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CONFORMATION AND CONFIGURATION Introduction Scientists have devised many models for treating the enormous number of conformations accessible to a flexible chain molecule. Most of these models pay little attention to the real covalent structure of the chain. Therefore they are ill suited to the development of a thorough understanding of why one polymer behaves differently from another. The rotational isomeric state model is unique in that it describes the conformation-dependent properties of the chain in terms of the real covalent structure, and does so in a computationally efficient manner. The data fed into the model include structural information (lengths of bonds, angles between successive bonds, values of the preferred torsion angles) and energetic information (contributions of short-range intrachain interactions to the preferences for specific values of the torsion angles, and specific pairs of torsion angles at neighboring bonds, ie, the interdependence of the torsions). This information is processed in a manner that rapidly gives the average of chain properties, such as the mean square unperturbed end-to-end distance r2 0 , with the average being performed over all of the accessible conformations. The rotational isomeric state model is not among the newer models used in polymer science. Its earliest application to polymers was reported half a century ago (1), using mathematical techniques invented 10 years earlier (2). Applications of the model to polymers received a strong boost from Flory’s group in the 1960s, leading to the publication of his classic book on the subject at the end of that decade (3). Another book, published 25 years later, updates the techniques and applications (4). The literature now contains rotational isomeric state models for literally hundreds of polymers. A few hundred of the models that appeared in Encyclopedia of Polymer Science and Technology. Copyright John Wiley & Sons, Inc. All rights reserved.

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the literature up to the early 1990s have been tabulated in a standard format in lengthy reviews (5,6). The classic use of the rotational isomeric state model is for the rationalization of the mean square unperturbed dimensions of a long-chain molecule, as measured either in dilute solution in the absence of excluded volume (ie, in a  solvent) or in the bulk amorphous state (7). A much simpler model for such a chain uses random flight statistics to write a temperature-independent r2 0 as the product of a term N that is directly proportional to the degree of polymerization and another term L2  that specifies the mean square step length of a segment. r 2 0 = NL2 

(1)

The counterpart of equation (1) in the rotational isomeric state model also has a dependence on the degree of polymerization, through the use of a serial product over all n bonds in the chain. r 2 0 = Z − 1 F1 F2 · · · F2

(2)

Here Z is the conformational partition function, which contains information about the temperature T and the energies of all of the accessible conformations of the chain. The Fi are matrices that combine geometric information, namely the length of bond i, li , the angle between bonds i and i + 1, θ i , and the perferred torsion angles at bond i, φ i , with the energetic information contained in Z. The rotational isomeric state model is more realistic than the freely jointed chain model because it includes temperature, different energies for the various preferred conformations, and the geometry of these conformations. The present description of the rotational isomeric state model has several parts. First, we will describe the formulation and uses of Z, which initially restricts the focus to the thermodynamic (energetic) part of the rotational isomeric state model. Then we will describe how the structural information (li , θ i , φ i ) is incorporated in the model. Finally, we will combine the thermodynamic and the structural information, which will take us back to equation (2). Along the way we will mention a few other properties, in addition to r2 0 , that can be successfully rationalized with the rotational isomeric state model. Finally, several illustrative applications will be presented. Although developed primarily for the analysis of the conformation-dependent physical properties of flexible chain molecules in the  state (essentially a singlechain problem, due to the neglect of excluded volume), the rotational isomeric state model has important applications in other types of problems. It has been used for generation of detailed models for glassy atactic polypropylene at bulk density (8). The rotational isomeric state model is also instrumental in the simulation of the mixing of structurally similar polymers, as illustrated also by polypropylene, where melts of isotactic and atactic polypropylene are miscible, but melts of isotactic and syndiotactic polypropylene are immiscible (9,10). Space limitations will preclude extensive description of these more advanced applications of the rotational isomeric state model.

