MOLECULAR FORCES AND ELASTIC CONSTANTS OF
POLYETHYLENE SINGLE CRYSTALS

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
.Ioginder N. Anand
19.6.6

 

 

'mrcrs

 

 

 

ABSTRACT

MOLECULAR FORCES AND ELASTIC CONSTANTS
OF POLYETHYLENE SINGLE CRYSTALS

by J oginder N. Anand

Lamellar polyethylene single crystals have a folded-chain
structure in which planar zigzag segments of molecules take an
orthorhombic lattice. For these crystals, nine independent elastic
constants appear in the generalized Hooke's law. Thus, they are
inherently quite anisotropic.

Their anisotropy is enhanced further by the directional
intra- and intermolecular forces. Intramolecular forces are due
to the covalent C-C bonds forming the chains, and are much stronger
than the intermolecular London dispersion-type of van der Waals
forces. The latter have been approximated by a 6-12 Lennard-J ones
potential involving two unknown constants; the ratio of which is fixed
by the equilibrium separation to give a minimum in the potential.
Their values are determined by comparing the computed crystal
potential energyand the experimentally-determined cohesive energy,
and are found to be comparable with those of argon.

First and second nearest-neighbor interactions are considered
to derive finite difference expression for the components of the force
acting on a unit in terms of relative displacements and interaction
constants. These are converted into partial differential equations
and compared with the corresponding equations of motion obtained

from continuum theory to establish relationships between the elastic

J OGINDER N. ANAND

constants and the interaction constants. A limited central force
assumption is employed to reduce the number of interaction
constants from thirty to fourteen for the second nearest-neighbors
and from eleven to seven for the first neighbors only.

Interaction constants for units belonging to the same chain
are obtained in terms of the C-C bond stretching, bending and
repulsive force constants while others are obtained from the 6-12
potential in terms of the Lennard-J ones constants and the appropriate
separation distances.

Finally, by substituting values of the intra- and intermolecular
force constants and the geometric parameters, numerical values of
the interaction constants and the elastic constants have been obtained.
From these the values of Young's moduli E1, E2 and E3 obtained in
directions a, b and c, are found to be about 0. 38 x 10-6, 0. 27 x 10"6
and 2. 39 x 10-4 dyne/AZ respectively; while the constants C44, c55
and C66, identified as shear moduli, have been calculated to be 1.12
x10-4, O. 89 x 10-4 and 6. 52 x 10-7 dyne/Az. The magnitudes of
the constants c23, C13 and CM. are found to be equal to those of
C44, c55 and C66.

The value of E3 for the chain direction compares well with the
Young's modulus of oriented polyethylene obtained theoretiCally as
well as experimentally. Furthermore, the value of the Young's
modulus of bulk polyethylene lies between the values obtained for
moduli along and across the chain. It is interesting to note that
polyethylene single crystals are found to have shear resistance even

when only first neighbor interactions are considered and forces are

central.

MOLECULAR FORCES AND ELASTIC CONSTANTS
OF POLYETHYLENE SINGLE CRYSTALS

by

J oginder N. Anand

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Metallurgy,
Mechanics and Materials Science

1966

ACKNOWLEDGEMENTS

The author is very grateful to Dr. T. Triffet for his continued
guidance and assistance throughout the course of the work which made
this thesis possible. He wishes to thank him for acting as thesis
advis or and chairman of the guidance committee.

He also wishes to extend thanks to the members of the guidance

committee for their helpful suggestions and interest in this work.

ii

TABLE OF CONTENTS

Page
ACKNOWLEDGEMENTS ....................... ii
LIST OF FIGURES .......................... iv
LIST OF APPENDICES ........................ v
I. INTRODUCTION .......................... 1
l. l. Crystallinity in Polymers ................ 1
1. 2. Polymer Single Crystals and Related Structures . Z
l. 3. Polyethylene and Its Single Crystal Structure ...... 4
1. 4. Anisotropy of Polyethylene Single Crystals ....... 10
l. 5. Objectives ......................... 10
II.‘ THEORETICAL DEVELOPMENTS ........... . . . . . l3
2. 1. Elastic Constants of Polyethylene Single Crystals . . . 13
2. Z. Intramolecular and Intermolecular Force Constants . . 21
2. 3. Interaction Constants and Elastic Constants . . . .‘ . . . 31
Z. 4. Interaction Constants and Molecular Force Constants. . 50
2. 5. Numerical Values of Constants ............. 59
III. DISCUSSIONS OF RESULTS ................... 65
3. 1. Anisotropy of Polyethylene Single Crystals ....... 65
3. Z. Continuum Theory of the Orthorhombic Lattice ..... 65
3. 3. Molecular Forces .................... 67
3. 4. Interaction Constants ................... 70
3. 5. Elastic Constants ..................... 73
IV. CONCLUSIONS .......................... 76
V. FUTURE EXTENSION OF RESEARCH .............. 80
VI. BIBLIOGRAPHY ......................... 103

