A THEORY FOR CONFORMATIONS IN
ISOTACTIC POLYMERS;

A MARKO'V PROCESS

Thesis for The Degree OT pl]. D.
MICHIGAN STATE UNIVERSITY

Victor E. Meyer
1961

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A THEORY FOR CONFORMATIONS IN ISOTACTIC POLYMERS;

A MARKOV PROCESS

by

VICTOR E. MEYER

AN ABSTRACT

Submitted to
Michigan State University
in.partial fulfillment of the requirements
for the degree of

-DOCTOR OF PHILOSOPHY

Department of Chemistry

1961

Approved

 

ABSTRACT

A THEORY FOR CONFORMATIONS IN ISOTACTIC POLYMERS;
A MARKOV PROCESS

by Victor E. Meyer

Body of Abstract

An equation has been derived which.re1ates the mean square end-to-end
length of an isotactic vinylic hydrocarbonrtype chain to the detailed
geometry of the chain. The equation is restricted to long linear chains
with regular head-to-tail structure which can be represented by a diamond
lattice nodel. The chain bonds are all considered to have constant and
equal length, and are connected at the tetrahedral valence angle. The
rotational angle, as defined by any three consecutive bonds, is
restricted to the EEEEE and two gauche conformations. Furthermore, the
bonds are considered to be statistically independent, Which allows the
problem to be solved by Markov chain statistics. The mean square length
is expressed as a function of the probabilities of finding the bonds in
their respective trans and gauche conformations.

The mean square end-to-end length of a polymethylene-type chain
has also been calculated by the methods indicated above. The equations
obtained are discussed relative to pertinent eXperimental data. It is
concluded that "crystalline“ polymers tend to retain their crystalline

state "conformations" in their "unperturbed" states.

A THEORY FOR CONFORMATIONS IN ISOTACTIC POLYMERS;

A MARKOV PROCESS

by

VICTOR E. MEYER

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1961

ACKNOWLEDGEMENTS

The author is indebted to Dr. J. B. Kinsinger for the inspiration
and aid he provided during the course of this work. The author is
also indebted to Dr. P. M. Parker of the Physics Department, Michigan
State University for his contributions in all of the mathematical
aspects of this work.

Appreciation is extended to the Research Corporation and to the
National Science Foundation for the financial support provided by
these organizations.

The author also extends his gratitude to his wife for her patience

and understanding during the course of his studies and research.

11

VITA

The author was graduated from St. Louis High School, St. Louis,
Michig‘n in Jung, 1950, He entered Michigan State College in September,
1950 and studied history until June,1952. At that time he enlisted in
in the United States Army and spent the next three years as a member
of the Army Security Agency. After receiving an honorable discharge
in May, 1955, he re-entered Michigan State College as a chemistry

major. He received the Bachelor of Science degree in June, 1957.

iii

I.

II.

III.

TABLE or CONTENTS

mTRODUCTION C O O O O O O O O O O O O O O O 0 O O O O

A.generaleeeeeeeeeeeeeeeeeeeee

B. IntrOdUCtion to Theory. 0 e e e e e e e e e e e e

C. Root Mean Square End-to-End Length. . . . . . . .

CALCULATION OF THE MEAN SQUARE
POLYMETHYLENE CHAIN. . . . . .

A. IntrOdUCtion e e e e e e e

B. The Diamond Lattice model.

END-TO-END LENGTH OF A

C. Mean Square End-to'End Length. 0 e e e e e e e e e

D. Probability Matrix . . . .

E. Calculation of the Mean Square Length from the

Probability Matrix . . . .

CALCULATION OF THE MEAN SQUARE END-TO-END LENGTH OF A

VTNYLIC ISOTACTIC CHAIN. . . e

A. Introduction . . . . . . .

B. Mean Square End“tO“End Length. 0 e e e e e e e e e

C. Probability Matrix . . . .

D. Calculation of Mean Square
the Probability Matrix . .

CONCLUSION AND DISCUSSIONS e e
A. Generll e a a e e e e e o e
B . PO methylene e e e e e e e

C. Isotactic Macromolecules .

End-to-End Length from

D. Additional Lructural Problems . . . . . . . . . .

BI BLI WWI-H O O O O O O O O 0

iv

Page

.115

. 61

Figure

10
11

LIST OF FIGURES

Segment of a schematic chain showing bond
length (l), valence angle (6', and rotational
a-ngle(¢)eeeeeeeeeeeeeeeeeeeeee

Vector-bond l in the first quadrant of a
cartesian cooPdinate system. . . . . . . . . . . . .

Newman projections of the trans (a) conformation
(¢ - O), and the two equivalent gauche (b)
Conromtion3(¢'+120°)eeeeeeeeeeeee

The complete matrix of transition probabilities
) for a polymethylene chain. The meaning
Mt column and row heading are explained in

rible I. O O O O O O O O O O O O O C O O I O O O O O

Planar zigzag illustration of a segment of an
isotactic macromolecule. . . . . . . . . . . . . . .

Newman projection of R -M bond sequences
in the trans and two giucpid1 conformation . . . . . .

Newman projection of Hi
sequences in the trans 1and Btu; MLugfi conformations.

The submatrix of transition probabilities P .
The column and row headings(state) are explained
inTableI-Ileeeeeeeeeeeeeeeeeeee

The submatrix of transition probabilities PM'
The column and row headings are explained in
T‘bleIIIOOOOOOOOOOOOOOOOOOOOO

The matrix of transition probabilities . . . . . . .

PIOtOfF/nlgversus(‘)eeeeeeeeeeeeee

1O

15

17

27

28

28

3h

35

37
h9

LIST OF TABLES

Table Page
I The twenty four states of the polymethylene

chain..........................18

II Relation of the relabeled states to the original
states for the polymethylene chain . . . . . . . . . . . 20

III The forty-eight states of the isotactic vinylic chain. . 38

IV Relation of the relabeled states to the original
states for the isotactic vinylic chain . . . . . . . . . 39

V Magnitude of error in.the leading term approximation . . b8

VI Variation of the mean square length with changes
in magnitude of probabilities. . . . . . . . . . . . . . 5h

VII Polypropylene at the Flory thetaéeétemperatures. . . . . 55

VIII Classification of states of the "collapsed"
matrix of transition probabilities for a syndiotactic

Chaineeeeeeeeeeeeeeeeieeeeeeeeeéo

vi

APPENDIX A
APPENDIX B
APPENDIX C

APPENDIX D

LIST OF APPENDICES

Evaluation of the 1,1-Element of (1 + S)'1. . . . . 6h
Evaluation of the 1,1-Element of [(1 + S)2]’1S . .
Evaluation of the 1,1-Element of s‘“+1 [(1 + s)2]--JL . 68

Evaluation of the 1,1-Element of (1 - S'T')’1

(S.T' - S'). e e e a e e e e e e e e e e e e e e 0

vii

I. INTRODUCTION

A. General

The Nobel prizewinner (1953)H. Staudinger was the first to recognize
the dependence of physical properties of macromolecules on their
structural detail. Staudinger proposed that amorphous (glassy) polymers
could not be crystallized because of non-symmetrical arrangement along
the polymer chain. Staudinger1 suggested that the lack of crystallinity
in polymers such as polystyrene might result from the formation of a
great number of stereoisomers during polymerization. He advanced the
hypothesis that polymers which crystallize require repetitious and
symmetrical order to achieve the rigorous demands of chain packing for

the crystalline state.

Later M; L. Huggins2 suggested that the observed changes in the
solution properties of poly-CX-olefins (viz., polystyrene) polymerized at
different temperatures, resulted from the change in stereosequence of the
polymers. Specifically, he suggested that as the temperature of polymer-
ization of polystyrene is increased, the distribution of (d) and (1) con-
figurations becomes more random. That is, at lower temperatures of poly—
merization it is probable that a certain arrangement of asymmetric centers
along the backbone of the chain would be more favored. For example, in
polystyrene the alternate (d) and (l) configurations are most probable,
since this results in the maximum distance of separation of the bulky
pendant phenyl groups. As the temperature of polymerization increases,
the distribution of asymmetric centers may be expected to become more
random relative to this ”favored" arrangement of asymmetric centers. This
interpretation was proposed for the experimental results of T. Alfrey,

A. Bartovics, and H. Mark3. The recent nuclear magnetic resonance studies

h’5 on poly(methyl methacrylate) gives some very

of F. A. Bovey et.a1.
direct evidence in support of Huggins' suggestion. J. W. L. Fordham
et al.6 also show a relationship between polymerization temperature and
the degree of crystallinity in poly(vinyl chloride). The degree of
crystallinity increased with decreasing temperature, indicating an increase
in configurational order as the temperature of polymerization is decreased.
The successful search for catalytic processes which would produce
vinyl polymers with extremely regular configurational order was first

reported by C. E. Schildknecht7’8

et al., who in 19h? prepared isotactic
poly(vinyl isobutyl ether), using a boron fluoride etherate catalyst.
However, it was not until 1955 that the significance of stereospecific
polymerization caught the imagination and attention of the polymer
chemists when G. Natta9, reported the successful preparationIJf
crystalline, isotactic polypropylene and polystyrene of high molecular
weight using a Ziegler-type catalyst (usually a mixture of triethyl
aluminum and titanium tetrachloride in an inert solvent). The discovery
of stereoregular polymerization has been actively explored and at the
present time poly-O(-olefins and vinyl polymers with the.following
configurations have been reported and classified.

1. Isotactic macromolecules. The asymmetric carbon atoms of any

given.molecule have either all (d) or all(1) configurations.

H’ H H H H H H H
l I I I I I l I

- C - C*- C - 0*- C "' 6*- c " 6*- e e e
I I I I I I I I
H R H R H R H R

The asterisk indicates the pseudo-asymmetric centers.

2. Atactic macromolecules. The asymetric carbon atoms of am
given molecule have random (d) or (1) configurations.

o—u:
*

O’tfl
*

W-
I
m-o—m
I
war
I
I'll—0"”:
I
311 -O*-N
I
:n—o—m
I
w-ofm

3. Syndiotactic macromolecules.1 0’11 The asymetric carbon atoms

of am given molecule have alternate (d) and (l) configuartions.

III???
-c-c*-c-c*-c-c*-...
I I I I I I
H a H H H a

Chains of the type (CHE; -CHR3)x have recently been polymerized
stereospecificallyw to yield stereoregular polymers of the form: .'

h. Diisotactic macromolecules. The asymmetric carbon atoms of
any given molecule have either all (d) or all (1) configurations.

'" H. , “d

5. Disyndiotactic macromolecules. The asymmetric carbon atoms

have alternate (d) and (l) configurations.

II1 “I T”? ’I‘“ -
* * i * i I-
..c- -c-c-c- -,,,
I ‘f I I I 1
R111 R111 R111

B. Introduction to Theory

The mathematical analysis of the properties and structure of high
polymer molecules has developed through a statistical approach. To
begin, the chain must be denoted by various symbols which represent
important characteristics of the chemical structure.12 Referring to
Figure 1, the covalent chemical bonds joining the atoms of the "backbone"
are regarded as jointed vectors (vector-bonds) ”lo, l1, l2, ..., ln-i ,
numbered according to their position along the chain from the first to
the last atom. The angle formed by the two bonds connecting any three
consecutive atoms is called the valence bond angle ('6') . The rotational
angle ([6) is defined as the angle made by bond .l-i-m from the plane

formed through bonds 3.1.1 and I1.

 

Figure 1. Se ent of a schematic chain showing bond length
(l , valence angle (6') , and the rotational angle

(I5).

