Vol. 2

CONFORMATION AND CONFIGURATION

97

CONFORMATION AND CONFIGURATION

Introduction

Scientists have devised many models for treating the enormous number of con-
formations 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 poly-
mer 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 effi-
cient manner. The data fed into the model include structural information (lengths
of bonds, angles between successive bonds, values of the preferred torsion an-
gles) and energetic information (contributions of short-range intrachain inter-
actions to the preferences for specific values of the torsion angles, and spe-
cific 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

r

2



0

, with the average being performed over all of the accessible confor-

mations.

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). Applica-
tions 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.

98

CONFORMATION AND CONFIGURATION

Vol. 2

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

r

2



0

as the product of a term

N that is directly proportional to the degree of polymerization and another term
L

2

 that specifies the mean square step length of a segment.

r

2



0

= NL

2



(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

F

1

F

2

· · · F

2

(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 F

i

are matrices that combine geometric information, namely the

length of bond i, l

i

, 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 re-
stricts the focus to the thermodynamic (energetic) part of the rotational isomeric
state model. Then we will describe how the structural information (l

i

,

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

r

2



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 single-

chain 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 polypropy-
lene, 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.

Vol. 2

CONFORMATION AND CONFIGURATION

99

The Conformational Partition Function Z

Although not in any way restricted in application to polyethylene, it is nev-
ertheless 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,
CH

3

CH

2

CH

2

CH

3

. 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

30

20

10

60

120

180

240

300

360

0

0

Torsion Angle , deg

␾

U

( ),

kJ

ⴢmol

−1

␾

Fig. 1.

Torsion potential energy function for the central C C bond in n-butane. From

Ref. 4. Copyright c

1994 John Wiley & Sons, Inc. Reprinted by permission of John Wiley

& Sons, Inc.

100

CONFORMATION AND CONFIGURATION

Vol. 2

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

φ, deg

Maximum or minimum

U, kJ

·mol

− 1

U/kT at 300 K

Population

180

Minimum

0

0

1(1

+ 2σ )

− 1

∼(±121)

Local maximum

∼14

∼6

∼0

∼(±70)

Local minimum

∼3

∼1

σ (1 + 2σ )

− 1

0

Maximum

∼33

∼13

∼0

0.2

0.1

0

0

60

120

180

240

300

360

Torsion Angle , deg

␾

Nor

maliz

ed P

opulation

300 K
400 K
500 K

Fig. 2.

Normalized populations of the torsion angle at the central C C bond in n-butane

at 300, 400, and 500 K. From Ref. 4. Copyright c

1994 John Wiley & Sons, Inc. Reprinted

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 u

i

divided by the sum of

all statistical weights, ie, the conformational partition function p

i

= Z

− 1

u

i

,

Z

=

u

i

.

Extension of this concept to a longer alkane suggests three conformations for

each internal C C bond, or 3

n

− 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 conforma-
tions with g states of the same sign have conformational energies near 2E

σ

, 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 predic-
tion 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 2E

σ

+ E

ω

, with E

ω

about 7.5 kJ

·mol

− 1

. The conformational

Vol. 2

CONFORMATION AND CONFIGURATION

101

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 incorpo-

rated 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 some-
where within a long chain, U

i

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 U

i

.

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).

U

i

=






1

σ

σ

1

σ σω

1

σ ω σ






i

(4)

More generally, U

i

is formulated as the product of a diagonal matrix D

i

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 U

i

(13). The second-order interactions occur in a matrix V

i

(which has

dimensions given by points (1) and (2)). For polyethylene, the U

i

in equation (4)

can be generated as the product of the V

i

and D

i

defined in equation (5).

U

i

= V

i

D

i

=






1 1 1
1 1

ω

1

ω 1






i






1 0 0
0

σ 0

0 0

σ






i

(5)

Z for n-butane is given by the sum of the elements in the top row of U

2

, and

Z for n-pentane is the sum of the elements in the top row of U

2

U

3

. In general,

the contribution of bonds 2 through n

−1 to Z is given by the sum of the elements

102

CONFORMATION AND CONFIGURATION

Vol. 2

in the top row of the serial product U

2

U

3

· · · U

n

− 1

. This sum can be extracted by

appending an initial U

1

defined as [1 0

· · · 0] and a terminal U

n

that is a column

in which all elements are 1.