iii

an

"IV

.11

Figure
l. l

1.2.

2. 5.
2. 6.
2. 7.
2.8.
2.9.
2. 10.
2. ll.
II-1
IV-1

LIST OF FIGURES

Page
Fringed micelle model ................... 3
Configuration of linear polyethylene ............ 4
Tetrahedron formed by the four bonds of carbon ..... 5
(a) Planar zigzag conformation (b) chain folding ..... 5-6
Schematic diagrams (a) flat (b) hollow pyramid, lozenge-
shaped, platelike polyethylene single crystals having a
folded-chain structure (c) hedrites ............ 6-7
Schematic representation of the unit cell of polyethylene . 8
Entangled-rectangular cell structure (a) and (b) ..... 9-10
Reduction to simple orthorhombic lattice ......... 13
Equivalent lattice with (-C2H4—) units occupying lattice
pctnts ............................ 14
Lennard-Jones 6-12 potential curve (Mr) and force
curve F(r) ......................... 24
Schematic representation of two—dimensional
lattice structure ...................... 27
First nearest-neighbors in first quadrant only ...... 33
Second nearest-neighbors in first quadrant only ..... 33
Rotation of x'y'z with respect to x y z ........... 34
Rotation of x"y"z with respect to x y z .......... 34
Geometrical representation of axes xij ......... . 41
Geometry of two corresponding C-C bonds ........ 51
Deformation of C-C bonds ................. 52-53
Rotation of axes . . . . . . . ............ . 90
Detailed drawing of chains in lattice in plane (a, b) . . . . 101

iv

LIST OF APPENDICES

Appendix Page
I. EvaluationofSeries................ 82
II. Transformation of Partial Derivatives . . . . . . 90

III. Strain Energy Connection Between Elastic
and Interaction Constants . . . . . . . . . . . . . 96

IV. A Note on the Lennard-Jones Potential . . . . . . 100

 

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I. INTRODUCTION

1. l. Crystallinity in Polymers

It has been found that most polymers, perhaps all, are partially
crystalline, having co-existing ordered and disordered regions (1).
Their x-ray diffraction patterns show both sharp features associated
with regions of three-dimensional order, and more diffuse features
characteristic of molecularly disordered substances like liquids. The
degree of crystallinity can be estimated from changes in density,
specific heat, refractive index, transparency, x—ray diffraction
patterns and various other physical properties. In fact, many of the
unique physical properties of polymers are associated with their
ability to crystallize.

Stereoregular isotactic polymers whose molecules are chemically
and geometrically regular in structure, such as linear polyethylene, are
typically crystalline. Noncrystalline polymers, on the other hand,
include those in which irregularity of structure occurs, such as atactic
polymers or copolymers with significant amount of two or more quite
different monomer constituents.

Unit cell data and other information, such as the configuration
or chemical structure and the conformation of several crystalline
polymers, have been compiled by Miller and Nielson (2). In a number
of new isotactic polymers two or more conformations and unit cells
have been observed to depend upon the temperature and other conditions
of crystallization. The most commonly occurring unit cell structures

in various polymers may be classified as orthorhombic, pseudo-

orthorhombic, triclinic, hexagonal, monoclinic and rhombohedral.
l. 2. Polymer Single Crystals and Related Structures

Crystalline polymers usually crystallize from dilute solutions
in the form of thin lamellae called single crystals; and such crystals
of many polymers such as gutta -percha, polyethylene, polypropylene,
polyamides,cellulose and its derivatives have been reported since their
independent discovery and identification in 1957 by Till (3), Keller (4),
and Fischer (5). All polymer crystals have the same general appear-
ance, being composed of thin, flat or hollow pyramidal platelets.

