Lecture notes for the course given by Prof. David Andelman, Tel-Aviv University 2009
Original author: Guy Cohen, Tel-Aviv University
PDF version.
All diagrams.
Introduction and Preliminaries
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We will make several assumptions throughout the course:
- The physics in question are generally in the classical regime,
.
- Materials are "soft": quantitatively, this implies that all relevant energy scales are of the order of
.
- Condensed matter physics deals with systems composed of
particles, and statistical mechanics applies. We are always interested in a reduced description, in terms of continuum mechanics and elasticity, hydrodynamics, macroscopic electrodynamics and so on.
We begin with an example from Chaikin & Lubensky, the story of an H2O molecule. This molecule is bound together by a chemical bond which is around
at room temperature and not easily broken under normal circumstances. What happens when we put
water molecules is a container? First of all, with such large numbers we can safely discuss phases of matter: namely

Gas is typical to low density, high temperature and low pressure. It is generally prone to changes in shape and volume, homogeneous, isotropic, weakly interacting and insulating. This is the least orderedform of matter relevant to our scenario, and relatively easy to treat since order parameters are small. The liquid phase is typical of intermediate temperatures. It flows but is not very compressible. It is homogeneous, isotropic, dense and strongly interacting. Its response to external forces depends on the rate of its deformation. Liquids are hard to treat theoretically, as their intermediate properties make simple approximations less effective. The solid is a dense ordered phase with low entropy and strong interactions. It is anisotropic and does not flow, it strongly resists compression and its response to forces depends on the amount of deformation they cause (elastic). Transitions between these phases occur at specific values of thermodynamic
parameters (see diagram (1)). First order changes (volume/density "jumps" at the transition, and no jump in pressure/temperature) occur on the lines; at the critical liquid/gas point, second order phase transitions occur; at the triple point, all three phases (solid/liquid/gas) coexist. The systems we are interested in are characterized by several kinds of interactions between their constituent molecules: for example, Coulombic interactions of the form
when charged particles are present, fixed dipole interaction of the form
when permanent dipoles exist, and almost always induced dipole/van der Waals interaction of the form
. At close range we also have the "hard core" or steric repulsion, sometimes modeled by a
potential. Simulations often use the so-called
Lennard-Jones potential
(as pictured in (2)), which with appropriate parameters correctly describes both condensation and crystallization in some cases.
Sidenote
When only the repulsive potential exists (for instance, for billiard balls), crystallization still takes place but no condensation/evaporation phase transition between the liquid and gas phases exists.
Starting from a classical Hamiltonian such as
, we can predict all three phases of matter and the transitions between them. In biological systems, this simple picture does not suffice: the basic consideration behind this is that of effects which occur at different scales between the nanometric scale, through the mesoscopic and up to the macroscopic scale. Biological systems are mesoscopic in nature, and their properties cannot be described correctly when a coarse-graining is performed without accurately accounting for mesoscopic properties.
A few examples follow:
The most basic assumption we need in order to model liquid crystals is that isotropy at the molecular level is broken: molecules are represented by rods rather than spheres. Such a description was suggested by Onsager and others, and leads to three phases as shown in (3).
When molecules are interconnected at mesoscopic ranges, new phases and properties are encountered.
This kind of substance is approximately 95% agent, with the remainder water – yet it behaves like a weak solid as long as its deformations are small. This is because a tight formation of ordered cells separated by thin liquid films is formed, and in order for the material to change shape the cells must be rearranged. This need for restructuring is the cause of such systems' solid-like resistance to change.
Polymers or macromolecules in liquid state, liquid crystals, emulsions and colloidal solutions and gels display complex visco-elastic behavior as a result of mesoscopic super-structures within them.
Interfaces between fluids have interesting properties: they act as a 2D liquid within the interface, yet respond elastically to any bending of the surface. Surfactant molecules will spontaneously form membranes within the same fluid, which also have these properties at appropriate temperatures. Surfactants in solution also form lamellar structures - multilayered structures in which the basic units are the membranes rather than single molecules.
Books: Doi, de Gennes, Rubinstein, Doi & Edwards.
Natural polymers like rubber have been known since the dawn of history, but not understood. The first artificial polymer was made
. Stadinger was the first to understand that polymers are formed by molecular chains and is considered to be the father of synthetic polymers. Most polymers were made by petrochemical industry. Nylon was born in 1940. Various uses and unique properties (light, strong, thermally insulating; available in many different forms from strings and sheets to bulk; cheap, easy to process, shape and mass-produce...) have made them very attractive commercially. Later on, some leading scientists were Kuhn and Flory in chemistry (30's to 70's) and Stockmayer in physical chemistry (50's and 60's). The famous modern theory of polymers was first formulated by P.G. de Gennes and Sam Edwards.
Material composed of chains, having a repeating basic unit (monomer). Connections between monomers are made by chemical (covalent) bonds, and are strong at room temperature.
![{\displaystyle \left[A\right]_{N}\equiv {\overset {N{\mbox{ times}}}{\overbrace {A-A-A-...-A} }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc2300dcda92d0cd78cc1f8c245ec9637a40a5bf)
is the polymerization index.
Sidenote
More generally, this kind of structure is called a homopolymer. Heteropolymers – which have several repeating constituent units - also exist. These can have a random structure (
) or a block structure (
), in which case they are called block copolymers. These can self-assemble into complex ordered structures and are often very useful.
Sidenote
For an example, look up ester monomers and polyester, or polyethylene.
Polymerization is also the name of the process by which polymers are synthesized, which involves a chain reaction where a reactive site exists at the end of the chain. Some chemical reactions increase the chain length by one unit, while simultaneously moving the reactive site to the new end:
![{\displaystyle \left[A\right]_{N}+\left[A\right]_{1}\rightarrow \left[A\right]_{N+1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38b85722e7bb4c5a8861d4a6e3778aea93835033)
There also exist condensation processes, by which chains unite:
![{\displaystyle \left[A\right]_{N}+\left[A\right]_{M}\rightarrow \left[A\right]_{N+M},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2beecabd37133c17ce74df89216aa7f5d26772ff)
where
. A briefer notation, dropping the name of the monomer, is