Because high molecular weight polymers consist of a large number
of chain bonds or monomer units, the theoretical treatment, considering
now the entire chain, is limited to a statistical approach. One of the
important descriptive parameters is the mean square end-to-end length,
a quantity which is by definition, statistical. The magnitude of this

parameter is found to be dependent on the number of links in the chain,

the state of the system, the temperature, and the internal structure of
the polymer chain. Another related quantity, the mean square radius

of gyration, is defined as the average of the squares of the distances
from the center of gravity to each segment of the chain. Many physical
properties of polymer molecules depend on the mean square radius of
gyration and, for linear polymer chains whose mean square lengths have
a Gaussian-type distribution, the mean square radius of gyration is
equal to one-sixth.the mean square end-to-end length. Because of this
simple relationship between the two parameters the mean square end-to-
-end length has become the parameter most frequently used in conjunction
‘with physical properties.

Phenomena ihich follow Gaussian-type distribution curves normally
have certain random properties. W. Kuhn13 showed that the mean square
end-to-end lengths of real linear polymer chains obey Gaussian—type
distributions despite the fact that the directions of the connected
links(bonds) are severely limited by the valence angleée), and the
restricted angle of rotation (95) . Kuhn has demonstrated that a long
linear chain of n bonds may be considered as composed of (n/m)
segments , where m is selected such that because of the more or less
free rotations ebout the bonds, the (n/m) segments are directed
randomly relative to one another. Thus, unless real linear polymer
chains are small or unusually rigid, their mean square end-to-end
lengths can be expected to have Gaussian-type distributions, and the
simple relationship between.the mean.equare length and the mean square
radius of gyration will be valid. In.addition, experimental evidence1h
exists which demonstrates that within experimental error the mean square

end-to-end length is proportional to the molecular weight as should be

true for an I'ideal" Gaussian-type linear macromolecule.

C. Root Mean Square End-tannd Length.

The first theoretical treatment of the mean square end-to-end
length.was the “freely jointed“ chain, where each bond is considered
statistically independent and all angles of 69) occur with equal
probability. The model was obviously a vast oversimplification for
real macromolecules, however, with appropriate corrections real
polymer molecules do conform to certain distribution functions (viz.,
G‘fl'gi‘nPtyp. distributions) derived from this model.

subsequent authors introduced a.more realistic model with fixed
bond angles15, but with free rotations. This model can be treated
by Harkovian statistical methods and has been termed the ”freely
rotating chain”.

In recent years considerable attention has been focused on the
problem of taking into account the effects of hindered rotations on the
mean square end-to-end dimensions. One approach to this problem is to
treat the chain.with restricted rotations through averages of the
rotational angle (¢). However, more refined analysis shows that the
chain.bonds are most likely to fix themselves at specific rotational
angles depending on the nature of the near neighbor interactions. That
is, because of the near neighbor interactions along the backbone of the
chain, not all conformations are energetically equivalent, and indeed
some conformations are practically excluded.

The stereoregular polymers possess structural regularity;
therefore the problem of relating the mean square end-to-end lengths
to the detailed geometry of the chain is much simpler than the older
prdblem of atactic(random) structures.

Two different mathematical techniques have been developed to relate

the detailed geometry of the chain to the mean square end-to-end length.
‘The first of these, which is presently receiving much attention, was
originated by H. Eyring.15

In 1932, Eyring proposed a.method of solving for the mean square
end-to-end length of linear macromolecules. The principal features of
the method are as follows:

1. Assign to each bond a suitibly defined coordinate system.

2. By means of rotational matrices relate all vector-bonds to
a single coordinate system, usually the one associated with the first
bond of the chain.

3. Take the dot product of the end-to-end length, and consider

the averages of the rotational matrices as they occur in the dot

product.
The method.of rotational matrices is presently the subject of
investigation of a large number of workers.16-22 Several of these

workers have extended the method to consider the case of isotactic
and/or syndiotactic macromolecules. The results obtained by these
workers are frequently cumbersome, occasionally not obtainable in
closed form and are difficult to compare.

Tobolsky23 has suggested an alternative approach to this problem.
The principal features of his method are:

1. Describe the chain in.terms of a.diamond latticez’4

and a
matrix of transition probabilities which:relates to a “walk“ on a
diamond lattice. (This applies to all chains which take predominantly
staggered conformations).

2. Mathematically relate the mean square end-to-end length to

the “walk’I as described by the matrix of transition probabilities.

Specifically, Tobolsky considered the case of a polymethylene

chain with hindered rotations. The two gauche positions are considered
to be energetically equivalent, and different from.the trans positions.
The mean square end-to-end length is then expressed as a function of
the tendency for any three bonds to be found in the trans position.

Recently R. P. Smith25 demonstrated the equivalency of the
rotational matrix and the diamond lattice approach for the polymethylene
chain. However, his method relies heavily on the rotational matrix
technique and it is therefore not immediately obvious whether the
method.may be extended to treat structurally more complicated chains.

In the calculation.presented in this thesis, the diamond lattice
model has been utilized. However, it is not possible to deal with the
matrix of transition probabilities in the manner suggested 141the
treatment of the polymethylene chain by Tobolsky. It was therefore
necessary to develop a differentlnathematical technique in evaluating
the mean square end-to-end dimensions from the matrix of transition

probabilities.

II. CALCULATION OF THE MEAN SQUARE END-TO-END LENGTH OF A POLYMETHYLENE

CHAIN.

A. Introduction.

A Because the polymethylene chain contains only a single type of
chain bond, it is both concepflnlally and mathematically much simpler
than the isotactic chain. For this reason the polymethylene chain
calculation is presented first in its entirety, although it may be
treated alternatively as a special case of the isotactic chain. Because
the polymethylene Chain is simpler, this approach allows a more orderly
and clearer presentation of the salient mathematical and conceptual
aspects of the calculation, and then, when the isotactic case is
considered, particular attention and emphasis may be given to those
aspects of the calculation which are peculiar to the isotactic chain.
In keeping with this objective, the detailed mathematical evaluations

have been placed in appendixes which are collected at the end of this

m0813e

B. The Diamond Lattice Model.

It is desired to compute the mean square length of a polymetbylene-
type chain of n+1 carbon atoms connected by n vector-bonds of constant
length (10), with constant tetrahedral valence angle (99. Recently,

W. J. Taylor26 showed that the error involved in neglecting the variation
in bond lengths and bond angles due to vibrations,in the calculation of
the mean square end-to-end length,is of fine order of magnitude of a

few tenths of one percent at normal temperatures, and thus need not be

considered here.

10

The bonds are numbered consecutively as indicated in Figure 1.
Each bond has vector components (X(i), Y(i), 2(1)), i = O,1,2,..., n-1.
For simplicity of calculation, the magnitude of each component is taken
to be unity. If (10) is the actual bond length, then the calculated
mean square length must, in the last step of the calculation, be scaled
by the factor 13/3.

Each vector—bond is thus represented by one of the eight combinations
(:1, 11, t1) and successive bonds differ only in the change of exactly
gag algebraic sign. This representation of hydrocarbon molecules is
commonly referred to as the diamond lattice model.

First, it is desired to demonstrate how the components of each
bond.may be represented by 11. Since each vector-bond may have
components only as given.by one of the eight combinations (:1, 1'1, 31);
each vector-bond is thereby restricted to lie along one of the principal
diagonals of the eight quadrants of the cartesian coordinate system.

Thus, regardless of the quadrant in which a vector-bond lies, one obtains

for the dot product of the vector-bond into itself

 

 

;1.;i a 1: a (X(i) + Y(i) + 2(1))°(X(i) + Y(i) + 2(1)) (1)
'z
zw-‘
’ H . Yo)
: f 9'
x(i,) . l/
a, ------- 4'

Figure 2. Vector-bond l1 in the first quadrant of a cartesian

coordinate system.

11

From Figure 2 it may be shown by simple trigonometry that for the
case of a vector lying along the principal diagonal of a cartesian
coordinate system,all of the components along the coordinate axes

are equal, i.e.,

 

 

X(i) Y(i)| g ‘Z(1)\

 

(3)

Equation (2) may therefore be written as,

13 , 3(x(i))2 g 3(Y(i))2 , 3(z(i))2 (h)
or
“uhz,uu52,mu52,gfi (9
From which
(Km) 3 i 1 (6)
(13/3)1;2

For vectors which are restricted to lie along the principal
diagonals of a cartesian coordinate system it is permissible to replace
the components of the vectors by :1. Since in the final equation for
a polymethylene chain only the squares of the components occur, the
final equation must be multiplied by 13/3.

Next it is desired to demonstrate how the change in exactly one
algebraic sign is consistent with a hydrocarbon chain of constant
tetrahedral valence angle 6'. From Figure 1 it is seen that the cosine of the

angle between two consecutive vector-bonds e.g., l

by

1.;1 is given

1’ ~

cos(w --e) = 1/3 . (7)

Now, if according to the diamond lattice model},i is represented

12

by the set (-1,1,1)th an l1” according to the change cf any one algebraic
sign,is represented by the set (-1,1,-1). Then since the dot product
of two vectors is defined as the product of the magnitudes of the

two vectors multiplied by the cosine of the angle between them,

‘ti ° $1+1 ' Ilil‘li+1‘ 003(li: li+1) (8)
From.which
°°3(;i . li+1) , l1 . l1+1 (-1,1,1).(-1,1,-1)

 

 

|$i||l1+1| ' 31/2 31/2 -1/3 (9)
The result (1/3) is obtained for any of the possible combinations

of (*1, *1, *1) which may be considered, with the requirement that the
two successive bonds differ only in the change of exactly 223 algebraic
sign. Thus, the diamond lattice model preserves the property of
hydrocarbon chains that the bonds be connected at the tetrahedral
valence angle 69).

The diamond lattice model imposes another restriction on the
hydrocarbon chain. That is, due to the requirement that.successive
bonds may have only one sign change, the rotational angle (¢) is
therefore restricted to the trans (¢ . 0°) and two gauche (¢ - i120°)

conformationst.

*"Conformation" has two different meanings as used in this thesis. It
‘will sometimes be used in reference to "trans" or "gauche" conformation,
and alternatively to indicate the orientation of the total chain. The

meaning will be clear from the context.

13

0. Mean Square End-to-End Length.
The end-to-end dimension of a polymethylene chain is the sum of
the bond vectors
n-1
3'}.0+l'1+32+ ... +}n_1-i§0;i (10)

where
,1 , “(1), Y(i)’ 2(1)). (11)

The expected square length is given by

n-1 n-1 n-1 n-1
E? a 2 2'. If? - z E{x(i)((3)+r(i)x(3)+ 265(3)} (12)
i=0 3-0 3 i=0 3:0
where E {X(1)X(j)+ Y(i)Y(j)+ Z(i)Z(j)} is the eXpected value of the
quantity enclosed in the curly brackets. Written in the convenient

form of an array, the double sum (12) becomes:

 

 

 

 

~01 '..o + $021.1 +3-o°}2 " " 1031,14
11°10 " $1311 * $1512 + +2=1“5°Lr1-1

ӣ1

(13)

12-1 $0 +in-1 .11 + ~n-1.12 + "’ + ~n-11.1'n-1

 

 

 

If end effects are neglected i.e., lo °~]-'o - 3,1 32,1 - eee , and
£041 - 1717;; - ... , and $012-$113 ... , etc.; and also since
the dot product conunutes i.e., 3,003.1 - 3.1-3.0; then because of the

symmetry about the principal diagonal of the array (13) may be
simplified to:

1h

(11:)

 

 

B 2 _ 0 _ o .
n10 + 2(n1)_1.0;L“1 + 2(n 2)}032 + + 23o $1-1 (15)

When (11.1) is written in terms of its rectangular coordinates it becomes

57 = 3n + 2 n22 r1214 Eix(°)x(k) + Y(°)Y(k) + z(°)z(k)E(16)

i=0 k=1
Put
Lk . r{x<°>x<k> + r<0>r<k) + 505(k)) (11)
then (16) becomes
n-2 n-1-i
iii-n+2: 2 Lk, (18)
180 k=1

and analogous to (15), this may be further simplified to give

n-1
57 - 3n + 2 Z

-k 1
k1(n)1~k (9)

D. Probability Matrix.

The mquence of pairs (303:1) , (1.13%), ... , (2.114%!) is now considered.
Each pair is a state in the Markov sense with the matrix of transition
probabilities given in Figure 11. The coordinates may be chosen such that

lo - (1,1,1), and 11 - (1,1,-1). Now} may have the possible sets

2
(1,-1,-1), (-1,1,-1), and (1,1,1) with probabilities b, b, and a,

1S

CZ C1 C;
F’ F! (4 (1+3 (1+3 *4
f4 f4 f4 f4 F1 f4
Cii<3
b b

Figure 3. Newman projections of the trans (a) conformation
(¢ - 0), and the two equivalent gauche (b)

conformations (¢ - 3 120°).