Spiral growths of additional lamellae originating from screw dislocations
are usually present on their surface. Crystallization conditions such as
solvent, solvent concentration, temperatur'eand rate of cooling determine
the size, shape and regularity of these crystals. However, their‘thick-
ness depends mainly on the crystallization temperature and any subsequent
annealing treatment (6)- .

Electron diffraction analyses of these crystals indicate that the
polymer chains are normal or nearly normal to the plane of the lamellae
(3, 4, 5). The length of polymer chains being several times the thickness
of a lamella, Keller (4) points out that the molecules must be folded
back and forth on themselves several times. In polyethylene, for
instance, the molecules can fold in such a way that only five carbon
atoms are involved in the fold itself (6 ), as shown in Figure 1. 4 (b).

When the rate of growth during crystallization is slow, relatively
thick aggregates of single crystal lamellae having a common nucleus
and orientation, called hedrites (Figure l. 5 c), have been reported for

polyethylene and polyoxymethylene (7, 8, 6 ). They have a polygonal

appearance, and are the closest approach to a macrosc0pic single
crystal. But at faster growth rates numerous defects, such as
vacancies and interstitials, terminal groups, branches and improper
folds, are incorporated into the lattice.

During crystallization from the melt spherulites develop which
have a complex lamellar structure, as seen in the electron micro-
scope (9). Their nuclei have a random orientation, and growth
occurs radially outward from these until the entire volume is filled.
When two spherulites meet during crystallization they form a common
straight boundary in which the transition in orientation takes places (1).

This picture, known as the "crystal defect solid" model of
crystalline polymers, visualizes the matrix as an ordered region
having defects incorporated throughout (1). It is in direct contrast
to the old "fringed micelle" concept, in which a crystalline phase
consisting of crystallites is taken to be embeded in an amorphous
matrix forming a second phase. In the 1atter.molecular chains are
visualized to pass through several crystallites and, thus, to have
several straight and several disordered segments, as shown in

Figure 1.1.

 

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Figure 1.1. Fringed micelle model.

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A

ll
-1

f?

I

EX‘v I

‘d‘.

.
I
"" VA
NI ‘CV

Although from a macroscopic point of view it may appear that at low
crystallinities both concepts are identical, as far as a microscopic
description is concerned the two are completely different.
1. 3. Polyethylene and Its Single Crystal Structure
The chemical formula of polyethylene is CnH2n+2' Thus, a poly-

ethylene molecular chain consists of n, - CHZ- chemical repeat units,
neglecting the end ones. The configuration of a linear molecule is
shown below in Figure l. 2.

H H H H H H

I I I I I I

H-C -C -C -C-C-C—
l I I I I I
H H H H H H

Figure l. 2. Configuration of linear polyetheylene.

This may also be abbreviated and written as (-CH2-)n .

Covalent single bonds formed by the sharing of two electrons,
exist between two consecutive carbon atoms and between carbon and
hydrogen atoms. The length of the C-C bond is about 1. 54A, slightly
longer than the C-H bond length of about 1. 10A. Carbon is tetravalent
and its four bonds are directed in space in such a manner that it lies

at the center of a tetrahedron as shown in Figure 1. 3.

.H

 

1.54A

C

Tetrahedron formed by the four bonds

Figure l. 3.
of carbon.

The angle between any two bonds is approximately the tetrahedral

angle of 109°28', as shown. Carbon atoms forming the backbone of
a linear polyethylene molecule take up a planar zigzag conformation

mm crystal lattice, as shown in Figure l. 4 (a).

  

109°28'

    

  

 

T]

 

repeat distance

Figure l. 4(a). Planar zigzag conformation.

Figure 1. 4(b). Chain folding involving five C atoms.

The repeat unit is (~--CHz-CH2 -) or («C2 H4-) and the distance between
two alternate C atoms is the repeat distance; it is approximately equal
to 2. 55A.

Polyethylene single crystals crystallizing from dilute solutions
have well defined forms, which usually consist of lozenge-shaped
flat or hollow pyramids lOO-ZOOA in thickness and about 10-20 microns

in lateraltdimensions:

    

lOO-ZOOA

lO-ZOp

 

 

Figure l. 5(a). Schematic diagram of flat single crystal.