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The Conformational Partition Function Z Although not in any way restricted in application to polyethylene, it is nevertheless useful to adopt this polymer for purposes of illustration because the conformational properties of its small oligomers are likely to be known to the reader. The discussion conveniently begins with the central C C bond in n-butane, CH3 CH2 CH2 CH3 . The conformational energy, which is a continuous function of the torsion angle at this bond, has been written using the cosine of the torsion angle (11). U(φ) = 11.8 + 7.66cosφ + 4.64cos2φ + 8.8cos3φ

(3)

Here U(φ) is in kJ·mol − 1 and φ = 0 in the cis conformation. This function, which is depicted in Figure 1, has the three maxima and three minima summarized in Table 1. When the populations are distributed according to exp(−U(φ)/RT), and T has values that are likely to be of interest, there will be a significant population of φ near all three minima, but the populations in the vicinity of the maxima will be very small, as shown in Figure 2. This observation suggests a simplification in which the continuous range for φ is approximated by a set of three discrete values corresponding to the three regions of φ that have significant population. These three regions are named trans and gauche± in n-butane. They are often abbreviated as t and g± . The normalized populations of these three regions are given in the last column of Table 1. The statistical weight of g+ (or g − ) relative to t is denoted by σ . The value of σ is temperature-dependent. It is less than 1/3 at finite T because the regions near φ = ±70◦ are of higher energy than the region near φ = 180◦ . The term in parentheses in the last column of Table 1 (1 + 2σ ) is the conformational partition function (or sum of statistical weights for

U(␾), kJⴢmol−1

30

20

10

0

0

60

120 180 240 Torsion Angle ␾, deg

300

360

Fig. 1. Torsion potential energy function for the central C C bond in n-butane. From c 1994 John Wiley & Sons, Inc. Reprinted by permission of John Wiley Ref. 4. Copyright
& Sons, Inc.

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Table 1. Maxima and Minima in the Torsion Potential Energy Function for the Internal C C Bond in n-Butane φ, deg

U, kJ·mol − 1

U/kT at 300 K

Population

Minimum Local maximum Local minimum Maximum

0 ∼14 ∼3 ∼33

0 ∼6 ∼1 ∼13

1(1 + 2σ ) − 1 ∼0 σ (1 + 2σ ) − 1 ∼0

Normalized Population

180 ∼(±121) ∼(±70) 0

Maximum or minimum

300 K 400 K 500 K

0.2

0.1

0

0

60

180 240 120 Torsion Angle ␾, deg

300

360

Fig. 2. Normalized populations of the torsion angle at the central C C bond in n-butane c 1994 John Wiley & Sons, Inc. Reprinted at 300, 400, and 500 K. From Ref. 4. Copyright
by permission of John Wiley & Sons, Inc.

all conformations) for n-butane in the rotational isomeric state approximation. The probability of any state is its statistical weight ui divided by the sum of all statistical weights, ie, the conformational partition function pi = Z − 1 ui ,
Z= ui . Extension of this concept to a longer alkane suggests three conformations for each internal C C bond, or 3n − 2 conformations for a chain of n bonds. If the bonds were independent of one another, the conformational partition function would be (1 + 2σ )n − 2 . However, interdependence of neighboring torsions destabilizes some of the conformations, with important implications for their statistical weights. The most important interdependence in the n-alkanes is the interaction known as the pentane effect because n-pentane is the smallest alkane in which it is seen. If the two internal C C bonds of n-pentane adopt g states, only the two conformations with g states of the same sign have conformational energies near 2Eσ , which is the prediction based on the assumption of independent bonds. If the g states are of opposite sign, however, the conformational energy is higher than this prediction by nearly 7.5 kJ·mol − 1 because of the repulsive interaction of the terminal methyl groups in these two conformations. The conformational energy of these two conformations is 2Eσ + Eω , with Eω about 7.5 kJ·mol − 1 . The conformational

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partition function for n-pentane is not (1 + 2σ )2 , as would have been the case if the two internal bonds were independent, but is instead given by 1 + 4σ + 2σ 2 (1 + ω). The influence of the interdependence of neighboring bonds is easily incorporated in the formulation of the conformational partition function for long chains. The method employs a statistical weight matrix U for each bond. For bond i somewhere within a long chain, Ui has the following composition: (1) The number of columns is given by the number of rotational isomeric states at bond i, ν i ; (2) The number of rows is given by the number of rotational isomeric states at bond i − 1, ν i − 1 ; (3) Every element in column η contains the statistical weight required by the first-order (dependent on one torsion) interaction when bond i is in the state indexed by that column; (4) Every element in row ξ of column η also contains the statistical weight required by the second-order (dependent on two torsions) interaction when bond i is in the state indexed by that column and bond i − 1 is in the state indexed by that row. For polyethylene, the first two points specify dimensions of 3 × 3 for Ui . Assuming the order of indexing of rows and columns is t, g+ , g − , the third point specifies the presence of σ in all elements in the second and third columns. The fourth point specifies the appearance of ω in the 2,3 and 3,2 elements. Therefore the statistical weight matrix for polyethylene is the 3 × 3 matrix in equation (4) (12). 

