Standard Methods of Molecular Weight Measurement: Part 1
2.1 Membrane Osmometry
A photograph and a schematic of a traditional membrane osmometer are shown below;
this design is physically and conceptually simple, with the osmotic pressure  of a
polymer solution determined by the height difference h established between a vertical
compartment containing the solution and a second vertical compartment containing the
corresponding pure solvent. The two compartments are both open to atmosphere, so the
pressures exerted at their upper surfaces are equal and exactly atmospheric. A solventpermeable but solute-impermeable membrane separates the compartments at their
common base. The indicated cathetometer, essentially a viewing scope, offers an
accurate means for measuring h. In an alternative design, the compartments are filled
with the same liquids and closed, with a sensitive pressure transducer monitoring the
pressure difference  between compartments. In either design, membrane area is
enhanced so as to hasten the equilibration of solvent between compartments.
[from P. Flory, Principles of Polymer Chemistry,
Cornell Univ. Press, 1978, p 277]
[E. Schröder, G. Müller, and K.-F. Arndt, Polymer
Characterization, Hanser, 1989, p 37]
Following standard practice for solution chemistry, solvent will be designated component
“1” and solute will be designated component “2”.
The osmotic pressure  of a solution of a nondissociating polymer can be related to the
chemical potential 1 of the solvent via
1 = 1o + RTlna1 + V1
o
where i is the chemical potential of pure solvent, a1 is the solvent activity, and V1 is the
solvent molar volume. The third term is a contribution traceable to the ‘extra’ pressure of

1

the solution, necessarily imposed at equilibrium to prevent the influx of solvent from
across the membrane. Immediately after the solvent and solution are placed in their
compartments, i.e., when this pressure is absent, solvent flows from the solvent
compartment into the solution compartment down the gradient of 1. This flow creates a
growing height difference between compartments, which at the membrane face, generates
a higher mechanical pressure at the bottom of the solution compartment than at the
bottom of the solvent compartment. This pressure difference is given by straightforward
hydrostatics as gh. When the pressure difference gets large enough, net solvent flow
stops and equilibrium is achieved. At equilibrium, the pressure difference between
compartments at the membrane is .
At equilibrium, the solvent chemical potentials on each side of the membrane are equal,
o
so 1= 1 . Thus,
RT
   ln a1
V1
The lowering of solvent activity a1 in a dilute solution can be expressed through a virial
expansion in solute concentration c,
 1

 ln a1 = - cV1  A2c  ...
M

If the solution is ideal (i.e., no interaction between solute molecules), the second virial
coefficient A2 is zero, and second term vanishes.

Combining expressions,

cRT
=
1
 A2 c  ...
Mn
This form suggests Mn be obtained by extrapolating  vs. c data to zero c.
The preceding equation did not require any assumptions about solute properties other
than nondissociation, and the method applies to small molecules and polymers alike. A
dissociating solute is one that splits into smaller molecules upon dissolution, usually with
a greater number of small molecules generated at lower concentration.
There are two major issues always encountered with this method. One is the slow
approach to osmotic equilibrium. Since  for higher M substances becomes smaller
with c fixed, corresponding to h of a few millimeters or less, the driving force for the
initial liquid flow through the membrane is extremely small (Note that the illustrated
design has a large membrane and small chamber depth perpendicular to the membrane,
geometrical characteristics designed to reduce equilibration time. If not well supported,
the membrane may “balloon”, or bulge, toward the solution compartment, and as a result,
the membrane sometimes breaks.)
In the design shown, for reasonable M, equilibration takes a few days for each c.
Approach to equilibrium is a decaying exponential in time. Various strategies to lower
the equilibration time have been implemented, all based on an externally applied pressure
2
drop intended to speed up solvent equilibration across the membrane. These strategies
can reduce measurement time to a few minutes.
The second issue is the semipermeability of the membrane. Nearly always, some
polymers penetrate through the membrane, in violation of the method’s theoretical
underpinnings. Further, tightening the pore size to reduce this problem raises the
equilibration time. Typical membrane materials are regenerated cellulose (saponified
cellulose acetate) for organic solvents and cellulose acetate for aqueous solvents.
A lesser issue is temperature control: small temperature differences can produce volume
changes in the two compartments larger than those associated with . To suppress
surface tension effects on the rise of liquids in the capillaries of the sketched design, the
capillaries must be large in diameter, making the sample size appreciable (~2 ml)
Munk (Introduction to Macromolecular Science, John Wiley & Sons, 1989) describes
membrane osmometry as “experimentally very demanding, very slow, and prone to a host
of errors.” Its use in the polymer community is nowadays rare; I’ve never seen a
presentation in which its application was mentioned, and instruments aren’t commerically
available in the U.S. It is described in most textbooks because the theory is simple and
rigorous, and the implications of this theory extend to other methods.
Limitations:
 High permeability through membrane pores establishes the low molecular weight
limit (~20,000 g/mol); configurational flexibility allows even larger polymers to
penetrate through the membrane eventually. Preferential leakage of smaller
polymers leads to systematic errors in measurements on broadly distributed
polymers
 The upper molecular weight limit (~100,000 g/mol) is established by the accuracy
of measurement of .
 Concentration effects are large, so c<<c*.
 A different theory is needed for polyelectrolytes, which can only studied at high
salt conditions
 Error in M is large, 10-30% for higher M.
3
2.2 Vapor Phase Osmometry (VPO)
With a nonvolatile solute, the vapor pressure of a solution is lower than that of the pure
solvent, a reduction dependent on the number of solute molecules per volume of solution.
Vapor pressure lowering arises for essentially the same reason that a pressure drop
develops between compartments of a membrane osmometer: solutes reduce solvent
activity. Argued differently, central to both experiments is an interface, either membrane
or solution/vapor surface, crossed by solvent but not by solute. The solvent flux across
this interface from solution side to solvent side is reduced by the presence of solute,
perturbing the balanced back-and-forth solvent flux across the interface at equilibrium.
To reestablish balance, the thermodynamic state on one side of the interface must change.
Vapor pressure is not easily measured by any direct means. There are several indirect
means, all of which must be calibrated. The sketch below shows a common VPO design:
1=syringe in rest position; 2=syringe applying a drop; 3=measuring probe with matched
thermistors or thermocouples at its inserted end; 4=detachable head containing heat
elements for syringes; 5=metal block for thermostating syringes; 6=glass beaker with
solvent reservoir and filter paper; 7=observation windows; 8= matched thermistors or
thermocouples; 9=thermostated aluminum block; 10=casing.
[adapted from E. Schröder, G. Müller, and K.-F. Arndt, Polymer Characterization, Hanser, 1989, p 49]
The central compartment is a glass vessel containing a reservoir of the pure solvent and a
wick to keep the headspace above the reservoir saturated in solvent vapor. Positioned in
this headspace are two matched thermistors or thermocouples. At the start of a
measurement, one of the thermistors or thermocouples is immersed in a droplet of pure
solvent and the other is immersed in a droplet of solution; these droplets are applied via
the syringes shown. Since the vapor pressure of the solution droplet is lower than the
4
saturated vapor pressure of the compartment, solvent will condense on this droplet,
releasing its latent heat of vaporization L (units: energy per mass). Consequently, this
droplet will heat up until the solution’s vapor pressure matches that of the neighboring
pure solvent droplet, establishing a slight temperature T difference. The equilibrium
temperature difference T between the thermistors or thermocouples obeys
T = -
RT 2
ln a1
LM1
where the subscript “1” again references solvent properties. With T of the order 0.1ºC or
less, even slight temperature fluctuations are troublesome. For most solvents, the most
convenient T for measurements
is about 60ºC below boiling. In this instrument design,