 

 

 

 

 

 

I
I
(b)
lO-ZOp
(C)
100p. J
r“ '7
Figure l. 5. Schematic diagrams (b) hollow pyramid,

lozenge-shaped, platelike polyethylene single
crystals having a folded-chain structure (c) hedrites.

The molecular chains forming these crystals have a fold-length of
the order of the thickness of the crystals (4).

The unit cell structure of polyethylene is orthorhombic, and
i.
3 Shown in Figure l. 6 (a, b, c).

 

 

,’ (a)

 

 

 

c -axi

~axi s

 

a -axis

 

 

(b)

 

 

 

(C)

 

 

 

 

Figure l. 6. Schematic representation of the unit cell of polyethylene
showing (a) parameters a, b and c (b) location of
molecular chains along the c-axis and (c) setting angle
of 41° that the planes of chains make with the b-axis.

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n

—,.o -
has \

 

D'. '

'- 6.

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.8...‘

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us .C

Axes a, b, and c are orthogonal and the respective parameters a, b
and c are approximately 7. 41 A, 4. 94A and 2. 55A at room temperature,
as determined by x-ray diffraction by Cole and Holmes (10). Molecular
chains lie along the c-axis, thus, c is just the repeat distance. The
unit cell consists of four parallel planar zigzag chains running along
the c-axis at the four corners of the (a, b) rectangle, their planes
making an angle of (3 = 41 o, with the b-axis, as determined by

Bunn (11), and one running through the mid-point of the rectangle
having a different orientation from the other four. The chain through
the midpoint of the rectangle and three additional similarly-oriented
chains through the midpoints of adjacent rectangles may be considered
to form their own (a, b) rectangle. Thus, the entire crystal may be
considered to consist of chains having these two orientations, their
respective rectangles forming an entangled cell structure as shown

in Figure l. 7 (a and b):

41

 

 

 

 

 

 

Figure l. 7(a). Entangled-rectangular cell structure.

 

 

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Figure l. 7(b). Entangled-rectangular cell structure.

1. 4. Anisotropy of Polyethylene Single Crystals

For the class of crystals having an orthorhombic lattice, the
36 constants of the generalized Hooke's law are reduced to 9 independent
constants, as against 3 for crystals having cubic symmetry (12).
Anisotropy of such crystals is, thus, much more complex. Besides,
polyethylene single crystals have another feature, peculiar to poly-
meric crystals, which contributes to additional anisotropy. In the
direction of the c-axis, or along the molecular chains, the primary
covalent C-C bonds, are several times stronger than the secondary
bonds existing between any two molecular chains due to the weak
van der Waals forces (13, 6). This difference is superimposed on
the inherent anisotropy of the orthorhombic lattice structure.
1. 5. Objectives

A connection between the nine independent elastic constants,

 

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11

as they appear in the generalized Hooke's law, and the microscopic
intermolecular and intramolecular force constants is obtained for
single crystals of polyethylene in the present work. However, in
accomplishing this objective the continuum or macroscopic elastic
constants are obtained first in terms of interaction force constants,
for the forces existing between units occupying the present orthor-
hombic lattice, by following the von Kafrma'n cubic crystal structure :

J L“!
approach (14).

First nearest-neighbor and second nearest-neighbor inter-

 

actions are accounted for, and the effect of central force assumption ' J
is demonstrated. The interaction force constants are in turn obtained
in terms of the more basic constants, such as stretching, bending,
torsion and repulsive force constants for the primary C-C bonds that
exist along the molecular chain axis, and the intermolecular force
constants for the net attractive forces between adjacent chains due
to secondary bond forces. The strength of the C-C bonds under
various types of deformation is well-established, by Mizushima and
Simanouti (15). However, the strength of the secondary bonds is
not known for polyethylene.

The net potential existing between adjacent chains is
approximated by a‘: 6-12 Lennard Jones potential (6), which involves
two constants whose ratio is determined by the equilibrium distance of ”
the two chains. Their exact values are then determined by calculating
the crystal potential energy density and comparing it with the
experimentally-determined value of the cohesive energy density for

polyethylene (16 ) .

