Instantaneous MMD

Equation (5.55) has the important implication that if the appropriate circumstances are satisfied, then the instantaneous MMD is independent of conversion, and hence that the cumulative and instantaneous MMDs are the same (to within an arbitrary normalization constant). In this case, the number MMD is a single exponential, and that suggests that a most informative way to plot experimental MMDs is as In P(M) against M (recalling that if the GPC calibration curve is linear, then P(M) = GPC trace/Af ), The result should be a straight line whose slope yields the transfer constant. 

Figure 5.13 Instantaneous MMD (arbitraiy units) obtained by successive subtraction of the cumulative MMD for a styrene pseudo-bulk system at 50 C (unswollen seed radius 77 nm, initiator 10 mol dm persulfate) recalculated from data in [57]. The range of tVp (as a percentage) for each sample is shown. The uncertainty is high at low and high...

Equation (5.51) states that the instantaneous MMD can in principle be obtained by successive subtractions of experimental cumulative MMDs obtained over small changes in conversion. This procedure requires appropriate normalization of the MMDs to be subtracted [57], each being normalized so that... 

The instantaneous MMD can be found from a knowledge of the rate coefficients for chain-stopping events, which are transfer (to monomer, polymer and chain-transfer agent) and termination, and for chain growth, which is propagation. In addition, since in a zero-one system entry into a particle which already contains a growing chain results (by definition) in instantaneous termination, entry is also a chain-stopping event in a zero-one system. 

To test the predictions of the model for the MMD given in Section 5.2.5, an extensive series of experiments has been performed to obtain the instantaneous MMD in a seeded styrene emulsion polymerization (57). The instantaneous MMDs were obtained by successive subtraction, as discussed in Section 5.2.5. 

Consider the BB problem when molar mass sensitive detectors are employed. First, let us analyze the ideal case of a chromatograph fit with perfect detectors and not exhibiting BB, secondary fractionations, or interdetector volumes. In this case, the instantaneous MMD is strictly uniform, and any molar mass detector type would provide the same result ... 

In relation to Methods IV and V, the instantaneous MMDs were simulated with the aim of calculating the (noise-free) calibrations log M V) and log M (V) (Fig. IB). These functions were obtained from the (noise-free or non-truncated) mass chromatogram in order to illustrate their true shapes in the complete range of the measured chromatogram. The resulting ad hoc calibrations are non-linear, generally less steep than log M(V), and close to the effective linear calibration log Af(V)liv (Fig. IB). 

At each retention volume of w(V), the instantaneous MMDs in the detector cells were calculated by considering the contributions (toward that V) of aU the hypothetical molecular species in the distribution, as determined by the discrete w (V). From such instantaneous distributions, the ad hoc non-linear calibrations log M (V) and log M V) were calculated.From w(V), log M (V), and log M (V), the MMD estimates w(log M ) and w(log M ) were obtained their averages are presented in Table 2. As expected, wGog M ) accurately estimates M but underestimates M,v, while vv(log M ) accurately estimates but overestimates M . Thus, the global polydispersity is underestimated in both cases. 

In a zero-one system, the instantaneous number MMD is simply given by... 

Figure 5.14 The slope of the straight-line region of the instantaneous number MMD for 44 and 130 nm radii seeded emulsion polymerizations of styrene at 30 °C (one example of which is shown in Figure 5.12) rec culated from data in [57], The slope is plotted as A = —AfodlnP(A/)/

Band broadening correction in SEC is still not a totally resolved issue, even when the MMD of a linear homopolymer is determined with a mass detector and a molar mass calibration. Fortunately, modem SEC columns are highly efficient, and the correction for BB is mainly limited to the case of narrowly distributed polymers. Even in the presence of BB, if an instantaneous quality variable is accurately measured, its corresponding global average will also be accurate. 

Vicente et al. [30] used the heat of reaction and the open-loop observers developed in Section 7.2.5.3 to determine the concentration of monomer and CTA and hence to infer the instantaneous number-average molar masses during emulsion homo- and copolymerization reactions. In addition, the authors used the inferred values for online control of the molar mass distributions of copolymers with predefined distributions. They demonstrated that polymer latexes with unimodal MMD with the minimum achievable polydispersity index in free-radical polymerization (PI = 2) and bimodal distributions could be easily produced in linear polymer systems [15, 30]. 

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