the continual condensation of solvent will dilute the solution droplet over time,
eventually causes T to drop. Correct readings are obtained in a modern instrument after
about 1 min of equilibration. The expression for lna1 of the last section leads to the
operating equation,
T
c
exp  1

= Ke   A2 c
M

2
where K th
e = RT /L is the theoretical value of the calibration constant. In reality, other
effects impact T, most of these lowering T. Among such effects are heat conduction
th
in the gas phase, thermal radiation, and heat conduction in the wires. Thus, K e is
exp
replaced, in practice by K e , an experimental constant determined by measuring in the
same solvent the value of T for a molecular weight standard (for an organic solvent,
usually benzil). As in membrane osmometry, the value of M for the unknown is obtained
by extrapolating measurements to zero c.
Commercial VPO instruments are available from 2 or 3 suppliers in the U.S., but the
method is not common, especially in the polymer community.
Limitations:
 There is no lower M limit to the method, although the solute must be nonvolatile,
which may restrict low M operation
 The high M limit (~10,000 g/mol) reflects difficulties in measuring extremely
small temperature differences.
 Impurities in the sample can have a big effect on measurements; cleaning of
mechanically sensitive thermistors or thermocouples is necessary between runs;
this cleaning may take longer than the measurement itself.
 Error is relatively small, of the order 5-10% under optimal conditions.
 Empirical calibration is necessary; the method is not really absolute.
5
2.3 Ebullioscopy (boiling point elevation), Cryoscopy (freezing point depression),
and Isothermal distillation
Alongside vapor phase and membrane osmometry, the methods listed in this section’s
heading complete the set of colligative property measurements for M; these methods all
assess the limiting change in solvent activity for a dilute solution of polymers. Such
methods balance solvent activity of the pure solvent with solvent activity of a solution by
applying on the solution either increased pressure under isothermal conditions or
increased temperature under isobaric conditions. The preceding discussions for
membrane and vapor phase osmometry explain how this balance is achieved in practice.
I have not found any discussion isothermal distillation in the polymer literature, so I will
not consider the method. From the name, I suppose that one measures the pressure drop
necessarily imposed on a solution to make a solution boil at the same temperature as the
pure solvent.
Flory [Principles of Polymer Chemistry, 2nd ed., Cornell Univ. Press, 1973, p 272]
compares the magnitudes of the temperature changes of ebullioscopy and cryoscopy with
the magnitude of the osmotic pressure change of membrane osmometry:
M
10,000 g/mol
50,000
100,000
Tb/c, C/(g/100ml) Tf/c, C/(g/100ml)
0.0031
0.0006
0.0003
0.0058
0.0012
0.0006
/c,
2
(g/cm )/(g/100ml)
25
5
2.5
Values for ebullioscopy (column 2) and cryoscopy (column 3) are given in the limit c0
for benzene, and the value for membrane osmometry (column 4) is given as the pressure
head (i.e., h) in centimeters for a liquid of unit density.
Assuming c of several g/100ml and M=10,000 g/mol, Tb or Tf approach the practical
measurement precision for liquid temperature, about 0.001ºC. Thus, these methods are
limited to M less than 10,000 g/mol. According to the last column, for c = 2.5 g/100ml
the same value of M produces a 10-cm pressure head. Assuming a height measurement
error of 0.01 cm, the sensitivity of osmometry methods is seen to enjoy an advantage of
about 103. This comparison explains why osmometry has led to the virtual demise of
ebullioscopy and cryoscopy.
Notwithstanding these arguments, cryoscopy is sometimes still rarely used in molecular
weight measurements for monomers or small oligomers, those with M<2-3x103 g/mol. I
could not find a description of how the method is practiced.
6
2.4 Static Light Scattering
Light scattering is familiar to everyone – scientist and nonscientist alike – but the reason
for such scattering is not much discussed in introductory chemistry or physics courses.
Here, basic concepts of light scattering are introduced in a first subsection (2.4.1),
followed in the same subsection by a short physical description of why the method offers
a means to measure M. The last two light scattering subsections will present derivation
of the method’s governing equations (2.4.2) and outline experimental practice of the
method (2.4.3).
Strong motivations for understanding static light scattering emerge from even a casual
reading of the table of M methods given in the last handout: static light scattering is the
most universal method for obtaining M. It not only offers M on an absolute basis,
differently than several other tabulated methods, static light scattering can be applied
across the broadest of M range. As a consequence, nearly all of the relative M methods
are calibrated with M standards established through light scattering. In recent years, the
cost and complexity of instruments have also dropped, making the method one of the
most accessible.
2.4.1 Light Scattering Concepts
Light scattering from any medium fundamentally arises from the presence of optical
heterogeneities, i.e. dispersed regions of refractive index (or stated differently, optical
polarizability) different than that of the homogenous background. For a dilute polymer
solution, the heterogeneities are individual polymer molecules and the background is
solvent. In other contexts, these heterogeneities might be individual gas molecules (e.g.,
the atmosphere) or dispersed droplets of liquid (e.g., clouds of water droplets in the
atmosphere).
Scattering is characterized by the redirection of light from an incident light beam such
that redirected light maintains the incident light’s wavelength. These are the conditions
