Determining the absolute, chemical-heterogeneity-corrected molar mass averages, distribution, and solution conformation of random copolymers

Abstract

We present a method by which to obtain the absolute, chemical-heterogeneity-corrected molar mass (M) averages and distributions of copolymers and apply the method to a gradient random copolymer of styrene and methyl methacrylate in which the styrene percentage decreases from approximately 30% to 19% as a function of increasing molar mass. The method consists of separation by size-exclusion chromatography (SEC) with detection using multi-angle static light scattering (MALS), differential viscometry (VISC), differential refractometry (DRI), and ultraviolet absorption spectroscopy (UV) and relies on the preferential absorption of styrene over methyl methacrylate at 260 nm. Using this quadruple-detector SEC/MALS/UV/VISC/DRI approach, the percentage of styrene (%St) in each elution slice is determined. This %St is then used to determine the specific refractive index increment, corrected for chemical composition, at each elution slice, which is then used to obtain the molar mass at each slice, corrected for chemical composition. From this corrected molar mass and from the chemical-composition-corrected refractometer response, the absolute, chemical-heterogeneity-corrected molar mass averages and distribution of the copolymer are calculated. The corrected molar mass and intrinsic viscosity at each SEC elution slice are used to construct a chemical-heterogeneity-corrected Mark–Houwink plot. The slice-wise-corrected M data are used, in conjunction with the MALS-determined R _G,z of each slice, to construct a conformation plot corrected for chemical heterogeneity. The corrected molar mass distribution (MMD) of the gradient copolymer extends over an approximately 30,000 g/mol wider range than the uncorrected MMD. Additionally, correction of the Mark–Houwink and conformation plots for the effects of chemical heterogeneity shows that the copolymer adopts a more compact conformation in solution than originally concluded.

Appendix

Correcting the R _{G, z} of a bulk copolymer for chemical heterogeneity

In an SEC/MALS/DRI experiment, the z-average radius of gyration R _G,z of a bulk homopolymer (i.e., not the R _G,z of each SEC elution slice, to which we shall refer here for simplicity as r _i ) which, by definition, does not necessitate correction for chemical heterogeneity, is calculated according to [13]:

$$ {R_{{\text{G}},z}} = \frac{{\sum\limits_i {{c_i}{M_i}r_i^2} }}{{\sum\limits_i {{c_i}{M_i}} }} $$

(15)

We want to find the corrected R _G,z value of a copolymer which possesses chemical heterogeneity, given the corrected concentration c and molar mass M values. We have

$$ {R_{{\text{G,}}z\;{\text{uncorrected}}}} = \frac{{\sum\limits_i {\left( {{c_i} + {\delta_{{c_i}}}} \right)\left( {{M_i} + {\delta_{{M_i}}}} \right)r_i^2} }}{{\sum\limits_i {\left( {{c_i} + {\delta_{{c_i}}}} \right)\left( {{M_i} + {\delta_{{M_i}}}} \right)} }} $$

(16)

where c _i is the corrected c value at slice i, M _i is the corrected M value at slice i, ${\delta_{{c_i}}}$ is the amount of correction of c at slice i, and ${\delta_{{M_i}}}$is the amount of correction for M at slice i (with the realization that the various δ terms may be either positive or negative).

By expansion

$$ \sum\limits_i {\left( {{c_i} + {\delta_{{c_i}}}} \right)\left( {{M_i} + {\delta_{{M_i}}}} \right)} = \sum\limits_i {{c_i}{M_i} + {\delta_{{c_i}}}{M_i} + {\delta_{{M_i}}}{c_i} + {\delta_{{c_i}}}{\delta_{{M_i}}}} $$

(17)

Because $\sum\limits_i {{c_i}{M_i}}$is needed in the corrected R _G,z equation, let the right-hand side of Eq. 17 be written as

$$ \sum\limits_i {{c_i}{M_i} + {\varepsilon_i}} $$

(18)

where ${\varepsilon_i} = {\delta_{{c_i}}}{M_i} + {\delta_{{M_i}}}{c_i} + {\delta_{{c_i}}}{\delta_{{M_i}}}$.

Substituting Eq. 18 into Eq. 16 results in

$$ {R_{{\text{G,}}z\;{\text{uncorrected}}}} = \frac{{\sum\limits_i {{c_i}{M_i}r_i^2 + {\varepsilon_i}r_i^2} }}{{\sum\limits_i {{c_i}{M_i} + {\varepsilon_i}} }} $$

(19)

Manipulation of Eq. 19 results in

$$ \left( {\sum\limits_i {{c_i}{M_i} + {\varepsilon_i}} } \right){R_{{\text{G,}}z\;{\text{uncorrected}}}} - \sum\limits_i {{\varepsilon_i}r_i^2} = \sum\limits_i {{c_i}{M_i}r_i^2} $$

(20)

Dividing both sides by $\sum\limits_i {{c_i}{M_i}}$ yields

$$ \frac{{\left( {\sum\limits_i {{c_i}{M_i} + {\varepsilon_i}} } \right){R_{{\text{G,}}z\;{\text{uncorrected}}}} - \sum\limits_i {{\varepsilon_i}r_i^2} }}{{\sum\limits_i {{c_i}{M_i}} }} = \frac{{\sum\limits_i {{c_i}{M_i}r_i^2} }}{{\sum\limits_i {{c_i}{M_i}} }} $$

(21)

As per Eq. 15, the right-hand side of Eq. 21 is the corrected value of R _G,z , i.e., R _{G,z,corrected}. Simplifying Eq. 21 results in

$$ {R_{{\text{G,}}z\;{\text{corrected}}}} = \left( {1 + \frac{{\sum\limits_i {{\varepsilon_i}} }}{{\sum\limits_i {{c_i}{M_i}} }}} \right){R_{{\text{G,}}z\;{\text{uncorrected}}}} - \frac{{\sum\limits_i {{\varepsilon_i}r_i^2} }}{{\sum\limits_i {{c_i}{M_i}} }}. $$

(22)