Copolymer model

Copolymer Systems. The copolymer model development followed the homopolymer model development, properly accounting though for the presence of two monomers, two types of radicals and other implications that a comonomer system can give rise to. Information from (72) was found very helpful. Details on the copolymer model development can be found in (59). 

Since there was no experimental data available for the copolymer case, the copolymer model was run as a homopolymer one by setting one of the monomer inputs equal to zero. In this way one could test two extreme cases and see if there were any flaws in the model s logic. The results for a batch latex reactor are shown in Figure 5a. 

In conclusion, one could say that despite the lack of experimental data with which the copolymer model predictive powers could be tested, its trends when applied to the VCM/VAc system were very reasonable and in agreement with general experience from systems of this type. There are certain aspects of the model (e.g. the expression for the xc of the system) that should be refined in the near future, but at least the basic structure... 

Qu Y, Payne SC, Apkarian RP, ConticeUo VP. Self-assembly of a pol3rpeptide multi-block copolymer modeled on draghne silk proteins. J Am Chem Soc 2000 122 5014-5015. 

Figure 6.3 Representation of casein self-association structures according to die simple copolymer model (a) p-casein, (b) asi-casein. Reproduced from Home (1998) with permission.

Figure 8.5 Interaction potential for model whey protein layer consisting of densely packed brush-like tethered chains with small a fraction of the whev protein replaced by p-casein chains as represented by a copolymer model. The energy A d) calculated from SCF theory is plotted as a function of surface-surface separation d A, no p-casein B, 2.5% p-casein C, 5% p-casein D, 5% p-casein alone (without whey protein layer). Potentials A, B and D imply that the emulsion system is flocculated potential C implies a stable emulsion state. Reproduced from Dickinson (2006b) with permission.

To conclude this section on the dense random copolymer model, wc briefly discuss the spinodal criterion and ask whether critical or multicritical points can exist for a general parent distribution pW(a) (with p = 1). The criterion (53), applied to our one-moment free energy becomes... 

We now consider the random copolymer model in the presence of solvent— that is, for a copolymer volume fraction p0 < 1. We are not aware of previous work on this model in the literature, but will briefly discuss below the link to models of homopolymer/copolymer mixtures [57]. The excess free energy (86) then depends on two moment densities, rather than just one as in all previous examples. For simplicity, we restrict ourselves to the case of a neutral solvent that does not in itself induce phase separation this corresponds to X = 0, making the excess free energy... 

The fastest proteins fold amazingly quickly some as fast as a millionth of a second. While this time is very fast on a person s timescale, it is remarkably long for computers to simulate. In fact, there is about a 1000-fold gap between the simulation timescales and the times at which the fastest proteins fold. This is why the simulation of collapse kinetics is extremely computationally demanding. Thus, the current challenge lies in understanding how particular chemical sequences in coarse-grained copolymer models lead to particular collapse features. This is a fundamental issue in the problem. 

A diblock copolymer model material with blocks of methylmethacrylate (approximately 25%) and styrene was prepared since this system should be thermally stable. The diblock copolymer was prepared using a technique described by Rempp et al. (9,10) with slight modifications. The following amounts were used methylmethacrylate 50 g (0.5 mol), styrene 150 g (1.44 mol), solvent THF 1,000 mL, and n-butyl-lithium 2.5 10"3 mol (anionic catalyst). Reaction temperature, —55°C. 

These examples do not mean that copolymerization reactions defy understanding. The simple copolymer model described here accounts for the behavior of... 

To predict the course of a copolymerization we need to be able to express the composition of a copolymer in terms of the concentrations of the monomers in the reaction mixture and some ready measure of the relative reactivities of these monomers. The utility of such a model can be tested by comparing experimental and estimated compositions of copolymers formed from given monomer concentrations. Asa general rule in science, the preferred model is the simplest one which fits the facts. For chain-growth copolymerizations, this turns out to be the simple copolymer model, which was the earliest useful theory in this connection [1,21. All other relations which have been proposed include more parameters than the simple copolymer model. We focus here on the simple copolymer theory because the basic concepts of copolymerization are most easily understood in this framework and because it is consistent with most copolymer composition and sequence distribution data. 

