
Olle Inganäs: Polymers – structure and dynamics
lattice. If we consider the subclass of self-avoiding random walks (SAWs) on the lattice –
avoiding lattice nodes which have already been visited – we find that the number of SAWs is
less than z(z-1)N-1, which is the number of walks where the chain will not cross at closely
spaced positions along the chain. At long distances crossing may occur; therefore this is an
overestimate. From knowledge of the distribution function for the random walk we can
calculate the number of walks taking us from the start to the end of the polymer. This is
the number of paths that is allowed for a chain of length N+1 mers. We have already argued
the probability that the chain will start at and arrive at at a distance r, being an instance
of application of the central limit theorem.
We now notice that , the number of paths going from , is the product of the number
of walks on the lattice and the probability that a walk will start at and go
exp (- r2/Na2)z(z-1)N-1
[2.6]
We can now calculate the entropy S of the random coil polymer chain, as we know to be S=
kBln() according to statistical physics.
We estimate the entropy of the polymer chain as
S= S0 – kB (r2/ Na2)
[2.7]
where the S0 constant now includes all the non-exponential terms.
Therefore the Gibbs free energy is thus found to be
G=H-TS= G0 + kBT(r2/ Na2)
[2.8]
The funny consequence is that the free energy of the polymer chain is very much similar to
the energy of the harmonic oscillator, something that can be a helpful picture when
considering the dynamics of polymer chains.
Polymer statistics – measures of polymer chain length
Synthetic polymers are produced by polymerisation, which does not lead to one and the same
length of the polymer chain. The variation of chain lengths in a polymer causes variation of
many properties, as well as the modes of forming solid materials. Important measures of this
chain weight are the number and weight average molecular weights Mn and Mw. By definition,
in a collection of molecules of length I , with the molecular weight of the chain of length I
equal to Mi, the number average and the weight average molecular weights are
N
Mn =
1
N
N
i
Mi
/ Ni
1
[2.9]
Olle Inganäs: Polymers – structure and dynamics
N
Mw = Ni
N
MiMi
/ Ni
1
Mi
[2.10]
1
The Mw is therefore the average molecular weight of the sample, with respect to the weight
fraction of chains of length i and weight Mi
The Mn is the average molecular weight with respect to the number fraction of chains of
length i.
The ratio between these numbers, (Mn /Mw ), is called the polydispersity ratio, and is one of
the measures often used to characterise a polymer sample. The polydispersity is always larger
than 1, and can be correlated to the mode of polymerisation used to produce the sample.
Polymers in solution
There are very direct ways of measuring the average molecular weight of a polymer sample
using the properties of polymer solutions. The viscosity and the osmotic pressure of a polymer
solution give direct measures of these parameters; light scattering of polymer solutions can
also be used to obtain molecular weights. The theoretical background to these studies is the
observation that polymer chains in dilute solution are similar to the atoms in kinetic gas
theory, used to analyse the behaviour of ideal gases. This is the point of analogy to the theory
of ideal gases; consider the polymer chains as balls of an ideal gas. Chains form (almost)
random coils of dimensions R= aNb, and mass NM0 and these chains are far apart in the dilute
solution (on the average).
The viscosity of the solution is increased because of the friction with polymer coils, and is
in the first approximation linearly proportional to the concentration of these coils, as
originally analysed by Einstein. The viscosity of a polymer solution can then be expressed as
=0(1+ + ....)
[2.11]
Where the volume fraction of polymer coils is , and where the concentration of polymer
coils is c/M polymer chains of dimension
( 4/3)( aN)3= ( 4/3)( a(M/M0) )3
Therefore we find the viscosity to be related to the weight average molecular weight as
lim c0 (0)/c = K Mw(3-1)
[2.12]
This relation allow us to evaluate the character of the solvent, whether leading to expanded
((3-1) =0.8) or ideal chains ((3-1) =0.5).
The osmotic pressure is also related to the concentration of polymer coils, but now weighted
by another measure in the relation of the osmotic pressure to the concentration c of
polymer of number average molecular weight
R /
= = lim c0 /c
Olle Inganäs: Polymers – structure and dynamics
It is therefore possible to obtain the averaged molecular weights by simple measurements on
polymer solutions.
