Thermal Analysis of Polymeric Materials

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3.2.6 Matrix Reactions

The chain-reaction polymerizations in Sects. 3.2.4 and 3.2.5 are methods that can be used to make narrower molar mass distributions than are possible with step reactions or chain reactions with random termination. This problem of producing “pure polymers” was mentioned above when discussing the step-wise reaction in Sect. 3.1.

Matrix polymerization is known from the biopolymerization of globular proteins and represents the ultimate in precision of not only molecular length, but also composition. A messenger RNA, attached to a ribosome, serves as the matrix (template) for protein syntheses. A matrix is shown schematically in Fig. 3.34. The

Fig. 3.34

specific monomers, Mx, can be absorbed only on the matrix sites indicated in step 1. Once adsorbed to these specific sites, as depicted in step 2, polymerization can begin in a matrix-controlled sequence as indicated in step 3. The matrix places the monomers not only into proper sequence, it also activates them for reaction. Step 3, furthermore, indicates that on polymerization the adsorption is weakened due to the change in molecular structure, so that the completed molecule is easily removed as seen in step 4, freeing the matrix for a new polymerization. The protein synthesis is not spontaneous. It needs the driving force (loss of free enthalpy) of a coupled reaction of adenosine triphosphate, ATP, to adenosine diphosphate, ADP, to execute the polymerization. A schematic of the ATP structure is shown in Fig. 3.35. Matrix control of polymerization of synthetic polymers remains, at present, of little importance for the production of homopolymers and copolymers, but provides a goal in the quest for more precise macromolecules for special applications and specificity. Initial efforts to make use of biological matrices of bacteria which have been altered to polymerize a specific set of synthetic monomers have been successful.

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Fig. 3.35

3.3 Molecular Mass Distributions

3.3.1 Number and Mass Fractions, Step Reaction

In the present section, quantitative molecular mass distribution and the equations for their averages will be derived. The basics of the description of mass distributions are discussed in Sect. 1.3 with Figs. 1.24 to 1.26, and the experimental methods of measurement are treated in Sect. 1.4.

Figure 3.36 gives the schematic of the step reaction, as discussed in Sect. 3.1. It must be noted, that considering only the forward reactions is a gross simplification, as discussed in the LiPO3 example of Sect. 3.1.6. Even more important, the chain-to- ring, ring-to-chain, and ring-to-ring reactions should be included. When in time, all reactions have reached thermodynamic equilibrium, all monomers should have ended up in a broad distribution of large rings since a large ring must be more stable than an equivalent chain with two active ends. When simplifying step polymerization to the idealized kinetics by treating chain-to-chain reactions only, i.e., assuming that all other reactions are negligible, the mole fraction of mers of length x is easy to find. It is simply the chance that a molecule taken randomly out of the mixture of all molecules has the length x. The number of molecules N, that make up the mixture at a given conversion is defined in Fig. 1.25, and linked to the probability, p, in Fig. 3.36. The initial number of monomer molecules is given by No and the molar mass of the repeating unit is Mo, leading to a number-average molar mass of Mn = Mo /(1 p), as seen from Mn in Fig. 1.25 and insertion of the expression for N/No. Because of the time-dependent approach to equilibrium of all reactions, the distributions from Fig. 3.36 cannot be precise, and are in need of comparison with experiments.

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Fig. 3.36

The compound probability for the chance to randomly pick a specified molecule contains three factors in Fig. 3.36. a. The chance to find a molecule (= 1 since it is assumed that we have already picked one of the molecules—note that each molecule has the same chance of being picked, independent of size!). b. The probability that x 1 bonds are made between its constituent repeating units (= px 1, needed so that the molecule is an x-mer). c. The probability that the xth bond is not made [= (1 p), since one additional bond would lead to an (x + 1)-mer]. The probability to have a randomly chosen molecule which is an x-mer is the product of the three factors. It is also equal to the number or mole fraction of x-mers, nx. The mass fraction is calculated from the equation for the mass fraction in Fig. 1.26. The result in Fig. 3.36 is derived by inserting the appropriate expressions, knowing that Nx = nxN.