Assumptions are invoked whenever an attempt is made to reduce the complexities of the real world to a mathematically tractable model. Following are some of the assumptions which are implied in the simple copolymer model ... 

The simple copolymer model, with two reactivity ratios for a binary comonomer reaction, explains copolymer composition data for many systems. It appears to be inadequate, however, for prediction of copolymerization rates. (The details of various models that have been advanced for this purpose are omitted here, in view of their limited success.) Copolymerization rates have been rationalized as a function of feed composition by invoking more complicated models in which the reactivity of a macroradical is assumed to depend not Just on the terminal monmomer unit but on the two last monomers in the radical-ended chain. This is the penultimate model, which is mentioned in the next Section. 

Deviations from the behavior of the simple copolymer model have been noted for various systems and have prompted the development of alternative models, all of which use more parameters than the two reactivity ratios in Eq. (7-1. ). Such models will often fit particular sets of copolymerization data better than the simple copolymer model. It appears in retrospect, however, that many of the apparent deviations from this model may be accounted for by large uncertainties in reactivity ratio values. The inadequacy of the simple copolymer theory can be established only if deviations between calculated and observed copolymer compositions arc shown to be systematic as the feed composition or monomer dilution is varied. Random errors do not necessarily show that the basic model is inapplicable. 

The simple copolymer model is a first-order Markov chain in which the probability of reaction of a given monomer and a macroradical depends only on the terminal unit in the radical. This involves consideration of four propagation rate constants in binary copolymerizations, Eqs. (7-2)-(7-4). The mechanism can be extended by including a penultimate unit effect in the macroradical. This involves eight rate constants. A third-order case includes antepenultimate units and 16 rate coefficients. A true test of this model is not provided by fitting experimental and predicted copolymer compositions, since a match must be obtained sooner or later if the number of data points is not saturated by the adjustable reactivity ratios. 

This is the case, for example, in the copolymerization of carbon monoxide and ethylene where the CO will not add to itself but does copolymerize with the olefin monomer. General theoretical treatments have been developed for such cases, taking into account temperature and penultimate effects. Again, the superiority of these more complicated theories over the simpler copolymer model is not proved for all systems to which they have been applied. 

The copolymerization theory presented in Chapter is of limited applicability to processes involving heterogeneous Ziegler-Natta catalysis. The simple copolymer model assumes the existence of only one active site for propagation, whereas the supported catalysts described above have reaction sites that vary in activity and stereoregulating ability. In addition, the catalytic properties of the active sites may vary with polymerization time. The simple copolymer model can be used with caution, however, by employing average or overall reactivity ratios to compare different catalysts and monomers. 

Statistical Copolymers. The term statistical is used here to refer to copolymers in which the sequence distribution of comonomers can be inferred statistically from the simple copolymer model (Chapter 7) or alternative theory. In the present context statistical copolymers excludes block and graft structures and incorporates all other copolymers. It is useful first of all in this section to point out that statistical copolymers are not mutually miscible if the mixture involves abrupt changes in copolymer composition. Coatings chemists observe this phase separation as haze (internal reflections) in films. 

Since only four copolymer compositions were studied and particularly since the maximum trimethyl comonomer content available was only 20%, it is difficult to precisely determine the three binary interaction parameters, suggested by the "copolymer model", Equations 1 and 2, from the observed variation of B with trimethyl comonomer content in the PEC, Figure 8. It is none the less interesting, however, to qualitatively assess the... 