Consider the example of hydrocarbon polymers, where we have a monomer which is
(Check this...). As a larger number of such units is joined together to become polyethylene molecules, the material composed of these molecules changes drastically in nature:
|
phase
|
type of material
|
1-4
|
gas
|
flammable gas
|
5-15
|
thin liquid
|
liquid fuel/organic solvents
|
16-25
|
thick liquid
|
motor oil
|
20-50
|
soft solid
|
wax, paraffin
|
1000
|
hard solid
|
plastic
|
Types of polymer structures
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Polymers can exist in different topologies, which affect the macroscopic properties of the material they form (see (4)):
- Linear chains (this is the simplest case, which we will be discussing).
- Rings (chains connected at the ends).
- Stars (several chain arms connected at a central point).
- Tree (connected stars).
- Comb (one main chain with side chains branching out).
- Dendrimer (ordered branching structure).
Polymer phases of matter
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Depending on the environment and larger-scale structure, polymers can exist in many states:
- Gas of isolated chains (not very relevant).
- In solution (water or organic solvents). In dilute solutions, polymer chains float freely like gas molecules, but their length alters their behavior.
- In a liquid state of chains (called a melt).
- In solid state (plastic) – crystals, poly-crystals, amorphous/glassy materials.
- Liquid crystal formed by polymer chains (Polymeric Liquid Cristal or PLC)
- Gels and rubber: networks of chains tied together.
Ideal Polymer Chains in Solution
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Some basic models of polymer chains
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The simplest model of an ideal polymer chain is the freely jointed chain (FJC), where each monomer performs a completely independent random rotation. Here, at equilibrium the end-to-end length of the chain is
, where
is the contour length.
A slightly more realistic model is the freely rotating chain (FRC), where monomers are locked at some chemically meaningful bond angle
and rotate freely around it via the torsional angle
. Here,

Note that for
we find
that
and this is identical to the FJC. For very
small
, we can expand the cosine an obtain

This is the rigid rod limit (to be discussed later in detail).
A second possible improvement is the hindered rotation (HR)
model. Here the angles
have a minimum-energy value,
and are taken from an uncorrelated Boltzmann distribution with some
potential
. This gives

Sidenote
See Flory's book for details.
Another option is called the rotational isomeric state model. Here, a finite
number of angles are possible for each monomer junction and the state
of the full chain is given in terms of these. Correlations are also
taken into account and the solution is numeric, but aside from a complicated
this is still an ideal chain with
.
Calculating the end-to-end radius
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For the polymer chain of (5), obviously we will always have
.
The variance, however, is generally not zero: using
,

FJC
In the freely jointed chain (FJC) model, there are neither correlations
between different sites nor restrictions on the rotational angles.
We therefore have
,
and

Sidenote
The mathematics are similar to that of a random walk or diffusion process, where in 1D
.
Therefore,
.
FRC
In the freely rotating chain model, the bond angles are held constant
at angles
while the torsion angles
are taken from a uniform distribution between
and
.
This introduces some correlation between the angles: since (for one
definition of the
)
,
and since the
are independent and any averaging over a sine of cosine of one or more of them will result in a zero, only the
independent terms survive and by recursion this correlation has the simple form