Next, one can follow the development of the molecular mass distribution for the step reaction as the reaction proceeds from pure monomers, p = 0, reaching almost completion, p = 0.99. Figure 3.37 shows a sequence of steps in the reaction. As already suggested in Sect. 3.1, the monomer concentration decreases quickly, but macromolecules develop very slowly. By the time the reaction is to 60% completed, the dimer is most common, as indicated by the maximum in the distribution curve. As p reaches 80%, the maximum has moved to the tetramer and pentamer. Only when driving the reaction close to completion is the most common species a true macromolecule, as shown in Fig. 3.38. Typical step-reaction polymers, such as the polyesters and polyamides of Sect. 3.1 are often reaching higher molar masses only because of the relatively large molar mass of the monomer. Also, note that the distribution of molecular masses is rather broad, leading to low concentrations of each individual species. To represent Fig. 3.37, it was necessary to multiply the scale of the ordinate of Fig. 3.37 by a factor of 4×104. In Sect. 3.3.3 it will be shown that the polydispersity of the step-reaction approaches two on completion of the reaction.

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Fig. 3.37

Fig. 3.38

3.3.2 Number and Mass Fractions, Chain Reactions

The molecular mass distribution of chain reactions was treated in Fig. 3.31 for living polymers. All active chain ends have in this case the same chance to add a new monomer. If all initiations have occurred at the same time, the chances are equal for

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each molecule to reach an average length . The mole fraction nx is then approximated by the Poisson distribution given in Fig. 3.31. For all other types of chain reactions the details of initiation, transfer, and termination determine the distribution, as will be described below. The number of x-mers Nx and the mass fraction wx can be written as shown in Fig. 3.39 (Nx = Ninx, where Ni is the number of initiator molecules). In the derivations, nx refers to the mole fraction of molecules of x monomers. To account for the initiator, x must be replaced by x 1 since the presence of the initiator unit is certain, and the new x-mer definition agrees with the molecules of the step-reaction example in Fig. 3.36. The initiator mass may be taken equal to the monomer mass, neglected, or it can be corrected for.

Figure 3.39 displays the change in molecular mass distribution after 40% of the assumed 900 initially present monomer molecules have reacted on the 100 initiator sites, i.e., the average chain length is 4.60 chain repeating units. Note that it is also

Fig. 3.39

assumed that the initiator mass is negligible. Naturally, the residual monomer is the predominant species. Further reaction steps are displayed in Fig. 3.40. Besides the shift of the distribution of oligomers to longer molecule lengths, there is, in contrast to the step reaction of Fig. 3.37, a much slower decrease in monomer. As long as the reaction is incomplete (p < 1.0), considerable monomer remains. The amount of unreacted monomer is given by (1 p)No, and one can compute the average chain length, , as shown in Fig. 3.39. By disregarding the unreacted monomer, can be computed for polymerized molecules only (by summing from x = 2 and replacing No with the reacted monomer, Nop). Beyond the monomer concentration, the oligomer distribution develops as in the step reaction, but it is much narrower. One notices the initial, quick development of dimers and trimers, which, at later time, decrease in mass fraction as longer molecules begin to dominate. Note, that while the step reaction is

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3.3.3 Step Reaction Averages

The next two sections contain a discussion of the computations needed to evaluate average molar masses from the derived distributions of Sects. 3.3.1 and 2. The derivations are simple, but lengthy, so that it is given for the first example only, using an abbreviated summary. The characterization of the molar mass distributions is given by the polydispersity Mw /Mn as discussed in Sect. 1.3 (Fig. 1.26). It approaches 2.0 for the broad molecular mass distributions of the step reactions as p becomes 1.0, and should be close to 1.0 for living polymers of sufficiently high molar mass. In the latter case, the high molar mass is achieved by low initiator concentration and completion of the reaction. Actual samples may, however, show a much higher polydispersity. Particularly chain reactions with complicated initiation, termination, and transfer reactions may show polydispersities as high as 20 or more, as will be shown below.