12

Numerical values of the interaction constants are obtained
from the values of the C-C bond strength constants, the Lennard-
Jones potential constants and the geometric parameters. These are
then substituted in the expressions for the elastic constants to yield
their numerical values. The nine independent elastic constants of
the generalized Hooke's law are measures of the Young's and shear
moduli of polyethylene single crystals in various directions. These
are compared with known values of the constants, such as the Young's
moduli of oriented and bulk polyethylene. Shimanouchi, et al. (17)
have calculated the Young's modulus for oriented polyethylene,

consisting of infinitely long molecular chains, to be 3. 4 x 10P4

dyne/Az. This is somewhat higher than the value of 2. 6 x 10-4
dyne/A2 determined experimentally by Dulmage and Contois (18),
using x-ray diffraction and the relaxation technique.

The values of the Young's moduli along other directions should
be considerably lower than this, because the forces existing along
other directions are much weaker. The Young's modulus of bulk
crystalline polyethylene is of the order of 10.5 dyne/Az, and this
must represent some kind of an average of the moduli of polyethylene

single crystals along the three lattice axes. Like the Young's moduli,

the shear moduli too should be higher along the chain direction.

 

II. THEORETICAL DEVELOPMENTS

2. 1. Elastic Constants of Polyethylene Single Crystals

In the orthorhombic lattice structure of polyethylene, as
discussed in Section 1. 3 and illustrated in Figures 1. 6 a-c, four
molecular chains having parallel orientations and one having a
different orientation occupy, respectively, the four corners and
the center of the rectangle (a, b). This may be reduced to a simple
orthorhombic lattice by considering pairs of two chains, consisting

of one at the corner and the other at the center of the rectangle, to

 

occupy~ lattice points, as shown in Figure 2.1. This reduction.

 

 

 

 

 

 

 

 

 

 

 

Figure 2.1. Reduction to simple orthorhombic lattice. Lattice
points are occupied by units of two chains connected

by a natural fold.
facilitates the application of symmetry operations to a polyethylene .-
single crystal. . since then the crystal would have the same number
of symmetry elements as a simple orthorhombic lattice cell structure.
It may also be noted that a chain lying along the c-axis can be broken

into (-C2H4-) repeat units without any loss of generality and, as

shown in Figure 2. 2, the equivalent lattice structure is then an

orthorhombic structure with units occupying the lattice points.

13

 

n.‘
\
'H

'V

l4

 

 

 

 

 

 

 

 

”lag.

Figure 2. 2. Equivalent lattice with (-C2H4-) units occupying
lattice points.

We are now in a position to write Hooke's law with proper
symmetry operations for polyethylene single crystals. For small
strains ,Hooke's law states that stress is proportional to strain and
for an anisotropic medium its generalized form may be written

mathematically as
0'. = c. 6. 2.1

The constants of proportionality cij are called the elastic constants
or moduli of elasticity, while (Ti and ej respectively represent the
stress and strain components. Equation 2. 1 may also be solved for

strains in terms of stresses to obtain

 

15

and elements sij of the inverse matrix are called the moduli of
compliance.
The 36 elastic constants in Equation 2.1 are reduced to only
9 independent constants (12, 19) for the symmetry elements of an
orthorhombic lattice. These nine constants are shown in the matrix

given below

 

 

y—I —
C11 C12 c13 0 0 0
c21 czz c23 o o o
c c c 0 O O
Cij ___ 31 32 33 2.3
O 0 0 C44 0 0
o o o 0 CSS 0
0 0 0 0 0 C66
wh : : :
ere c:12 CZl’ c13 C31 and C.23 c32'
Equation 2.1 may now be written out by using this matrix:
“1 = “11'514r C12 €2 + C13 63
cr2 = C12 61* C22 E2 + C23 63
“3 = C13 61 + C23 '52 + C33 63
2.4
“4 = C44 64
“5 = c55 ‘5
‘6 = C66 66

L .
ettlng u, v and w be the components of displacement along the

ax
es 3:. y and 2, respectively, where these correspond sequentially

with
the parametric axes a, b and c, we may write the strains Ei

 

 

 

.-

(i.