The end-to-end radius is
![{\displaystyle {\begin{array}{lcl}R_{0}^{2}&=&\sum _{ij=1}^{N}\left\langle \mathbf {r} _{i}\cdot \mathbf {r} _{j}\right\rangle \\&=&\sum _{i=1}^{N}{\overset {\scriptstyle =\ell ^{2}}{\overbrace {\left\langle \mathbf {r} _{i}^{2}\right\rangle } }}+\ell ^{2}\sum _{i=1}^{N}{\overset {\scriptstyle k=i-j}{\overbrace {\sum _{j=1}^{i-1}\left(\cos \vartheta \right)^{i-j}} }}+\ell ^{2}\sum _{i=1}^{N}{\overset {\scriptstyle k=j-i}{\overbrace {\sum _{j=i+1}^{N}\left(\cos \vartheta \right)^{j-i}} }}\\&=&N\ell ^{2}+\ell ^{2}\sum _{i=1}^{N}\left[\sum _{k=1}^{i-1}\left(\cos \vartheta \right)^{k}+\sum _{k=1}^{N-i}\left(\cos \vartheta \right)^{k}\right].\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0af4a76883c82c6c8832d6701f0be599e9a5a959)
At large
we can approximate the two sums in
by the series
, giving

To extract the Kuhn length
from this expression, we rewrite in in the following way:


To go back from this to the FRC limit, we would consider a chain with a random distribution of
angles such that
.
Consider once again the polymer chain of (5). Define:

The unprimed coordinate system is refocused on the center of mass,
such that
. Now, it is easier to work with
the following expression:

We will calculate
for a long FJC. For
we can replace the sums with integrals, obtaining
![{\displaystyle {\begin{array}{lcr}\left\langle R_{g}^{2}\right\rangle &=&{\frac {1}{2N^{2}}}\sum _{ij}{\overset {\scriptstyle {\left|i-j\right|\ell ^{2}}}{\overbrace {\left\langle \left(\mathbf {R} _{i}-\mathbf {R} _{j}\right)^{2}\right\rangle } }}\\&=&{\frac {1}{2N^{2}}}\int _{0}^{N}\mathrm {d} u\int _{0}^{N}\mathrm {d} v\,\ell ^{2}\left|u-v\right|\\&=&{\frac {2}{2N^{2}}}\int _{0}^{N}\mathrm {d} u\int _{0}^{u}\mathrm {d} v\,\ell ^{2}\left(u-v\right)\\&=&{\frac {\ell ^{2}}{N^{2}}}\int _{0}^{N}\mathrm {d} u\left[u^{2}-{\frac {1}{2}}u^{2}\right]\\&=&{\frac {1}{6}}N\ell ^{2}.\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f80eda4239c225483876b9ac446265cf6c39eb4b)
This gives the gyration radius for an FJC:

Polymers and Gaussian distributions
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An ideal chain is a Gaussian chain, in the sense that the end-to-end
radius is taken from a Gaussian distribution. We will see two proofs
of this.
Random walk proof
One way to show this (see Rubinstein, de Gennes) is to begin with
a random walk. For one dimension, if we begin at
and at each
time step
move left or right with moves
and the final displacement

then

We define
as the number of configurations of
steps with a final displacement of
.
is the associated normalized probability.

In fact, for
the central limit theorem tells
us that
will have a Gaussian distribution for any
distribution of the
. This can be extended to
dimensions
with a displacement
:

To find the normalization constant
we must integrate over all dimensions:


Some notes:
- An ideal chain can now be redefined as one such that
is Gaussian in any dimension
.
- This is also true for a long chain with local interactions only, such that
.
- The probability of being in a spherical shell with radius
is
.
- The chance of returning to the origin
is
.
is typical of an ideal chain.
- For any dimension
,
.
Formal proof
Another way to show this follows, which is also extensible to other
distributions of the
.
Sidenote
This proof can be found in Doi and Edwards.
In general, we can write

In the absence of correlations, we can factorize
:

For example, for a freely jointed chain
.
The normalization constant is found from
,
giving

We can replace the delta functions with
,
leaving us with
![{\displaystyle {\begin{matrix}P_{N}\left(\mathbf {R} \right)&=&{\frac {1}{\left(2\pi \right)^{3}}}\int \mathrm {d} \mathbf {k} e^{i\mathbf {k} \cdot \mathbf {R} }\int \mathrm {d} \mathbf {r} _{1}...\int \mathrm {d} \mathbf {r} _{N}\prod _{i}\left[e^{-i\mathbf {k} \cdot \mathbf {r} _{i}}\psi \left(\mathbf {r} _{i}\right)\right]&=&{\frac {1}{\left(2\pi \right)^{3}}}\int \mathrm {d} \mathbf {k} e^{i\mathbf {k} \cdot \mathbf {R} }\left[\int d\mathbf {r} e^{-i\mathbf {k} \cdot \mathbf {r} }\psi \left(\mathbf {r} \right)\right]^{N}.\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/492ec59f8a7a1cec6ef4d3861d203d9363634ceb)
In spherical coordinates,