of elastic scattering, a process in which the scattered entities (here, photons) maintain
their energy. Although a scattering medium may also absorb light or even fluoresce, light
scattering proceeds by its own distinct mechanism. When the heterogeneity size
approaches or exceeds the incident radiation wavelength, as sometimes occurs when light
is scattered by large polymer coils, the radiation scattered from one part of a
heterogeneity may interfere with the radiation scattered from other parts of the same
heterogeneity, creating complex interference effects at the position of a distant observer.
Or, for the same observer, the light scattered from different heterogeneities might
interfere. The intensity of scattered light from a polymer solution can be measured as a
function of direction of scattered light propagation, polymer concentration, temperature,
etc. to understand polymer size, structure, and molecular weight.
We are all familiar with light scattering, as the phenomenon has a strong influence on
how we “see” our world. For example, on a foggy night, the light from a car headlamp
strongly scatters away from the desired forward direction, decreasing illumination in
front of a car but increasing illumination to the side. In this case, the optical properties of
water, present as finely suspended droplets, depart from those of the neighboring air,
7
causing light incident on them to scatter. In another example, the blue color of the sky
manifests the scattering of sunlight by gas molecules in the upper atmosphere. The
optical properties of these molecules are in contrast to those of the vacuum that surrounds
them. In absence of this scattering, the daytime sky would be dark; light would be noted
only when looking directly at the sun. The blue color of the sky tells us that, of the
sunlight reaching the earth and visible to our eyes, blue wavelengths scatter most
strongly. When heterogeneities are larger than the wavelength of light, as with the water
droplets in a cloud in the sky, all wavelengths scatter about equally. Equal wavelength
scattering explains why clouds appear white.
The following sketch shows scattering of light from a medium containing many dispersed
solutes much smaller than the wavelength of light. For such systems, within the plane of
scattering (the xy plane of the bottom sketch), scattered light emerges uniformly in all
directions.
scattered
incident
transmitted
scattered
At a molecular level, light propagates as an oscillating electric field that imposes
oscillating forces on the electron clouds of solutes to which the light impinges. These
forces cause the solutes’ electron clouds to themselves oscillate, creating transient dipoles
that oscillate at the same frequency as the incident radiation. Just like a radio transmitter,
these oscillations of charge generate electromagnetic radiation of the frequency of their
oscillation. However, there is no reason that the emitted radiation from the oscillating
dipoles should propagate in the same direction as the incident beam, or indeed, in any
8
other preferred direction within the plane of scattering. Thus, scattering from a single
small solute is isotropic, propagating equally in all directions within this plane.
The scattering of light is sensitive to the optical frequency polarizability of the solutes’
electron clouds. “Polarizability” assesses the ease at which charge can be moved about
within a solute, and at optical frequencies, this property is directly related to refractive
index. Refractive index is the bulk manifestation of molecular level polarizability, and
mismatch of refractive index, or equivalently of polarizability, provides the contrast
mechanism for light scattering.
The polarizability of a homogeneous molecule or particle is proportional to its volume,
and scattered intensity is proportional to the polarizability squared. Thus, larger objects
scatter light much more intensely, approximately as object size raised to the sixth power.
A few stray “dust” particles of colloidal size will scattering much more light than all the
more common polymer molecules present in a dilute polymer solution.
If the scattering sites are very large, very dense, and/or highly mismatched in
polarizability with the surrounding environment (all factors that increase contrast), light
traversing a heterogeneous medium may be multiply scattered, the light redirected before
reaching an observer by interaction with many scattering sites. With significant multiple
scattering, little incident light is transmitted through the medium to emerge on the
opposite side along the original beam path. Multiple scattering makes a medium appear
turbid or milky. Indeed, the opacity of milk is due to multiple scattering of light from a
high concentration of dispersed fat droplets.
Why does light scattering from a polymer solution offer a means to measure M?
Being a little more precise than before, light is scattered from a solution according to the
mismatch  between optical properties of solute and solvent. This mismatch may be
positive or negative, reflecting which of the two components is more optically polarizable
(i.e., has the higher refractive index). However, the scattered intensity arising from
positive and negative mismatches are of equal magnitude, and so to first approximation,
scattered intensity must be proportional to 2, a dependence making the sign of 
unimportant. Further, the scattered intensity from volume Vs is proportional to the
number of scattering solutes contained in this volume. Ignoring the prefactor, these
physical arguments provide the following relationship for the scattered intensity It,
It ~  2
N
Vs
Because of the optical mismatch, solution refractive index n shifts from the pure solvent
value no as polymer is added. This shift is captured in the refractive index increment dn/dc,