The composition of a copolymer thus cannot be determined simply from a knowledge of the homopolymerization rates of the two monomers. The simple copolymer model described here, however, accounts for the behavior of many important systems and the entire process is amenable to statistical calculations which provide a good deal of useful information from few data. Thus, it is possible to calculate the distribution of sequences of each monomer in the macromolecule and the drift of copolymer composition with the extent of conversion of monomers to polymer. 

According to the reaction scheme of the simple copolymer model. Mi—Mi bonds are formed only by reaction (7.2). The probability that a propagating species MJ (that is, ending in an Mi unit) adds an Mi unit is equal to the rate of this reaction divided by the sum of the rates of all reactions available to this propagating species. This is the probability Pn given by [see Eq. (P7.1.3)] ... 

The effects on the B15 and V15 chain conformations of the incorporation of one or two 3HV and 3HB units, respectively, have been investigated. First, the central (8th) unit of B15 model chain is replaced by an HV unit, this copolymer model is called B14V1[8], and the conformation was optimized starting again from the X-ray structure of P(3HB). The rotational angles of the optimized conformation are shown in Fig. 21.6. 

The copolymer models rich in 3HV units have been examined in the same way. For example, the result for the V14B1[9] model, in which the central (9th) 3HV unit of V15 is replaced by a 3HB unit, is shown in Fig. 21.8. 

Li.gru.n-VeAd.ved Potyi ocqanaXeA Efforts to increase the incorporation of lignin into polyurethane products have concentrated on transforming polymeric lignins into polyisocyanates useful for reacting with polyols. Two alternative reaction pathways have been explored with the three lignin-like model compounds shown in Figure 3. These models were vanillic acid or a derivative thereof (Model Type A) a derivative of tetralin di-carboxylic anhydride (Model Type B) and a derivative of a styrene-maleic anhydride copolymer (Model Type C). 

The adsorption of proteins at fluid interfaces is a key step in the stabilization of numerous food and non-food foams and emulsions.1 Our general goal is to relate the amino acid sequence of proteins to their surface properties, e. g. to the equation of state or other structural and thermodynamic properties. To improve this understanding, the effect of guanidine hydrochloride (Gu HC1) on /1-casein adsorption is evaluated in the framework of the block-copolymer model for the adsorption of this protein. At first the main features of the model are presented, and then the effect of Gu HC1 is interpreted using the previously introduced concepts. 

Fig. 48a. Normalized inverse scattering intensity NS l(q, e) observed in the Monte Carlo simulation of a block copolymer model on the simple cubic lattice (see Fig. 44) plotted vs the normalized inverse temperature eN. b Reciprocal structure factor S (q ) -l(cxrcfes, left scale) and q ( squares, right scale) plotted vs temperature for a nearly symmetric diblock copolymer of polystyrene/poly (cis— 1,4) isoprene (Mw = 15 700). Filled symbols refer to cooling, open symbols to heating runs. The straight tine indicates the extrapolation to a spinodal temperature (T,) that occurs above the actual transition temperature (Tmst). where the data show a jump. From Stuhn et al. [323],...

Simultaneous polymerization of two monomers by chain initiation usually results in a copolymer whose composition is different from that of the feed. This shows that different monomers have different tendencies to undergo copolymerization. These tendencies often have little or no resemblance to their behavior in homopolymerization. For example, vinyl acetate polymerizes about twenty times as fast as styrene in a free-radical reaction, but the product obtained by free-radical polymerization of a mixture of vinyl acetate and styrene is found to be almost pure polystyrene with hardly any content of vinyl acetate. By contrast, maleic anhydride, which has very little or no tendency to undergo homopolymerization with radical initiation, readily copolymerizes with styrene forming one-to-one copolymers. The composition of a copolymeir thus cannot be predicted simply from a knowledge of the polymerization rates of the different monomers individually. The simple copolymer model described below accounts for the copolymerization behavior of monomer pairs. It enables one to calculate the distribution of sequences of each monomer in the macromolecule and the drift of copolymer composition with the extent of conversion of monomers to polymer. 

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