Z

= U

1

U

2

· · · U

n

(6)

Application of this equation yields Z

= 1 + 2σ for n-butane and Z = 1 + 4σ +

2

σ

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 U

i

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 − E

0

= kT

2

(

∂ ln Z/∂T)

(7)

S

= (E − E

0

)

/T + k ln Z

(8)

The probability that bond i will be in state

η, denoted by p

η;i

, is calculated

with a matrix U

η;i



that is obtained from U

i

by zeroing out all of the elements

except those in the column indexed by state

η at this bond (16).

p

η;i

= Z

− 1



i

− 1

j

= 1

U

j

U



η;i



n

k

= i+1

U

k

(9)

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 p

η;i

that have 1 as

their sum.

p

t;i

= 0.596

(10)

p

g

+

;i

= p

g

−

;i

= 0.202

(11)

Vol. 2

CONFORMATION AND CONFIGURATION

103

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 p

g

+

;i

= p

g

−

;i

. Knowledge of all of the p

η;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 U

ξη;i



that is obtained from U

i

by zeroing out all of the elements

except the single element at row

ξ, column η.

p

ξη;i

= Z

− 1



i

− 1

j

= 1

U

j

U



ξη;i



n

k

= i+1

U

k

(12)

The numerical values of p

ξη;i

are conveniently presented in the form of a

matrix that has the same dimensions as U

ξη;i



. For the same case described in

equations (10) and (11), this matrix has elements with the numerical values pre-
sented in equation (13).

p

ξη;i

=




0

.321

0

.138

0

.138

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 p

tg

+

;i

= p

tg

−

;i

= p

g

+

t;i

= p

g

−

t;i

arises from the

symmetry of the torsion potential energy function and the absence of a distinguish-
able 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, p

g

+

g

+

;i

= p

g

−

g

−

;i

> p

g

+

g

−

;i

= p

g

−

g

+

;i

.

Another useful probability can be derived from the previous equations. Given

that there is state

ξ at bond i − 1, q

ξη;i

is the probability for finding state

η at

bond i.

q

ξη;i

= p

ξ;i

/p

ξη;i − 1

(14)

The numerical results in equation (13) specify nine values that can be pre-

sented in matrix form.

p

ξη;i

=




0

.538 0.231 0.231

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 con-
formation

κ in a long chain (where κ is a subscript that uniquely defines that

104

CONFORMATION AND CONFIGURATION

Vol. 2

conformation) is given by the product of the appropriate p

η;i

for bond 2 and the

appropriate q

ξη;i

for all subsequent bond pairs.

p

k

= p

α;2

q

αβ;3

q

βγ ;4

· · · q

ψω;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

r

2



0

(where the average is over all

conformations), it is desirable to formulate r

2

(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

= l

1

+ l

2

+ · · · + l

n

(17)

The form adopted in equation (17) assumes that r and all of the l

i

are ex-

pressed in the same coordinate system. Specification of the elements in any one
of the l

i

requires knowledge of the length of the bond and its orientation in the

coordinate system common to all l

i

. The rotational isomeric state model uses a

different approach. Each l

i

is expressed in a local Cartesian coordinate system

with axis x

i

along bond i, axis y

i

in the plane of bonds i

− 1 and i, with a positive

projection on bond i

− 1, and z

i

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 l

i

is easily written in

terms of the length of this bond.

l

i

=




l

i

0
0




(18)

The l

i

can all be expressed in the coordinate system of the first bond by pre-

multiplication by a serial product of transformation matrices, T

1

· · · T

i

− 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).

T

i

=






− cosθ

sin

θ

0

− sinθcosφ − cosθcosφ − sinφ

− sinθsinφ − cosθsinφ

cos

φ






i

(19)

Vol. 2

CONFORMATION AND CONFIGURATION

105

1

2

0

y

2

y

1

x

1

x

2

Fig. 3.

The local coordinate systems defined by two consecutive bonds. From Ref. 19.

Copyright c

1976 John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons,

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

. T

1

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 l

i

and T

i

.

r

= l

1

+ T

1

l

2

+ T

1

T

2

l

3

+ · · · + T

1

T

2

· · · T

n

− 1

l

n

(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



n

− 1



l

1



n

(21)

A more compact notation writes each of the matrices on the right-hand side

of equation (21) as A

i

.

r

= A

1

A

2

· · · A

n

(22)

If the bond vector is replaced by the dipole moment vector for bond i, m

i

, 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



n

(23)

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

= G

1

G

2

· · · G

n

(24)

106

CONFORMATION AND CONFIGURATION

Vol. 2

The internal G

i

are 5

× 5 matrices which can be written in block form with

dimensions 3

× 3.