16

respectively, corresponding to their respective conformations, with
b + b + a - 1. In.Figure h it is seen.that these sets correspond to
transitions from state (32!) 1 to states (columns) h, 5, and 6,
respectively. Each of these events is treated as though it has occurred,
and the subsequent possibilities are considered. For example, if
12 - (1,-1,-1) has occurred, corresponding to the transition from
state (32!) 1 to state (column) b, with probability (b) (Figure h), as
just discussed, then, the next transition originates from state (32!) h,
and may proceed to states (columns) 13, 1b, and 15, with probabilities
(conformations) b, a, and b, respectively. The next transitions then
originate from states (£232) 13, 1h, and 15, depending on the outcomes
of the preceeding trials. Nomi};1 may also have the sets (1,-1,1),
and (-1,1,1) as well as the set (1,1,-1), given.above. Therefore,
these possibilities and all possible subsequent events must also be
considered. This procedure is then continued until the matrix of
transition probabilities, Figure h has been obtained. It is then
discovered that all subsequent bond pairs return to some previously
considered state of Figure h.

Since the coordinate system has been chosen such that}:O has unity

for its components, it is seen that Lk (17) simplifies to

Lk - ngm + rm + 20:); . (20)

The definition is now made

uk - xo‘) + rm + 2“" . ' (21)

Next, it is desired to consider the sequence of pairs (UoU1)’ (U1U2),

e e e , (UH-11111).

\oco~10~\nJ='w--

NNNN—b—D-fidd—Id—D—b—a
gUNdOWmNOKU'lC‘UN-‘O

17

Z 8 9 1O 11 12 13 1h 15 16117 18 19 20 21 22 23 2h

 

 

bba

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure h. The complete matrix of transition.probabilities (Pz4x24)
for a polymethylene chain. The meaning of the column and

row headings are explained in Table I.

 

18

Table I

The twenty four states of the polymethylene chain.

 

 

 

State Magnitude of Components* oi) State Magnitude of Components of
U3 Uj+1 Uj U3+1 U3 U3+1 Uj U3+1
1 3 1 3 13 13 -1 1 -11 12
2 3 1 3 13 1h -1 1 -41 13
3 3 1 3 11 15 -1 -3 -11 -3
h 1 -1 13 ~11 16 -1 1 -13 11
5 1 -1 13 -13 17 -1 1 -12 13
6 1 3 13 3 18 -1 -3 -12 -3
7 1 -1 13 -13 19 -1 1 -13 12
8 1 -1 13 -11 20 -1 1 ~13 11
9 1 3 12 3 21 -1 -3 ~13 -3
10 1 -1 11 -13 22 -3 -1 -3 -12
11 1 -1 11 --12 23 -3 -1 -3 —13
12 h 1 3 11 3 2h -3 -1 -3 —11

 

 

 

 

 

 

 

 

1}
(1,1,1) I 3; (1,1,-1) 8 13; (1,-1,1) I 12; (-1,1,1) I 11; (-1,-1,-1) I -3;

(-1,-1,1) a -13; (-1,1,-1) 3 -12; (1,-1,-1) I ~11.

Since only the magnitude of U is needed for the calculation,

.1
the 211 x 214 matrix of transition probabilities (Figure 14) is more
elaborate than need be, and the twenty-four states can be combined to
give only eight states of the desired Markovian character. This

”collapsing" process involves combining states of the same value of

U j , and of the same transition probability, with the exception of those

19

states of the 8 x 8 matrix of transition.probabilities (22) which have
unity as their transition probability. The unities result from.the

fact that a bond may differ by only one algebraic sign from the components
of an adjacent bond. Thus, if Uj I 3, then 03;, I 1, and U3+1
necessity, and a (1,3)-state has probability of unity of going to a

I 1, of
(3,1)-etate, and the states with unity as transition probability

result from combining a + b + b I 1. By this ”collapsing” technique

Figure h. reduces to the matrix P of transition probabilities (22).

3 1 -1 1 -1 1 -1-3 -3-1 1-1 1-1 1 3

 

1 3 1 0 b b a
2 -1 1 ' O b a b
3 -1 1 O a b b
h ~1-3 1 O 0 O O S
P,- __. (22)

 

S (3

8 1 3 1 O O 0

 

 

 

 

The relation of the relabeled states to the original states are
given in Table II. The matrices of transitions prdbabilities ch x 2h
and (22) both have the prOperty that the sum of the elements of any
row or column.add to unity. This property is preserved in the higher
powers of ch x 2h and (22) and in probability theory the matrices are
referred to as'doubly stochastic" (see reference 27, p. 358). This

property has a very simple physical interpretation. If one proceeds

20

down a hydrocarbon Chain in a diamond lattice, then after each “step",
the appropriate bond has a total probability of unity of being found

in one of its possible positions (conformations). This condition
results in a simple stochastic matrix (only the rows add to unity).
However, if the probability of proceeding in a forward direction is the
same as proceeding in a reverse direction, then, this condition results
:1n a doubly stochastic matrix of transition probabilities. In the
example just discussed, if the transition‘UoU1 I'U1U2 corresponding

to (1,1,1), (1,1,-1), (1,1,1) has probability (a), and if the probability
is also (a) for the transition U2U1 I»U1Uo corresponding to (1,1,1),
(1,1,-1), (1,1,1), (this being true in general for all transitions),
then, the resultant matrix of transition probabilities will be doubly

stochastic.

Table II
Relation of the relabeled states to the original states

for the polymethylene chain.

 

 

(Relabeled State Original State Value
Number Number Relabeled State
1 1, 2, 3, 3 1
2 13, 17, 20 -1 1
3 1h, 16, 19 -1 1
h 15, 18, 21 -1-3
5 22, 23, 2h -3-1
6 h, 7, 11 1-1
7 S, 8, 10 1-1
8 6, 9, 12 1 3

 

 

 

 

 

21

The power and utility of the matrix of transition prdbabilities
approach to statistical problems may be demonstrated by the following
considerations. Let the problem be posed, “What is the probability
of ending up in state two (-1 1) after two steps, if the starting point
is state one (3 1) of Equation (22). Otherwise worded, ”If four
consecutive bonds are considered with the first two bonds in state one
(3 1), what is the probability that the last two bonds will be in
state two (-1 1)?” From (22), it is seen that there are two possible
paths by which state two may be reached; path one: state one-+ state
six 9 state two, with probability of b2, and path two: state one +
state seven-+ state two, with probability be. The transition from
state one to state six, and from state six to state two are independent

events 28

and therefore the probability is b“, and by the same reason
the probability of path two is be. Now,"the probability that some one
or other of the set of mutually exclusive events will happen in a
single trial is the sum of their separate probabilities of happeningJ'
therefore the total probability of going from state one to state two
in two steps is b2 + ab.

Now, if the p::) element of P2 is calculated, the result,p§§) I
b2 + be is obtained, and indeed, it is found that the elements p§§) of
P1‘ automatically give the desired probability of transition from.state
(i) to state (3) in (k) steps. For example, it would be an almost
impossible task to calculate the probability of going from state one
to state two in twenty steps by the method of considering all different
paths. However, to find pggo) of P20 by matrix methods can be done in
a relatively Short period of time and, indeed, the method gives all

of the transition probabilities from any state one through eight to

22

any other state one through eight. Thus by the use of a matrix of

transition probabilities, a great deal of information is expressed

in.a very simple and useful manner. Naturally, some of the ”information"

has been lost by collapsing from the twenty-four by twenty-four matrix

of transition probabilities, however, the above approach does demon-

strate some of the useful features of this formulation of the problem.
From the above discussion it is seen.that.all states are assumed

to be statistically independent. If strong near neighbor interactions

are present, then the assumption of statistical independence represents

an approximation to the actual physical situation, since the state of

any given bond depends on the states of its near neighbor bondszz.

E. Calculation of the Mean Square Length from.the Probability Matrix.
The coordinate system has been chosen such that U(O) I(1+1 +1) I 3

therefore, L0 becomes simply

Lo . 3 (20a)

and (19) may be written as

n-1
it I -3n + 2 Z (n-k) L

k-O K (19a)

Let the matrix of transition probabilities (22) appropriate for
01+ k transition in the sequence (U0 U1), (U1 U2), ... , (Uk-1 Uk)’

be denoted by Pk. If the first row of PR has elements p§$), p12)’ ... ,
(k

P18), the probability that U(k) I 3, -3, 1, -1 equals, respectively

p11), at), it) +1011) with p12) wt” + p113,

23

k k k k) (k) (k) (k) (k)
Lk - 3(pg1) - 13%)) + (13:6) ” 12$? ” P18 "‘ p12 'p13 ' P11. )

(23)
.. -3 pg) + (p93) + pg) + 13105)) for 1: odd, and (2h)
. 3 pg?) ..(p$§) + pg?) + p$§)) for k even. (25)
From.the stochastic property,
(1:) (1:) (k) ,_ _ 00
p12 + p13 + p1h 1 p11 for k even, and (26)
pig) + pg?) + pgg) - 1 -p$§) for k add. (27)
it follows that (23) becomes simply:
(k) k
Lk - (L. s" -1)(-1) (28)
where es?) stands for the 1,1-element of the S -submatrix.
Therefore Equation(19a) becomes:
n-1 k (R)
1‘17- -3n + 2 z (n-k)(-1) (14s11 -1) , (29)
k-O
11-1 k 11-1 I)“ k (k)
B7 - -3n-2n z (-1) + 2 z (-1)k'k + 8 z (-1) (n-k)s11 (3o)
ksO klo k'O .
Evaluation of the first two summations of (30) yields
-bn n-1
a: . + a z (-1)k(n-k)s§’f) (31)
-hn-1 k-O

where the upper term.in the curly brackets is for n even and the lower
term is for n odd. The summation term of (31) is, exclusive of the

factor 8
n-1 k k
5- z (-1) (n-k) s11 (32)
k-O

2h

where Sk stands for the 1,1-element of the matrix Sk.

11
The expansion of (32) gives

n-1 n-1
. n z (-1)ks‘1‘1 - z (-1)kk sf, (33)
k-O k-O

- n(1 - s + 52 ..53 + ...) + (1 - 28 + 332 - us3 + ...) s

t (1 -2s + 352 - LS3 + ...) S“+1 (3h)

where again the upper sign is for n even and the lower sign is for n
odd, and the series of (3h) are infinite series. Now (1 + S)w

(1 -s + s2 - s3 + ...) s 1, and 1f (1+s)'1 exists, then

1-3 + 52 - 33 + ... - (1 + s)-1 (35)

Similarly (1+s)3(1-2s + 333 — us3 + ...) = 1, and 1t [(1 + s)=‘]'1 exists,
then

(1 - 23 + 352 - us3 + ...) - [(1 + s)2] -1 (36)

With these assumptions (31) becomes:
-hn - n+1
a? - + 8n(1+s)-1 + 8[(1+S)2]’1 S + 88 [(1+S)2]'1 <37)
-hn-1
and the 1,1—elements of the above matrix expressions are desired.