dn
~
dc

9

N
Vs
c
On the right-hand-side, the numerator is the refractive index change produced by addition
to solvent of polymer at mass concentration c. Note that dn/dc varies as the first power of
, wheras It varies as the second: this difference is the entire key to M determination.
Knowing c and number density N/Vs, M can be found simply as the ratio
M =
N Ac
N 
 
Vs 
Combining these three equations,

dn 2
It ~   cM
dc 
In M measurement, a known value of dn/dc is combined with measurements of It across a
range c. We do this under conditions that suppress the impact of light interference. The
next subsection provides this
 light scattering analysis in a more exact manner, one that
offers the as-yet unspecified prefactor in the preceding equation.
2.4.2 Theory of Light Scattering
2.4.2a Vectorial Description of Light and Light Scattering
Light scattering as a method to determine M for polymers is best approached by
envisaging a collimated light beam incident on a dilute polymer solution. In treating the
polymer-light interaction, this beam is viewed as a transverse traveling wave, one that
oscillates in both position and time while propagating in a single, well-defined direction.
Oscillations of such a wave are orthogonal to the direction of propagation. Exploiting the
Euler identity exp(i)=cos+isin, the beam propagating in the solution can be written in
terms its oscillating electric field vector Ei,
Ei = Eo expi(t - ki  x 
where both Ei and Eo are perpendicular to the direction of wave propagation (i.e., to the
beam path), defined by the light’s wave vector ki . (Remember that only the real part of a
complex exponential is ultimately important when using the Euler identity to model
oscillation. Applying this knowledge, there is no need to write Re in front of the
exponential of the preceding equation, as sometimes done in mathematics textbooks.)
The magnitude of ki depends only on the light’s wavelength  in the solution,
2
k = k =

When such a beam is vertically polarized, nearly always the case for M measurements by
light scattering, Ei and Eo point back-and-forth out of the plane in which scattering is
monitored; the optical arrangement within this plane is shown in the following sketch.
10
2.4.2b Physical Description of the Light Scattering Event
When incident light interacts with a solute at position x within a solution, the light’s
oscillating electric field imposes an in-phase oscillation of the solute's electron density.
In the linear optical limit, and for an isotropically polarizable solute or scattering site, the
imposed oscillation generates a time-dependent dipole P proportional to the instantaneous
value of Ei,
P = E = Eoexp i(t - ki  x)
where  is the solute’s excess polarizability.
The excess polarizability conveys how easily a solute’s electron cloud is distorted by an
imposed electric field, the level of these distortions assessed relative to level of
distortions caused by the same field in the surrounding solvent. If s is the polarizability
of solvent and ´ is the polarizability of solute, then
 =  - s
to good approximation.
Note that in the cgs units system chosen for this presentation polarizability has
fundamental units of length3. Modeling a solute as a homogeneous molecular object,  is
proportional to solute volume, or stated different, the more solute electrons moved by the
field, the larger is the ensuing solute dipole. Also note that  can be either positive or
negative. In all instances, spatial fluctuation (i.e., heterogeneity) in polarizability is the
ultimate cause of light scattering. If  is zero, no light is scattered as there is no contrast:
such is the case when the refractive indices (equivalently, polarizability) of solute and
solvent match.
11
A key principle of electromagnetism is that accelerating charge generates electromagnetic
radiation. Driven by an incident beam’s electric field, the oscillating solute or scattering
site dipole – with it charge accelerating back-and-forth - will itself behave as an
electromagnetic source, propagating radiation of frequency  in directions away from the
original beam path. When the incident light is vertically polarized, light propagated in
this manner site is also vertically polarized. Static light scattering involves an elastic
interaction, with equal  for incident and scattered light.
To measure the intensity of scattered light from a solution, scattered photons are collected
with a photomultiplier tube or photodiode positioned at location r. Variations in r
typically correspond to variations in scattering angle , as shown in preceding sketch. For
some laboratory instruments,  variation is achieved by rotating a goniometer arm that
supports a single optical detector, but in other instruments, to avoid the need for rotation,
several detectors are simultaneously positioned at different values of . The propagation
direction of the scattered beam monitored at the detector is given r-x.
Bold-faced representations for Es, P, and Eo are now dropped, since each vector is
vertically polarized, i.e., has an orientation perpendicular to the scattering plane. The
scattered electromagnetic wave at  has a form calculated through Maxwell’s equations,
Es =
1 2P
rc 2 t 2
=-
E o 2
rc 2