First, Fig. 3.42 shows how to start from the equations of Fig. 3.8 to evaluate the simple equation for the number average molar mass of a step reaction. More difficult is to establish the mass average molecular mass. Starting from the mass-distribution

Fig. 3.42

equation of Fig. 3.36, one needs to sum over all species of x, as shown in Fig. 3.42. This sum is a geometric series that can be written as:

Mw = Mo (1 p)2 [ 1 + 4 p + 9 p2 + . . .] .

The sum in brackets, x2 px 1, can, however, only easily be summed for large values of x and without the factor x2. To achieve the latter, one writes the differential of the integral of x2 px 1 as shown in the second line of the derivation. The integration constants, c, disappear when ultimately the differentiations are carried out. The first

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integration reduces the factor x2 to x and yields, after reorganization and a second differentiation and integration, the sum px 1 which can be evaluated easily for large values of x, as shown in the boxed part of Fig. 3.42. Next, the two differentiations are carried out. The resulting equation is quite simple, proving the approach of the polydispersity, Mw /Mn, to the value of two as soon as the degree of completion of the reaction approaches one. This polydispersity of two was suggested above as a characteristic of the step reaction.

3.3.4 Chain Reaction Averages

To assess molecular mass distributions of chain reactions, considerably more detail must be known about the mechanism than in the case of step reactions. As the reactions progresses, the length distribution of the polymer molecules changes increasingly from the simple case of fixed number of initiator molecules without termination, described above for the living polymers. Constantly some initiator is newly activated and growing molecules are terminated. One can compute a kinetic chain length only during the short time period of steady state in Fig. 3.29, as illustrated in Fig. 3.43. The chain length is the ratio of propagation to termination rates as

Fig. 3.43

defined in Sect. 3.2. The probability of termination, 1 p, is shown for the case of termination by coupling of two molecule radicals. It is the termination rate divided by all reaction rates at the given moment. Analogous equations must be written for other termination and transfer reactions. The key point remains that even if the mechanism is known, the averages will change with time. One can approach a steady state that produces a fixed molecular mass distribution only in a flow reactor with constant monomer addition, polymer removal, and initiator replenishment. The

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computation of the distribution of the mole fractions of molecules of length x is, again, based on a probability argument, as displayed in the last two lines of Fig. 3.43 and can be compared to the similar equations for the step reaction in Fig. 3.36.

The top equation in Fig. 3.44 lists the number and mass averages for chain reactions terminated by the combination of two growing molecule-radicals as described in Fig. 3.43. The derivation of the equation must be done in the same way as illustrated for the step reaction in Fig. 3.42. The next example, shown in Fig. 3.44, applies to termination by chain transfer (see Sect. 3.2). In this case the termination

Fig. 3.44

removes one polymer molecule but starts at the same time a new one. Again, the derivations of nx and wx are done as before and the summation is the same as for the step reaction in Fig. 3.42. The final averages are those derivable from the Poisson distribution which describes living polymers. Again, the limitations of the just derived equations must be kept in mind.

3.4 Copolymerization and Reactions of Polymers

The multiplicity of possible linear macromolecules is enormously increased by copolymerization, i.e., by polymerizing more than one monomer. Because it is possible to vary concentration as well as placement of the various monomers, as discussed in Sect. 1.2, there are unlimited numbers of different polymers, and it is impossible to ever produce all polymers of promise to check their properties and pick optimum materials. One must develop the ability to predict polymer properties before making the actual molecules. This prediction capability is one of the key goals of the study of polymers. It must start with a critical evaluation of data of the physical