 1 σ σ   Ui =  1 σ σ ω  1 σω σ

(4)

i

More generally, Ui is formulated as the product of a diagonal matrix Di with dimensions given by point (1), and with the statistical weights for the first-order interactions on the main diagonal, in the order required by the indexing of the columns of Ui (13). The second-order interactions occur in a matrix Vi (which has dimensions given by points (1) and (2)). For polyethylene, the Ui in equation (4) can be generated as the product of the Vi and Di defined in equation (5).   1 0 0 1 1 1    Ui = Vi Di =  1 1 ω   0 σ 0  0 0 σ 1 ω 1 

i

(5)

i

Z for n-butane is given by the sum of the elements in the top row of U2 , and Z for n-pentane is the sum of the elements in the top row of U2 U3 . In general, the contribution of bonds 2 through n−1 to Z is given by the sum of the elements

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in the top row of the serial product U2 U3 · · · Un − 1 . This sum can be extracted by appending an initial U1 defined as [1 0 · · · 0] and a terminal Un that is a column in which all elements are 1. Z = U1 U2 · · · Un

(6)

Application of this equation yields Z = 1 + 2σ for n-butane and Z = 1 + 4σ + 2σ 2 (1 + ω) for n-pentane, as expected. It correctly incorporates the first-order interaction and the pentane effect for longer chains. Interactions of higher order (depending on three or more successive bonds) are not included. This reliance on short-range (first- and second-order) interactions means that the expression cannot be expected to apply when longer range interactions play an important role, as they do under circumstances where excluded volume is important. The treatment based on short-range interactions is appropriate for the  state, where excluded volume is not important. The serial product in equation (6) implies that the Ui need not all be identical. The only restriction on their relationship is that all pairs must be conformable for matrix multiplication. Conformability is ensured by the requirements expressed in the first two points before equation (4). Equation (6) can be used with chains in which different types of bonds are present, as in polyoxyethylene (14). It can also be used for chains in which not all bonds have the same number of rotational isomeric states, as in the polycarbonate of bisphenol A (15). The standard operations of statistical mechanics permit extraction of useful information from Z. The temperature dependence of Z gives the amount by which the average conformational energy and conformational entropy exceed their zero values. E − E0 = kT2 (∂ ln Z/∂ T)

(7)

S= (E − E0 )/T + k ln Z

(8)

The probability that bond i will be in state η, denoted by pη;i , is calculated with a matrix Uη;i  that is obtained from Ui by zeroing out all of the elements except those in the column indexed by state η at this bond (16).  i − 1   n

U (9) U U pη;i = Z − 1 j k η;i j=1 k = i+1 Equation (9) evaluates the ratio of the sum of the statistical weights for all conformations in which bond i is in state η to the sum of the statistical weights of all conformations, ie, Z. If bond i is in the middle of a long polyethylene chain, the combination of equations (4), (6), and (9), with σ = 0.543 and ω = 0.087 (as appropriate for a temperature of about 423 K), yields three pη;i that have 1 as their sum. pt;i = 0.596

(10)

pg+ ;i = pg − ;i = 0.202

(11)

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The t state is preferred in the melt at this temperature. The symmetry of the torsion potential energy function in equation (3) for the C C bond in polyethylene, U(φ) = U(−φ), produces pg+ ;i = pg − ;i . Knowledge of all of the pη;i is often useful in the interpretation of conformation-dependent spectral properties that have a local origin, such as the chemical shift and coupling constants in nmr spectroscopy (17). The probability that bonds i and i − 1 are in states η and ξ , respectively, is computed with a Uξ η;i  that is obtained from Ui by zeroing out all of the elements except the single element at row ξ , column η. pξ η;i = Z − 1

 i − 1

  n

U U U j k ξ η;i j=1 k = i+1

(12)

The numerical values of pξ η;i are conveniently presented in the form of a matrix that has the same dimensions as Uξ η;i  . For the same case described in equations (10) and (11), this matrix has elements with the numerical values presented in equation (13). 

