(1)Characterization and the Solution State:
Polymers can not display a gas or vapor state but polymers can dissolve in some solvents which simplify determination of molecular characteristics since isolated molecules can be observed. Polymers are rarely used in the solution state with some notable exceptions in the oil drilling and additive fields and a few other cases such as adhesives. Characterization of polymers must involve measurement of the primary features of importance to predicting properties:
Chemical Composition Techniques common to organic chemistry: IR Spectroscopy, NMR Spectroscopy, Raman Spectroscopy, UV spectroscopy to a limited extent and mostly for additives.
Chain length Techniques somewhat specific to Polymers: GPC, x-ray and neutron scattering, light scattering, osmotic pressure
Dynamic Features Techniques specific to Polymers: DMTA, Dynamic torque rheometer, Dynamic dielectric thermal analysis
Transitions Techniques common to metals and ceramics but generally lower temperature: TGA, DSC, XRD
Crystallographic structure Mixture of Techniques reflecting hierarchy of structure in polymer crystals: XRD, SAXS/SANS, SALS, TEM, SEM, Optical Microscopy, AFM
Distribution of properties:
Generally polymers display a distribution of properties rather than a single value. This is true of the melting point, glass transition, modulus. Many of these distributions are base in a distribution in the molar mass of the polymer chains. In order to understand polymers it is important to understand population distributions in general and specific measures of molecular weight distributions that have been adopted by the polymer community. NIST web page on distribution functions.
Flory Most Probable Distribution
What is a distribution function.
A distribution function describes how a population of items is distributed. Distributions can be in terms of number, surface area, volume, mass or other measures depending on the application. For instance particle size distribution in catalysis is generally given as the surface area distribution with size. This is related to the number distribution by multiplying the number distribution by R2 and normalizing the distribution. A normalized distribution is a distribution multiplied by a constant that makes the integral of the distribution equal to 1. There are a number of functions that describe common distributions, the most popular being the Gaussian or Normal distribution which describes a random distribution through a mean, maximum probability and standard deviation or breadth of the distribution. The Gaussian distribution then is a three parameter distribution function which is basically the smallest number of parameters needed to describe a symmetric distribution. (A symmetric distribution has the same values for the same distances above and below the mean.) It is possible to construct an asymmetric distribution commonly seen in a wide variety of situations by using a Gaussian distribution on a logarithmic scale. This is called the log normal distribution and it is by far the most commonly observed of asymmetric distributions. Another asymmetric distribution functions with 3 parameters is the Schultz distribution. The Lorentzian distribution is asymmetric and requires 4 parameters with a free parameter describing the asymmetry of the distribution. Generally, we consider that molecular weights are distributed with a single mode (single peak in the distribution function) and a Gaussian number distribution.
Moments of a Distribution:
A distribution function can be used to obtain moments by integration. For instance the number average can be obtained from the number distribution of chain lengths, P(n), through,
(1)
where the denominator is 1 for a normalized number distribution. Higher order moments are obtained through integration of higher powers of n,
(2).
The n'th root can be taken of these moments to yield a comparable value to the number average in terms of units. The higher order the moment the higher the value for a given distribution, so the square root of the second order moment is almost always larger than the first order moment.
A common method to match the units of the number average is to use the ratio of two moments separated by one multiple of n. For instance the weight average molecular weight is the ratio of the second moment to the first,
(3).
this is termed the weight average since n is proportional to the mass of the chain and nP(n) is the weight distribution function.
Since higher moments of the distribution have higher values for most distributions, the ratio of a higher order moment to a lower order moment can be a useful measure of polydispersity. For polymers nw/n1 defines a unitless polydispersity index for molecular weight, PDI,
(4).
The polydispersity index increases from 1 but does not have an upper limit. Typically commercial polymers display a PDI between 2 and 20 depending on the synthesis method with nw of about 100 kDa.
In addition to n1 and nw higher order moments are often reported such as the z average, n3/n2 and the viscosity average which is a fractional moment for most polymers.
The polydispersity index is directly related to the standard deviation of the molecular weight distribution, s,
(5).
through,
(6).
Determination of molecular weight:
Molecular weight for a polymer is generally determined using gel permeation chromatography, GPC as described in the above link. A GPC curve for a narrow distribution polyethylene standard sample from the National Institute of Standards and Technology (NIST) and a commercial HDPE sample form Equistar are given in the linked Excel file. Notice that the commercial sample is bimodal while the standard sample is monomodal and of narrower distribution. You should determine the moments of these two distributions.
Other average molecular weights used in polymer science,
z-average
Viscosity Average
w
L. H. Sperling Web Page on GPC.
Modeling of Molecular weight distributions.