The rotational isomeric state theory

Flory (1969 1971 1974) has developed the rotational isomeric state theory for predicting the conformation of polymer molecules. This incorporates the interdependence of the bond rotational potentials. This theory is fully explained in Flory s comprehensive monograph (Flory, 1969) and only the barest details will be mentioned here. 

The rotational isomeric state theory is essentially a matrix procedure that, in princi e, allows all of the myriad of conformations accessible to a polymer molecule to be appropriately averaged. The end-to-end length r is simply the sum of the individual bond vectors I, 

These bond vectors are readily expressed in their own coordinate system as the column vector col (/ 0 0). To find r, however, the vectors must all be expressed in the same coordinate system, which we will take to be that of the first bond (see Fig. 4.6). This can be accomplished by using the transformation matrix T,. Premultiplication by this matrix transforms a vector expressed in reference frame (/ +1) into its representation in reference frame i of the preceding bond  

The above expression can be written in a much more compact form by introducing generator matrices A,-, the serial multiplication of which generates the unwieldy sum in equation (4.19). Thus 

Here i denotes the serial index of the matrix T and bond vector I within the generator matrix. The superscript (n - 2) in equation (4.20) denotes the serial multiplication of (n—2) matrices beginning with A2. The initial matrix A[i is defined by the first row of Ai and the final matrix A ) is correspondingly the last column of A . 

---

A valuable reference on the subject of single chains is the work by Flory, described in his book Statistical Mechanics of Chain Molecules. A new book on the subject is of interest. The reader may also wish to study the 1974 paper by Flory in Macromolecules, which serves as an excellent introduction to the rotational isomeric state theory.Another classic book is Hopfinger s Conformational Properties of Macromolecules... 

Yoon and Flory > have compared the experimental data of Gawrisch et al. to the scattering of unperturbed chains calculated on the basis of the rotational isomeric state theory. They found good agreement up to the highest values of the scattering vector corresponding to distances of approximately one chain diameter... 

The valence angle model, though more realistic than the freely jointed model, still underestimates the true dimensions of polymer molecules, because it ignores restrictions upon bond rotation arising from short-range steric interactions. Such restrictions are, however, more difficult to quantify theoretically. A simpler procedure is to assume that the conformations of each sequence of three backbone bonds are restricted to the rotational isomeric states that correspond to the potential energy minima such as those shown for n-butane in Fig. 2.3. For the simplest case of polyethylene and for vinylidene-type polymers, the application of the rotational isomeric state theory yields the following equation... 

P. J. Flory, Macromolecules, 7, 381 (1974). Foundations of the Rotational Isomeric State Theory and General Methods for Generating Conformational Averages. 

Equations (83) and (84) provide a molecular interpretation of the thermoelastic data through equation (89). This equation establishes the relationship between the purely thermodynamic quantity f /f and its molecular counterpart of dlno/dT, which can be interpreted in terms of the rotational isomeric state theory of chain configurations. It permits comparison of the change of the unperturbed dimensions o obtained by thermoelastic measurements on polymer chains in the bulk (in network structures) with that obtained by viscosity measurements on chains of the same polymer, essentially isolated in dilute solution. 

The conformational energies are largest (2000 cal/mol) for the 1276-1264-cm bands and range from 1080 to 1283 cal/mol for the 1195-1168-and 1152-1140-cm bands. These data have been interpreted in terms of the rotational isomeric state theory. The IR spectroscopic approach is straightforward and is useful for obtaining information of this type. 

W.L. Mattice and U.W. Suter, Conformational Theory of Large Molecules The Rotational Isomeric State Model in Macromolecular Systems, Wiley, New York, 1994. 

Mattice WL, Suter UW (1994) Conformational theory of large molecules the rotational isomeric state Model in macromolecular systems. Wiley, New York... 

Mattice WL, Suter UW (1994) Conformational theory of large molecules The rotational isomeric state model in macromolecular systems, Wiley, New York Moritani T, Fujiwara Y (1973) J Chem Phys 59 1175 Nagai KJ (1959) J Chem Phys 31 1169 Nagai KJ (1962) J Chem Phys 37 490 Natta G, Corradini P, Ganis P (1962) J Polym Sci 58 1191 Suter UW (1981) Macromolecules 14 523 Viswanadhan VN, Mattice WL (1987) Macromolecules 20 685 Volkenstein M (1958) J Polym Sci 29 441 Volkenstein MV (1951) Dokl Akad Nauk SSSR 78 879... 

Extremely detailed models for unperturbed flexible linear homopolymers can be constructed using rotational isomeric state theory.[14,20 In the limit as n — °°, these detailed models yield... 

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-... 

There are two broad classes of dihedral space models for studying and building flexible chain molecules. The first, known as Rotational Isomeric State Theory (RIS) [25], was developed to study industrial polymers. The second, known as Chain Buildup (CB) [26], was... 

From consideration of molecular models and the geometric requirements for excimer formation, we have concluded that the trans, trans meso rotational dyad is the predominant EPS in the aryl vinyl polymers. [5,22] The identification of the EPS trap with a particular rotational dyad state was a critical factor because it opened the way to utilize the powerful rotational isomeric state theory of Flory [23] to calculate the EPS population for the isolated PS chains. This trap population is relatively small in polystyrene. 

---