Chapter 5

Characterization of Molecular Weight

5.1 Introduction

With the exception of a few naturally occurring polymers (such as proteins and DNA), all polymers consist of molecules with a distribution of chain lengths. Compared to molecules such as water that have a definite structure (H₂O) and molecular weight (18), the polymer chains making up a sample consist of molecules with a range of molecular weights. For example, poly(ethylene glycol) or PEG is often used at a range of molecular weights, around 400 (which is actually just an oligomer of approximately nine repeat units), PEG is a viscous liquid, while with increasing molecular weight, PEG becomes waxy (around 600), and can be formed into a solid powder (around 3000). Because the molecular weight is strongly related to the mechanical and chemical properties of polymers, it is necessary to characterize the entire distribution quantitatively, or at least to define and measure average chain lengths or molecular weights for these materials. Extensive reviews are available [1, 2] concerning the effects of molecular weight and its distribution on the mechanical properties of polymers.

First, the molecular weight of a single polymer chain, M, is easily calculated, if the chain length, x, and chemical composition of the repeat unit (and thus the molecular weight of the repeat unit, M_r) are known:

(5.1)

Note that the terms chain length and degree of polymerization can be used interchangeably, and both are represented by x.

When considering a collection of polymer chains, an average molecular weight must be considered. However, the method for determining this average depends on how you count. In normal averaging (e.g., your grades in a class), you sum up each of the individual chain molecular weights (or scores) and divide by the number of chains (or assignments):

(5.2)

This is a perfectly valid way to calculate molecular weight, but we must consider statistics and distribution patterns to get a more accurate picture of how polymers behave, so this chapter will discuss different ways of calculating molecular weights, introduce distribution profiles, and discuss laboratory methods used to measure or estimate the average molecular weight of polymer samples.

5.2 Average Molecular Weights

Since nearly every collection of polymer molecules contains a distribution of molecular weights, a number-average molecular weight may be defined in an analogous fashion to Equation (5.3):

(5.3)

Although this equation appears complicated, if you work with it, you will see that it reduces to the common equation used to average grades, except that it gives the average length of all the chains in a polymer sample. Any analytical technique that determines the number of moles present in a sample of known weight, regardless of their size, will give the number-average molecular weight.

Rather than count the number of molecules of each size present in a sample, it is possible to define an average in terms of the weights of molecules present at each size level. This is the weight-average molecular weight, . places an emphasis on the larger molecular weight chains, which is particularly useful when considering the mechanical properties of polymers, as the larger chains contribute more to a polymer's strength than smaller-sized chains.

Consider a bowl of fruit that contains three apples, five oranges, and a bunch of grapes. If you consider each grape as a separate fruit, the average size (“number average molecular weight”) will be much closer to the size of a grape, but the majority of the mass in the bowl is due to the larger apples and oranges, so a weight-average size will represent that the larger fruits make up more of the mass. For a polymer, the weight-average molecular weight, , is

(5.4)

Analytical procedures that, in effect, determine the weight of molecules at a given size level result in the weight-average molecular weight. Thus, is greater than and would be closer to the size of the apples and oranges.

The number-average molecular weight is the first moment of the molecular weight distribution, analogous to the center of gravity (the first moment of the mass distribution) in mechanics. The weight-average molecular weight, the second moment of the distribution, corresponds to the radius of gyration in mechanics. Higher moments, for example, , the third moment, may also be defined and are used occasionally.

It is sometimes more convenient to represent the size of polymer molecules in terms of the degree of polymerization or chain length x, rather than molecular weight, as related by Equation (5.1). Thus, and can also be used to represent average polymer sizes, and

(5.5a)

(5.5b)

(5.5c)

where m = molecular weight of a repeating unit (also sometimes called M_r)

= number-average degree of polymerization or chain length

= weight-average degree of polymerization or chain length

(These relations neglect the difference between the end groups on the molecule and the repeating units. This is perfectly justifiable in most cases since the end groups are an insignificant portion of a typical large polymer molecule.) It is also possible to use these relations for copolymers, but m should be modified to reflect the average composition of a repeat unit in the chain. In terms of chain lengths, Equations (5.3) and (5.4) become

(5.3a)

(5.4a)

It may be shown that ≥ ( ≥ ). The averages are equal only for a monodisperse polymer, where all of the chains are exactly the same length. The ratio / = / is known as the polydispersity index, PI. PI is a measure of the breadth of the molecular weight distribution. A monodisperse sample would have a PI of 1.0, but typical values range from about 1.02 for carefully fractionated polymers synthesized by anionic addition reactions to over 50 for some commercial polymers.

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Example 5.1 Measurements on two essentially monodisperse fractions of a linear poly(methyl methacrylate) (PMMA), A and B, yield molecular weights of 100,000 and 400 000, respectively. Mixture 1 is prepared from one part by weight of A and two parts by weight of B. Mixture 2 contains two parts by weight of A and one of B. Determine the weight- and number-average molecular weights of mixtures 1 and 2, as well as their weight- and number-average degrees of polymerization. The repeat unit for PMMA is CH₂C(CH₃)COOCH₃, which has a molecular weight, m, of 100.

Solution. For mixture 1:

For PMMA, m = 100, thus

from Equations (5.5b) and (5.5c):

For mixture 2:

Again, for PMMA, m = 100, thus

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Example 5.2 Two polydisperse samples are mixed in equal weights. Sample A has = 100,000 and = 200,000. Sample B has = 200,000 and = 400,000. What are and of the mixture?

Solution. First, let us derive general expressions for calculating the averages of mixtures

where subscript i refers to the various polydisperse components of the mixture. Now, for a given component,

(5.6)

(5.7)

where (w_i/ΣW_i) is the weight fraction of component i in this mixture. In this case, let W_A = 1g and W_B = 1 g. Then

Note that even though the polydispersity index of each component of the mixture is 2.0, the PI of the mixture is greater, that is, 2.25.