which gives

We are left with the task of evaluating the integral. This can be
done analytically with the Laplace method for large
, since the
largest contribution is around
: we can approximate
by
.
The integral is then
![{\displaystyle {\begin{array}{lcl}P_{n}\left(\mathbf {R} \right)&=&\left({\frac {1}{2\pi }}\right)^{3}\int \mathrm {d} \mathbf {k} e^{i\mathbf {k} \cdot \mathbf {R} }e^{-{\frac {k^{2}\ell ^{2}N}{6}}}\\&=&\left({\frac {1}{2\pi }}\right)^{3}\int \mathrm {d} k_{1}\mathrm {d} k_{2}\mathrm {d} k_{3}\exp \left[\sum _{\alpha }\left(ik_{\alpha }R_{\alpha }-{\frac {Nk_{\alpha }^{2}\ell ^{2}}{6}}\right)\right]\\&=&\left({\frac {1}{2\pi }}\right)^{3}\prod _{\alpha }\int \mathrm {d} k_{\alpha }\exp \left(ik_{\alpha }R_{\alpha }-{\frac {Nk_{\alpha }^{2}\ell ^{2}}{6}}\right)\\&=&\left({\frac {3}{2\pi N\ell ^{2}}}\right)^{\frac {3}{2}}\exp \left\{-{\frac {3R^{2}}{2N\ell ^{2}}}\right\}.\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69a9ee05f8680ba318e7377b7ec6dbde66bb79c8)
This is, of course, the same Gaussian form we have obtained from the
random walk (we have done the special case of
, but once again
this process can be repeated for a general dimension
).
03/26/2009
Rigid and Semi-Rigid Polymer Chains in Solution
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In considering the
limit of the freely rotating
chain, we have seen that
.
This is of course unphysical, and this limit is actually important
for many interesting cases of stiff chains (for instance, DNA). If
we take the
limit along with
and start over, we can make the following change of variables:
![{\displaystyle {\begin{matrix}\left\langle \mathbf {r} _{i}\cdot \mathbf {r} _{j}\right\rangle &=&\ell ^{2}\left\langle \cos \vartheta _{ij}\right\rangle &=&\ell ^{2}\left(\cos \vartheta \right)^{\left|i-j\right|}&=&\ell ^{2}\exp \left[-{\frac {\left|i-j\right|\ell }{\ell _{p}}}\right],\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc1d5cf05a3b1f6f787c1bf229db6d505fa90589)
which defines the persistence length
. For the FRC
model,

This is a useful concept in general, however: it defines the typical
length scale over which correlations between chain angles dies out,
and is therefore an expression of the chain's rigidity.
At small
we can expand the logarithm to get


Taking the continuum limit carefully then requires us to consider
and
such that
is constant. Now, we can calculate the end-to-end length
at the continuum limit using out the new form for the correlations:

To simplify the calculation, we can define the dimensionless variable
,
and
.
With these replacements,

The final result (known as the Kratchky-Porod worm-like-chain or WLC)
is

Importantly, is does not depend on
or
but only on
the physically transparent persistence length and contour length.
We will consider the two limits where one parameter is much larger
than the other. First, for
we encounter the
rigid rod limit: we can expand the previous expression into

The fact that
rather than
is a result of the long-range correlations we have introduced, and is an indication that at this regime the material is in an essentially different phase. Somewhere between the ideal chain and the rigid rod, a crossover regime must exist.
For
we can neglect the exponent, obtaining

This therefore returns us to the ideal chain limit, with a Kuhn length
. The crossover phenomenon we discussed occurs
on the chain itself here as we observe correlation between its pieces
at differing length scales: at small scales (
) it behaves
like a rigid rod, while at long scales we have an uncorrelated random
walk. An interesting example is a DNA chain, which can be described
by a worm-like chain with
and
:
it will therefore typically cover a radius of
.
Free Energy of the Ideal Chain and Entropic Springs
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We have calculated distributions of
for Gaussian chains
with
components,
. Let's consider
the entropy of such chains:

The logarithm of
is the same as that
of
, aside from a factor which does
not depend on
. Therefore,

The free energy is

since
for an ideal
chain.
What does
mean? It represents the
energy needed to stretch the polymer, and this energy is
like a harmonic spring (
) with
.
Note that the polymer becomes less elastic (more rigid) as
the temperature increases, unlike most solids. This is a physical
result and can be verified experimentally: for instance, the spring
constant of rubber (which is made of networks of polymer chains) increases
linearly with temperature.
Consider an experiment where instead of holding the chain at constant
length, we apply a perturbatively weak force
to its
ends and measure its average length. We can perform a Legendre transform
between distance and force: from equality of forces along the direction
in which they are applied,

To be in this linear response (
) region,
we must demand that
,
and to stress this we can write

Numerically, with a nanometric
and at room temperature the
forces should be in the picoNewton range to meet this requirement.
A more rigorous treatment which works at arbitrary forces can be carried
out by considering an FJC with oppositely charged (
) ends
in an electric field
. The chain's
sites are at
with
.
The potential is

