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It is important to discuss the methods for the characterisation of branched polymers as an understanding of the theory is required to comprehend discussions from literature sources referenced within this chapter. One of the most widely used methods of characterisation for polymers is SEC, also known as gel permeation chromatography (GPC). Polymers are dissolved in a suitable solvent and injected on to a column packed with porous beads, commonly polystyrene (PS). Large molecules cannot permeate all of the mixed pore sizes and are excluded from travelling through this large network of pores and their retention time in the column is short. Smaller molecules permeate more of the pores and so their retention time in the column is longer, hence polymer separation occurs according to molecular size.182, 213

The most common method for the analysis of polymers is conventional SEC, whereby a concentration detector, such as refractive index (RI) is used to measure the varying concentration of polymer samples as they elute from the column using Equation 2.1.

ࡾࡵ=࡮(ࢊ࢔ ࢊࢉ⁄ )ࢉ

Equation 2.1: Where RI is refractive index, B is an instrument constant, dn/dc is the change in RI of the solution as a function of concentration (this value varies for each polymer-solvent pair under constant conditions, common dn/dc values can be found in literature)183, and c is the polymer concentration in solution

The elution time of the polymer sample is related to molecular weight by the calibration of the column with narrow polymer standards, of a known molecular weight. However, this method of analysis does not yield a universal curve for all polymer samples, leading to limitations in the use of conventional SEC analysis.214As column separation is based on hydrodynamic volume of the molecules and not molecular weight directly, accuracy of results requires, ideally, use of narrow standards of the same polymer type to the analyte, although this is not usually possible due to the low variety and number of commercially available polymer standards. Another issue arises in the analysis of non-linear polymers; again this is due to separation being based on hydrodynamic volume. Underestimation of the molecular weight occurs in the analysis of non linear polymers by conventional SEC, as a reduction in hydrodynamic volume is observed for

branched polymers compared to their linear counterparts of the same molar mass (Figure 2.1).215, 216

Figure 2.1: Illustration of the difference in hydrodynamic volume of a linear and branched polymer in solution

As this branching level increases the discrepancy caused by the reduction in hydrodynamic volume becomes more prominent. Another factor to consider is whether the sample is heterogeneous, not only molecular weight and architecture but also composition, such as copolymers, microstructure and tacticity, all of these factors can contribute to elution time. As conventional SEC is only accurate in a small number of cases, and cannot be considered an accurate means of measuring molecular weight for branched polymers, combinations of multiple detectors including viscometry, and multi angle light scattering (MALS) can be used.214, 215, 217, 218

2.1.2 Viscometry Detection in SEC - Universal Calibration

Universal calibration uses a combination of SEC with RI and viscometry detectors and is based on the hypothesis that elution volume is solely dependent on hydrodynamic volume, taking advantage of the fact that IV and molecular weight are related to the hydrodynamic volume by Equation 2.2:

V

[η] = K

M

Equation 2.2: Where [η] is intrinsic viscosity, V is hydrodynamic volume, a measure of molecular

size, M is molecular weight and k is a constant, whose value is independent of polymer structure.214

Due to the relationship between hydrodynamic volume and IV, and the fact that SEC separates molecules by size, the generation of a calibration curve for a set of

of the standards used, (Figure 2.2).

Figure 2.2 Example of universal calibration in size exclusion chromatography. Image copied from reference without editing214

Deviations in the universal calibration can arise when analysing macromolecules of differing shape factors, such as rod like and spherical polymers,219, 220as hydrodynamic volume is also proportional to the third power of the mean-square radius of gyration according to the Flory-Fox equation,221, 222 although a wealth of literature exists of cases for the applicability of universal calibration.116, 223-225

The use of a viscometry detector allows the determination of the IV for the molecular weight distribution of polymer samples. These values can be used in the formation of Mark-Houwink plots using the Mark-Houwink equation (Equation 2.3).

M = K[η]α

Equation 2.3 Where M is molecular weight, [η] is intrinsic viscosity with K and α as constants.

The contraction of a branched polymer of constant branching in solution will be

compared to its’ linear counterpart.226 If branching is not consistent throughout the sample, both the gradient and IV of the Mark-Houwink plot will contrast with its’ linear counterpart.3, 213The use of universal calibration will be expanded in further detail later in this chapter.

2.1.3 Triple Detection

Triple detection is a further SEC technique widely used for the characterisation of branched polymers. This method consists of a combination of RI, Light scattering (LS) and viscometry detectors, obtaining the highest level of information from the samples. LS detection measures absolute molecular weight utilising the Rayleigh equation (Equation 2.1):

ࡾ(ࣂ)│ࣂ→૙≅ࡷ࡯ࡹ

Equation 2.4: The amount of light scattered by a dilute solution of analyte at a given angle θ which is termed the Rayleigh ratio, R(θ), where K is an optical constant, C is concentration and M is

molecular weight

The Rayleigh equation requires the intensity of scattering at zero angle. Measurement directly in the incident beam is not possible, hence several methods to minimise the error are utilised to gain an accurate measure of molecular weight. Measurement of scattering at low angles (LALS), such as 10 degrees or less, error is minimal and is disregarded. Measurement at multiple angles, MALS, and extrapolation to zero can also be used.

In conjunction with LS and RI detection for the determination of absolute molecular weight, viscometry and RI detection are utilised in the measurement of IV, all of which can all be achieved without the need for column calibration.213, 218, 227Triple detection is based on the theory, originating from Zimm and Stockmayer in 1949, that chain size, or the mean squared radius of gyration (Rg2), is affected by branching.228, 229 As with

hydrodynamic volume, the more branched a polymer, the greater the reduction in the Rg2. This reduction in Rg2for branched polymers can be defined as a contraction factor,

g, by Equation 2.5:

ࢍ= (ࡾࢍ

૛₎ ࡮

(ࡾࢍ૛)ࡸ

information on the chain length contraction factor, g’ by Equation 2.6:230

ࢍᇱ₌[ࣁ]࡮

[ࣁ]ࡸ

Equation 2.6: where [η] is intrinsic viscosity and subscripts B and L refer to branched and linear

structures respectively

Although this method cannot give an absolute branching number, it can give an indication of the extent of branching when comparing a series of branched polymers.120,

218

It is noted that applicability of triple detection to low molecular weight polymers is poor, as light scattering detection is insensitive to low molecular weights, hence, universal calibration and the use of g’ plots are often the preferred method of characterisation.227