exp it  k i  x  (r  x)
where r is the distance from solute or scattering site to detector (i.e., r is the magnitude of
r-x), and in this equation alone, c designates the speed of light in the scattering medium
(elsewhere,
 c represents mass concentration of solute). Just as for the incident light, the
scattered light's electric field at position x is in-phase with the oscillating dipole P at the
same location. As the light propagates from solute or scattering site to detector, it gains
the phase shift given by the last term in the exponential’s argument. The wave vector of
scattered radiation is designated ks, and the magnitude of this vector matches that of ki.
However, the two wave vectors have different directions. Defining the scattering vector
q as the difference between wave vectors,
q = ki - ks
and replacing c with the factor o/2no, where o is the wavelength of light in vacuum,
the formula for Es becomes
  4 2 no 2 
Es = 2  exp i(t - ks  r - q  x)
r 
 o 
The magnitude of q is readily found by trigonometry, qq (4/)sin(/2). In the
solvent medium, wavelength  is reduced from o in inverse proportion to the solvent
refractive index no.
2.4.2c Intensity of Scattered Light
The average intensity I of an electromagnetic wave is proportional to the time average of
the square of the amplitude of the wave’s electric field,
12
n  
I   0 2 0  Et 2
 8 
where <...> denotes time averaging. For sinusoidal E(t),
< E(t)2 > =
Eo 2
2
and the time-average intensity of the incident light Io becomes
Io =
no oEo 2
2
16
Likewise, the mean scattered intensity per dipole or scattering site can be written
noo  2 Eo 216 4 no 4 
Io16 4 2 no 4
Is =
 =
2 
2
2
8 
2r o4
r o4



This equation is the celebrated formula for Rayleigh scattering; in a slightly different
form, this equation was used in the first course handout.
The  dependence predicted by the Rayleigh equation explains why the overhead daytime
sky is blue; each gas molecule in the atmosphere acts as an independent scattering site,
with a polarizability different than the surrounding vacuum, and the more highly
scattered blue light has the shortest wavelength generated by the sun and readily
perceived by the human eye. The same argument explains why the color of the directly
observed sun is shifted toward the red, particularly when the sunlight must penetrate a
hazy sky. Scattering during passage of the sunlight through the atmosphere has
preferentially removed the shorter wavelengths from transmitted light.
2.4.2d Light Scattering from a Distribution of Scattering Sites
The total intensity It of light scattered from N identical, point-like solutes dispersed
randomly in a scattering medium is simply the sum
N
16 4 2no4
1
r2 o4
It =  Is = NIs = Io N
Constructive and destructive interference effects of the scattered radiation exactly cancel
for this important special case, so they can be ignored here.
As shown in this subsection’s first sketch, the solution region both illuminated by the
laser and “seen” by the detector is called the scattering volume Vs. For most instruments,
this volume varies as 1/sin. The same  dependence thus holds for N, and It. In the
absence of constructive or destructive interferences, scattering is isotropic within the
scattering plane, light propagated equally in all directions by scattering.
13
The Rayleigh ratio or Rayleigh factor R, sometimes called the reduced scattered
intensity, is defined to eliminate trivial geometric factors and normalize the scattered
intensity with respect to the incident intensity:
R =
It r2
16 42 no4 N
=
IoVs
o 4
Vs
Note that R has units of inverse length.
2.4.2e Interference Complications
As stated several times, for randomly positioned and point-like molecules R is
independent of . Unfortunately, conditions necessary for such simple scattering are
rarely met in real polymer experiments.
Even when solutions are dilute, because polymers attract or repel each other, chains are
not randomly positioned except under special thermodynamic conditions (in a ‘theta’
solvent). Because of different path lengths, and thus different phase shifts, accumulate
during passage to detector, light scattered from different polymers can constructively or
destructively interfere at the detector, affecting the measured intensity, as sketched
below. Interference clearly depends on , and at low  and in a good solvent, scattered
intensities are reduced, i.e., destructive interference dominates.


Also, since polymer chains are usually large compared to o, the assumption of point-like
solute size is poor. As sketched second, interference at the detector occurs for light
scattered from different parts of the same molecule.
14
The two neglected effects, to be revisited later, cause intraparticle and interparticle
interference of the scattered radiation, and as a consequence, R for dilute polymer
solutions does not accurately follow the Rayleigh equation except at very low  and c,
where these interferences are lessened or eliminated.
Note that the factor2 causes scattering of small spherical particles to be proportional to
the sixth power of their radius; this dependence explains why light scattering is useful for
the characterization of polymer chains only in a solution free of large particulates or dust.
2.4.2f Refractive Index Measurements to Determine 
To ascertain M from R requires knowledge of , a parameter sensitive to the chemistry
of both polymer and solvent, reflecting their electronic structures. As originally proposed
by Debye, by taking a set of refractive index measurements on polymer solutions varying
in c,  can conveniently be obtained via the Lorenz-Lorentz formula. Derived for a
suspension of small, randomly positioned spheres, this formula provides the refractive
index n for such a suspension in terms of the solvent refractive index no and the product
of the number of spheres per volume N/V with the excess polarizability per sphere ,
N 4
n 2  no 2
=
V 3
2no 2  n 2