Since
, we can write the potential as

with
. The
partition function is

The function
is separable into product of functions
.
Now,

In spherical coordinates
we can solve the integral:
![{\displaystyle {\begin{array}{lcl}Z_{N}\left(\mathbf {f} \right)&=&\left[\int _{0}^{\infty }\mathrm {d} r{\frac {r^{2}}{4\pi \ell ^{2}}}\delta \left(r-\ell \right)\right]^{N}\times \left[\int _{0}^{2\pi }\mathrm {d} \varphi \right]^{N}\times \prod _{i}\int _{0}^{\pi }\mathrm {d} \vartheta _{i}\sin \vartheta _{i}e^{{\frac {f\ell }{k_{B}T}}\cos \vartheta _{i}}\\&{\underset {\scriptscriptstyle x=\cos \vartheta }{=}}&\left({\frac {1}{4\pi }}\right)^{N}\left(2\pi \right)^{N}\left[\int _{-1}^{1}\mathrm {d} xe^{{\frac {f\ell }{k_{B}T}}x}\right]^{N}\\&=&{\frac {1}{2^{N}}}\left[{\frac {2k_{B}T}{f\ell }}\sinh \left({\frac {f\ell }{k_{B}T}}\right)\right]^{N}\\&=&\left[{\frac {k_{B}T}{f\ell }}\sinh \left({\frac {f\ell }{k_{B}T}}\right)\right]^{N}.\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba8a951c85623dcccc30f52e5ff8d419175750e0)
The Gibbs free energy (Gibbs because the external force is fixed)
is then
![{\displaystyle G_{N}\left(\mathbf {f} \right)=-k_{B}T\ln Z_{N}\left(\mathbf {f} \right)=-k_{B}TN\ln \left[\sinh \left({\frac {f\ell }{k_{B}T}}\right)\right]+k_{B}TN\ln \left({\frac {f\ell }{k_{B}T}}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9272e22cba23ceafb2f7e830a2081946bd67fca)
and the average extension
![{\displaystyle {\begin{matrix}\left\langle R\right\rangle _{f}&=&-{\frac {\partial G_{N}\left(f\right)}{\partial f}}&=&-k_{B}TN\coth \left({\overset {\scriptstyle \equiv \alpha }{\overbrace {\frac {f\ell }{k_{B}T}} }}\right){\frac {\ell }{k_{B}T}}+k_{B}TN{\frac {1}{f}}&=&N\ell \left[\coth \alpha -{\frac {1}{\alpha }}\right]\equiv N\ell {\mathcal {L}}\left(\alpha \right)\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a65ce55569e91ade6c407f62b4ec96c18bb3e9f)
The Langevin function
is also typical of spin magnetization in external magnetic fields
and of dipoles in electric fields at finite temperatures.
04/02/2009
Polymers and Fractal Curves
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Introduction to fractals
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Book: B. Mandelbrot.
A fractal is an object with fractal dimensionality ,
called also the Hausdorff dimension . This implies a new definition
of dimensionality, which we will discuss.
Consider a sphere of radius
. It is considered three-dimensional
because it has
and
for
. A plane has by the same reasoning
for
,
and is therefore a
object. Fractals are mathematical objects
such that by the same sort of calculation they will have
,
for a
which is not necessarily an integer number (this definition
is due to Hausdorff). One example is the Koch curve (see (7)): in
each of its iterations, we decrease the length of a segment by a factor
of 3 and decrease its mass by a factor of 4. We will therefore have

Note that a fractal's "real" length is infinite, and its approximations
will depend on the resolution. The structure exhibits self-similarity:
namely, on different length scales it will look the same. This can
be seen in the Koch snowflake: at any magnification, a part of the
curve looks similar to the whole curve. There's a very nice animation
of this in Wikipedia.
The total length of the curve depends on the ruler used to measure
it: the actual length at iteration
is
.
Another definition for the fractal dimension is

Linking fractals to polymers
edit
Sidenote
The Flory exponent is defined from
such that
.
Consider the ideal Gaussian chain again. It has
.
Since
is proportional to the mass, we have an object with a fractal
dimension of 2 no matter what the dimensionality of the actual space
is. We can say that a polymer in
-space fills only
dimensions of the space it occupies, where
is 2 for an ideal
polymer Gaussian and
in general. Flory
has shown that in some cases a non-ideal polymer can also have
,
in particular when a self-avoiding walk (SAW) is accounted for. The
SAW as opposed to the Gaussian walk (GW) is the defining property
of a physical rather than ideal polymer, and gives a fractal dimension
of
. A collapsed polymer has
and fills
space completely. Note that two polymers with fractal dimensions
and
do not "feel" each other statistically if
.
Polymers, Path Integrals and Green's Functions
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Books: Doi & Edwards, F. Wiegel, or Feynman & Hibbs.
Local Gaussian chain model and the continuum limit
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This model is also known as LGC. We start from an FJC in 3D where
and
.
By the central limit theorem
will always be taken from a Gaussian distribution when the number
of monomers is large (whatever the form of
, as long as it
is symmetrical around zero such that
):

