Experimental methods and data evaluation procedures for the determination of radical copolymerization reactivity ratios from composition data (IUPAC Recommendations 2025)

1 Introduction

Several kinetic representations have been proposed to describe the incorporation of monomers into copolymer chains during radical copolymerization. ¹ Of these, the terminal model, where only the last unit in the chain affects the reactivity of a chain-end radical, is by far the most widely applied to adequately relate the copolymer composition to the composition of the monomer in the reacting mixture. Other models include the penultimate model, ² in which the penultimate unit also affects the reactivities, and the non-terminal model, where there is no difference in reactivity for the monomers towards the propagating species, which thus only applies to the special case of ideal copolymerization. ³ Still other representations consider complexation between monomers (the complex participation model) or complexation between monomer and copolymer chain end (the bootstrap model). ⁴ Copolymerization models serve to create mechanistic understanding of copolymerization reactions, but are ultimately most important in modelling of composition in manufacturing of copolymers. A high conversion is typically pursued for commercial manufacturing of copolymers where then composition drift can occur, i.e. monomer and copolymer compositions shift with increasing conversion due to the differences in monomer consumption rates.

The core assumption of the widely-used terminal copolymerization model is that the reactivity of the growing chains is entirely determined by their final monomer unit. Thus, a copolymerization of two monomers, M₁ and M₂, contains two types of growing chains, and a total of four propagation reactions, as shown in Scheme 1. ⁵ Reactivity ratios, ¹ r _i , are defined as the ratio of the rate coefficients of propagation k _ii and k _ij , corresponding to homopropagation and crosspropagation of chains containing a terminal unit M _i .

Scheme 1:

Terminal model for copolymerization of two monomers M₁ and M₂, thus i = 1 and j = 2.

Defining f _i as the mole fraction of M _i in the monomer mixture ( f i = M i M i + M j ) and F i inst as the mole fraction of M _i that is instantaneously being incorporated into the copolymer F i inst = d M i d M i + M j , gives the well-known Mayo–Lewis ⁶ two monomer copolymer composition equation (eq. (1)):

Differentiating f ₁ with respect to the total monomer mole fraction (eq. (2)) and integration after separation of variables leads to the Skeist equation ⁷ (eq. (3)) relating total monomer conversion, X, to the change in monomer composition:

The Skeist equation may be solved numerically. Alternatively, an analytical solution to this equation was provided by Meyer and Lowry ⁸ (eq. (4)). It relates the conversion X to the current monomer composition f _i and the initial monomer composition f _i0 (f ₁₀ for monomer 1 and f ₂₀ for monomer 2).

It should be noted that this equation contains singularities at r ₁ = 1, r ₂ = 1 and r ₁ + r ₂ = 2, and these can complicate its utilization (for specific solutions at the singularities, see Autzen et al. ⁹ ). Finally, the cumulative monomer composition, F 1 cum can be obtained from eq. (5).

If a sample of copolymer taken during a copolymerization is analyzed, it is the cumulative copolymer composition that has been determined, which from now on will be denoted as F _i . A key point to understand is that eq. (1) only rarely gives so-called azeotropic conditions, which are that F ₁ ^inst = f ₁ = δ, and thus both stay the same throughout a copolymerization. Far more commonly it is the case that F ₁ ^inst ≠ f ₁, and thus both values start changing as soon as any copolymerization occurs, which is termed composition drift. The magnitude of this drift is described by eqs. (4) and (5) above, and it is pivotal in determining how data should be analyzed.

Reactivity ratios play a central role in eqs. (1)–(4), but as the equations are non-linear, it is not immediately evident how to determine these reactivity ratios from experimental data. Several methods have been proposed over the last 75 years, many of which involve linearization of the copolymer composition equation. Widely used linearized methods such as Fineman-Ross ¹⁰ and Kelen-Tüdős ¹¹ distort the error structure of the experimental data, however, and can lead to biased and imprecise results. ⁹ For this reason, non-linear least-squares fitting (NLLS) ⁹ or visualization of the sum of squares space (VSSS) ^{9 , 12} is greatly preferred. For more background on the above preferences we refer to the basis for this recommendation in the full research paper from the same authors. ⁹ Further problems are encountered when the assumptions of the models are violated: for example, by applying eq. (1) to copolymerizations with non-negligible conversion, where eq. (4) would be more appropriate. These incorrect procedures can lead to significant errors in the estimation of reactivity ratios.