Although derived for a dilute suspension of optically isotropic spheres with diameters
small compared to the wavelength of light, the Lorenz-Lorentz formula works reasonable
well for polymeric solutes, modeling each chain as a linear assembly of point-like
polarizable scattering units, each surrounded by a large sea of solvent.
Applying the Lorenz-Lorentz equation to a polymer solution and redefining N/V as the
number density of polymer chains and  as the excess polarizability per chain is
convenient; the redefinition doesn’t change the product (N/V), the key parameter of the
Lorenz-Lorentz formula. For c low enough to restrict n to values near no, an expansion of
n in c can be truncated at the second term,
n = no +
dn 
c + ....
dc 
[remember that c here is the concentration (mass/volume), not the speed of light]. Using
the mathematical approximations below,
2
dn 
2
n  no 2 + 2no
c and n + 2no 2  3no 2
dc 
and substituting into the Lorenz-Lorentz formula,
dn c
dc 
 =
N
2 no
V
15
an experimentally useful expression for finding the product N/V.
2.4.2g Governing Equations for Light Scattering in the Absence of Interference
Inserting the preceding formula into the Rayleigh scattering equation and substituting M
for the parameter combination NAcVs/N,
R =
4 2 no2 dn 2
cM
o 4 N A  dc 
Defining an optical constant K conveniently combines several of the fixed parameters
into a single prefactor
4 2no2 dn 2
K =
o4 N A dc 
permitting a simple form
Kc
1
=
R
M
THIS IS THE KEY EQUATION FOR POLYMER MOLECULAR WEIGHT
MEASUREMENT BY STATIC LIGHT SCATTERING. IT SHOWS THAT THE
INTENSITY OF SCATTERED LIGHT FROM A SOLUTION IS PROPORTIONAL
TO THE PRODUCT OF c AND M. THE PROPORTIONALITY CONSTANT K IS
A SYSTEM-SENSITIVE PARAMETER INDEPENDENT OF c AND M, AND THE
VALUE OF K DEPENDS ONLY ON KNOWN EXPERIMENTAL PARAMETERS
AND THE SYSTME’S REFRACTIVE INDEX INCREMENT, WHICH CAN BE
MEASURED SEPARATELY, CALCULATED BY THEORY, OR READ FROM A
TABLE. THIS EQUATION ASSUMES NEGLIGIBLE INTERFERENCE OF
SCATTERED LIGHT, A CONDITION EXPERIMENTALLY REALIZED IN THE
LIMIT OF SMALL  AND c.
In experimental practice, background scattering due to solvent density fluctuations,
ignored in the preceding analysis but often significant, can be removed by subtracting the
solvent Rayleigh ratio Rsolv from the solution Rayleigh ratio R before doing any of the
analysis described
R = R  Rsolv
The parameter R then replaces R in all previous formulae.
The unknown solvent Rayleigh ratio Rsolv can be determined by calibrating the apparatus
with a solvent such as toluene for which Rsolv is known; assuming the scattering volume
is fixed (an assumption discussed later), the ratio of measured scattered intensities for the
two solvents at any angle is just the ratio of their Rayleigh ratios.
16
2.4.2h Interference Corrections
Returning to the interference effects that complicate scattering from real polymer
solutions, it should be understood that, except at theta conditions, polymer chains in a
solution always interact with each other. If the interactions are repulsive, as in a good
solvent, the chains will tend to avoid close approach. Since interchain separations for this
condition are no longer random, but statistically biased toward a finite mean separation,
destructive (and possibly) constructive interference of the light scattered by different
chains occurs.
The direct cause is easy to understand: enhanced interchain spacing increases phase
differences between light waves scattered from neighboring chains. The effect is large
enough to allow determination of the second virial coefficient A2 through measurements
of c-dependent scattering intensity at zero :
Kc
1
=
1  2A2 Mc  ...
R
M
A2 is positive for polymer chains in a good solvent, so intrachain interference causes a
decrease in scattered intensity.
Next, consider that light scattered from different parts of a single molecule may interfere.
Using the same interference arguments as the previous paragraph, larger chains – those
with their scattering sites distributed over a larger volume - will scatter less light than
smaller chains of equal M. Although conceptually similar, this phenomenon is distinct
from the one outlined in the previous paragraph, being rooted in intrachain rather than
interchain interference.
However, the method of analysis is similar.
In the
intramolecular case, the scattered intensity will depend on scattering angle , or
equivalently, on the scattering vector magnitude q:
2

Kc
1  Rg 2
=
1 
q  ...
R
M 
3



where Rg2 is the mean-square radius of gyration for chains of molecular weight M. At
low q (i.e. low  or large ), this limiting equation is universal; it holds irrespective of
solvent condition or solute type. At high q, according to theory the expansion must be
modified to account for details of solute structure, which add terms in higher powers of q.
Fortunately, for reasonable M, these corrections are negligible.
When the two preceding equations are combined, a realistic equation for scattering from
a dilute polymer solution results:
 Rg 2 2

Kc
1
=
1  2A2 Mc  ...1 
q  ...