In the LGC approximation we exchange the rigid rods for Gaussian
springs with
and
, by
setting

We can then obtain for the full probability distribution

where
.
describes
harmonic springs with
connected
in series:

An exact property of the Gaussian distributions we have been using
is that a sub chain of
monomers (such as the sub chain starting
at index
and ending at
) will also have a Gaussian distribution
of the end-to-end length:

At the continuum limit, we will get Wiener distributions : the
correct way to calculate the limit is to take
and
with
remaining constant. The length
along the chain up to site
is then described by
,
. At this limit we can also substitute derivatives
for the finite differences
,
such that

If we add an external spatial potential
(which is single-body), its contribution to the free energy will amount
in a factor of

to the Boltzmann factor.
04/23/2009
Functional path integrals and the continuum distribution function
edit
Books: F. Wiegel, Doi & Edwards.
Consider what happens when we hold the ends of a chain defined by
in place, such that
and
. We can calculate the probability
of this configuration from

At the continuum limit the definition of the chain configurations
translates into a function
and the product
of integrals can be taken as a path integral according to
.
The probability for each configuration with our constraint is a functional
of
. The partition function is:

and we can normalize it to obtain a probability distribution function,
given in terms of this path integral:

We now introduce the Green's function
which
as we will soon see describes the evolution from
to
in
steps. We define it as:

Note that while the nominator is proportional to the probability
,
the denominator does not include include the external potential.
has several important properties:
- It is equal to the exact probability
for Gaussian chains in the absence of external potential.
- If we consider that the chain might be divided into one sub chain between step
and
and a second sub chain from step
to step
, then 
We can use this property to compute expectations values of observables. If we have some function of a specific monomer
, for instance:

- The Green's function is the solution of the differential equation (see proof in Doi & Edwards and in homework):

- The Green's function is defined as 0 for
and is equal to
when
in order to satisfy the boundary conditions.
Relationship to quantum mechanics
edit
This equation for
,
is
very similar in form to the Schrödinger equation. To see this, we
can rewrite it as:
![{\displaystyle \left[{\frac {\partial }{\partial N}}-{\overset {\scriptstyle \equiv {\mathcal {L}}}{\overbrace {{\frac {\ell ^{2}}{6}}{\frac {\partial ^{2}}{\partial \mathbf {R} ^{2}}}+{\frac {U\left(\mathbf {R} \right)}{k_{B}T}}} }}\right]G\left(\mathbf {R} ,\mathbf {R} ^{\prime };N\right)=\left[{\frac {\partial }{\partial N}}-{\mathcal {L}}\right]G\left(\mathbf {R} ,\mathbf {R} ^{\prime };N\right)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7690dadb58ed020dc992d9176bdddf7f9929cbde)
If we make the replacement
,
and
this is identical
to
.
Like the quantum Hamiltonian the Hermitian operator
has eigenfunctions such that
,
which according to Sturm-Liouville theory span the solution space
(
)
and can be orthonormalized (
).
The solution of the non-homogeneous problem is therefore

where the
are solutions of the homogeneous equation
.
Example
A polymer chain in a box of dimensions
:
The potential
is
within the box and
on the edges.
The boundary conditions are
if
or
are on the boundary. The
function is also separable in Cartesian coordinates:

Let's solve for
(the other
functions are
similar):

If we separate variables again with the ansatz
we obtain

With the boundary condition

This gives an expression for the energy and eigenfunctions:

The Green's function can finally be written as

Since with the Cartesian symmetry of the box the partition function
is also separable and using

we can calculate

We can now go on to calculate
, and
we can for instance calculate the pressure on the box edges in the
direction:

Two limiting cases can be done analytically: first, if the box is much larger than the polymer,
and

This is equivalent to a dilute gas of polymers (done here for a single
chain). At the opposite limit,
, the polymer
should be "squeezed". The Gaussian approximation will be no
good if we squeeze too hard, but at least for some intermediate regime
we can neglect all but the first term in the series:

There is a large extra pressure caused by the "squeezing" of
the chain and the corresponding loss of its entropy.
04/30/2009
The same formalism can be used to treat polymers near a wall or in
a well near a wall, for instance (see the homework for details). In
the well case, like in the similar quantum problem, we will have bound
states for
(where the critical temperature is defined by
a critical value of
, and
describes the condition for the potential well to be "deep"
enough to contain a bound state).
Dominant ground state
edit
Note that since

where
is positive and the
are real and ordered (assuming
no degeneracy,
), at large
we can neglect
all but the leading terms (smallest energies) and