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5.3 Determination of Average Molecular Weights

Everything would be quite easy if we knew the molecular weight of each chain in a polymer sample, but this is quite difficult to determine. This challenge has led to the development of several different techniques that attempt to measure or at least estimate the molecular weight of polymers [3]. This section covers some of the more common ways to determine molecular weight, including an overview of the operation of the instruments and (hopefully) giving an appreciation for the associated advantages and limitations.

In general, techniques for determination of average molecular weights fall into two categories: absolute and relative. In the former, measured quantities are theoretically related to the average molecular weight; in the latter, a quantity is measured that is in some way related to the molecular weight, but the exact relation must be established by calibration with one of the absolute methods.

Another important way to categorize these techniques is the type by which the average molecular weight is determined: or (or a viscosity-average molecular weight, ). Another type of molecular weight, , molecular weight between crosslinks, is only applicable for polymer networks and is discussed in later chapters. The techniques covered in this chapter are listed in Table 5.1, along with the type(s) of molecular weight determined by each.

Table 5.1 Methods of Determining Average Polymer Molecular Weights [5]

Method	Type of Molecular Weight Determined
End group analysis
Boiling-point elevation
Freezing-point depression
Osmotic pressure
Light scattering
Ultracentrifugation (sedimentation)
Viscometry
Gel permeation chromatography	, , MWD, and PI

5.3.1 Absolute Methods

These methods allow the determination of the total number of moles of chains in a sample, N. If the total sample weight, W, is known,

(5.8)

5.3.1.1 End-Group Analysis

If the chemical nature of the end groups on the polymer chains is known (as in many condensation polymers), standard analytical techniques may sometimes be employed to determine the concentration of the end groups and thereby of the polymer molecules, giving directly the number-average molecular weight. For example, in linear polyesters formed from a stoichiometrically equivalent batch of monomers, there are, on average, one unreacted acid group and one unreacted –OH group per molecule. These groups may sometimes be analyzed by appropriate titration. If an addition polymer is known to terminate by disproportionation (see Chapter 9), there will be one double bond for every two polymer molecules, which may be detectable quantitatively by halogenation or by infrared measurements. Other possibilities include the use of a radioactively tagged initiator that remains attached to the end of each chain.

These methods have one drawback in addition to the necessity of knowing the nature of the end groups. As the molecular weight increases, the concentration of end groups (number per unit volume) decreases, and the measurement sensitivity drops off rapidly. For this reason, these methods are generally limited to smaller polymers in the range of <10,000.

5.3.1.2 Colligative Property Measurements

You may recall from an introductory chemistry course that addition of salts to a liquid can lead to boiling-point elevation or freezing-point depression. When a solute is added to a solvent, it causes a change in the activity and chemical potential (partial molal Gibbs free energy) of the solvent. The magnitude of the change is directly related to the solute concentration. For example, when pure water is boiled, the chemical potentials of the liquid and the vapor in equilibrium with it are the same. If, now, some salt is added to the water, it lowers the chemical potential of the liquid water. To reestablish equilibrium with the pure water vapor above the salt solution, the temperature of the system must be raised, causing a boiling-point elevation. In a similar fashion, the addition of ethylene glycol antifreeze depresses the freezing point of water.

Freezing-point depression, boiling-point elevation, and, a third technique, osmotic pressure, may be used to determine the number of moles of polymer per unit volume of solution and thereby establish the number-average molecular weight. The following are the relevant thermodynamic equations for the three techniques:

(5.9)

(5.10)

(5.11)

where

c = solute (polymer concentration), mass/volume

T = absolute temperature

R = gas constant

ΔH_v = solvent enthalpy of vaporization

ΔH_v = solvent enthalpy of fusion

ΔT_b = boiling-point elevation

ΔT_f = freezing-point depression

π = osmotic pressure

ρ = density

It is important to note that the thermodynamic equations apply only for ideal solutions, a condition that can be reached only in the limit of infinite dilution of the solute. For real situations, a dilute solution can be assumed in most cases if the concentration, c, is less than 1 wt%. It should also be noted that for these techniques to work, the polymer must be fully dissolved in a good solvent, whereas polymers precipitate in a bad solvent. If not dilute enough, polymer chains may interact in the solution (Figure 5.1), causing serious error in molecular weight estimation. In each of these equations, data collected for different concentrations can be plotted versus concentration, with the value at c = 0 used to determine .

Figure 5.1 (a) Dilute polymer chains that have no interchain interactions (though they may have intrachain interactions), and (b) concentrated polymer solution showing significant interchain interactions.

Freezing-point depression and boiling-point elevation require precise measurements of very small temperature differences. Although they are used occasionally, the difficulties involved have prevented their widespread application. For example, a Δ T_f of 0.0002 °C might be the only observed change for a polymer with molecular weight 25,000. Osmotic pressure, on the other hand, is the most common absolute method of determining . A schematic diagram of an osmometer is shown in Figure 5.2. The solution and solvent chambers are separated by a “semipermeable” membrane, one that ideally allows passage of solvent molecules but not solute molecules. The solvent flows through the membrane to dilute the solution. This is a natural consequence of the tendency of the system to increase its entropy, which is accomplished by the dilution of the solution. This dilution continues until the tendency toward further dilution is counterbalanced by the increased pressure in the solution chamber. At this point, the chemical potential of the solvent is the same in both chambers and the pressure difference between the chambers is the osmotic pressure, π. By making measurements at several concentrations, plotting (π/c) versus c and extrapolating to zero concentration, the number-average molecular weight is established through Equation (5.11).

