Lecture notes for the course given by Prof. David Andelman, Tel-Aviv University 2009
Original author: Guy Cohen, Tel-Aviv University
Introduction and PreliminariesEdit
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 ordered
form 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
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)
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.
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
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.
Soft 2D membranesEdit
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.
What is a polymer?Edit
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.
is the polymerization index.
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.
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:
There also exist condensation processes, by which chains unite:
where . A briefer notation, dropping the name of the
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:
||type of material
||liquid fuel/organic solvents
Types of polymer structuresEdit
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 matterEdit
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 SolutionEdit
Some basic models of polymer chainsEdit
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
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 radiusEdit
For the polymer chain of (5), obviously we will always have .
The variance, however, is generally not zero: using ,
In the freely jointed chain (FJC) model, there are neither correlations
between different sites nor restrictions on the rotational angles.
We therefore have ,
The mathematics are similar to that of a random walk or diffusion process, where in 1D .
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
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
This gives the gyration radius for an FJC:
Polymers and Gaussian distributionsEdit
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
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
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:
- 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 , .
Another way to show this follows, which is also extensible to other
distributions of the .
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 ,
We can replace the delta functions with ,
leaving us with
In spherical coordinates,
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
The integral is then
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 ).
Rigid and Semi-Rigid Polymer Chains in SolutionEdit
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:
which defines the persistence length . For the FRC
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)
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.
While an ideal chain has and a rigid rod has , in general polymer chains can have a scaling law . The power need not be an integer.
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 SpringsEdit
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
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 .
In spherical coordinates
we can solve the integral:
The Gibbs free energy (Gibbs because the external force is fixed)
and the average extension
The Langevin function
is also typical of spin magnetization in external magnetic fields
and of dipoles in electric fields at finite temperatures.
Polymers and Fractal CurvesEdit
Introduction to fractalsEdit
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 the ruler used to measure
it: the actual length at iteration is .
Another definition for the fractal dimension is
Linking fractals to polymersEdit
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 FunctionsEdit
Books: Doi & Edwards, F. Wiegel, or Feynman & Hibbs.
Local Gaussian chain model and the continuum limitEdit
This model is also known as LGC. We start from an FJC in 3D where
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
and , by
We can then obtain for the full probability distribution
where . describes
harmonic springs with connected
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 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 ,
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.
Functional path integrals and the continuum distribution functionEdit
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 mechanicsEdit
This equation for , is
very similar in form to the Schrödinger equation. To see this, we
can rewrite it as:
If we make the replacement ,
and this is identical
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
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
If we separate variables again with the ansatz
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
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.
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 stateEdit
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 WalksEdit
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
To make the calculation easy, consider a potential even simpler than
the 6-12 Lennard-Jones:
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 )
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
A connection between SAWs and critical phenomena was made by de Gennes
in the 1970's. 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: .
|| (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
Here is a local density such that its average value
If we neglect local fluctuations in , then
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.
Field Theory of SAWEdit
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.
Scattering and Polymer SolutionsEdit
The form factorEdit
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.
Modeling the monomers as points is reasonable when considering probing on the scale of the complete
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.
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 scatteringEdit
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,
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 functionEdit
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 , .
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 correlationsEdit
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 and counting
which other monomers are at a distance from it, averaging
over all chain configurations. If we now average over all monomers
, we obtain
Fourier transforming it,
The fact that the structure function is the Fourier transform of the
scatterer density correlation function is, of course, not unique to
the case of polymers.
At large , it can be shown (homework) that if
then . We can therefore
determine the fractal dimension of the chain from the large tail
of the structure factor (see table).