G

i

=




1 2l

T

T l

2

0

T

l

0

0

1




i

, 1<i<n

(25)

G

1

is given by the top row of G

i

, and G

n

is given by the last column.

The squared dipole moment

µ

2

is obtained by the same formalism upon

substitution of m for l and m

2

for l

2

.

If all of the atoms in the chain are of the same mass, the squared radius of

gyration is generated as a matrix product that uses the same information required
for r and r

2

, but processing that information differently.

s

2

= (n+1)

− 2

H

1

H

2

· · · H

n

− 1

H

n

= (n+ 1)

− 2

n

i

= 1

H

i

(26)

The internal H

i

are 7

× 7 matrices when written out element by element,

but they can also be written in a more compact blocked form.

H

i

=




1 G

[

l

2

0 G G

]

0

0

1




i

(27)

Here G

[

denotes the top row of G, G

]

denotes the last column of G, H

1

is

formulated as the top row of H

i

, and H

n

is formulated as the last column of H

n

.

Other conformation-dependent properties can be formulated in a similar

manner (4,20).

Averages over All Conformations

The basis for computing the averages over all conformations is illustrated for
the case of the mean square unperturbed end-to-end distance

r

2



0

(20). Let r

κ

2

denote the squared end-to-end distance of conformation

κ. The probability of this

conformation is given by equation (16). The desired result is the weighted sum
over all conformations.

r

2



0

=

p

κ

r

2

κ

(28)

The probability is the ratio of the statistical weight for conformation

κ, de-

noted by w

κ

, to the sum of the statistical weights for all conformations, Z.

r

2



0

= Z

− 1

w

κ

r

2

κ

(29)

Vol. 2

CONFORMATION AND CONFIGURATION

107

The w

κ

is the product of statistical weights w

κ;i

, that appear in the appro-

priate elements of the U

i

, and r

κ

2

is the serial product of G matrices, according to

equation (25), where the appropriate

φ

i

is used in each G

i

. The w

κ;i

can be grouped

with the appropriate G

κ;i

.

r

2



0

= Z

− 1

κ

n

i

= 1

w

κ;i

G

κ;i

(30)

The numerator is merely the expression for Z, with each statistical weight

expanded by multiplication onto the appropriate G matrix. Therefore the desired
result is obtained by equation (2) with supermatrices that contain the geometric
information in G and the energetic information in U.

F

i

=










u

11

G

φ

1

u

12

G

φ

2

· · ·

u

1v

i

G

φ

vi

u

21

G

φ

1

u

22

G

φ

2

· · ·

u

2v

i

G

φ

vi

..

.

..

.

. ..

..

.

u

v

i

− 1

1

G

φ

1

u

v

i

− 1

2

G

φ

2

· · · u

v

i

− 1

v

i

G

φ

vi










i

, 1<i<n

(31)

The first bond is treated with F

1

= [G

[

0]

1

and a column of i

−1 copies of G

[n

is used for the last bond.

The mean square unperturbed end-to-end distance is usually reported as

the dimensionless characteristic ratio, defined as the ratio of the true

r

2



0

to the

value that would have been predicted for the same chain, using the freely jointed
chain model (eq. (1)).

C

n

= r

2



0

/nl

2



(32)

The values of C

n

approach a limit as n increases if the chain is flexible. The

limiting value is denoted by C

∞

.

A perfectly analogous approach permits computation of

r

0

and

s

2



0

, sub-

stituting the expressions in equation (22) or equation (26) for the one in equation
(24).

The FORTRAN source code for a simple yet versatile computer program

for calculation of

r

2



0

by equation (2) is available in Appendix C of Reference 4.

This source code, along with sample input and output files, can be downloaded
from www.polymer.uakron.edu/˜wlm. The program executes quickly on any mod-
ern computer that has a FORTRAN compiler.

Uses of the Expressions

This section provides several illustrative examples of questions that can be ad-
dressed with the rotational isomeric state model.

108

CONFORMATION AND CONFIGURATION

Vol. 2

Often useful information is accessible from Z itself. For example, one can

inquire into the average number of bonds in a sequence of t placements in a
polyethylene melt at 450 K. The solution is obtained as the ratio of the probability
of a t state to the probability for the initiation of a sequence of t states. This ratio
is p

t;i

( p

g

+

t;i

+ p

g

−

t;i

)

− 1

, where bond i is in the interior of a long chain, far enough

from either end so that it is not subject to end effects. In practice, the chain might
have n

= 20 and i = 10. Z is computed from equation (6), using the U given by

equation (4). The probabilities are obtained from equations (9) and (12). The ratio
has a value of about 2.4, for reasonable assignment of the numerical values of E

σ

and E

ω

.