 0.321 0.138 0.138 pξ η;i =  0.138 0.0591 0.00516  0.138 0.00516 0.0591 i

(13)

Summation of the elements in the individual columns reproduces the results in equations (10) and (11). The result ptg+ ;i = ptg − ;i = pg+ t;i = pg − t;i arises from the symmetry of the torsion potential energy function and the absence of a distinguishable direction along the polyethylene chain. The interdependence of the bonds makes the probability for two successive g states dependent on the relationship of their signs, pg+ g+ ;i = pg − g − ;i > pg+ g − ;i = pg − g+ ;i . Another useful probability can be derived from the previous equations. Given that there is state ξ at bond i − 1, qξ η;i is the probability for finding state η at bond i. qξ η;i = pξ ;i / pξ η;i − 1

(14)

The numerical results in equation (13) specify nine values that can be presented in matrix form. 

 0.538 0.231 0.231 pξ η;i =  0.682 0.292 0.026  0.682 0.026 0.292 i

(15)

The sum of the three elements in each row is 1 in equation (15), whereas the sum of all nine of the elements is 1 in equation (13). The probability for conformation κ in a long chain (where κ is a subscript that uniquely defines that

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conformation) is given by the product of the appropriate pη;i for bond 2 and the appropriate qξ η;i for all subsequent bond pairs. pk = pα;2 qαβ;3 qβγ ;4 · · · qψω;n − 1

(16)

Equation (16) can be employed to generate representative samples of chains for purposes such as the evaluation of the distribution function for the end-to-end distance (18).

Geometry of Individual Chain Conformations Equation (6) writes Z as a serial product of n statistical weight matrices, one matrix for each bond in the chain. The information in Z specifies the probability for each and every conformation of the chain, via equation (16). In preparation for the computation of properties such as r2 0 (where the average is over all conformations), it is desirable to formulate r2 (for a single conformation) as a serial matrix product, just as Z was expressed as a serial matrix product in equation (6). The end-to-end vector r for a chain is often written as a sum of bond vectors. r = l1 + l2 + · · · + ln

(17)

The form adopted in equation (17) assumes that r and all of the li are expressed in the same coordinate system. Specification of the elements in any one of the li requires knowledge of the length of the bond and its orientation in the coordinate system common to all li . The rotational isomeric state model uses a different approach. Each li is expressed in a local Cartesian coordinate system with axis xi along bond i, axis yi in the plane of bonds i − 1 and i, with a positive projection on bond i − 1, and zi completing a right-handed Cartesian coordinate system. The x and y axes for the local coordinate systems of the first two bonds are depicted in Figure 3. In its own coordinate system, each li is easily written in terms of the length of this bond.   li li =  0  0

(18)

The li can all be expressed in the coordinate system of the first bond by premultiplication by a serial product of transformation matrices, T1 · · · Ti − 1 , where the form of each T depends on the conventions adopted for expressing the bond angle and torsion angle. A common form is presented in equation (19). 

− cosθ  Ti =  − sinθ cosφ − sinθsinφ

sinθ − cosθcosφ − cosθ sinφ

 0  − sinφ  cosφ

i

(19)

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y1 x2

1

0

2

x1 y2

Fig. 3. The local coordinate systems defined by two consecutive bonds. From Ref. 19. c 1976 John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Copyright
Inc.

The torsion angle at bond i, defined as 0 for a cis placement, is denoted by φ i , and the angle between bonds i and i + 1 is θ i . T1 is the special case where the undefined φ 1 is taken to be 180◦ . With this notation, equation (17) can be written as a sum of n terms that involve li and Ti . r = l1 + T1l2 + T1 T2l3 + · · · + T1 T2 · · · Tn − 1ln

(20)

This sum can be generated as a serial product of n matrices, one for each bond in the chain (20).  r = [ T l ]1

T l 0 1



 ··· 2

T l 0 1



 l 1 n− 1 n

(21)

A more compact notation writes each of the matrices on the right-hand side of equation (21) as Ai . r = A1 A2 · · · An

(22)

If the bond vector is replaced by the dipole moment vector for bond i, mi , an equivalent serial product of matrices produces the dipole moment for this chain.  µ = [ T m ]1

T m 0 1



 ···

2

T m 0 1



n− 1

m 1

(23) n

The squared end-to-end distance is generated as a matrix product using the same information that was required for r in equation (21), but processing that information differently (20). r 2 = G1 G2 · · · Gn

(24)

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The internal Gi are 5 × 5 matrices which can be written in block form with dimensions 3 × 3. 

 1 2lT T l2 Gi =  0 T l , 0 0 1 i

1