Figure 5.2 Schematic diagram of an osmometer.

One of the major difficulties with membrane osmometry is finding suitable semipermeable membranes. Ordinary cellophane or modifications of it are commonly used. Unfortunately, if the membrane is sufficiently tight to prevent the passage of low molecular weight chains, the rate of solvent passage is slow and it takes longer to reach equilibrium. In practice, all membranes allow some of the low molecular weight polymer in a distribution to sneak through. Also, as the average molecular weight increases, π increases, so the measurement precision also decreases. These factors usually limit the applicability of osmometry to 50,000 < < 1,000,000.

High-speed, automated membrane osmometers monitor the fluid volume in one of the chambers and externally apply the osmotic pressure to the solution chamber to prevent flow. Since no flow through the membrane is necessary, they can reduce measurement time from days to hours or even minutes.

A related colligative technique is vapor-pressure osmometry. Two thermistors are placed in a carefully thermostatted chamber that contains a pure-solvent reservoir so that the atmosphere is saturated with solvent vapor. A drop of solvent is placed on one thermistor and a drop of polymer solution is placed on the other. Because of the solvent's lower chemical potential in the solution, solvent vapor condenses on the solution drop, giving up its heat of condensation, warming the solution drop relative to the pure solvent drop. In principle, the equilibrium ΔT is thermodynamically related to the molar solution concentration, thereby allowing calculation of . In practice, heat losses (mainly along the thermistor leads) require that the instrument be calibrated for precise results, really making it a relative technique (as opposed to absolute). On a routine basis, commercial instruments are probably limited to maximum values of 40,000–50,000 [4].

5.3.1.3 Weight-Average Techniques

The absolute techniques discussed to this point establish the number of molecules present per unit volume of solution, regardless of their size or shape. Other methods measure quantities that are related to the average mass of the molecules in solution, thereby giving the weight-average molecular weight. One of the more common of these is light scattering, which is based on the fact that the intensity of light scattered by a polymer molecule is proportional, among other things, to the square of its mass. A light-scattering photometer measures the intensity of scattered light as a function of the scattering angle (Figure 5.3). Measurements are made at several concentrations. By a double extrapolation to zero angle and zero concentration (known as Zimm plot) and with a knowledge of the dependence of the solution refractive index versus concentration, is established. This technique also provides information on the solvent–polymer interaction and on the configuration of the polymer molecules in solution, since the quantitative nature of the scattering depends also on the size of the particles. Light scattering is generally applicable over a wider range 10,000 < < 10,000,000. The weight-average molecular weight can also be obtained with an ultracentrifuge, which distributes the molecules according to their mass in a centrifugal force field.

Figure 5.3 Light-scattering experiment to determine of polymers in dilute solution.

It must be noted that for light-scattering and sedimentation (by ultracentrifuge) techniques (as well as most relative methods, below) to give an accurate measurement of molecular weight, dilute solutions are required. If two polymer molecules are entangled, they will appear as a single molecule and skew the results to higher molecular weights (that is why many techniques extrapolate to zero concentration).

5.3.2 Relative Methods

The molecular weight determination techniques discussed to this point allow direct calculation of the average molecular weight from experimentally measured quantities through known theoretical relations. Sometimes, however, these relations are not known, and although something is measured that is known to be related to molecular weight, one of the absolute methods above must be used to calibrate the technique.

A case in point is solution viscosity. It has long been known that relatively small amounts of dissolved polymer could cause tremendous increases in viscosity (ever made Jell-O®?). It is logical to assume that, other things being equal, larger molecules will impede flow more than smaller ones and give a higher solution viscosity. The solution viscosity will also depend on the solvent viscosity, temperature, solute concentration, as well as the particular polymer and solvent, because the interactions between the polymer and the solvent influence the conformation of the polymer molecules in solution and entanglements between the polymer molecules. In functional form, we may write

We can get rid of the effect of solvent viscosity by calculating the fractional increase in viscosity caused by the added solute (polymer) through the specific viscosity η_sp:

(5.12)

where η_r = η/η_s is known as the relative viscosity. Similarly, we can normalize for concentration by dividing the specific viscosity by concentration to get the reduced viscosity, η_red:

(5.13)

To get rid of the influence of entanglements on viscosity, we extrapolate the reduced viscosity to zero concentration to get the intrinsic viscosity [η]:

(5.14)

The intrinsic viscosity, then, should be a function of the molecular weight of the polymer in solution, the polymer–solvent system, and the temperature. If measurements are made at constant temperature using a specified solvent for a particular polymer, it should be quantitatively related to the polymer's molecular weight.

Huggins proposed a relation between reduced viscosity and concentration for dilute polymer solutions (η_r < 2):

(5.15)

Interesting enough, k' turns out to be approximately equal to 0.4 for a variety of polymer–solvent systems, providing a convenient means of estimating dilute solution viscosity versus concentration if the intrinsic viscosity is known. By expanding the natural logarithm in the definition of inherent viscosity, η_inh, into a power series it may be shown that an equivalent form of the Huggins equation is

(5.16)

where

Equation From (5.14) it is seen that an alternative definition of intrinsic viscosity is

(5.17)

Plots of the reduced and inherent viscosities are linear with concentration, at least at low concentrations, in accord with Equations (5.16) and (5.17), and have a common intercept, the intrinsic viscosity (Figure 5.4). Exceptions occur with polyelectrolytes, where the degree of ionization and, therefore, the chemical nature of the polymer changes with concentration.

Figure 5.4 Plot to determine intrinsic viscosity (Example 5.3).