The detailed calculation of the above matrix expressions is given in

APPENDIX A, B, C, respectively. The results of these calculations are

8n(1 + S)’1 . 2n.%%—£}%3~

8‘“ * W‘s ' “(31:25? '1)

and

8 Sn+1[(1 + S)2]’1 - 1/2 for large n.

25

Substitution of these quantities into (37) and rearranging, yields

— 2n(1+3a) (3+z1)(5at-1)+1/2

ha . (1 - a) - (1 - a)a <38)

 

This result must be multiplied by 13,/3 to give the ”desired” mean
square length

-— 13 2n (1+3a) (3+a)(Sa-1)

h2 . 3 (1 - a.) _ (1 - 832! +1/2 (39)

 

 

 

26

III. CALCULATION OF THE MEAN SQUARE END-TO-END LENGTH

OF A VINYLIC ISOTACTIC CHAIN.

A. Introduction

The vinylic isotactic chain (Figure 5) is more complicated than
the polymethylene chain in two respects:

1. The asymmetric carbon atoms of a given chain.may be either
of all (d)- or of all (1)-configuration. However, since both types
of molecule can be expected to have the same mean square end-to-end
lengths (if they have the same degree of polymerization), this
consideration is of no further concern in this particular calculation.

2. The second difficulty encountered is that a given chain
possesses two different types of bond. In the manner of S. Lifson18,
the distinction is made between the clockwise sequence of bonds
(attached groups) about the asymmetric carbon atoms.

If the asymmetric carbon atom C1 is viewed from.the<iirection

of,§i_1, the clockwise sequence of attached groups is: H, C R.

i+1’
Now, if the same asymmetric carbon atom is viewed.from the direction
of §5, the clockwise sequence of attached atoms is : H, R, Ci-1’ or
the counterclockwise sequence is: H, Ci-1’ R. Thus, although each
bond is between a methylene carbon atom and an asymmetric carbon atom,
because of the diffebences in the relative orientations of the bonds
to their asymmetric carbon atom, the isotactic chain is composed of
two alternating types of bond as is indicated in Figure 5.

It is desired to consider a vinylic isotactic,chain containing

27

 

Figure 5. Planar zigzag illustration of a segment of an

isotactic macromolecule.

28

P H H (2+3 (2+3 R
H H H H H H
Ci+3 R H

A B C

Figure 6. Newman projections of‘fii -Mi+1 EEi+2 bond sequences

in the trans and two gauche conformations.

Ci“ C241 C£+1
H H Q+4 H H Ci+4~
H R H R H F?
(1.. H H
a b C
Figure 7. Newman projections of‘lflii‘P1 tBi+2 7yi+3 bond

se uences in the trans and two auche conformations.
__ 5——

29

n+1 backbone carbon atoms connected by n vector-bonds of constant

length (10), at the constant tetrahedral valence angle to). Each

(new), if“), 2(1)),

vector—bond has rectangular vector components

9

, 1: ~-: ..., “"1- S 111‘: H20 r‘>(.).'lj.r'nu;:'l,h{Mme (3115*.i.r1,f'()r' f-t1i211'g‘:].j(t'l’(,y

1
C.

(.7:

i =
of calculatior,the magnitude of each component is taken to be unity.
If (10), is the actual bond lenqth, the calculated mean scuare length
must therefore, as the last step of the calculation, be scaled by

the factor 12/3. :ach vector-bond is again represented by one of the
eight combinations (11, *1, *1) and successive bonds differ only in
the change of one algebraic sign.

If an’fii-bond is in position (1,1,1) and the succeeding fli+1
bond is in position (1,1-1), then the probability of occurrence of
the following Bi+2-bond in the(1,1,1) position is taken to be A; in
the (-1,1,-1) position is taken to be B; and in the (1,-1,-1)
position is taken to be C. These correspond to the trans and two
gauche conformations (Figure 6) respectively, with, A + B + C = 1.
The conformations and positions of B and C will be reversed depending
on whether one is dealing with a macromolecule of (d)- or (1)-
configuration. Therefore care must be taken to consider the molecule
as depicted in Figure 5, and in particular the number sequence
of the subscripts on the backbone carbon atoms must be rigorously
observed. To be definite, the molecule is selected such that when
the above mentioned three bonds are in the trans conformation, the

pendant R group of carbon atom Ci+ lies in the (1,-1,-1) position.

2
If the above three bonds are in the trans conformation, then the

30

‘Mi+3-bond has probabilities a, b, and c of being found in positions
(1,1,-1), (-1,1,1) and (1,-1,1) respectively. The probabilities

a, b, and c correspond to the trans and the two gauche conformations
(Figure 7) in that order; and a + b + c - 1. It should be carefully
noted that these conformations are uniquely determined by the last
three bonds which have been added to the chain. From this it is

seen that A, B, and C refer to E,- M - E,three-bond-combinations;

and a, b, and c refer to M.- E,- M three-bond-combinations.

B. Mean Square hnd-to—tnd Length.

If a chain commences with an fi-bond the end-to-end length is the

magnitude of

DR 'cBo + $1 + 32 + ”3 + .00 + Mn-1 (hO)

and if the chain commences with an.M-bond the end-to-end length is

the magnitude of

QM=%*E1+M2+B3*°°°*Bn-1 (in)

The a 23323} probabilities of occurence of these two situations

is assumed to be one-half for each of the two cases, since polymers

are formed by pairs of chain atoms. Alternatively, one may only
consider a molecule such as is given by (LO)° The mean square distance

is then given by the array,

 

B1'43'0 + 1:11 "7‘4

 

Or, if a molecule represented by (h1) is selected, the mean square

distance is then given by

T.
hM

Moog-30 +
...

 

From either (h2) or (h3)

term of (37), i.e. [(1+S)-1] is Obtained.

Bn-1°w0

in»

130% +
k1-g1 +

Mn--1 130 ” Nn-1 5‘31 " Lin-1 33/2 + + yin-1 «Mn-1

31

sir—sew

“ W

* I311'3’2 + " M1 ”31:14 *

“am"

*1 I32 + +B1Bn-1+

+ ”n4 £91 + En-1 :32 + + Bn-1 Bn-1

 

 

However, if either (h2)

(L12)

(L13)

the leading term corresponding to the leading

or (h3) is expanded separately, it will be discovered that the non-leading

terms are extremely irregular, and can not be easily eXpressed as simple

series as obtained in the polymethylene case.

may be rearranged to give

However, (b2) and (h3)

32

 

 

 

 

 

 

 

 

 

 

hams " 91‘1'21‘1 ' (1‘2) B043‘s * 30% + l30°32 + + 3044M + .3

(1‘3) 315% + ~‘-1°“1 * 51% + * 842,14
gkht)

(1‘2) B2°50 + 1231 * B2932 + + E2 n-1 i

' 3

. I;

1.

._______ ._______ ...—.... _______. (

(hB) Bn-VgO + I«in-1.51 + I"in-4.932 + "' + Bn-VBn-1-

“3) 1@0930 + 130% + 1' “132 + + 1363114 I:

1

(1‘2) 31’30 ” iii-1'31 * 31% " " ”1'Mn-1 )1

. 1?
+ . ’1 KhS)

' 1

(112) tin-1'3; +m_1° + _1- + + 4-,,

 

 

where the origins of each row are indicated by the numbers in parentheses
along the left-hand side of the arrays. As a consequence of "mixing"
(h2) and (hB) in this manner it is now possible to write general
eXpressions for the resulting series which are obtained. The use of

two molecules i.e., (hO) and (h1) may be further justified on the basis
that one may, when selecting a molecule "grab" one end one-half of the
time or the other end one-half of the time. However, both approaches
lead to the same approximate expression and the advantage of simila
taneously considering (to) and (h1) is that considerable symmetry is
introduced into the intermediate steps. Returning to (hO) and (h1) the

over-all mean square length for both situations will be h: (where the bar

33

indicates average quantities), and

‘7 - 1/2 ha" mi 4- 1/2 EM’DM (1.6)
If end effects are neglected, 31°31, ~1+k 51 +3 +1: , vi 1,11 +3
31 +k°Ei + j +1: 3 then (1.16) can be arranged in the form corresponding to
(1111) and (115)
n-1n-1
-1/2[3n + 2 Z (n-3)R oN ] +1/2[3n + n21 (n--k)'§'im-u-o 0N 3] (147)
3‘1 k"
where N, - 3,3 for j even, 1&3. = Ij‘lvj for 3 odd; Nk - 3k for k odd and
NR - 35k for k even.
Let L 3 be defined as the expected value of the scalar product term

occurring in (117), i.e.,

E {gown ,, Yoko) , gaze)? , (1,8)
then,
n-1 n- '1
. 1/2 [3,, + 2 2 (11-3) L3 ]R + 1/2 [3n + 2k; (n-k) Lkln (119)

(o o) o)
where in this expression L pertains to terms for which (X ), Y( , Z( )

j ( )
is an 5-bend, and Lk pertains to terms for which (1: ° , Y(°), z(°)) is an
Iii-bond. The coordinate system is now chosen such that (X(°), Y(°), 2(0)) =

(1,1,1), in which case

L3 , E (In) ,Yo) , 2(3)) (50)

and clearly Lo - 3. However, (X(°), Yb), 2(0)) - (1,1,1) may still be

an «lg-bond or an §;bond.

C . Probability matrix.

Again as in the polymethylene chain, the sequence of bond-pairs

3h

12 3 b 5 6 7 8 91011121311. 15161718192021 2223211

,4

(DH O‘iUIC'UN-e

9
1O

11
12
13
1h
15
16
17
18
19
20
21
22
23 '

211 c b a.

 

Figure 8. The submatrix of transition probabilities PR. The

column and row headings (state) are explained in

Table III e

35

25 26 27 2829 3031 32 33311 35 36373839110 111112 133111-1115 116117118

bca
ca

c a
A

 

Figure 9. The submatrix of transition probabilities PM' The

column and row headings are explained in Table III.

36

(565,), @5132), ... , (§n_1hh) is considered. (Stick and ball models

will be found helpful in the following considerations.) The coordinates
“R4 may then have
the sets of components (1,-1,-1), (-1,1,-1), and (1,1,1), with probabilities

may be chosen such that 50 . (1,1,1), £11 - (1,1,-1).

0, B, and A, corresponding to their respective conformations. In Figure 8
it is seen that these sets correspond to transitions from state(r_o_w) 1 to
states (columns) 11, 5, and 6, respectively. Now, each of these events

is treated as though it had occurred, and the subsequent possibilities

are considered. For example, if It, 8 (1,-1,-1) has occurred, corres-
ponding to the transition state (£9!) 1 to state (columnH-t with
probability C (Figure 8), as just discussed, then, the next transition
originates from state (32!) h, and may proceed to states (columns) 13,

1h, and 15, with probabilities (conformations) b, a, and 0, respectively.
The next transitions then originate from states (39313) 13, 1b, and 15,

depending on the outcomes of proceeding trials. Now, 4!} may also have

1
the sets (1,-1,1), (~1,1,1) as well as the set (1,1,-1), given above.
Therefore, these possibilities and all possible subsequent events must
also be considered. This procedure is then continued until the submatrix

of transition probabilities P Figure 8, has been obtained. It is then

R’
discovered that all subsequent bond pairs return to some previously
considered state of Figure 8.

Now, the bond with coordinates (1,1 ,1) may also be any-bond.
By methods analogous to those given above, the case where No - (1,1,1)
leads to the submatrix PM given in Figure 9. The two submatrices are
then combined to give (51), which describes all possible transitions.

13118th ' (S1)

 

 

37

9101112 131141516

nu no AU AU
DOCO
Cbao
0001..

12314567890
1

1
1

2
1

3
1

In.
1

S
1

 

6
1

The matrix of transition probabilities P.

Figure 10.

38

The submatrices P and P are given in Figures 8, and 9, respectively.

B M

Table III. The forty-eight states of the isotactic vinylic chain

 

 

State U 3 U 3 +1 State U 3 U 3 +1
1 £3 ll 13 13 3.11 g; 12
2 £3 33 12 ”1 £41 5 1a
3 g3 g5 11 15 3-11 )5-3
11 1113 3-11 16 3-12 5 11
5 1413 5-12 17 5-12 :1 1a
6 M13 B, 3 18 3-12 g-B
7 £112 3-13 19 3-13 1:1 1a
8 M12 3-11 20 lit-13 £1 11
9 9112 B 3 21 3-13 5-3
10 £11 3-»13 22 g-B 3-13.
11 £1, an; 23 21-3 £43
12 3111 B, 3 2h u-B 11-11

 

 

 

 

 

 

 

 

 

The remaining twenty-four states are numbered from twenty-five to

forty-eight in the order above and with 3‘; and g interchanged.