With the incorporation of the matrix expressions for the properties of indi-

vidual chains, more information becomes accessible. For example, how sensitive
are the components of

r

0

to the interdependence of the bonds in a polyethylene

melt at 140

◦

C? Using the U in equation (4) with

ω = 1 (independent bonds) and

again with

ω = 0 (strongly interdependent bonds), along with equation (6) for Z

and a simple modification (substitution of A for G) of equation (2) for

r

0

, yields,

in the limit for long chains,

r

0

=



0

.61nm

0

.52nm

0



for interdependent bonds and

r

0

=



0

.34nm

0

.29nm

0



for independent bonds.

How quickly does C

n

reach its asymptotic limit at large n? The upper curve

in Figure 4 depicts the C

n

calculated from the U in equation (4) with

σ = 0.43

0.0

0.1

0.2

0.3

1/

n

0

1

2

3

4

5

6

7

⬍

r

2

⬎

0

/nl

2

,⬍

s

2

⬎

0

/nl

2

⬍r

2

⬎

0

/

nl

2

⬍s

2

⬎

0

/

nl

2

Fig. 4.

The dependence of

r

2



0

and

s

2



0

on 1/n for unperturbed polyethylene chains. From

Ref. 4. Copyright c

1994 John Wiley & Sons, Inc. Reprinted by permission of John Wiley

& Sons, Inc.

Vol. 2

CONFORMATION AND CONFIGURATION

109

0.0

0.1

0.2

0.3

1/

n

5

6

7

8

9

⬍

r

2

⬎

0

/⬍

s

2

⬎

0

Fig. 5.

The dependence of

r

2



0

/

s

2



0

on 1/n for unperturbed polyethylene chains, using

the data from Figure 4. From Ref. 4. Copyright c

1994 John Wiley & Sons, Inc. Reprinted

by permission of John Wiley & Sons, Inc.

and

ω = 0.034, using Z from equation (6) and r

2



0

from equation (2). The final

approach of C

n

to C

∞

is linear in 1/n.

C

n

= C

α

− a/n

(33)

Here a is a positive constant. C

n

is within about 5% of C

∞

in Figure 4 when n

reaches 100. The form of equation (33) is common to flexible chains, but the value
of a depends on the nature of the chain, with a tendency for a to become larger as
the stiffness of the chain increases.

Is the rotational isomeric state model consistent with

r

2



0

/

s

2



0

= 6 for long

flexible chains? The lower curve in Figure 4 depicts the values of

s

2



0

, calculated

in the same manner as the

r

2



0

in the upper curve, but with substitution of H

i

for

G

i

and division by (n

+ 1)

2

, as required by equation (26). The ratio

r

2



0

/

s

2



0

is

depicted in Figure 5. It reaches 6 in the limit n

→ ∞, but the ratio is larger than

6 at small n, because

r

2



0

/nl

2

reaches its limit faster than does

s

2



0

/nl

2

.

The rotational isomeric state model can be used to rationalize the

temperature-dependence of conformation-dependent properties, through the de-
pendence of the statistical weights on temperature. How strong is the temperature
dependence of the dimensions of the chains in a polyethylene melt, assuming that
we remain above the temperature at which crystallization occurs? Calculations
of

r

2



0

from equations (2), (4), and (6), assuming that the only temperature de-

pendence is from the Boltzmann factors

σ = exp(−E

σ

/kT) and

ω = exp(−E

ω

/kT),

yields

∂ ln r

2



0

/

∂ T = −0.001 deg

− 1

.

A common use of the rotational isomeric state model is to learn how specific

structural properties (at the local level) affect conformational properties at the
level of a long chain. Polyisobutylene provides a good example. The bond angles
in the backbone of polyisobutylene alternate between 110

◦

(for CH

2

C CH

2

) and

124

◦

(for C CH

2

C). How strongly, and in what direction, does this alternation in

bond angles affect the mean square dimensions of the chains in a polyisobutylene

110

CONFORMATION AND CONFIGURATION

Vol. 2

melt? Using Z from equation (6) and

r

2



0

from equation (2), along with the U

and

φ from the rotational isomeric state model for polyisobutylene described in

Reference 21, replacement of the two different

θ by their average (θ = 117

◦

for all

bond angles) increases

r

2



0

by 21%.