2 Experimental Methods

2.1 Copolymer composition vs comonomer composition at low conversion (f ₀ − F)

The most common method for determination of reactivity ratios involves collecting copolymer composition data (F) (for example with ¹ H NMR, gas chromatography or infrared spectroscopy) at low conversion across a range of initial monomer composition (f ₀). At sufficiently low conversion, the change in monomer composition (f) during copolymerization is negligible, and the cumulative and instantaneous copolymer compositions can be assumed to be equivalent, allowing direct fitting of the Mayo–Lewis equation (eq. (1)).

There is, however, no unique interpretation of “low conversion” in the f ₀ − F method, as with disparate reactivity ratios (i.e., r ₁ >> 1 and r ₂ << 1), strong composition drift can occur even at conversions below 5 %. This may introduce significant errors in F (see Fig. 1).

Fig. 1:

Cumulative copolymer composition (F ₁) versus conversion (X) at different monomer-1 starting fractions (f ₁₀). The blue dotted line represents the cumulative copolymer composition at 100 % conversion, which equals f ₁₀. This data has been calculated using r ₁ = 23 and r ₂ = 0.02, as these exemplify well the discussed ideas. Indicated in the graph: low conversion experiments (*, f ₀ − F) with the f ₀ − F curve in dashed green, which is method 2.1 of the text; following conversion and f (*, f ₀ − f − X), (f not shown in graph), which is method 2.2; starting from several f ₀ values and monitoring the copolymer composition with conversion (*, f ₀ − X − F), which is method 2.3, the IUPAC-recommended method. Reproduced with permission from Autzen et al. ⁹

The working group investigated several methods to correct for shifts in F at lower conversions including, amongst others, using the average monomer composition over the conversion range instead of f ₀. However, all these approximate correction strategies require knowledge of the conversion, suggesting that direct application of an integrated form of the copolymer equation such as the Meyer-Lowry equation (eq. (4)) is then possible. In such a case application of the integrated expression is preferable.

3 Data Evaluation

A thorough consideration of measurement errors is important in three ways: (1) They can be used to weight the data; (2) They determine the size of the joint confidence interval (JCI) (when the errors are well known); and (3) They can be used to determine whether the fit is adequate by comparing the actual fitting residues with the estimated errors.

Both F and X may be expected to be subject to experimental errors, and as such the error in variables method (EVM) is preferred. ¹⁵ However, in many cases it is likely that errors in determination of X will be small relative to errors in determination of F, as the measurement of X is more straightforward. In these cases, X can be treated as the error-free independent variable for the purposes of fitting. Looking at the individual fit residuals (calculated F minus measured F i.e. F _calculated − F _measured) it is possible to detect trends (e. g. deviations at high conversion, deviations at low or high f ₀).

Besides comparing the errors per datapoint, the overall error estimated by the user and the overall error obtained from the fit (s _F ) (eq. (6)) can also be compared using a Fisher test ¹²

where S _min(r ₁, r ₂) is the sum of squares of the residuals at the minimum, and n is the number of datapoints. The theoretical overall error is obtained when replacing S _min(r ₁, r ₂) in eq. (6) by the summation of the squares of the estimated errors by the user ¹² which can be compared to the total fit error.

In the case where f is monitored as a function of conversion (for example with in-situ NMR ¹⁶ ), these values can easily be converted to f ₀ − X − F data via mass balance (eq. (5)). In this conversion, the errors assumed in f and X are then converted to errors in F through Gaussian error propagation (note, a random error in f ₀ is not included in this equation) (eq. (7)):

We recommend that errors in the measurements be expressed in absolute terms rather than as a percentage of the measured value (relative error) ⁹ because a relative error structure is not often seen in experimental copolymerization data, and the results obtained using a relative error structure differ depending on how the monomers are labelled. ⁹

We suggest that the calculations are made with f ₀ − X − F data (not directly with the f ₀ − f − X data), because in the end we are interested in using the reactivity ratio estimates for predicting copolymer compositions, and this is the more robust approach in the data evaluation. Another advantage is that in the conversion from f ₀ − f − X to f ₀ − X − F both the random errors in f (and if needed f ₀) and in X can be propagated to give a well estimated random error in the calculated F (eq. (7)), and thus more realistic error estimates for the reactivity ratios. A potential problem is that some of the monomer can evaporate if the reactor is not a closed system, so even if f data are converted into F data it is advised to measure at least the final average composition of the copolymer to check for internal consistency.