However, for the purpose of calculating the mean square length,
)
, x(a) ,, Y<a + 2(3),

the bond components are not needed. Since [1(3)

only the sum of the components is neededo Therefore, the 118 x 118
matrix of transition probabilities (S1), is more elaborate than need

be, and by a collapsing process similar to that used in the polymethylene
chain, (51) may be collapsed to Figure 10, corresponding to the sequence

of pairs (U001), (U1U2), (0203), , (U ). Again each member

U
n-1 n
of this sequence is a state in the sense of the Markov formalism and

39

and the sequence of pairs forms a Markov chain with matrix of transition
probabilities given in Figure 10. Thesiates are relabeled from one to
sixteen, and the relationship of the original to the relabeled states is

given in Table IV.

Table IV. Relation of the relabeled states to the original

states for the isotactic vinylic chain.

 

 

 

 

Relabeled State Numbers Original State Number
Number Relabeled State

1 3 1 1, 2, 3 ‘

2 -1 1 13, 17, 20

3 -1 1 1h, 16, 19

h -1 -3 15, 18, 21

S -3 -1 22, 23, 2h

6 1 -1 h, 7, 11

7 1.1 5, 8,10

8 13 6, 9, 12

 

 

 

 

The.remaining eight states (states nine through sixteen) are
obtained from the original states twenty five through forty-

eight by a similar scheme in analogous order.

The matrix of transition probabilities P in terms of the relabeled
sixteen states is given in Figure 10. The states one through eight

and nine through sixteen constitute "closed sets"27. That is, from any
of the states one through eight it is not possible, regardless of the
number of "steps" taken, to end up in any of the states nine through

sixteen and vice versa.

 

ho

D. Calculation of Mean Square End-to-End Length from the Probability
Matrix.
From the selection of U(o) as (1+1+1) viz., 3, and from (50) it

follows that if the k-th power of P is denoted by Pk and its matrix
(R)

elements by pij , then

(k) (k) (k) (k) (k) (k) (k) (k)
Lk ' 399,9 ' (99,10 + p9,11 + p9,12)'3P9,13 + (P9,1h + p9,15+ p9,16)
(52)
and
L, . 312$?) - (p33) + pg) + p353) sag.) + egg) + p33) + :13): )
53

For convenience, the complete matrix of transition probabilities

may be denoted by

 

 

o s o 6”

P _ T o o o (Sh)
o o o v
_p 0 U 9,
PR 0'7

s (55)

o P
L. M.

 

 

Since PR and PM constitute closed sets, they may be considered separately
(see reference 27, p. 3h9). Only the leading term will be calculated

for the isotactic case (this is an excellent approximation for large n).
A straightforward application of the method of calculation used in the

polymethylene case generates matrix inverses of the form (1-ST)’1. These

inverses, however, do not exist, since det (1-ST) a O, and the calculation

h1

gives a leading element of the form 0/0.
To avoid this difficulty it is necessary to set
0 K O S'
P - + (56)

K O T' O
whereflK is the four by four matrix all of whose elements are 1/11 (see
Appendix 0), and s' - s - x, and T' - T - K. Observing that Kn - K
for all n; that the sum of the elements of any row or column of S' or T'
is equal to zero; and that this null property is preserved for any

product matrix from T' and 5'; it is found that

K O S'T' O

P; e PRPR - + (57)
O K O T'S'
1— ‘ r. "‘
0 K O S'T'S'

pa . PEP; - + (58)
K 0 T'S'T' 0
L. _ L.

 

 

 

 

and so on. P; is, of course, the unit matrix of order eight

As before, L is given by (53). The K matrix gives 23 contribution to

3
L3. Therefore, for purposes of computing ER one may'write

O S' S'T' 0
Pfi-‘l’ Pi. , Pfi- , 000 (59)
T' 0 O S'T'
and, in general
0 Q(3)1
Pg - for 3 odd, (60)

 

h2

and

,
Qm 0

Pg - for 3 even,
0 Ql(j)

 

he

and the meaning of Q(j), and Q'(j) is clear from the context. For

(61)

1
all positive 3, Q(i (and Q'(j)) has the property that each of its rows

and columns sums to zero. Denoting the elements of Q(j)
becomes

L. - w [as - «as us +11%:
Since

( ) ( ) ( ) (')
Q13 ”11% ”1113 '“qfi-J

it follows that

Lj-(-1)3hq$~;‘) , “£0 ,and Lo=3

Therefore
n-1 ( )
- - 3n + 2 Z n-j L
E. 3.. j
n-1 ,
- 3n + 21 {-1)3(n-j) q§¥)
j:

The leading term of the summation in (66) is

n-1 .
m' - z (-1)3nq1($)
3'1

which is the element in row one and column one of the matrix sum
n-1
)

[ml . n z (_1)j q(j
3-1

by qég), (53)

(62)

(63)

(6h)

(65)

(66)

(67)

(68)

113

01'
M' - n(-S' + S'T' - S'T'S' + ...) (69)

where the convergent finite series (68) has been replaced by an

infinite series (69). Now,

1

3M'-s'+9r-sww'+u. um

and upon solving for M' , one secures
H' I n(1-S'T')'1(S'T'-S‘) (72)

The needed inverse now does exist. The construction of n1'1 is a
straightforward but laborious task (APPENDIX D). After m1'1 is found

ER follows readily.* One obtains
E - (n/D) [8(AB+AC+BC)(ab+ac+bc) + h(Ab+Ac+Bc)(Ab+Ac+Cb) + A] (73)
where

D - (ab+ac+bc)(B3+BC+Cz) + (Ab+Ac+Bc)(b3+bc+cz) (7h)

1: A better approximation to ER can be found in principle by adding to
the ER given by (h?) the element of row one and column one of

16 S'T'[(1-S'T')3]’1 (S'-1) + 8(1-S?T')‘15' (this corresponds to the
second term of Equation 27, because of the null property of (S')°°
there is no third term). The resulting ER in that case would be of
the order of approximation that corresponds to the one for the special
case of polymethylene.

A - 2 (A3+BZ+02)(a3+b=+o2) - 2(Aa+Bb+Cc)2
+ (Ab-Ba)(Bb+Cb-3Bc)
+ (Ac-Ca)(Bc+Cc-30b) (75)
+ (Be-ob)(Bb-Ce+2Be-2Cb)

+ 2(BCa3 - Aabc)

If A-a, B-C-b-c-1/2(1-a), then A-0, and the remaining terms of (75)
reduce readily to the leading term of (38).

Comparison of PM with PR shows that EH can be obtained from ER
by making the following interchanges in ER; A with a, B with c, and
c with b. With EM and ER known, E” follows from (h9), and one obtains

h" _ 1m (1- oz)(1-2'.3) + 20322-2( 014 + A4 . 1.94 3.3..
(1- 03)(B2 + BC + ca) + (1-22)(b3 + be + a?) 3

where

0.2 . 8.2 ... b2 4. c2
22 . A2 ... Ba ... cz
ox“ - (la-Bc)(a‘-bc)
p‘ - (anACXba-ac)
8“ - (Ca-ABXcz-ab)

and within the curly brackets, the mean square length is scaled by the

factor 12/3.

hS

IV. CONCLUSIONS AND DISCUSSIONS

A. General.

Applications of equations (39) and (7o) are restricted to polymer
molecules whose conformations are undisturbed by interactions with
solvent molecules. Thus the equations are useful for polymers only in
the ”unperturbed" state or in bu1k(no solvent). Solvent interactions
with the polymer molecules cause conformational changes which result
in eXpansion of the polymer coil. In addition, excluded volume of a
polymer becomes appreciable with resulting changes in the mean square
dimensions.

However, in analogy with the Boyle point of a gas, real polymer
molecules can be investigated in an "ideal solution” where the solvent—
polymer interactions, which cause eXpansion, are precisely counterbalanced

by the intramolecular interactions which favor contraction of the chain.
Under these conditions, called the "unperturbed” state, the polymer

coils in a ”random flight" manner, the excluded volume is gero, and the
chain dimensions are minimal. The exact temperature at which the
unperturbed state Obtains is designated as the Flory 699-temperature.
According to Flory this condition also obtains in the ”melt" state of
polymer molecules.

The crystallization of polymer molecules involves the minimization
of the potential energy due to rotational barriers in.the chain backbone.
Calculations by P. J. Florth indicate that near neighbor intramolecular

forces are of paramount importance in determining crystal structure.

If this is true, then the preferred conformation for a polymer molecule

to

in its unperturbed state should also tend towards the crystalline state
conformation. The recent infrared spectral studies of M. Takeda and
coworkers35 on isotactic polystyrene indicates that the helical
crystalline state structure of isotactic polystyrene is partially
retained in solution.

Polymer molecules, however, are unable to crystallize in the
continuous and orderly manner of small molecules. Since the molecules
are so large it becomes extremely difficult for long segments of
different polymer molecules to become extensively ordered (within
experimentally feasible time limits) such that perfect crystals are
obtained. Crystalline polymers, as normally obtained, contain
extensive amorphous regions of imperfect order intermixed with crystallites
of high order. This L'picture" of polymer crystallization has recently
come under attack, and the alternate ”picture“ of considering the
amorphous character as resulting from exaggerated lattice defects is
favored by some authors36. However, at the=Ortemperature the polymer
molecules are in the liquid state and.the difficulty of aligning many
macromolecules disappears. Thus, all bonds of a particular macromolecule
are free to tend towards their equilibrium positions as are indicated by
their crystalline state conformations.

The fbllowing restrictions apply to the equations derived in (II)
and (III);

1. The polymer chains must be long (large n), linear macromolecules
of regular head-to-tail structure.

2. The valence angle has been restricted to the tetrahedral valence
angle.

3. The "excluded volume" effect has not been taken into account

h?

(see reference 12, chap. 12). That is, the chain is allowed to return

to positions in space which it has already occupied, and more seriously,
also to positions which are excluded because of intramolecular repulsions.
However, this difficulty essentially disappears at the Flory-Oatemp-
erature.

h. The bonds are assumed to be statistically independent. This
means that the conformation of any given bond is got influenced by the
conformations of neighboring bonds.

5. Any three successive bonds are restricted to the trans or

one of the two gauche conformations.

B. Polymethylene
It is desired to investigate the behavior of Equation (39) when
only the leading term is considered. To be specific, the calculation

will be performed for the case corresponding to free rotation, a - 1/3.

Equation (39) .ecomes
52'. (2n - 3/2) 13 (77)

In Table V, the magnitude of the error involved in considering
only the leading term of (77) is indicated in column three. It is
seen that the error becomes very small with increasing n. Since even
low molecular weight polymers normally have n many times greater than
one hundred, it is seen that the leading term eXpression is an excellent

approximation.

118

TABLE V. Magnitude of error in the leading term approximation.

 

 

 

 

 

 

 

 

..2
n 52/1: 2n;;5(: [18) x 100
h /.Lo
5 8.5 17.5
10 18.5 8.1
20 38.5 3.9 L
50 98.5 1.5 “
100 198.5 .7
In Figure 11, the leading term (large n) of (76) is plotted LT-

 

against increasing values of (a). For the case of free rotation a a 1/3
and (BE/n13) - 2.00. At a - .8, (hi/n12) - 1o.o, which is the experimen-

tally determined value of (FE/n13) for polymethylene chains in their
"unperturbed”states. The value (hi/n13) - 10.6 was calculated from
second virial coefficient data by'w. R. Krigbaum29 from the experimental
results of Q. A. Trementozzih2.

