Cyclization theory, Jacobson and Stockmayer

The molar cyclization equilibrium constants, Kx, of PDMS are measured. Using the Jacobson and Stockmayer equilibrium theory of macrocyclization, the dimensions of PDMS chains with 40-80 chemical bonds in the bulk polymer at 383 K are deduced. Dilution effects in the PDMS systems are contrasted with predictions of the Jacobson-Stockmayer theory, and the experimental molar cyclization equilibrium constants of the smallest siloxane rings are discussed in terms of the statistical properties of the corresponding oligomeric chains using tire RIS model of PDMS of Flory, Crescemi, and Mark [S 116]. 

In their classic paper published in 1950, Jacobson and Stockmayer (21) presented a theory giving a precise expression for the concentrations of macrocyclics Af, in ring-chain equilibrates. The molar cyclization equilibrium constants Kx were given by... 

Experimental molar cyclization equilibrium constants for cyclics in the PDA n%lt and the P1 melt at 423 K are own dotted as log gainst Ic x in Fig. 12 and 13. They are compared with theoretical values calculated by the Jacobson and Stockmayer ex esrion Eq. (6) with - 2jc- Values of < / >q required by this expression were comfHited by the exact mathematical methods of Flory and Jemi-gan 37,30) using the rotational isomeric state models for the polyesters set up by Flory and Williams 27,128). Agreement between experiment and theory is excel-... 

Fig. IS. Experimental molar cyclization constants Kx (in mol dm ) for cyclics (CO C6H4 CO O CHj CHj 0) t with X = 2-9 in the melt at 543 K (delated o) are plotted as log against logx. They are compared with theoretical values calculated by the Jacobson and Stockmayer theory assummg that the corresponding open chain molecules obey (Jausrian statistics (these ralues are denoted )...

The theory of cyclization developed by Jacobson and Stockmayer describes the following equilibrium ... 

The Jacobson-Stockmayer cyclization theory is based on the assumption that all rings are strainless and that the conformational probability of ring closure is given by Eq. 3-2, i.e. conformational restrictions or preferences are absent. This assumption apparently cannot be fulfilled in real systems for small rings (as shown in Sect. 3.2.2). In some systems this leads to a lowering of the concentration of cydics when compared with their equilibrium concentration. This is because the probability of small ring closure is reduced due to the strain caused not only by bond angle deformation but also by bond opposition and transannular interactions. In such a case, as in the discussed earlier l,3-dioxolane-BF3 system, the concentration of small cycles (up to 25-30 bonds) is lower than the calculated one 141... 

Fig. 11. Experimental molar cyclization equilibrium constants Kx (in mol dm-3) in undiluted (o) and solution ( ) equilibrates of poly( 1,3-dioxolane) at 333 K compared with values calculated (x) by the Jacobson and Stockmayer theory...

Jacobson and Stockmayer analyzed such types of equilibria and formulated a theory describing dependence of the cyclization equilibrium constants on the ring size. 

Jacobson and Stockmayer analyzing the density of distribution of segments of linear polymer formulated a theory relating the distribution function of the end-to-end veaor r for linear polymer composed of x units (Wx(r)), per unit range in r, with the equilibrium constant of cyclization for systems without ring-strain and other enthalpic factors (equilibrium governed only by entropic factors) ... 

These and related phenomena can be explained in terms of the thermodynamic theory of macrocyclics distribution, formulated by Jacobson and Stockmayer9) and its kinetic extension 10). The Jacobson-Stockmayer theory, relating the distribution of cyclic oligomers to the conformational probability of ring closure, does not take into account kinetic limitations and has mostly been used as a convenient tool for studying the conformation of macromolecules in solution s). A number of papers appeared in which distribution of cyclic oligomers was studied with this aim and which ignored mechanistic and kinetic aspects of the cyclization processes. 