Note that the intrinsic viscosity has dimensions of reciprocal concentration. For some strange reason, concentrations were traditionally given in g/dL (=100 mL), although g/mL is now becoming more common. In fact, the relative, specific, reduced, intrinsic, and inherent viscosities are not true viscosities, and do not have dimensions of viscosity. More appropriate terminology has been proposed, but not widely adopted. Table 5.2 summarizes the various quantities defined and typical units.

Table 5.2 Solution Viscosity Terminology [5]

Now that the intrinsic viscosity has been established, how is it related to molecular weight? Studies of the intrinsic viscosity of essentially monodisperse polymer fractions whose molecular weights have been established by one of the absolute methods indicate a rather simple relation (Figure 5.5):

Figure 5.5 Intrinsic viscosity-molecular weight relations for polyisobutylene in cyclohexane at 30 °C and diisobutylene at 20 °C. Reprinted from P.J. Flsory, Principles of Polymer Chemistry. Copyright 1953 by Cornell University. Used by permission of Cornell University Press.

(5.18)

where the subscript x refers to a monodisperse sample of a particular molecular weight. Equation (5.18) is known as the Mark–Houwink–Sakurada (MHS) relation.

What about an unfractionated, polydisperse sample? Experimentally, the measured intrinsic viscosity of a mixture of monodisperse fractions is a weight average

(5.19)

A viscosity-average molecular weight, , is defined in terms of this measured intrinsic viscosity as

(5.20)

With 0.5 < a < 1.0 for polymers, < < , but is closer to than . If the molecular weight distribution of a series of samples does not differ too much, that is, if the ratios of the various averages remain nearly the same, approximate equations of the type [η] = K′()^a′ may be applicable.

Why bother with calculating if it is different from either of the molecular weight averages that we defined (Eqs (5.3) and (5.4a) for and )? Viscosity is an easily measured property of polymer solutions, and can be used for quality control in polymerization processes, so it is still quite useful.

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Example 5.3 The following data were obtained for a sample of poly(methyl methacrylate), PMMA, in acetone at 30 °C:

ηr	c (g/100 mL)
1.170	0.275
1.215	0.344
1.629	0.896
1.892	1.199

For PMMA in acetone at 30 °C, [η] = 5.83 × 10⁻⁵ ()^0.72. Determine [η] and for the sample and K′, the constant in the Huggins equation.

Solution. For the data above, calculations give

In Figure 5.4, η_red and η_inh are plotted against concentration and extrapolated to a common intercept at zero concentration, the intrinsic viscosity, [η] = 0.580 dL/g.

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The regression slope of the upper line is 0.137. From Equation (5.17), k′ = slope/[η]² = 0.408 (dimensionless).

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Example 5.4 For the PMMA fractions in Example 5.1, calculate for mixtures 1 and 2 in acetone at 30 °C and compare with and (a is given in Example 5.3).

Solution. From Equation (5.20), for mixture 1:

In mixture 2:

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Viscosities for molecular weight determination are usually measured in glass capillary viscometers, in which the solution flows through a capillary under its own head. Two common types, the Ostwald and Ubbelohde, are sketched in Figure 5.6. (Since polymer solutions are non-Newtonian, intrinsic viscosity must be defined, strictly speaking, in terms of the zero-shear or lower-Newtonian viscosity (see Chapter 14). This is rarely a problem, because the low shear rates in the usual glassware viscometers give just that. Occasionally, however, extrapolation to zero-shear conditions is required.)

Figure 5.6 Dilute-solution viscometers: (a) Ostwald; (b) Ubbelohde.

In these types of viscometers, fluid is drawn up through a capillary into bulbs, with timing lines marked on the glass. As the fluid flows back down through the capillary (the right side of both viscometers in Figure 5.6), the flow time is recorded. These flow times are related to the viscosity by an equation of the form

(5.21)

where a and b are instrument constants (found through calibration), ρ is the solution density, and ν is the kinematic viscosity (with typical units of cm²/s). The last term (b/t), the kinetic energy correction, is generally negligible for flow times of over a minute, and since the densities of the dilute polymer solutions differ little from that of the solvent, the ratio of the viscosity of the polymer solution to the solvent viscosity can be approximated by the flow times recorded for these solutions, respectively:

(5.22)

The Ubbelohde viscometer has the distinct advantage that the driving fluid head is independent of the amount of solution in it; hence, dilution can be carried out right in the instrument.

The equipment necessary for intrinsic viscosity determination is inexpensive, and the measurements straightforward and rapid. The MHS constants K and a in Equations (5.18) and (5.20) are extensively tabulated for a wide variety of polymer–solvent systems and temperatures (Table 5.3) [6]. Even more simple devices (such as cup viscometers–which are metal cups with fixed holes in the bottom) are commonly used in industry for rapid viscosity estimation, this is particularly true in paint formulations. Automated viscometers, such as rotational viscometers, are also commonly used in analytical laboratories and offer rapid and accurate results.

Table 5.3 K and a Values of the MHS Equation for Selected Polymer–Solvent Systems [6]

5.4 Molecular Weight Distributions

Before continuing to one additional molecular weight characterization technique (gel permeation chromatography), let us delve a bit deeper into the concept of molecular weight distributions. A typical synthetic polymer might consist of a mixture of molecules with degrees of polymerization x ranging from one to perhaps millions. The complete molecular weight distribution specifies the mole (number) or mass (weight) fraction of molecules at each size level in a sample. (Actually, moles or masses could be specified, but they vary linearly with sample size, that is, are extensive quantities, while the mole or mass fractions are intensive, independent of sample size, and are therefore preferable.)