This is possible because the exponent is decreasing rather than oscillating,
as it is in the quantum mechanics case. Taking only the first term
in this series is called the dominant ground state approximation .
Polymers in Good Solutions and Self-Avoiding Walks
edit
So far, in treating Gaussian chains, we have neglected any long-ranged
interactions. However, polymers in solution cannot self-intersect,
and this introduces interactions
into the picture which are local in real-space, but are long ranged
in terms of the contour spacing – that is, they are not limited to
. The importance of this effect depends on dimensionality:
it is easy to imagine that intersections in 2D are more effective
in restricting a polymer's shape than intersections in 3D.
The interaction potential
can in general
have both attractive and repulsive parts, and depends on the detailed
properties of the solvent. If we consider it to be due to a long ranged
attractive Van der-Waals interaction and a short ranged repulsive
hard-core interaction, it might be modeled by a
Lennard-Jones
potential. To treat interaction perturbatively within statistical
mechanics, we can use a virial expansion (this is a statistical-mechanical
expansion in powers of the density, useful for systematic perturbative
corrections to non-interacting calculations when one wants to include
many-body interactions). The second virial coefficient is
![{\displaystyle v_{2}=\int \mathrm {d} ^{3}r\left[1-e^{-{\frac {V\left(r\right)}{k_{B}T}}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/247a0ae391d1e1231f1cb07716d22b380a5f2e0f)
To make the calculation easy, consider a potential even simpler than
the 6-12 Lennard-Jones:

This gives
![{\displaystyle {\begin{matrix}v_{2}&=&{\overset {\scriptstyle ={\frac {4\pi }{3}}\sigma ^{3}\equiv V_{0}}{\overbrace {\int _{r<\sigma }\mathrm {d} ^{3}r\left[1-e^{-{\frac {V\left(r\right)}{k_{B}T}}}\right]} }}+{\overset {\scriptstyle ={\frac {4\pi }{3}}\left[\left(2\sigma \right)^{3}-\sigma ^{3}\right]\left(1-e^{\beta \varepsilon }\right)}{\overbrace {\int _{\sigma <r<2\sigma }\mathrm {d} ^{3}r\left[1-e^{-{\frac {V\left(r\right)}{k_{B}T}}}\right]} }}&=&8V_{0}-7V_{0}e^{\beta \varepsilon }.\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97fcbb6fc5ade4168ed8f0b24eb3057b88769965)
This can be positive (signifying net repulsion between the particles)
at
or negative (signifying
attraction) for
. While
the details of this calculation depend on our choice and parametrization
of the potential, in general we will have some special temperature
known as the
temperature (in our case
)
where

This allows us to define a good solvent: such a solvent must have
at our working temperature. This assures us (within
the second Virial approximation, at least) that the interactions are
repulsive and (as can be shown separately) the chain is swollen .
A bad solvent for which
will have attractive interactions,
resulting in collapse . A solvent for which
is
called a
solvent, and returns us to a Gaussian chain
unless the next Virial coefficient is taken.
A common numerical treatment for this kind of system is to draw the
polymer on a grid and make Monte-Carlo runs, where steps must be self-avoiding
and their probability is taken from a thermal distribution while maintaining
detailed balance. This gives in 3D
where
.
Renormalization group
edit
A connection between SAWs and critical phenomena was made by de Gennes
in the 1970s. Some of the similarities are summarized in the table
below. Using renormalization group methods, de Gennes showed by analogy
to a certain spin model that

This gives in 3D a result very close to the SAW:
.
Polymers
|
Magnetic Systems
|
,
|
(critical temperature) is a small parameter.
|
.
|
Correlation length – critical exponent .
|
Gaussian chains (non-SAW).
|
Mean field theory.
|
.
|
|
For , .
|
MFT is accurate for (Ising model: ).
|
This is a very crude model which gives surprisingly good results.
We write the free energy as
.
For the entropic part we take the expression for an ideal chain:
,
. For the interaction, we use the second virial
coefficient:
![{\displaystyle {\begin{matrix}{\frac {F_{int}\left(R\right)}{k_{B}T}}&=&{\frac {1}{2}}\nu _{2}\int \left[c\left(r\right)\right]^{2}\mathrm {d} ^{3}r.\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a5abb6cffea77838fc07e153892fe4a590599b1)
Here
is a local density such that its average value
is
.
If we neglect local fluctuations in
, then
![{\displaystyle {\begin{matrix}\int \left[c\left(r\right)\right]^{2}\mathrm {d} ^{3}r&=&V\left\langle c^{2}\left(r\right)\right\rangle \approx V\left\langle c\left(r\right)\right\rangle ^{2}=R^{2}\left({\frac {N}{R^{d}}}\right)^{2},{\frac {F_{int}}{k_{B}T}}&\approx &{\frac {1}{2}}v_{2}N^{2}R^{-d}.\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/683dadab3c2dae9dbcaecee4903befa94fdbee51)
The total free energy is then