For linear macromolecules the conformational potential energy is
given by:

V(¢o. ¢1, .... ¢n_1). (78)

That is, the conformational potential energy is a simultaneous

function of all the rotational angles. If the bonds are statisti-

cally independent, the potential energy may be eXpressed as the sum ofthe
individual potential energies:

n-1

W.» 1251, 15.4) - 130 mi). (79)

\”;J

10.6

2
n1o

.67

119

 

 

 

Figure 11.. Plot of hz/nl: versus (a).

 

50

‘With this simplification, the probability (a) of occurrence of

the trans position is given‘by:23’ 30

a - [1 + 2e "NJ/RT)“1 (80)

where AE is the energy difference between.the trans and gauche
conformations. For a - .8, R - 2 cal. mole'ldeg:1, and T - 363°K,
the-Obtemperature for polymethylene, the energy difference between the
trans and gauche conformation is calculated to be: AE - 1.51 kcal. mole-1.
This value is of the correct order of magnitude since from small

30 to be approximately .8 kcal. mole'1.

29

molecules, AE has been estimated
Since the value of (hi/n13) - 10.6 reported by Krigbaum was calculated
from measurements made in a good solvent, the results obtained are only
approximate. However, the result a - .8 does indicate considerable
retention<3f crystalline state structure in the unperturbed state for
the case of polymethylene. For statistically independent bonds this
implies that eight out of every ten bonds are in the trans conformation
corresponding to the planar zigzag crystalline structure.

Equation (39) may also be expressed in terms of average rotational
angles by the following considerations. The average of the cosine of

the rotational angle (¢), where (¢) is measured from the trans position

(see Figure 3), is

cos y - a cos (0°) + b cos (120°) + b cos (2h0°), and (81)
Era—F'- a sin (0°) + b sin (120°) + b sin (2L10°) - 0. (82)

Since b - (1/2)(1-a), Equation (81) becomes

cos - 2353—1 , or (83)

51

a - 2 0°83¢ + 1 (8’4)

Substitute (8h) into (39). The leading term eXpression then becomes:

 

 

32.2.12 “6°”? (85)
1+-cos¢
Utilizing a rotational matrix approach, H. Benoit;31 and 0.B.
Ptitsyn and M. V. Volkenstein32 obtained the eXpression
fig . n12 1 + cos(nd6) 1 + cos Z (86)

 

0 1 - cos(wd09 1 _ Egg—a
When'6-- 109.3°, i.e., the tetrahedral valence angle, (86) becomes the
same as (85). Comparison of (86), (85), and the leading term of (39),
shows that the expression for 57' corrected for any angle of (09, and
expressed as a function of the probability (a) becomes:

h: : n13 1 + cos («~69 1 + 3a (87)

3 1 - cos (nae) 1 - a
Since polymethylene in the crystalline state has a valence angle (09
of 111°, a better approximation of (a) and AE can be obtained by (87).
In this case a - .77, and AE - 1.38 kcal. mole‘l. (The number of
”significant" figures retained do not imply degree of accuracy, but are
only useful for comparative purposes.)

In a recent publication (publiShed after the completion of this

calculation) R. P. Smith25 has shown the rotational matrix method and
the method of transition probabilities ileads to identical equations

for the simple polymethylene-type chain. By expressing cos a and sin 6

in a manner analogous to (81) and (82) he obtained by a.rotational matrix

52

approach, the expression

 

2
h? - “3° (2 + 6“) -3/2 " —————8 (3° ’1) 12 (88)
3 (1 - a) 3 (1 - a)2 °

which by appropriate rearrangement is found to be exactly_the same as

(39).

C. Isotactic Macromolecules.

As previously state , isotactic polymers are more complex in that
two types of bond constitute the repeating mer of the chain. This type
of molecule has periodicity two and requires four independent parameters
to describe the chain.

Both isotactic polystyrene and isotactic polypr0pylene crystallize
in a 31 helical structure37; a Chain where trans (a) and gauche (B)
conformations alternate. If there is a strong tendency to retain flnis
structure in the unperturbed state, then the assumption may be made
B . a, and A - C - b - c. The calculated.values of (h?/ n13) using
Equation (76) for various assumed values of the probabilities are given
in column four of Table VI. This case will be discussed later relative
to pertinent experimental data.

An additional regular conformational chain of interest is one
consisting of rotations of each bond in the same direction into alternate

gauche positions. In this case, the assumption is made C - b, and
all other probabilities are assumed equal. This type of conformation
generates a bl helix and results in a tightly packed polymer chain.
Polyoxymethylene has been shown to possess a structure of this type.
Although the unperturbed dimensions of this polymer have not been

measured it is possible that its dimensions could be less than that

53

calculated for the free rotation case, i.e., less than 2nlg. The values
calculated according to Equation (76) for a chain of this nature is
given in column five of Table VI. It may be noted that the chain
dimensions eXpand as more trans conformations are introduced or if

high retention of the gauche conformations obtains the chain

dimensions pass through a minimum which is less than 2n12.

Column six of Table VI summarizes increasing values of (fig/n13)
for increasing values B - b. The extreme case B - b - 1, corresponds
to a chain which follows a path as described by the "chair" form of
cyclohexane. Since real polymer chains can not occupy any position
more than once, this situation is obviously unrealistic in terms of any
physical model. However, it does serve as a useful check on the behavior
of the derived equation. Since, the equation, if properly derived (does
not contain computational errors), must predict (FE/n13) 4’0, as B -
b‘+'1.

For macromolecules which tend to crystallize in a planar zigzag
conformation (21 helix37), the perfect crystalline state may be
characterized by'A - a - 1, corresponding to an all trans sequence of
bonds. If there is a strong tendency to retain this structure in the
unperturbed state, then the assumption.may be made A - a, and B - C -
b - c. The calculated values of (ha/n13) using Equation (76) for
various assumed values of the probabilities are given in column three
of Table VI. The polymethylene chain is an example of a polymer
chain which rigorously corresponds to the above mentioned assumptions.
This chain was discussed in the preceding section.

Ball and stick models of the structures corresponding to A - a - 1,

all trans; B - a - 1, gauche-trans, and, C - b = 1, gauche-gauche,

58

clearly demonstrate that the mean square end-to-end length Should, for
. the same number of bonds, and for the same value of the respective
parameters, decrease in the same order as is indicated in Table VI.
Comparison of columns three, four, and five shows that the mean
square lengths as calculated by the derived equation, i.e., (76), does

predict this relative behavior.

Table VI. Variation of:mean.square length with change in

magnitude of probabilities.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(Hz/n13)

2.. A.. Baa Cdb B-b
Jr’r’ 47'——' Bac-b-c A-Ceb-c AsB-a-c A-C-a-c
0.0 0.5 .67+ 1.99* 8.33*:] 8.67
0.2 0.8 1.38 1.96 2.73 2.29
1/3* 1/3 2.00 2.00 2.00 2.00
0.8 0.3 2.85 2.12 1.88 1.78
0.6 0.2 7.18 2.96 1.81 1.08
0.8 0.1 11.8 5.96 3.10 0.87
0.88 .08 18.8 7.55 8.61 0.38
0.9 .05 28.7 12.88 6.19 0.23

 

 

 

 

 

 

 

ifCorresponds to a tetrahedral macromolecule with free rotations.
37

+Indicative of macromolecules which tend to form 21 helices .

1Indicative of macromolecules which tend to form 31 helices37.
i 37

Indicative of macromolecules which tend to form h1 helices .

55

There are only a few measured unperturbed dimensions reported

for isotactic polymers,

for the atactic configuration.

since most of the data in the literature is

In Table VII the experimental data

of J. B. Kinsinger and R. A.Vbssling are summarized for atactic and

isotactic polypropylene.

Table VII. Polypropylene at the Flory thetaCGQtemperatures.

39.80.83

 

 

 

1
Config- Ghsolvent are) 1"19:10“ 11:11 (figflflyz 613/11}, 2 (EV/PWé 83/111?
uration X 1011 x 1011
atactic 1-chloro- 78 18.2 0.50 875 868 2.02 8.15
naphtha-
lene

atactic cyclo- 92 17.2 0.50 875 820 1.96 7.67
hexanone

atactic phenyl- 153.8 12.0 0.50 875 571 1.75 6.12
ether

isotactic phenyl- 185.2 16.2 0.50 875 778 1.93 7.88

. ether

 

 

 

 

 

 

 

 

 

 

15 is taken as 2.1 x 1021
'0'
+ [7‘] - 101‘

As a measure of "stiffness” of the polymer chain the unperturbed

polymer dimensions (HE) are frequently compared with the free rotation

dimensions (5%). The square root of the ratio of unperturbed to freely

rotating dimensions are given in column eight;

order of increasing temperature.

in column nine in

It is seen that the unperturbed

dimensions of the atactic polymer decrease with increasing temperature.

This is consistent with an exponential dependence of the probabilities

as is indicated in the discussion of the polymethylene chain.

The retention of crystalline state structure for isotactic
polypropylene may be estimated from the experimentally determined
unperturbed dimensions given in Table VII. Since isotactic polypropylene
has a 31 helical structure corresponding to a trans (a)-gauche (B)

sequence of bond conformations, from column four of Table VI, one has

B - a - .88, and

A - C = b a c - .08
This implies approximately 80% retention of crystalline state conformation
in the unperturbed state. From the data of W. R. Krigbaumy1 et al. for

isotactic polystyrene, the quantities

B = a - .91

A e C u b I c - .OhS

are obtained, indicating an even greater degree of retention of crystal-
line state structure in the unperturbed state.

It is desired to investigate the error involved in assuming
A - c - b - c. For this purpose, (HY/n13) is calculated for B - a - .88
but for C - c - O, and A - b - .16. This represents a maximum is
deviation from the equality A - C - b - c. For these values of the
parameters the result is obtained (fig/n13) ' 7.07, in contrast to the
value (Egynlg) ' 7.88 for the case where A - C a b - c.

Another source of error is the assumption ofthe tetrahedral valence

33 is

angle. Actually the valence angle for isotactic polypropylene
69-- 118'), and for isotactic polystyrene CS - 116‘) as measured by
X-ray diffraction in the crystalline state. An indication of how these
deviations from the tetrahedral valence angle effect the calculation

can be obtained by observing the behavior of the polymethylene chain

57

as-e-increases. This may be accomplished.by utilizing Equation (87)
which applies to polymethylene chains with constant bond length, but

arbitrary bond angle. For-9-- 11h°, Equation (87) becomes

(HZ/n13) - 2:32“ :33) (89)
) - a

Comparison of (89) with the equation for the tetrahedral valence angle

viz.,

 

(EV/n13) - 2 (1 * 3a) , (9o)
3(1 - 8)

indicates that the ”retention of crystallinity" as calculated with the
assumption of tetrahedral valence angle, is probably too large. From
these considerations it is seen that the calculation of "retention of
crystallinity” is approximate in nature.

If the rotational angle (¢) is defined such that 0 - 0, for the
trans conformations, and (0) is measured counterclockwise (see figures
6 and 7), then the mean square length may be expressed in terms of

average angles as follows

cos ¢MRM - a cos (0°) + b cos (280°) + c cos (120°) . a - 1/2 (b + c)
(91)

sin pMRM I a sin 0° + b sin (2h0°) + 0 sin (120°) .‘i;: (c - b),

(92)

and further

EBE‘ZhMR - A - 1/2 (B + c) , (93)

and

mm J; (B - C) (98)

58

By means of (91), (92), (93), and (98), and the stochastic relationships
A + B + C - a + b + c - 1, the mean square length (78), may also be
expressed as a function of average angles. With this modification,

the mean square length’equation (76), then applies to isotactic vinylic
chains with constant tetrahedral valence angle (0) and statistically
independent bonds, but with arbitrary (0). That is, the chain is no

longer restricted to staggered conformations.

D. Additional Structural Problems.

The other important configurational chain of regular asymmetric
order is the syndiotactic-type vinyl polymer chain. As previously
stated, this chain is composed of two mer units (four chain bonds) with
the two pseudo asymmetric centers in alternate d and 1 configuration.
‘With these structural restrictions this type chain has periodicity four
and thus requires a separate solution.