Distributions are often presented in the form of a plot of mole (n_x/N) or mass (w_x/W) fraction of x-mer versus either chain length, x, or chain mass, M_x. Since x and M_x differ by a constant factor, the molecular weight of the repeating unit M_r (Eq.(5.1)), it makes little difference which is used. Because x can assume only integral values, a true distribution must consist of a series of spikes, one at each integral value of x, or separated by M_r molecular weight units if plotted against M_x. The height of the spike represents the mole or mass fraction of that particular x-mer. No analytical technique is capable of resolving the individual x-mers, so distributions are drawn (and represented mathematically) as continuous curves, drawn through the tops of the spikes as sketched in Figure 5.7.

Figure 5.7 Molecular weight distributions illustrating the actual discrete distribution and the usual continuous approximation to it.

The averages may be calculated from either the mole- or the mass-fraction distributions. For continuous distributions, the summations in Equations (5.3) and (5.4) must be replaced by the corresponding integrals (the subscript x has been dropped for clarity and because of the continuous nature of the distributions).

From the mole (number)-fraction distribution (n/N),

(5.23)

Since = m and M = mx:

(5.24)

Note that the integrals replace the summation symbols used in the definition of (Eq.(5.3)). Also,

(5.25)

that is, the mole fractions must sum to 1. Also,

(5.26)

And, since = m,

(5.27)

If the mass (weight)-fraction distribution (w/W) is known, the average molecular weights and chain lengths can be determined:

(5.28)

(5.29)

(5.30)

(5.31)

Note that

(5.32)

that is, the mass fractions must sum to 1.

Given the mole (number)-fraction distribution, the mass (weight)-fraction distribution may be calculated, and vice versa. Since w = nM and, by definition of (Eq.(5.3)), W = N,

(5.33)

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Example 5.5 One analytic representation of a distribution of chain lengths that finds some practical application (as we will see later) is

where C is a constant.

a. Determine for this distribution.

b. Determine and the polydispersity index, PI = /, for this distribution.

c. Obtain the expression for the weight-fraction distribution (w/W).

d. Sketch the number- and weight-fraction distributions in the form of (n/N) and (w/W) versus x/.

Solution.

a.

Fortunately, for most of us, the above definite integral is tabulated in standard references. The result is

= C

b.

This integral is also tabulated, and gives the simple result

= 2

that is, PI = / = 2 for this distribution.

c.

d. The distributions are shown in Figure 5.8. Note that while the number of molecules decreases exponentially with x, the weight of a molecule increases linearly with x. The weight fraction therefore goes through a maximum in this case.

This distribution is derived for a particular kind of polymerization reaction in Chapter 9.

Figure 5.8 Molecular weight distribution of Example 5.5 plotted in reduced form.

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5.5 Gel Permeation (Or Size-Exclusion) Chromatography (GPC, SEC) [7]

Until fairly recently, the approximate molecular weight distribution of a polymer could be determined only by laborious fractionation of the sample according to molecular weight followed by determination of the molecular weights of the individual fractions with one of the techniques previously discussed. The bases for these fractionation techniques are discussed in Chapter 7. Suffice it to say that they tend to be difficult and time-consuming and so, in general, are avoided whenever possible.

Now, however, gel permeation chromatography (GPC, also known as size-exclusion chromatography, SEC) has been firmly established as a means of rapidly determining molecular weight averages (, , ) and the corresponding distributions. GPC makes use of a column, or series of columns, packed with particles of a porous substrate (known as the stationary phase). The term gel in gel-permeation chromatography refers to a crosslinked polymer that is swollen by the solvent used. This is perhaps the most common type of substrate, but others, for example, porous glass beads, are also used. The column is maintained at a constant temperature, and the solvent (known as the mobile phase) is passed through it at a constant rate. At the start of a run, a small amount of polymer solution is injected into the flowing mobile phase just before it enters the column. The solvent flow carries the polymer through the column. The smaller molecules in the sample have easy access to the substrate pores and diffuse in and out of the pores, creating a highly tortuous, lengthy pathway from the entrance to the exit from the column. The larger polymer chains simply cannot fit into the pores in the stationary phase and pass around the outside, and because they have a shorter pathway from entrance to exit, they elute from the column first. Thus, polymer chains of different molecular weights are effectively separated, with the molecular weight of the chains exiting the column decreasing with time.

A concentration-sensitive detector is placed at the outlet of the column. The most common detector is a differential refractometer (which measures the difference in refractive index between the pure solvent and that of the eluting fluid, containing the polymer), but ultraviolet (UV) or infrared (IR) detectors are also useful as long as the polymer has a functional group that can be detected in these light ranges. Regardless of the type of detector, it is essential that it measure some quantity Q that is proportional to the mass concentration of polymer at the column outlet (Q must be independent of M):

(5.34)

Thus, a GPC curve consists of a plot of Q (usually in arbitrary scale divisions, but related to the quantity of chains eluting at a particular time) versus v, the volume of solvent that has passed through the detector since sample injection, called elution or retention volume. The x-axis can easily be converted to time by dividing the elution volume by the volumetric flow rate of the solvent.

An example of a GPC curve is shown in Figure 5.9. Such a curve is often immediately useful for quality-control purposes, as the width of the distribution indicates sample polydispersity. The graph can also be compared directly to the graph for a known standard material, revealing a quick qualitative comparison. This may be enough to take corrective action in processing, but the usefulness of GPC extends well beyond qualitative comparisons, because, with appropriate calibration, it can determine all types of average molecular weights (, , ), and the PI of polymer samples. An additional benefit is that GPC can also be useful in detecting small molecular weight impurities or additives (such as plasticizers or stabilizers––see Chapter 18). These materials show up as peaks at the low molecular weight (high v or long t) end of the spectrum.