The free parameter here is
, but we do not know how it relates
to
. For constant
the minimum is at

which gives the Flory exponent

This exponent is exact for 1, 2 and 4 dimensions, and gives a very
good approximation (0.6) for 3 dimensions, but it misses completely
for more than 4 dimensions. For a numerical example consider a polymer
of
monomers each of which is about
in length.
From the expressions above,

This difference is large enough to be experimentally detectable by
the scattering techniques to be explained next.
The reason the Flory method provides such good results turns out to
be a matter of lucky cancellation between two mistakes, both of which
are by orders of magnitude: the entropy is overestimated and the correlations
are underestimated. This is discussed in detail in all the books.
Books: Doi & Edwards, Wiegel
The seminal article of S.F. Edwards in 1965 was the first application
of field-theoretic methods to the physics of polymers. To insert interactions
into the Wiener distribution, we take sum over the two-body interactions
to the continuum limit
.
This formalism is rather complicated and not much can be done by hand.
One possible simplification is to consider an excluded-volume (or
self-exclusion) interaction of Dirac delta function form, which prevents
two monomers from occupying the same point in space:

The advantage of this is that a simple form is obtained in which only
the second virial coefficient
is taken into account. The
expression for the distribution is then

With expressions of this sort, one can apply standard field-theory/many-body
methods to evaluate the Green's function and calculate observables.
This is more advanced and we will not be going into it.
05/07/2009
Scattering and Polymer Solutions
edit
Materials can be probed by scattering experiments, and for dilute
polymer solutions this is one way to learn about the polymers within
them. Laser scattering requires relatively little equipment and can
be done in any lab, while x-ray scattering (SAXS) requires a synchrotron
and neutron scattering (SANS) requires a nuclear reactor. We will
discuss structural properties on the scale of chains rather than individual
monomers, which means relatively small wavenumbers. It will also soon
be clear that small angles are of interest.
Sidenote
Modeling the monomers as points is reasonable when considering probing on the scale of the complete
chain.
If we assume that the individual monomers act as point scatterers (see (8)) and consider a process which scatters the incoming wave
at
to
, we can define a scattering
angle
and a scattering wave vector
(which becomes smaller in magnitude as the angle
becomes
smaller). We then measure scattered waves at some outgoing angle for
some incoming angle as illustrated in (9), where in fact many chain
scatterers are involved we should have an ensemble average over the
chain configurations (which should be incoherent since the chains
are far apart compared with the typical decoherence length scale).
All this is discussed in more detail below.
Sidenote
For this kind of experiment to work with lasers or x-rays, there must
be a contrast : the polymer and solvent must have different
indices of refraction. X-Ray experiments rely on different electronic
densities. In neutron scattering experiments, contrast is achieved
artificially by labeling the polymers or solvent – that is, replacing
hydrogen with deuterium.
Within a chain scattering is mostly coherent such that that the scattered
wavefunction is
.
The intensity or power should be proportional to
).
If we specialize to homogeneous chains where
, then

This expression is suitable for a single static chain in a specific
configuration
. For an ensemble
of chains in solution, we average over all chain configurations incoherently,
defining the structure factor
:

The normalization is with respect to the unscattered wave at
,
. Note that in an
isotropic system like the system of chain molecules in a solvent,
the structure factor must depend only on the magnitude of
.
Inserting the expression for
into the above equation gives

We now switch to spherical coordinates with
parallel
to
with the added notation
.
Since in these coordinates
,
we can write


The gyration radius and small angle scattering
edit
For small
(which at least in the elastic case implies small
),
we can expand the above expression for
in powers
of
to obtain

The last equality is due to the fact
.
If the scattering is elastic,
and

With this expression for
in terms of the angle
,
the structure factor is then

From an experimental point of view, we can plot
as a function
of
and determine the polymer's
gyration radius
from the slope.
The approximation we have made is good when
,
and this determines the range of angles that should be taken into
account: we must have
.
For laser scattering usually
(about enough
to measure
) while for neutron scattering
(meaning we must take only very small angles into account to measure
, but also allowing for more detailed information about correlations
within the chain to be collected).
Debye scattering function
edit
Around 1947, Debye gave an exact result (the Debye function )
for Gaussian chains:


At the limit where
we can expand
around
, yielding the
limit we have encountered earlier.
For
,
.
Sidenote
Another way to observe GW behavior is to use a
-solvent.
This also works very well for non-Gaussian chains in non-dilute solutions,
where a small percentage of the chains is replaced by isotopic variants.
This gives an effectively dilute solution of isotopic chains, which
can be distinguished from the rest, and these chains are effectively
Gaussian for reasons which we will mention later. An example from Rubinstein is neutron scattering from PMMA as done
by R. Kirste et al. (1975), which fits very nicely to the Debye function
for
. In general, however, a SAW in a dilute
solution modifies the tail of the Debye function, since
and
for a SAW.
The structure factor and monomer correlations
edit
Consider the full distribution function of the distances
.
This is related to the correlation function for monomer
:

This function is evaluated by fixing a certain monomer 