The matrix of transition probabilities has been derived for this
type chain but the solution has not been completed. The process of
solution should be identical to that for the isotactic chain, but
more laborious to perform.

In the case of syndiotactic macromolecules the four different

d,
matrix of transition probabilities for a Mbrkov chain composed of the

types of'bond may be denoted by R' Md’ R1, M1. The "collapsed”

above indicated sequence units is:

59

 

 

 

 

 

 

 

 

 

 

Ad
C1
I)1
Ba
P ' D (95)
1
Bo
C1
A8
where
.- T )- q
C B A o c b a
B A C o b a c
A- B.
doacs’d. b.
(1000) L1 0_
'0 B c A" '0 b c a7 (96)
Cl'OCAB D1.0cab
OABC Oabc
1000‘ L1000

 

 

 

 

The states of (95) are numbered 1 through 32, and are classified in

Tab10,VIII.

Table VIII.

60

Classification of states of the "collapsed"

matrix of transition probabilities for a

syndiotactic chain.

 

 

 

 

 

 

 

State Uj Uj+1 State Uj Uj+1
1 Rde 3 1 17 Mle 3 1

2 Rdul -1 1 18 Mlad -1 1

3 11de -1 1 19 Mlnd -1 1
8 11de -1-3 20 M112 d -1-3
5 1“!de -3-1 21 Rde -3-1

6 Mde 1-1 22 Rde 1-1
7 ‘Mde 1-1 23 Rde 1-1
8 MdR1 1 3 28 1%de 1 3
9 . as 31 25 Ma. 31

. 10 RlMl -1 1 26 Mdal -1 1
11 R1M1 -1 1 27 1“!de -1 1
12 R1111 -1-3 28 14de -1-3
13 11le -3-1 29 R1711 -3-1
1h Mle 1-1 30 RlMl 1-1
15 Mle 1-1 31 RIMl 1-1
16 Mle 1 3 32 RlMl 1 3

 

 

Again, the limiting value matrix K must be subtracted from each of the
four by four matrices in order to avoid obtaining inverses of an
indeterminate form. The solution then follows in a manner analogous

to that given for isotactic macromolecules.

1.

7.

8.

9.
10.

11.

12.

13.
18.
1S.
16.
17.

61

V. BIBLIOGRAPHY

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Press, Ithaca, N. Y., 1953, Chap. 10.

w. Kuhn, Kolloid 2., 16, 258 (1936); 81, 3 (1939).
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20.
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62

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P. R. Rider, ”College Algebra", The MacMillan Comparv, New York,

 

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_7_8_, 657 (1951).

G. Natta, and P. Corradini, J. Polymer Sci., _2_O_, 251 (1956).

P. J. Flory, Proc. Roy. Soc., 238 A, 60 (1956).

A. Takeda, K. Iimura, A. Uamada, and Y. Imamura, "Studies on the

 

Configuration and Conformation of Polystyrene", to be published
in the Bulletin of the Chemical Society of Japan, September 1960.
A. Peterlin, F. Krasovec, E..Pirk1nayer, and I. Levstek, Makromol.

Chem., 3_7_, 231 (1960).

63

37. R. E.Hughes, and J. L. Lauer, J. Chem. Phys., 39, 1165 (1959).

38. Q. A. Trementozzi, J. Polymer Sci., 39, 113 (1958).

39. R. A. Wessling, and J. B. Kinsinger, J. Am. Chem. Soc., _8_1_, 2908
(1959).

80. R. A. Wessling, "The Dependence of Phase Equilibria on the
Configuration of Polypropylene", Master"s Thesis, Michigan State
University, Department of Chemistry, 1959.

81. W. R. Krigbaum, D. K. Carpenter and S. Newman, J. Phys. Chem.,
9.2.. 1586 (1958).

82. Q. A. Trementozzi, J. Polymer Sci., 36, 113 (1959)

83. Presented at the 135th Meeting of the American Chemical Society,
Boston, April 5-10, 1959.

EVALUATION OF

. Determinant of (1+3)

One has that

det (1+5) -

Adding rows two, three and four

det (1+3) .

 

 

O
O

1

2
O
O

1

APPENDIX A

<68

THE 1,1-ELEMENT 0F (1+3)‘1L

b b
1+b a
a 1+b
0 0

2 _2
1+b a

a 1+b
0 0

 

to row one,

2
b
b

1

 

Adding column two and three to column four,

det (1+3) 3

Multiplying thefburth row

one, and exPanding by the method of pivotal elements yields

det (1+5) -

det (1+3) - 3(1-a)2

 

 

0
0
0.
1

2 - 2
1+b a

a 1+b

0 0

2 2'
1+b a

a 1+b
0 0

6
2
2
1

 

(i)

(ii)

(iii)

by 2 and subtracting from row

8

’2

2

 

--8

 

1 “1 1
1+b a 1

a 1+b 1

 

(iV)

(V)

Adjoint of (1+s)1,1

One has that

adj (1+5) - adj

65

 

~1 b b
0 1+b a
0 a 1+b
1 0 0

1
a
b
b

1

 

d

The 1,1-element of the above adjoint matrix is

adj (1+S)1’1 .

 

1+b a b
(-1)2 a 1+b b
O O 1

=(3/h)(3+a)(1-a)

Combining (v) and (viii), one obtains

-1
(1+S)1,1 -

31/14

8n (1+3);i -

adj (1+5)1,1

 

det (1+s)'

(3+a)
(1-a)

 

2n (3+3)

 

(1-a)

and

 

- (1+b)2-a2

(v1)

(vii)

(viii)

(ix)

(x)

(xi)

66

APPENDIX B

EVALUATION OF THE 1,1-ELEMENT 0F [(1+s)2]'ls
Let [(1+s)2]-1s - v - TS, where
T = [WSW]:1 (1)
Since the 1,1-element is desired, then by matrix multiplication:

W11 ' tll 311 + t12 321 + t13 331 + tl4 341
(ii)
- 0 + 0 + 0 + t1,

Thus one needs to obtain only the (1,8)-element of [(1+S)2]’1.
Now, det (1+5)2 - det (1+8) det (1+3) - [det(1+s)]2. But det (1+s)

has already been evaluated, hence

det (1+8)2 = [3(1-a)2]2. (iii)
Also,
(1+b)2-a2 ab-b(1+b) ab-b(1+b) 2+5a2-7a-sb2
b(1+b)-ab (1+b)+b2-a(1+b) aa-a-bz ab-b(1+6)
adj(1+S)-
b(1+b)-ab -a-b2+a2 (1+b)+b2-a(1+b) ab-b(1+6)
L:-.(1+h)3+az b(1+b)-ab -ab+b(1+b) (1+b)2-a2

 

 

The 1,8-element of [adj(1+S)]2 is
1:14 - [(1+b)2-a2]2 + [ab-b(1+b)]2 + [ab-8(1+b)]2 + [(1+b)2-a2][1+5a2-7a-5b2]
= [(1+b)2-a3][(1+b)2-a2+2+5a2-7a-5b2] + 2[ab-b(1+b‘-]2 (iv)

- 2(3/8)2[(1-a)‘-(3+a)(Ea-1)]. (V)

67

From (iii) and (v) one obtains

8[(1+s)2]'~ls . gig—3.}:— - 1; (”‘18:“) , (vi)

6’8”

I

APPENDIX 0
EVALUATION 01‘ ms 1,1-mr 0F 5“"1 [(1+s)3]"1

Since Sn+1 is not easily obtained for arbitrary n, the approximation

33% s"+1 - 3(a) - K (i)

is made, where K is a four by feur matrix all of whose elements are
equal to 1/8.
The isotactic chain has periodicity two. Markovian.processes of

periodicity two have as their limiting value (see reference 27, page

361)
1/8 if in s

(n2) ,
.1138 3k 0 (11)

otherwise.

Thus one immediately obtains

(a9 0 1/8 0 K
p . .. for n odd, and (iii)
1/8 0 K 0
1/8 0 K 0
a a for n even. (17)
0 1/8 0 K
Therefore
9,“ Sn+1 - K for all n. (v)

The.1,1-element of s(°°) [(1+s)"1‘]"1 - [1/8][(1+S)3]"1, is then 1/8
times the sum of the first column of [(1+s)=]"1.

Since

[1/h][(1+S)‘]‘1 - (1/81 ‘d “*S” . (vi)

et +

69

the problem reduces to securing the first column of adj (1+s)2.
If the elements of adj (1+3) are denoted by 313, then the sum of

the first column of adj (1+5)2 is

‘dJ (1+S)3" ' 311 + 312 321 + 313 331 + 314 341 +
321 311 + 322 321 + 323 331 * 324 341 *

(vii)
331 311 * 332 321 * 333 331 * 334 341 +

341 311 + 342 321 + 343 331 + 344 341

' 311 (311 + 321 + 331 + 341) +
321 (312 + 322 + 332 + 342) +
331 (313 + 323 + 333 * 343) +

341 (314 + 324 + 334 + 344)

(viii)

' 311 31 + 321 32 + 331 33 * 341 34 (ix)
When the indicated summations are performed, one obtains
31 T 32 ' 33 ' 34 ' 6b2 (1)
Therefore (ix) becomes
‘d3 (1+5)? ' 6b2 (311 + 321 + 331 + 341) (Xi)

and when this sum.is evaluated, one obtains

adj (1+5): - 683-682 - 6284 - 9/8 (1-a)‘ (xii)

The det(1+S) has already been calculated (A-v), and therefore

det (1+5)a - [3(1-a)3]2 - 9(1-a)4 (xiii)
Substitution of (xii) and (xiii) gives then

s(°°)[(1+s)2]-1l - 1/16 , (xiv)

70‘

and finally

8 511+].
[(1+S "
)2] 1 4’ 1/2
(IV)

.71

APPENDIX D

EVALUATION OF THE 1,1-ELENENT 0F (1-s'T')-1(s'T'-s')

Needed are:
(a) First row of (1-S'T')‘1

(b) First column of (S'T'-S')

Now

where A' - A -1/8, etc.

SIT...

Further,

(1-51r1)'-

 

 

 

 

 

 

:1/8 C' B' A') —-1/8 c' b' an
-1/8 8' A' C' -1/8 b' a' c'
S'T' - -1/8 A' 0' B' x -1/8 a' c' b' (i)
L3/8 -1/8 -1/8 -1/8 J L 3/8 -1/8 -1/8 -1/8 J
Matrix multiplication.yields:
PA -1/8 Cb + Ba -1/8 Ca + Bo -1/8 Cc + Bb -1/8-)
C -1/8 Bb + Aa,-1/8 Ba + Ac -1/8 Bc + Ab -1/8 .
B -1/8 Ab + Ca -1/8 Aa + Co -1/8 Ac + Cb -1/8 (11)
L.-1/8 c -1/8 b -1/8 a -1/8 ‘7
The first column of S'T'-S' is therefore:
A -1/8 + 1/8 - A
C -1/8 + 1/8 - C
B -1/8 + 1/8 - B (iii)
-1/8 - 3/8 - -1
-1-(Ae1/8) -(Cb + 3. -1/8) -(Ca + Bc -1/8) +(0c~+ 3b -1/8)1
-(C-1/8)‘ 1-(Bb + Aa -1/8) -(Ba + Ac -1/8) -(Bc + Ab -1/8)
-(B-1/8) —(Ab + Ca -1/8) 1-(Aa + Co -1/8) -(Ac + Cb -1/8)
L + 1/8 -(c -1/8) -(b -1/8) 1-(a -1/8) 1

 

 

fiv) I

’72

To obtain the determinant of (1-S'T'), in order,one may:
1. Add columns one, two and three to column four.
2. Add rows two, three, and four to row one.
3. subtract row one from row four.