R
M
3




This formula is known as the Zimm equation, after the first scientist who applied it to
polymer solutions in about 1945. A 'Zimm plot' displays values of Kc/R against
sin2(/2)+k´c, where curves of constant  and c are fitted by lines, which are then
17
extrapolated to their zero values (i.e., a curve of constant  is extrapolated to c=0 and vice
versa). k´ is an arbitrary constant chosen to spread the data and make graphical
extrapolations easier. Extrapolating constant  data to zero c generates a line of the form
2
Kc
1  Rg 2 
=
1 
q 
R
M 
3



whereas extrapolating constant c data to zero  generates a line of the form
Kc
1
=
1  2A2 Mc
R
M
The two extrapolated lines meet at the y-axis, where the intercept equals the reciprocal of
M. Such extrapolations are easily done in software.
2.4.2i Comments on Derivations
Students find derivations of the equations for M measurement by light scattering
challenging. Actually, the mathematics are simple, and the main difficulty is keeping
track of the large number of symbols/variables. Further, although partial derivations are
presented in even elementary polymer textbooks, most textbook discussions contain
errors. The errors have several causes: (1) confusion is applying CGS vs. SI unit systems
(leading to errors of factors of 4), (2) complexity in describing scattering when the
incident beam is unpolarized (an unnecessary complication that doesn’t conform to
modern experimental practice), and (3) deriving the equations in vacuum and then trying
to transform these equations in an ad hoc manner to the solution environment.
18
As noted in the first handout, modeling polymers as linear assemblies of point-like,
isotropic scattering sites requires a deeper justification than presented here. The main
argument against providing such a justification in these course notes, other than physical
and mathematical complexity, is that the chosen, simplest depiction actually works in real
life. In discussions of depolarized light scattering from polymer solutions, a better
polymer scattering model is needed; fortunately, depolarized scattering doesn’t
significantly influence actual M measurements.
2.4.3 Practice of M Measurement by Static Light Scattering
As the preceding discussions of light scattering demonstrate, raw light scattering data for
the measurement of M are typically obtained in the form of It versus  and c, all other
experimental parameters held constant. In modern instruments, the light source is a
vertically polarized laser, as assumed.
The three parameters that define proportionality between R and It (see page 14) are
almost never measured. Instead, they are bypassed by taking It measurements on a pure
solvent for which R is known. The most common calibrating solvent is toluene. Unlike
from polymers, light scattered from a simple solvent has comparable polarized and
depolarized components. Thus, the correct value of R for the calibrating solvent must
be chosen, a value typically given the symbol R,v, the added subscript implying a value
particular to vertically polarized incident radiation with all polarizations of scattered
radiation collected. Also, solvent scattering is isotropic in the scattering plane, so
calibration can be established at any value of .
This preceding calibration of reduced intensity has a potential, yet rarely mentioned
drawback: the size of the scattering volume is affected by the solvent refractive index. If
the refractive index of the solvent for polymer is not the same as the refractive index of
the solvent for calibration, Vs for the two are different; the deduced calibration constant
must therefore be corrected. Unfortunately, there is no universal correction, as the effect
depends on the way the optical elements are arranged. In an optical system with both a
cylindrically symmetric incident beam and a detection path that has similar cylindrical
symmetry (i..e., a detection optical path defined through ordinary lenses and circular
slits), R is corrected by multiplying by the ratio (no/nc)2, where nc is the refractive
index of the calibrating fluid.
Most modern instruments vary  over an angular range of roughly 20º to 150º by use of a
goniometer (a device that rotates the detector about the sample) or by the fixing of
multiple detectors at discrete angles across this range. Both “flare” from imperfections of
optical elements and diffraction from edges of slits/pinholes make it difficult to reduce 
below 20º. (We have placed optical elements in a bath fluid to make measurements at
angles as low as 10º, but this strategy is extremely inconvenient.)
19
Assuming the optical arrangement similar to the one described above (cylindrical
incident and detection paths), Vs is proportional to (sin)-1. Thus, for an isotropic
scatterer, such as the calibrating fluid, the product of Is and sin is constant. An “Isin”
plot presents this product vs. . Constancy of the product across all  within 2% must
be obtained before attempting measurements for high molecular weight polymers if M
accuracy within 5-10% is desired. Errors in “Isin” affect the accuracy of the
extrapolation of data to zero .
Source lasers produce light of optical wavelength (450 nm<o<800 nm). One wants as
much light intensity as possible without appreciable heating of solution. It is generally
desirable to avoid wavelengths that overlap polymer adsorption bands, but some
adsorption is acceptable (heating or degradation are the problems here).
Data are most often interpreted through Zimm plots, but depending on the polydispersity,
other plotting techniques are sometimes preferred (Berry plots). On a Zimm plot, lines of
constant  and constant c should be linear. Nonlinearity of the constant c lines suggests
alignment difficulties, although the early literature gives other, far less likely reasons:
form factor effects and polydispersity. By static light scattering, one cannot determine
polydispersity or measure the form factor for synthetic polymers. Nonlinearity of the
constant  lines can have several sources: concentration-dependent aggregation or
incorrect concentrations the most common.
Issues:
 There is no significant upper M limit to static light scattering. Although often
suggested otherwise, nonlinear form factor (intramolecular interference) effects
for dissolved coiled polymers are not significant until Rg is of the order 250 nm.
The product of the scattering vector q with Rg for such molecules can be as large
as 5 before these nonlinearities become manifested in Zimm plots as curves
rather than lines fitting data at constant c. The difficulty with such samples is in
solution preparation – it is difficult to filter or sediment dust particles
preferentially to polymer when the polymer molecules themselves are very large
(>100 nm). Also, large polymers are susceptible to flow-induced degradation
during filtration through small pore membranes.
 Dust is a continual source of error. Remembering that spherical particles scatter
as the 6th power of their radius, a few large dust particles can totally overwhelm
the signal of a much more concentrated but smaller polymer. Water is a
troublesome solvent, as its high surface tension “traps” dust particles from the
air.
 There is no significant lower molecular weight limit to static light scattering.
Small molecules don’t scatter much light, so it becomes difficult to “see” them
with ordinary laser illumination in ordinary optical arrangements. However, such
samples can be studied at higher mass concentrations than samples of higher
molecular weight. Even with small lasers (2 mW), the molecular weights of
small molecules (sugar in water) have been measured in special scattering
devices. In an ordinary commercial scattering apparatus, molecules with M
20





