Which average of copolymer composition does NMR provide?

1 Introduction

Polymers are heterogeneous at least in molar mass. In the case of copolymers or polymer blends chemical heterogeneity exists besides molar mass heterogeneity. Due to the heterogeneities, values assigned to experimentally determined molecular properties of polymer samples are usually averages. Different averages (number, weight, viscosity, z-average) are commonly used in polymer science. These averages differ in the statistical weights given to the properties of the individual chains in the sample. The average derived for a particular property depends on the experiment performed. For example, molar masses resulting from static light scattering are weight average molar masses, while determination of molar mass using colligative properties or by end-group analysis, e.g. by nuclear magnetic resonance (NMR), yield a number average. When dealing with molar masses the different averages are critically distinguished. However, less care is usually taken in regard to the composition of polymers.

NMR allows characterizing the type and quantifying the relative number of different structural units. This allows determination of the number average molar mass based on end-group analysis. However, as will be shown below, the assumption that counting structural units by NMR reveals also a number average composition is incorrect.

2 Results and discussion

Let us consider a binary copolymer composed of monomers A and B having molar masses M_A and M_B, respectively.

The total number n_A of A units in the sample can be calculated via

where N_i,j is the number of copolymer molecules having a degree of polymerization i containing j units of type A. Thus, xi, j=ji is the mol ratio of monomer units A in the molecule of degree of polymerization i.

Copolymer molecules of degree of polymerization i have the same number of repeating units. However, their molar mass depends on the ratio of the monomer units, if these differ in their molar masses (M_A≠M_B). The molar mass of a copolymer molecule having a degree of polymerization i containing j units of type A is given by:

where M_i,j is the average molar mass of a repeating unit in a molecule of degree of polymerization i containing j units of type A.

Therefore, the number of A units in the sample is given by:

Here m₀ is the total mass of the sample. m_i,j are and w_i,j are the mass and the weight fraction, respectively, of all copolymer molecules of degree of polymerization i containing j units of type A.

NMR determines the number of A units relative to the total number of monomer units in the sample. Consequently, the mol fraction of A units as determined by NMR is given as

The weight fraction of type A monomer units derived by NMR, w_ANMR, is calculated via

where w_Ai,j is the mass fraction of A units in a copolymer molecule of degree of polymerization i contain j units of type A.

According to Equation [4] the mol fraction derived by NMR is neither a number nor a weight average. Only if M_i,j (i.e. the average molar mass of the repeating unit) does not depend on the degree of polymerization, the mol fraction derived by NMR equals a weight average composition. According to Equation [2] M_i,j is independent of the degree of polymerization, if either M_A=M_B, i.e. both monomer units are of identical molar mass, or if the fraction x_i,j does not change with the degree of polymerization i and thus with molar mass. In this case, differentiation of number and weight averages is of no concern, however.

However, the mass fraction derived by NMR yields, a weight averaged mass fraction of monomer units of type A, as seen from Equation [5].

The above analysis shows that particularly the frequently-encountered assumption of NMR yielding a number average composition is incorrect. The following simple example calculation will prove this.

Let us assume to have a mixture of 0.5 g poly(ethylene oxide) (PEO) of molar mass 5 kg/mol and 0.5 g poly(styrene) of molar mass 500 kg/mol. The NMR experiment will determine 0.5/44=114 mmol EO units and 0.5/104=48 mmol styrene units. Thus, the mol fraction of EO units derived by NMR is 70 mol%.

Let us now calculate the number and weight average mol percentages of the EO units using the definitions of number and weight average properties.

The number average mol fraction of EO units is given by

where x_PEO is the mol fraction of EO units.

The weight average mol fraction of the EO units is

Neither the weight nor the number average mol fraction results in the mol fraction determined by NMR. However, if the mol fraction is calculated according to Equation [4] we find

which corresponds to the mol fraction derived by NMR.

If the poly(ethylene oxide) is replaced by poly(methyl methacrylate) (PMMA), both monomer units have very similar molar masses. NMR will determine a mol fraction of 51 mol% MMA units. In an analogy to Equation [6] and Equation [7] one calculates x_n=99% and x_w=50%, for the number and weight averaged mol fractions of MMA, respectively. Thus, the weight averaged mol fraction is very close to the mol fraction derived by NMR, as predicted above. This is a consequence of the very similar molar masses of the repeating units.

Although the above discussion correlated determination of chemical composition with NMR measurements, the same is true for all experiments providing the number or mass fractions of monomer units.

Often the number average molar mass of a block copolymer is calculated based on the number average molar mass of the precursor block and the mass fraction of the block copolymer determined by NMR (1), (2), (3), (4), (5), (6), (7), (8). This raises the question whether the fact that the number average molar mass is calculated using a weight average composition is a contradiction to the above results.

The number average molar mass of the block copolymer, M_n,block, is calculated from the number average molar mass of the precursor, M_n,A and the mass fraction of A units from NMR using:

As ∑​NA, i=nA=∑i∑j=0iNi, j=n it follows

Thus, the correct number average molar mass results based on the precursor number average molar mass and the NMR derived mass fraction, despite the fact that the latter is obtained as an weight average.

Gradient chromatography can separate statistical copolymers by composition (9), (10), (11), (12), (13), (14). Thereby, often nearly linear relationships are observed between composition and retention time. If the detector signal is proportional to the mass concentration of the eluting fraction and a linear relation exists between elution volume and composition, the median of the chromatogram corresponds to the weight average mass composition. Consequently the mass fraction determined by NMR can be directly assigned to the median of the chromatogram to establish a calibration between composition and elution volume.

Another consequence of the fact that the mass fraction determined by NMR is closer to the weight than to the number average follows for SEC separations using chemically sensitive detection. Even if the chemical composition varies with elution volume and thus with molar mass, there is no need to correct for the molar mass dependence of composition when comparing the mass composition from SEC with the composition derived by NMR. If the composition is calculated from the mass concentration at each elution volume and the corresponding mass fraction of the copolymer using a chemical sensitive detector, the resulting weight averaged mass fraction can be directly compared to the mass fraction from NMR.

3 Summary

It was shown theoretically and by model calculations that the mol fraction derived by NMR or other spectroscopic techniques provides neither a number nor a weight average if the repeating units differ in molar mass. If both monomer units are of identical molar mass, the derived mol fraction is a weight average. Therefore, the frequently encountered assumption of NMR yielding a number average composition is incorrect. In contrast, the mass fraction calculated based on NMR is a weight averaged mass fraction.

The fact that a weight average mass fractions results from NMR eases comparison with compositions derived, e.g. from gradient chromatography or from size exclusion chromatography with chemical sensitive detection.