The rotational isomeric state model can also tell us how stereochemical com-

position affects the conformations of a chain. When selecting from isotactic and
syndiotactic poly(2-vinyl pyridine), and isotactic and syndiotactic poly(N-vinyl
carbazole), which of the four chains has the smallest and the largest unperturbed
dimensions? Using the rotational isomeric state models proposed for these two
polymers (22,23), the values of C

∞

at 300 K are 4.9 and 7.7, respectively, for iso-

tactic and syndiotactic poly(2-vinyl pyridine), whereas isotactic and syndiotactic
poly(N-vinyl carbazole) have much larger values of approximately 15 and 39, re-
spectively. The most compact unperturbed chain is isotactic poly(2-vinyl pyridine),
and the most extended chain is syndiotactic poly(N-vinyl carbazole).

ACKNOWLEDGMENTS

The preparation of this manuscript was supported by National Science Foundation grant
DMR 9844069. The text was completed in April 2000, while WLM was on sabbatical leave
at Sandia National Laboratory. He thanks Dr. John G. Curro for his hospitality during this
leave.

BIBLIOGRAPHY

“Conformation and Configuration” in EPST 1st ed., Suppl. Vol. 1, pp. 176–195, by A. E.
Tonelli, Bell Laboratories; “Conformation and Configuration” in EPSE 2nd ed., Vol. 4,
pp. 120–144, by A. E. Tonelli, AT&T Bell Laboratories.

1. M. V. Volkenstein, Dokl. Akad. Nauk SSSR 78, 879 (1951).
2. H. A. Kramers and G. H. Wannier, Phys. Rev. 60, 252 (1941).
3. P. J. Flory, Statistical Mechanics of Chain Molecules, Wiley-Interscience, New York,

1969; reprinted with the same title by Hanser, Munich, 1989.

4. W. L. Mattice and U. W. Suter, Conformational Theory of Large Molecules. The Rota-

tional Isomeric State Model in Macromolecular Systems, Wiley-Interscience, New York,
1994.

5. M. Rehahn, W. L. Mattice, and U. W. Suter, Adv. Polym. Sci. 131/132, 1 (1995).
6. J. D. Honeycutt, in J. E. Mark, ed., Physical Properties of Polymer Handbook, American

Institute of Physics, Woodbury, N. Y., 1996, p 39.

7. P. J. Flory, J. Chem. Phys. 17, 303 (1949).
8. D. N. Theodorou and U. W. Suter, Macromolecules 18, 1467 (1985).
9. T. Haliloglu and W. L. Mattice, J. Chem. Phys. 111, 4327 (1999).

10. T. C. Clancy and co-workers, Macromolecules 33, 9452 (2000).
11. A.-C. Tang, J. Chinese Chem. Soc. 19, 33 (1952).
12. A. Abe, R. L. Jernigan, and P. J. Flory, J. Am. Chem. Soc. 88, 631 (1966).
13. P. J. Flory, P. R. Sundararajan, and L. C. DeBolt, J. Am. Chem. Soc. 96, 5015 (1974).
14. A. Abe, K. Tasaki, and J. E. Mark, Polym. J. 17, 883 (1985).
15. M. Hutnik, A. S. Argon, and U. W. Suter, Macromolecules 24, 5956 (1991).
16. R. L. Jernigan and P. J. Flory, J. Chem. Phys. 50, 4165 (1969).
17. A. E. Tonelli, F. C. Schilling, and R. E. Cais, Macromolecules 15, 849 (1982).
18. J. E. Mark and J. G. Curro, J. Chem. Phys. 81, 6408 (1984).
19. A. Abe, J. W. Kennedy, and P. J. Flory, J. Polym. Sci., Polym. Phys. Ed. 14, 1337 (1976).

Vol. 2

CRITICAL PHASE POLYMERIZATIONS

111

20. P. J. Flory, Macromolecules 7, 381 (1974).
21. L. C. DeBolt and U. W. Suter, Macromolecules 20, 1424 (1987).
22. A. E. Tonelli, Macromolecules 18, 2579 (1985).
23. A. Abe and co-workers, Macromolecules 21, 3414 (1988).

W

AYNE

L. M

ATTICE

C

ARIN

A. H

ELFER

The University of Akron

CONTACT LENSES.

See H

YDROGELS

.

COORDINATION POLYMERS.

See M

ETAL

-

CONTAINING POLYMERS

.

CREEP.

See V

ISCOELASTICITY

.