Figure 5.9 A depiction of typical results from GPC. Three GPC chromatograms showing the refractive index (which is proportional to the quantity of sample eluted) against elution volume for (a) four monodisperse polystyrene standards used to calibrate the GPC, (b) a sample that contains a 2:1 polymer blend of narrowly disperse polymers with 80,000 and 20,000, and (c) a sample that contains a 1:1 polymer blend of widely polydisperse polymer sample 80,000 and 20,000. The elution volumes match the calibration curve shown in Figure 5.10.

GPC chromatograms might look like one of the curves plotted in Figure 5.9. Nearly monodisperse polystyrene standards are used to calibrate a GPC. If run through the column as a mixture, the elution curve will show sharp peaks for each molecular weight standard, with the highest molecular weight eluting first. After the column has been calibrated, polymers with unknown molecular weight distributions can be tested. In Figure 5.9, one sample with two narrowly dispersed samples are shown in curve B (note that the curves do not overlap and that the area under the curves is proportional to the amount of polymer of each molecular weight), while a mixture of highly polydisperse polymer samples will have very broad peaks that overlap (curve C).

The total area under a chromatographic peak is proportional to the amount (mass) of polymer eluted. Thus, for polymer mixtures, it is possible to use GPC to estimate the weight fractions of different components, as long as their peaks do not overlap on the elution (x) axis.

It should be emphasized that GPC requires dilute solutions, so that each polymer molecule passes through the porous stationary phase with no entanglements. Because the hydrodynamic volume occupied by grafted or branched polymers is somewhat less than that occupied by a linear polymer with the same molecular weight, GPC typically underpredicts and for nonlinear polymers.

Conventional GPC is a relative method. Thus, to provide quantitative results, the relation between M and v (or elution time, t) must be established by calibration with monodisperse polymer standards. A calibration curve is shown in Figure 5.10. Typically, when the data are plotted in the form log M versus v, the curve is linear over much of the range, but sometimes turns up sharply at low v (high M).

Figure 5.10 GPC calibration curve. Narrow polydispersity (/) < 1.1 polystyrene samples in tetrahydrofuran at 22 °C.

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Example 5.6 Discuss the physical significance of an upturn in the calibration curve at low v.

Solution. The point at which the curve begins to shoot up is set by the largest pore size in the column. All of the chains that are too large to fit in those pores will pass through the column at the same rate, giving the infinite slope; that is, the packing material in a particular column cannot discriminate between molecules above that size. If the polymer sample to be analyzed contains significant material at high molecular weight, a column with larger pores should either replace the original column or be added in series to extend the calibration and obtain a good separation.

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The calibration curve, strictly speaking, applies only to the particular polymer, solvent, temperature, flow rate, and column for which it was established. Change any one, and the calibration is no longer valid. Most SEC calibrations are obtained with polystyrene, because the necessary monodisperse standards are readily available at reasonable cost. What do you do when you want to analyze another polymer? If absolute values are not needed, the molecular weights can be reported with respect to the polystryene (or other polymer) standards used to calibrate. A more sophisticated approach was suggested by Grubisic et al. [8]. When they plotted log([η]M) (the product [η]M is proportional to the hydrodynamic volume of the molecules) versus v, they obtained a single, universal calibration curve for a variety of different polymers. Thus, if you measure intrinsic viscosity along with SEC elution volume for your polystyrene standards (or calculate [η] for them with Eq. (5.18) and literature values for K and a), you can plot the universal calibration curve for your column.

To back out a calibration curve for a different polymer, you would need to know K and a under the new conditions:

(5.35)

(5.36)

where the subscript 0 above refers to the calibration conditions, that is, values from the universal calibration.

Balke et al. [9] have proposed a calibration method that requires only a single polydisperse sample of known and . The method assumes that the relation between log M and v is linear, and therefore it can be characterized by two parameters, a slope and an intercept. Given the GPC curve and the calibration, and can be calculated (using techniques outlined below). With the GPC data on the standard, a computer program adjusts the two calibration parameters until the known and are obtained, in effect using the two known average molecular weights to solve for the two unknown calibration parameters. If the calibration curve is not linear (and you would have no way of knowing), serious errors can result.

Most GPC packages include the software necessary to calculate molecular weight averages and the complete distribution from the data and the calibration, so one needs only to push the appropriate buttons. Nevertheless, the equations are outlined below to provide an understanding of what the software is doing and what its limitations might be.

Molecular weight averages may be approximated directly from the GPC curve and calibration by breaking the curve into arbitrary volume increments Δv. Usually, Δv is taken as 5 cm³, but smaller Δv's improve accuracy. The number of moles of polymer in a volume increment Δv is n_i:

(5.37)

where c_i = polymer mass concentration in the ith volume increment

M_i = molecular weight of polymer in the ith volume increment (assumed essentially constant over small Δv)Because c_i = Q_i/k (Eq.(5.34)),

(5.38)

If the volume increment Δv is constant, inserting Equation (5.38) into Equations (5.3) and (5.4) gives

(5.39)

(5.40)

Note that the proportionality constant k between concentration and GPC readout cancels out, and therefore need not be known. Q_i is normally read from the GPC curve as the distance above the baseline (in any convenient units) at a given v. The baseline presumably represents the detector output with the pure solvent. Establishing a good baseline is not always a trivial procedure, and the results can be quite sensitive to its location. M_i is read from the calibration curve at the same v. For greater accuracy, if necessary, the integral analogs of Equations (5.39) and (5.40) may be used, with the integrals evaluated numerically:

(5.41)

(5.42)

In most cases, the averages as calculated above and the qualitative information about the distribution provided by the GPC curve are all that are necessary. However, the technique is capable of providing the true distributions, if desired. The necessary calculations are outlined below:

(5.43)

(5.44)

(5.45)

Note:

The preceding calculations assign all the moles of polymer in a range m of the continuous GPC curve to a single spike (see Section 4) to calculate the correct height of the distribution:

(5.46)

(5.47)

(5.48)

(5.49)

(5.50)

Again, k cancels out in these equations. Once the distributions are known, the exact averages may be calculated by the techniques in Section 4. For example, insertion of Equation (5.47) into Equations (5.23) and (5.26) gives Equations (5.41) and (5.42).