Note that this is not a full errors-in-variables method as the result is only optimized on the copolymer composition F. Proper weighting of those data can however take place. ¹² In case of using EVM there is no significant difference between the two approaches. It is likely that the analysis of f ₀ − f − X data and the analysis of those data converted into f ₀ − X − F might give slightly different results if EVM is not used. ⁹ This is due to the fact that in the f ₀ − f − X approach fitting is often of the conversion data, X, while in the f ₀ − X − F approach, fitting is on the composition data, F.

For parameter estimation the best experiments are those that are most sensitive to parameter variation, for example according to the well-known criteria of Tidwell and Mortimer for the terminal model at low conversion. ¹⁷ However, this assumes that the model is known. Thus, this IUPAC method strongly advises variation in initial monomer compositions (f ₀) as well as conversion (X), in order to check for deviations from the terminal model ⁵ as well as systematic errors in the measurements. In other words, we have combined parameter estimation with investigation of whether the (terminal) model is adequate for compositional data in this IUPAC recommended method.

A requirement of the technique is that the conversion is measured for each experiment, which largely improves on the quality of the data in all cases. In the case that only low conversion data is used, it is again important to carefully compare the estimated errors in F with the fit residuals (looking at individual datapoints and also utilizing the Fisher-test described in eq. (6)). If in doubt, the f ₀ − X − F method should be used.

In the event that the errors are known, for example through an error propagation exercise or through replicate measurements, the errors can be used to construct the joint confidence interval using the χ ² distribution ¹⁸ with S(r ₁, r ₂) _z , the boundary of the JCI at level z (for example a 95 % probability) (eq. (8)):

Here σ ² corresponds to the average absolute variance of the dependent variable (in this case F) and is calculated from the known errors as entered by the user. The S _min(r ₁, r ₂) is the sum of squares of residuals at the minimum and with p degrees of freedom (p equals two in the present cases). If the errors are only estimates (which is often the case), the JCI at level z is constructed through the following equation (eq. (9)):

Here F _z (p, n − p) represents a value from the Fisher-distribution at level z (for example at 90 or 95 % probability) with p and n − p degrees of freedom (p equals two in the cases at hand), n data points and S _min(r ₁, r ₂) sum of squares of residuals at the minimum. (Note that this use of F is distinct from its use for copolymer composition).

It is important to note that these procedures are developed assuming random errors in the data. As soon as systematic errors appear, the JCI will no longer give a useful reflection of reality. In particular, if the true f ₀ is significantly different from the reported value, all the data from that experiment are systematically biased. So although the error in f ₀ is random, it results in a systematic error in that particular series of f ₀ − X − F data (see eq. (5)). For this reason, it is recommended that several different starting values for f ₀ are used. If each of these f ₀ sets (e.g. the different curves in Fig. 1) have different systematic errors, the overall fit with all the different f ₀ values and associated systematic errors is more likely to transpose to a random error. ⁹ The sum of squares of residuals at the minimum will be larger than for the individual f ₀ sets, and S _min(r ₁, r ₂), through eqs. (8) or (9), will increase the size of the JCI. ⁹

This effect has been shown clearly in a simulated X vs f dataset generated with r ₁ = 0.4, r ₂ = 0.6 and f ₁₀ = 0.5. ⁹ When only changing the f ₁₀ value (i.e., introducing a systematic error) in fitting this simulated data, the effect on output reactivity ratios is shown in Fig. 2.

Fig. 2:

Results obtained from fitting of simulated X vs f data obtained with r ₁ = 0.4, r ₂ = 0.6 and f ₁₀ = 0.5. Random noise of ±0.005 in X was applied to the simulated data, which was then fitted using different values of f ₁₀, as per the abscissa, in order to mimic the effect of systematic error. Output values of r ₁ (blue), r ₂ (orange) and S _R (sum of squares of residuals; grey) are shown. For f ₁₀ = 0.5 the true results are obtained (r ₁ = 0.4, r ₂ = 0.6, S _R = 0.0539), but otherwise there is distortion of r ₁ and r ₂. Reproduced with permission from Autzen et al. ⁹

It can be seen that a change in f ₁₀ of −0.001 shifts r ₁ from 0.4 to 0.424 and r ₂ from 0.6 to 0.64, while larger changes in f ₁₀ give progressively stronger distortions. The typical random error in NMR for f ₁₀ is most likely larger than 0.001, so the effect is significant. This means that even with a very accurate value of f ₁₀ one might wish to optimize this value rather than assume it is rigidly correct. Then, as per our recommendation, combining different sets with different f ₁₀ values is further mitigating this issue as discussed before.