above 5,000 to 10,000 g/mol can be studied with relative ease, although Rg can’t
be determined until M is of the order 50,000 g/mol.
dn/dc is not always easy to measure, and its value sharply affects M. Most
tabulated values of dn/dc correspond to different  than used in modern light
scattering instruments. Fortunately, dn/dc is not much wavelength dependent for
solutions that don’t absorb visible light, so it is usually permissible to use these
values with negligible error. Modern refractometers for chromatography,
although not designed for this purpose, offer an excellent way to determine
dn/dc. One simply runs a solution of known c through the refractometer and
measures the change in n.
Errors in the light scattering measurement of M are usually traced to dust
contamination. Assuming no dust (almost never possible in water), other
common errors are inaccurate dn/dc, inaccurate c (mainly due to solvent
evaporation), and noise in the scattering signal. In the ideal case, M can be
determined to within 5%.
The value of Rg can be determined to higher accuracy than M since no
extrapolation is necessary and the value is independent of dn/dc. For reasons I
don’t understand, values of A2 seem less accurate, although I haven’t seen a
formal comparison between methods.
Volume of sample: Traditional goniometer-based scattering instruments require
on the order of 1 ml of sample. Newer fixed angle instruments can handle
sample volumes of the order 10 l.
One can estimate the value of dn/dc by dividing the difference between the
refractive index of pure polymer and the refractive index of pure solvent by the
density of pure polymer. One can get an even better indication of dn/dc by using
standard, but more complicated formalisms relating refractive index to molecular
structure (these are outlined in undergraduate physical chemistry books).
Refractive indices for organic compounds vary from about 1.35 to 1.60, so the
typical range of dn/dc is 0.05 to 0.2 ml/g. Since the calculation of M depends on
(dn/dc)2, one wants to maximize dn/dc, thereby maximizing contrast. If one is
not careful, it is possible to pick systems for which dn/dc is near zero, thus
making the M measurement widely inaccurate.
21
2.5 End Group Analysis
This name is highly descriptive of the methods encompassed, each providing the number
average molecular weight of a polymer sample from an experiment that “counts” the
number nend of end groups in a known polymer mass m,
Mn =
zm
nend
where z is the known number of end groups per chain. For a linear homopolymer with
two identical end groups, z=2; if the two ends of such a polymer are not identical and the
counting method detects justone end, then z=1; and finally, if the polymer is regularly
branched with identical end groups, then z>2. Obviously, end group analysis is
inapplicable to situations when z is not known. Thus, some knowledge of sample
chemistry is a requisite to the application of the approach.
There are many ways to count end groups, which can generally be grouped into three
classes: chemical methods, radiochemical methods, and physical methods.
Chemical methods rely on the complete reaction of reactive end groups with a low
molecular reagent, the consumption of this reagent measured by a standard “wet
chemistry” analytical method for small molecules. To perform the reaction efficiently
(i.e., to react every end group), the polymer is usually present in dilute solution during
reaction, the analysis of products done in the same medium. End groups that can be
assess in this manner include -COOH, -NH2, and –SH. Indirect chemical methods allow
for assessment of –OH.
The difficulty with all end group analyses, chemical methods included, is quantitative
detection of end groups, which become less concentrated as the degree of polymerization
rises. At some M, the analytical method for the end group loses sensitivity, placing a
limit on the values of M that can be obtained. For chemical methods, the upper limit is
usually about 10, 000-50,000 g/mol, although the exact limit depends on the chemistry
employed.
Some examples are the colormetric titration of –COOH end groups of polyesters with
cationic dyes or the potentiometric titration of -NH2 end groups of polyamides with
strong acids (or the functionalization of these groups with fluoro-2,4-dinitrobenzene
which leads to a colored polymer product in an appropriate solvent).
Another problem with chemical methods is completeness of the reaction of the end
group. If incomplete, the methods find too little of the end group, and the calculated M is
too high. Thus, in my experience, chemical methods always suffer from systematic
overestimation of M.
Radiochemical methods, which attach radioisotopes to the chain end, are usually more
sensitive than chemical methods because radiation detection is more sensitive than
chemical detection. Values of M up to 106 g/mol have been obtained, although
techniques are more specialized and expensive.
22
Physical methods typically detect nend by spectroscopy, as exemplified by UV/Vis, IR, or
NMR. The first two of these spectroscopies, UV/Vis and IR, are predicated on high
molar absorption coefficients for end groups. Upper limits to the measurement of M are
in the range of chemical methods, but the errors in M in the applicable range are
sometimes higher, since thee adsorption coefficients may be affected/interfered by
contaminants and scattering.
NMR detection of end groups is nowadays by far the most popular approach to end group
analysis. Most polymers have end groups with distinct NMR signatures that are readily
assessed. However, with ordinary effort, the detection sensitivity is not so high as with
UV/Vis or IR, so the upper M limit is lower.
An ‘extra’ functionalization of native end groups to optimize the chemical,
radiiochemical, or physical detection of end groups is common. However, the
inefficiency of such functionalizations tends to reduce the number of end groups
detected, spuriously raising M. Because of the inefficiency of –OH end group
functionalization, results from indirect chemical methods may be presented only as
“hydroxyl value”, not M.
23