GPC columns can be selected that have a wide spectrum of pore sizes (and thus separate a wide range of molecular weights–from thousands to millions). Alternatively, columns with more narrow pore size distributions can be used to get a better separation of polymers that have similar molecular weights—say from 1000 to 20,000. These columns are designed to be used with organic solvents (required for most polymers) or water that flow through as the mobile phase. The selection of which columns to use depends on the polymer structure and the expected molecular weight distribution.

A multiangle laser light-scattering photometer is often added to the usual concentration detector. This combination performs an online light-scattering determination of the M of the polymer in the eluting stream. This makes GPC an absolute technique, eliminating the need for calibration. Similarly, a differential viscometer is commercially available which performs an online intrinsic viscosity measurement on the column effluent [10]. Recall that the ordinate of the universal calibration is [η]M. This device requires simple calibration (in effect locating the universal calibration curve for the particular column, temperature, solvent, etc.) to provide absolute molecular weight distributions and also provides information on the hydrodynamic volume of the molecules in solution and therefore on branching, because a branched molecule in solution has a smaller hydrodynamic volume than a linear molecule of the same molecular weight. As mentioned earlier, an important caution regarding GPC/SEC is as follows: because the molecules are separated based on their size, linear polymers will flow differently through the stationary phase (the packed column) than branched or grafted polymers, so molecular weight estimation by this technique is normally less accurate for branched, grafted, or dendritic (very highly branched) polymers.

5.6 Summary

Molecular weight has an enormous impact on polymer properties, as we will discover in later chapters. Because polymers are made of many copies of the same repeat unit, traditional methods to determine absolute molecular weights are not applicable. Thus, several techniques are discussed in this chapter to measure or estimate polymer molecular weights. Depending on the techniques used, the number-average or weight-average molecular weights are determined, with most sensitive techniques requiring that the polymer be in dilute solution so that physically entangled polymers are not mistakenly recorded as one large polymer. Of the techniques discussed so far, gel permeation chromatography offers the greatest amount of information about a polymer sample, giving the size distribution of a polymer sample, which allows calculation of and and the polydispersity index.

Problems

1. Given the following chain lengths (x's) for poly(ethylene terephthalate), determine the number average molecular weight, weight average molecular weight, and polydispersity index. The system consists of 10 polymer chains, with repeat unit lengths of 12, 25, 175, 186, 192, 194, 199, 200, 202, and 212.

2. The weights of the (extremely) offensive linemen of the MIT (Monongahela Institute of Technology) football team are:

Cussler	Split end	180 lb
Miller	Left tackle	270 lb
Westerberg	Left guard	256 lb
Brenner	Center	285 lb
Anderson	Right guard	260 lb
Jain	Right tackle	305 lb
Prieve	Tight end	250 lb

Calculate the number- and weight-average line weights.

3. Using end group analysis to find molecular weight, 6.50 moles of ethanediamine (H₂N–CH₂CH₂–NH₂) were reacted with 6.50 moles of adipic acid (HOOC–CH₂–CH₂–CH₂–CH₂–COOH). After the reaction, a titration was performed using NaOH (which reacts with COOH) and an indicating dye to determine the number of acid groups remaining free after the polymerization. If 73.0 mL of 3.0 M NaOH were required to neutralize the polymer, what is the average molecular weight (M_n) of the polymer?

4. An osmometry experiment was done to estimate the molecular weight of a sample of polystyrene. The concentration of polymer in toluene was varied and the height difference between the solution and the solvent was measured. The temperature was 27 °C, where the specific gravity of toluene is 0.8667. Based on the following data, determine the molecular weight of PS. (A graph is required.) Is this or ?

Concentration, CA g/mL	Osmotic Pressure, πA, cm of Solvent
0.0320	0.700
0.0660	1.82
0.100	3.10
0.140	5.44
0.190	9.30

5. Six students have taken viscometry data for a monodisperse sample of atactic polystyrene at six different concentrations in toluene. Unfortunately, the techniques were not all identical, so they reported different versions of viscosity. The density of toluene at 30 °C is 0.8665 g/mL, 0.8656 g/mL at 40 °C. The viscosity of toluene is 0.5260 cP at 30 °C, 0.4710 cP at 40 °C.

Given the data below, resolve the experiments so that you can determine the viscosity-average molecular weight, , of the polymer.

Experiment 1: Polymer concentration in toluene C = 0.1000 g/dL; inherent viscosity = 0.060254 dL/g, 30 °C.

Experiment 2: C = 0.0500 g/dL; used an Ostwald viscometer at 30 °C to determine a flow time of 4 min, 36.50 s; in the viscometer calibration equation η/ρ = at + b/t, a = 0.001340, and b = 65.760 (where time is in seconds, density in g/mL, and viscosity in cP)

Experiment 3: C = 0.1100 g/mL; solution viscosity = 1.07360 cP at 40 °C.

Experiment 4: C = 0.1650 g/dL; determined the relative viscosity solution to be 1.0130 at 30 °C.

Experiment 5: C = 8.0 g/mL; determined specific viscosity was 10.896 at 30 °C

Experiment 6: C = 0.0290 g/dL; used a cone-and-plate viscometer at 30 °C to find a solution viscosity of 0.52655 cP

MHS constants for atactic polystyrene at 30 °C are: K = 0.2 × 10⁻³ mL/g; a = 0.72. Verify the validity of each experiment for estimating .