Design of experiments can also be applied on the IUPAC recommended method. In the case of low conversion data, we recommend use of at least three different f ₀ values, where two of them ( f 10 ′ , f 10   ″ in eq. (10)) can be chosen through the Tidwell-Mortimer D-optimal design criteria, ¹⁷ requiring an initial estimate of the reactivity ratios (eq. (10)). We realize that the Tidwell-Mortimer approach is only applicable to low conversion experiments and cannot be extended to higher conversions

The IUPAC recommended method has been tested extensively with a large body of experiments and the results are also compared with other methods. ¹⁹ An extended discussion on the importance of the knowledge of the experimental errors for this method is also published. ²⁰

4 Summary of Recommendations

The following is a summary of the recommendations coming out of our work:

1. Either use low conversion f ₀ − F data or conversion dependent data in the form of f ₀ − f − X or f ₀ − X − F, in all cases with at least three different starting monomer compositions f ₀.

2. Obtain the best possible information about the errors in the measurements, and utilize weighting in the fit according to the errors in the dependent variable (in most cases F).

3. Use only non-linear regression or the visualization of the sum of squares space.

4. If the independent variable (usually f) has considerable error, use non-linear regression combined with EVM.

5. If using f ₀ − f − X data without EVM, convert the f ₀ − f − X data into f ₀ − X − F with proper error propagation, taking errors in f (also f ₀ if needed) and X into account.

6. If using low conversion f ₀ − F data, check that no significant (i.e., more than the expected random error) change in F has occurred due to composition drift. This can be done by using the estimated reactivity ratios to calculate the predicted change in F with conversion. If this indicates too much composition drift over the range of X used experimentally, then one should go back and use the f ₀ − X − F method instead.

7. Be aware of errors in f ₀, especially in conversion-dependent experiments.

8. Mitigate errors in f ₀ through (1) measuring f ₀ (eg through NMR), and/or (2) investigating limited variations in f ₀ though fitting f ₀ − f − X single experiments, and/or (3) looking at the residuals in a set of experiments and detecting systematic patterns – if there are such patterns, then vary f ₀ again, i.e., step (2).

9. Investigate if the fit residuals exceed the expected errors; if they do, this usually indicates that the terminal model is not valid for the copolymerization system under investigation and/or that systematic errors are present.

10. The obtained reactivity ratios should be reported with the correct number of significant digits (typically 2) and an indication of the uncertainty in those values (preferably a joint confidence interval).

5 Conclusions

The IUPAC working group on “Experimental Methods and Data Evaluation Procedures for the Determination of Radical Copolymerization Reactivity Ratios” has established a robust method to determine reactivity ratios from composition data following the terminal model. The method is based on measuring conversion (X) and (cumulative) copolymer composition (F) in a few copolymerization reactions at different starting monomer compositions (f ₀), although a set with only low conversion can also be used (f ₀ − F). We make freely available the analysis software for this method, and we strongly recommend that it be used for reactivity ratio determination. ^{21 , 22 , 23} The method not only provides parameter estimates but can also reveal deviations from the terminal model and systematic errors in the dataset. It is shown that error estimation for the F-values is important for weighting the data, determining the size of the joint confidence interval (in case of accurately known errors) and discerning whether the fit with the terminal model is adequate. In principle previous experiments measuring f ₀ − F (if conversion is known or sufficiently low) can still be analyzed with the IUPAC recommended method. Special attention has been given to the occurrence of systematic errors in the f ₀ − X − F and f ₀ − f − X experiments. It is shown that the current statistical treatment is not able to properly accommodate systematic errors occurring within such experiments. However, with the analysis of the residuals space (f ₀ − X − F) these errors can be identified and where possible corrected through optimization of f ₀ as a third parameter.