Polymer solids - structure
The formation of solid state materials from polymer chains is in many ways different from the
formation of crystalline or amorphous solids from elemental compounds, such as Al or Si
forming crystals, such as glass formed from SiO2. The main difference resides in the fact that
collections of polymer chains are normally collections of different polymer chains – some of
them short and some of them long, some of them branched and some crosslinked ( meaning
that two long chains are somehow bonded), some of them carrying chemical or structural
deviations. There is therefore not the simple element of repetition that forms the lattice in a
crystal, or at best, there is a lot more besides this repeating element. This has a decisive
influence on the structure of polymeric solids, which invariably are very disordered, as
compared to crystalline materials. This disorder is crucial to understand electronic transport
and optical processes in polymer electronics. The limitations and possibilities of polymers are
due to the consequences of disorder and solubility; melt and solution processing of polymer
materials are the normal methods, and are also possible for materials suitable for polymer
electronics.
If a polymer sample is sufficiently regular – that is that by all chemical and geometrical
measures the chains are (almost) identical- then there is a possibility of forming polymer
crystals. Such crystals consist of polymer chains forming thin lamellae, where polymer are
aligned in parallel for short distances, but then switch directions to once more go back in the
lamellae(Fig) . Chain imperfections – such as ends, chemical defects – accumulate outside the
polymer crystal, and form part of the amorphous disorded phase that is (almost) invariably
found close to the polymer crystals. One and the same polymer chains can therefore be part of
the crystal and the amorphous phase. In highly regular polyethylene it is possible to have
crystallinity approaching 98 % (High Density PolyEthylene, or HDPE for short as the
crystallinity and density are directly correlated), in Low Density PE (LDPE) the crystallinity
is lower. Mechanical properties vary much between these two different grades.
Olle Inganäs: Polymers – structure and dynamics
Fig. 2.3 Lamellaes are found in semicrystalline polymers, and contain polymer chains which
fold back to form a highly ordered structure.
Supramolecular ordering of crystalline and amorphous domains are found in spherulites,
where ordering of the polymer chains in lamellae (30-50 nm thickness, microns of side) and
segregation between crystalline and amorphous domains give rise to structures 1-100 μm
large, causing rotation of plane polarised light and therefore easily visible in the optical
(polarising) microscope.
Olle Inganäs: Polymers – structure and dynamics
Fig. 2.4 Organisation of polymer chains in both crystalline and amorphous regions leads to
structuring of a polymer solid.
The mechanical properties of a polymer are much influenced by temperature. At low
temperatures, lower than the Tg of the polymer, all polymers are rather brittle and glass like.
Above the glass transition temperature Tg, at a point where the polymer chains found in
amorphous regions are somewhat more free to move, to respond to mechanical and thermal
energy, the polymer will be softer and flexible. Above the melting temperature range of
polymers – indeed a range, as all polymers are polydisperse in molecular weight and the
melting temperature depends on molecular weight – the polymer will flow on stress, and
behave like a fluid. It is in this temperature range that melt processing of polymers is
typically done. If polymers are crosslinked, by chemical or physical means, the behaviour
above the melting range is somewhat more complex- we then have a rubber, that will pull
back after stretching.
Olle Inganäs: Polymers – structure and dynamics
Web resources
http://www.uwsp.edu/chemistry/polyed/
http://www.psrc.usm.edu/pslc/index.htm
Some references to polymer materials science
Some references to polymer physics
L.H:Sperling, “Introduction to Physical Polymer Science”, John Wiley second edition 1992
U. W. Gedde : “Polymer Physics” Chapman&Hall, second ed. 2002
Daniel I. Bower: “An Introduction to Polymer Physics”, Cambridge University Press, 2002
More advanced is
P.G.de Gennes:” Scaling concepts of polymer physics”, Cornell University Press, 1991
G. Strobl:”The Physics of Polymers. Concepts for understanding their structures and
behaviour”, Springer , second ed. 1997