Note: Discount experiments if they were not properly conducted.

6. Consider a copolymer with repeating units A and B in which the mole fraction of A is y_A. The molecular weights of the repeating units are m_A and m_B. Obtain the analog of Equation (5.5a) for the copolymer, in which x represents the total number of repeating units of both kinds.

7. A discrete distribution of chain lengths has the same number of moles of each species over the range 1 ≤ x ≤ a and nothing outside of that range. Obtain expressions for , , and the polydispersity index for this distribution. Hints: Start with Equations (5.3a) and (5.4a). You may have to look up some series sums in a math table.

8. Consider the continuous distribution of chain lengths in which the number of moles of each species is constant over the range 0 ≤ x ≤ a and zero outside that range.

a. Obtain expressions for , , and the polydispersity index.

b. What is the constant value of (n/N) between 0 and a?

c. Obtain an expression for the weight-fraction distribution (w/W) as a function of x and sketch it.

d. Show that your answers to Problem 5.7 above agree with these when a » 1.

9. Crud Chemicals is using GPC to analyze polystyrene samples plasticized with low molecular weight mixed hydrocarbon oil, = 400. Their calibration curve is identical to the one in Figure 5.10 and from past experience they know that the unplasticized polymer gives a GPC curve just like Figure 5.9c.

a. Sketch what you think a GPC curve for a plasticized material would look like.

b. If you know that k_oil = 0.70 k_PS (in Equation (5.34)), could you use the GPC curve to determine the weight fraction plasticizer in the sample? If so, clearly describe how. If not, what additional information would you need?

10. Given the hypothetical distribution of chain lengths:

a. Determine the value of the constant C.

b. Calculate , , and the polydispersity index for this distribution.

c. Find the expression for the weight fraction of chains as a function of molecular length x, w/W, and roughly sketch this distribution as w/W versus x. Sketch n/N versus x on the same plot.

11. Some authorities prefer to represent molecular weight distributions in cumulative (integral) form rather than the differential form used here. Cumulative distributions are the number or weight fraction of material in the sample with chain length ≤ x (or molecular weight ≤ M), so that the maximum value of the y-axis is 1.

For the distribution of chains in Example 5.5, obtain expressions for the cumulative mole and weight distributions in terms of x/ and sketch them.

12. A polystyrene sample has the same distribution of chain lengths as given in Example 5.5. A light-scattering measurement gives a molecular weight of 208,000. Using the result from Example 5.5(c), calculate the mass of material with x = 100 in a 100 g sample of this polystyrene.

13. Many commercially used polymers are actually blends of two or more different molecular weights, which gives a net composition with a bimodal molecular weight distribution.

a. If an equal weight of polymers A, B, and C (below) are blended, what are the new , , and polydispersity of the blend?

b. Draw (sketch) a graph to represent the results of running this blend through a gel permeation chromatograph. The y-axis is an indicator of intensity (refractive index or absorbance) and is proportional to the mass of polymers that are leaving the GPC column. Label the x- and y-axes and label the parts of the chromatograph to show the different polymers in the blend (A, B, and C). Consider that the A, B, and C peaks may overlap.

14. The people at Crud Chemicals eagerly await the first GPC analysis of the polystyrene from their new polymerization process. There is a hushed silence as the first sample is injected into the GPC column. Suddenly, the chart pen begins to move, and to everyone's amazement, the GPC curve in Figure 5.11 emerges:

Figure 5.11 Raw data from GPC Chromatogram of Polystyrene.

Not knowing what to make of this, they call you in as a consultant. Their GPC operates at 22 °C with THF solvent. Its calibration curve is identical to Figure 5.10.

a. Sketch the number-fraction molecular weight distribution for their sample, including quantitative molecular weight extremes.

b. Calculate and for their sample. You can do this approximately using the discrete equations but really impress them and do it exactly by treating the continuous distribution.

15. Consider the polymer mixture formed in Example 5.2. Each component of the mixture follows the distribution of chain lengths given in Example 5.5. For simplicity, the molecular weight of a repeat unit may be taken as m = 100. What are the mole fractions of material with x = 10 (n₁₀/N) and (w₁₀/W) in the mixture?

References

1. Martin, J.R., J.F. Johnson, and A.R. Cooper, J. Macromol. Sci. Rev. C8(1), 57 (1972).

2. Nunes, R.W., J.R. Martin, and J.F. Johnson, Polym. Eng. Sci. 22(4), 193 (1982).

3. Collins, E.A., et al., Experiments in Polymer Science, Wiley, New York, 1973.

4. Burge, D.E., American Laboratory Raw data from GPC Chromatogram of Polystyrene 9(7), 41 (1977).

5. Billmeyer, F., Textbook of Polymer Science, 2nd ed., Wiley, New York, 1971, Chapter 3.

6. Kurata, M. and Y. Tsunashima, Viscosity-molecular weight relationships and unperturbed dimensions of linear chain molecules, Chapter VII/1 in Polymer Handbook, 3rd ed., J Brandrupand E.H. Immergut (eds), Wiley, New York, 1975.

7. Provder, T. (ed.), Size Exclusion Chromatography: Methodology and Characterization of Polymers and Related Materials, ACS Symposium Series No.245, American Chemical Society, Washington, DC, 1984.

8. Grubisic, Z. et al., J. Polym. Sci. B-5, 753 (1967).

9. Balke, S.T., et al., Ind. Eng. Chem., Prod. Res. Dev. 8, 54 (1969).

10. Yau, W.W. and S.W. Rementer, J. Liquid Chromatogr. 13(4), 626; Wang, P.J. and B.S Glassbrenner, ibid, 11(16) 3321 (1988).