In the traditional gelation theory formulated by Flory (1) and Stockmayer (2), it is assumed that like functional groups are equally reactive and all reactions occur intenaolecularly before the gel point. Subsequently, beyond gelation, finite species formed in the sol portion are limited to acyclic trees. This is not correct because intramolecular reaction leading to the formation of ring structures in a random polycondensation must occur. The effect of cyclization has been treated by various approaches, e.g. JacObson-Stockmayer ring-chain factors (3), cascade theory (4), and rate theory... 

EXAMPLE 32.2 Polymer cyclization (Jacobson-Stockmayer theory). What is the probability that the two ends of a polymer chain come close enough together for them to react with each other This probabihty is useful for calculating the rate and equilibrium constants for cyclization processes in which linear chains form circles (see Figure 32.8). Here we follow the random-flight model of H Jacobson and WH Stockmayer (4). 

Cyclic oligomers with x - 2-9 are found to be present in poly(1,3-dioxolane) samples prepared by monomer-polymer-equilibrations using boron trifluoride diethyl etherate as catalyst. The molecular cyclization equilibrium constants 7fx are measured and the values are in agreement with those calculated by the Jacobson-Stockmayer theory, using an RIS model to describe the statistical conformations of the corresponding chains and assuming that the chains obey Gaussian statistics. 

A theoretical model to determine the probability of loop formation, based on an elaborated form of the Jacobson-Stockmayer theory of cyclization equilibria, is developed and used on RNA chains of homogeneous puckering and lengths up to 2 residues. 

This apparent duality has led to certain misunderstanding and confusion and for the reader less acquainted with this field the relation between theoretical distribution predicted by Jacobson-Stockmayer theory and the distributions observed in real systems (frequently different from those predicted by the theory) could be not clear. Thus, in this section we describe the theories and then explain major experimental results in terms of the thermodynamics and kinetics of cyclization. 

In such a system concentration of each oligomer should pass through a maximum (provided that cyclization is fast enough to compete with chain growth) and then decreases approaching the equilibrium concentration predicted by the Jacobson-Stockmayer theory 10). On the other hand, when random back-biting is the only... 

The ring-opening polymerization of D4 is controlled by entropy, because thermodynamically all bonds in the monomer and polymer are approximately the same (21). The molar cyclization equilibrium constants of dimethylsiloxane rings have been predicted by the Jacobson-Stockmayer theory (85). The ring—chain equilibrium for siloxane polymers has been studied in detail and is the subject of several reviews (82,83,86—89). The equilibrium constant of the formation of each cyclic is approximatdy equal to the equilibrium concentration of this cyclic, Kn [(SiRjO) J. Thus the total concentration of cyclic oligomers in the equihbrium is independent of the initial monomer concentration. As a consequence, the amount of linear polymer decreases until the critical dilution point is reached, at which point only cyclic products are formed. 

In Fig. 20, experimental molar cyclization equilibrium constants Kx for sodium metaphosphates (NaPC>3)x in molten sodium phosphate at 1000 K are compared with theoretical values (2 76). The latter were calculated by the Jacobson-Stockmayer theory (21) assuming that chains of all lengths obey Gaussian statistics and using... 

Figure 32.9 (a) For long chains the DNA cyclization probability diminishes as predicted by the Jacobson-Stockmayer theory. Shortening the chain stiffens it, reducing the cyclization probability below the Jacobson-Stockmayer value, (b) At higher resolution the cyclization probability depends on periodicities in the polymer, neglected in the random-flight treatment. Source D Shore and RL Baldwin, J Mol Biol 170, 957-981 (1983). 

This is an important result, because an agreement between the observed distribution of the cyclic oligomers with a distribution predicted by the Jacobson-Stockmayer theory (a slope equal to 2.5 for the plot of log K on log X, where K is the molar cyclization-equilibrium constant for macrocycles with a polymerization degree equal to x) strongly indicates that polymerization proceeds with a linear active species, forming macrocycles by back-biting and end-to-end closure. 

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