In this manner one obtains

‘ -8 -1 -1 -1. \
80-1 (Bb + Aa -1)-1/8 Ba + Ac -1/8 -21
88-1 Ab + Ca -1/8 (Aa + Cc-1)-1/8 -1
-8 8c -1 8b -1 -8
(v)

8. Subtract column four from column one, and multiply row four

det (1-S'T') - 1/16

 

 

by 1/8 and column one by 1/8 and obtain

1 0 -1 -1 -8
C (Bb + Aa -1)-1/8 Ba + Ac —1/8 -1

det (1-S‘T') -
B Ab + Ca -1/8 (Aa + Cc -1)-1/8 -1

 

 

)0 c b 0
(vi)

5. Expand (vi) by minors of the.first row
6. Let 1 - (A + B + C)(a + b + c) in the expression obtained

from (vi)

7. Rearrange and Obtain

det (1-S'T') . 2[(1-0‘2)(Bz + BC + 02) + (1-112)(b2 + be + 62)]
(vii)

where:
02-a3+b2+c2

73

The adjoint numerators standing in the first row of Adj (1-S‘T') are:

(Bb + Aa -1)-1/8 Ba + \c -1/8 Bc + Ab -1/8‘

[1,1] - (-1) Ab + Ba .1/8 (Aa + Cc -1)-1/8 Ac + Cb -1/8

 

 

c -1/8 b -1/8 (a-1) -1/8
(viii)
Cb + Ba -1/8 Ca + Bo -1/8 Co + Bb -1/8‘

[1,2] ' (+1) Ab + Ca -1/8 (Aa + Ca -1) -1/8 Ac + Cb -1/8

 

 

c -1/8 b -1/8 (a-1) -1/8.‘
(ix)
Cb + Ba -1/8 Ca + Bo -1/8 Co + Bb -1/8‘

[1,3] I (-1) (Bb + As -1) -1/8 Ba + Ac -1/8 Bc + Ab -1/8

 

 

c -1/8 b -1/8 (a-1) -1/8
(I)
Cb + Ba -1/8 Ca + Bo -1/8 Co + Bb -1/8-

[1,8] - (+1) (Bb + Aa -1) -1/8 Ba + Ac -1/8 Bc + Ab -1/8

Ab + Ca -1/8 (Aa + Co -1) -1/8 Ac + Cb -1/8

 

 

J
(xi)

Adding columns one and two to columns three in each of the above, one

obtains:
(Bb + Aa -1) -1/8 Ba + Ac -1/8 -c -3/8‘
[1,1] . (-1) Ab + Ba -1/8 (Aa + Ce -1) -1/8 -B -3/8
c -1/8 b -1/8 ’3/8

 

 

 

'7 (xii)

[1:2] ' (+1)

[1.3] ' (-1)

[1,8] - (+1)

 

 

78

’ Cb + Ba -1/8

Ab + Ca -1/8
c -1/8

Cb + Ba -1/8

(n+Aan)4m

c -1/8

Cb + Ba -1/8

(Bb + As -1) -1/8

 

Ab + Ca -1/8

Ca + Bo ~1/8

(Aa + Co -1) -1/8
b -1/8

Ca + Bo -1/8
Ba + Ac -1/8
b -1/8

Ca + Bo -1/8
Ba + Ac -1/8

(Aa + Co -1) -1/8

1-AaA‘
-B -3/8
-3/8

 

(xiii)

.1- A -3/8
- c -3/8
-3/8

 

(xiv)

1 - A -3/8
- C -3/8
- B -3/8 4

(XV)

 

Now, multiplication of the first row of adj (1-S'T') into the first

column of (S'T'-S') gives:

A[1,1] + C[1,2] + B[1,3] - [1,8] I (A[1,1] - [1,8]) +-(C[1,2] + B[1,3]).

In detail:

A[1.1] ' (-1)

'[19h] ‘ (’1)

 

1

 

(Bb + Aa -1) -1/8
Ab + Ca -1/8
Ac -(1/8)A

Cb + Ba -1/8

(Bb + As -1) -1/8

Ab + Ca -1/8

Ba + Ac -1/8
(Aa + Cc -1) -1/8
Ab -(1/8)A

Ca + Bc -1/8
Ba + Ac -1/8

(Aa + Co ~1) -1/8

-0 -3/8'"
-B ;3/h
-(3/8)A.‘

 

(xvi)

1-A3N
- c -3/L
- B we

 

(xvii)

C[1,2] - (+1) C(Ab + Ca -1/8) [(Aa + Co -1) -1/8]C C(-B -3/8)

75

A Cb + Ba -1/8 Ca + Bo -1/8 1 -A -3/8

 

 

 

 

c -1/8 b -1/8 -3/8
(xviii)
Cb + Ba -1/8 Ca + Bo -1/8 1 -A -3/8
B[1,3] - (-1) B[(Bb + As -1) -1/8] 3(3. + Ac -1/8) B(-C -3/8)
c -1/8 b ~1/8 -3/8
(xix)

Combining (xvi) 4- (xvii) and (xviii) + (xix) one obtains

(-1)

(+1)

 

 

 

 

(Bb + Aa -1) -1/8 Ba + Ac -1/8 -c -3/8
Ab + 02. -1/8 (Aa + Co -1) -1/8 .13 -3/8 +
Ac + Cb + Ba -(1/8)A -1/8 Ab + Ca + Bc -Q/8)A -1/8 1-(7/8)A -3/8
(xx)
Cb + Ba -1/8 Ca + Bo -1/8 1-A-3/8
C(Ab+Ca)‘-B(Bb+Aa)+B+(1/L)(E C) C(Aa+Cc)-B(Ba->Ac)-C-(1/8)(B-C) (3/8)(B-C)
c -1/8 I b -1/8 -3/8
(xxi)

Expression (xx) will be evaluated first. The procedure is as

follows :

1.
2.
3.
8.

Multiply column three by (-1).

Add rows one and two to row three.

Multiply column one and colunn two by 3.

Add column three to column one, and add column three to

column two. This gives:

'76

3(Bb + As -1) +0 " 3(Ba + Ac) + C C + 3/8

(xx) I 1/9 3(Ab + Ca) + B 3(Aa + CO -1) + B B + 3/8

 

 

 

 

 

 

 

 

 

 

-3. (31+ 0) -3b(B + C) (3/8)A ‘+ 9k
(xxii)
3(Bb + As -1) + C 3(Ba + Ac) + C 80 + 3
I 1/12 3(Ab + Ca) + B 3(Aa + Co -1) + B 88 + 3
-c (B + C) -b (B + C) A + 3
(xxiii)
Equation (xxiii) man be split into:
3(Bb + As -1) + C 3(Ba + Ac) + C 80
- 1/12 3(Ab + Ca) + B 3(Aa + Ce -1) + B 8B -+
-c(B + C) 4b(B + C) A
(xxiv)
3(Bb + Aa -1) + C 3(Ba + Ac) + C 3
+ 1/12 3(Ab + Ca) + B 3(Aa + Co -1) + B 3
-c(B + C) -b(B + C) 3
(m)
3(Bb + Aa -1) + C 3(Ba + Ac) + C C
I 1/12 3(Ab + Ca) + B 3(Aa + Co —1) + B B
-h(B + B)c -8(B + C)b A
(mi)
3(Bb + Aa -1) + c + c(B +10) 3(Ba + Ac) + c + b(B + c) ‘ 0

+ 1/12 3(Ab + Ca) + B + c(B + c) 3(Aa + Co -1) + B + b(B + 0) 0

 

-c(B + C) -b(B + C) 3

(xxvii)

 

77

Equation (xxvi) is obtained by factoring out a 8 from column three of
(xxiv) and multiplying row three by 8. Equation (xxvii) is obtained‘
from (xxv) by subtracting row three from.row one and from row two.
The two determinants (xxvi) and (xxvii) are now evaluated in a
straightforward manner. As a consequence of the modifications performed
on them it is found.that a number of terms will cancel when (xxi) is
treated in a similar manner. The evaluation of (xxi) proceeds as
follows:

1. Multiply column one and column two by 3.

2. Subtract column three from columns one and two, and substitute

(1-A) '(3 + C) to obtain:

3(Cb + Ba) -(B + c) 3(Ca + Bo) -(B + C) (B + c)-3/8

(xxi) - 1/9 3C(Ab+Ca)-3B(Bb+Aa)+3B 3C(Aa+0c)-3B(Ba+Ac)3c (3/8)(B-C)

3c 3b -3/h
(xxiX)

 

Factor out a 3 from the third row, 1/8 from the third column:

3(Cb+Ba)-(B+C) 3(Ca+Bc)-(B+C) 8(B+C)-3
1/12 30(Ab+0a)—38(Bb+Aa)+38 3C(Aa+Cc)-3B(Ba+Ac)-3C 3(B-C)

c b -1

 

 

(xxx)
Split (xxx) into two determinants as follows:

3(Cb + Ba) - (B + c) 3(Ca + Bc) - (B + c) 8(B + c) -3
0/130 3 (Ab + Ca) 3 (Aa + Co -1) -3

c b -1

 

 

(xxxi)

 

 

(IMF

 

7B

 

3(Cb + Ba) -(B + c) 3(Ca + Bo) - (B + c) 8(B + c) -3
-3(Bb + As -1) -3(Ba + Ac) 3
c b -1
(xxxii)

From row two of (xxxi) subtract 3 times row three, and to row two of

(xxxii) add 3 times row three to obtain

(V12)3

(V133

3(Cb + Ba) - (B + C)

 

 

3(Ca + Bc) - (B + C) 8(B + C) -3

 

 

,3(Ab + Ca) -3c 3(aa + Co -1) -3b 0
'c b -1
(xxxiii)
3(Cb + Ba) - (B + C) 3(Ca + Bo) - (B + c) 8(B + c) -3
3(Bb + As -1)) + 3c -3(Ba + Ac) + 3b 0
c b -1
(xxxiV)

Determinants (xxxiii) and (xxxiv) are now evaluated in a straightforward

manner. ‘When determinants (xxvi),(xxvii), (xxxiii) and (xxxiv) are-

evaluated

1/8

and combined, one obtains for their sum

3(a2 - bc)(A2 + B3 + 02 - AB - AC - Bc)

+3 C (Cb + Ba) + 3 B (Ca + Bo) - 3 A (Cc + Bb + 2Aa) + 3 A

+ 9 [(A2 - Bc)(a2- be) - (Cc + Bb + 2Aa) + 1]

+ 9 [82(b2 - ac) - AB (c2 -ab) -Bb]

+ 9 [c2(.2 -ab) - Ac(b2 -ac) - Cc]

+ 8 [-AC(b -a) + 03(c - a) + B2(b - a) — AB(C - a) a'(B + 0)]

(XXXV)

79

In order to obtain (xxxv) it is necessary to use the relationship.

A3 + 33 + 03 - 3ABC I A2 + B2 + C2 - AB - AC I BC (xxxvi)

By the use of (xxxvi) it is possible to obtain an expression whose
highest power terms are of the fourth power in the parameters. Further
simplifications are possible by writing E terms of this expression as
fourth power terms. This can be done by multiplying third power terms
by a + b + c I A + B + C I 1, second power terms by (a + b + c)21-
(A + B + C)(a + b + c) I 1, or (A + B + C)2 I 1, etc. The advantage
of this form is that the expression is then algebraically unique,
whereas if lower power terms occur, there are many different possible
ways to write the same expression.

When all terms are raised to the fourth power as explained above,

one obtains for the sum of (xxvi), (xxvii), (xxxiii), and (xxxiv):

3 A3(b2 + c2 + be) + cz(aa + 2b2 - 2ac — be) + B3(a2 + 2c2 -»2ab - bc)

+ AC(-c3 + b2 + 2ab + 8ac + 3bc) + AB(c2 - b2 + 8ac + 2ab + 8bc)

+ Bc(na - 2ba - 2c2 + 2ab + 2ac - bc).

cal-2mm? ”BM”

 

111111)

”181

E H
MHHWN
M II".
N .)
A [If]
m II
H l

l
1

.881)

77

3 1293