I
-
‘3624

lher

ZI‘

Q
C.‘ {v
\.

l6

 

as
-3_u - 2! -E’fl
6l - 3x ' 62— 8y ’ 63 -32
and 2.5
8w 3v Bu 6w _8v Q3
64-3??? 93-575? 66-3.. *3,
Also, it may be noted that a
64 z Y23 E
65 : l’13
66 : le 2.6

Where the Yi-'8 are shear strains. Young's modulus E and the

Shear modulus G are defined by

Where 0' and E are the longitudinal stress and strain, and

2.8

Where 7 and Y are the shear stress and strain.
Applying this to the particular case in hand, we obtain three

Young's moduli El' E2, E3, and three shear. moduli G23. G13.

and G12 as shown below.

Rewriting the first of the six expressions in 2. 4 as for
1111' -
lab-(13,1 stress 0'1 only

E161: or1 : C1161+¢12€z ”1363 2.9

and -
dlviding both sides by cl. we get

’ESS

D

1!:

s

17

€ €

_ 7- __3_

I51 ’ C11+C12 61+C13 e1

Similarly, by repeating the above operations for the other five

expressions in 2. 4, we get

6 6
E2 = C125+‘322J'023E;
2.10
6 6
_ _l_ .3.
E3 “ C13s +C235 “:33
3 3
and
E4 : C44 z 23
5 = C55 = (:13 2.11
E6 = C66 = G12

6.
v..=-—-1-=--—1—=-1—— 2.12
13 e. e. V.
J _.1 11
6.
1
to get six vi.'s, such that
V - :1 V _ _ 5.2.
12 62 21 61
6 E
3 1
U : - — , V : - -— 2.13
31 61 13 63
6 6
”23 = ‘ _Z ' ”32 = ' El
63 2
$11
IbST—‘ituting 2.13 in 2.10, we obtain:
E1 : C11 ”‘12 ”21 ' C13 ”31
E2 = C22 ‘C12”12 ' C23 ”32 2'14

 

”F1‘L'i:~.ikl-mmmnu ‘
. - .
.

3 ’0

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on

3 J
‘3 .
U)

a.
RI.

cs. ec

The Vij's may be evaluated in terms of the cij's by using

simple uniaxial stre 3 se 3 .

0’

Solving the last two equations of 2.15

and similarly by considering uniaxial

21

31

C

c €+c €+c E

C

dlre<:1:ions, we obtain:

 

32

12

13

23

ll

12

136

m
HN

m m
u.)

p—a

m

m
NU)

m

"‘l
N H

m

"‘l
oar—-

m

m
CON

€+c €+c E

l

l

1

18

Thus, considering a uniaxial stress

1 along the x-direction, 2.4becomes

122133

222 233

+ C23 62 + C33 63

C

 

C33 C12 C23 13
‘ , 2
C33 C22 " C23

(:22 C13 " C12 C23

 

" 2

C33 C22 (:23

C11C23 ' C13 c12

 

 

_ c 2
C11 33 '°13
C33 €12 ' c23 C13

' 2
C33 C11"°13

C22 °13 c12 c23

 

22 C11“ c12

c11 C23 C12 c13

 

11 22' 12

2.15

simultaneously, it follows that:

stresses in the x and y-

2.16a

 

a-

O

u “4

1“-
TV},G

l9

' Substituting 2. l6.in 2. 14 gives:

°12(°23°13 ' °33°12) + °l3(°12°23 ' °22°13)
1:31 = °11+

1 7‘)

°22°33 ' °23

2 2 _ 2
C12 °13 °23 ' °12 °33 °13 °22

= c +

11 (C C _ c 2)

22 33 23

+ 2 2 2 _ C 2

°11°22°33 °12°23°31 ' °12°33 " °23°11 31°22

2
(°22 °33 " °23)

 

 

2.17
Similarly, E2 and E3 may be obtained by considering uniaxial
stresses 0'2 and 63intle y and z-directions; they are:
+ 2 Z c 2 - c 2c
E °11°22°33 °11°23°31 ' °12°33 ‘ 23°11 31 22
2 _ 2 '
(C11 °33 " °13)
2.18
ccc +2ccc --C2 c2 -c2
E 11 22 33 11 23 31 12°33 23°11 31°22
3 - .

2
(C11 C22 ' °12)
2.19

It Should be noted that
(c c -c2)E=(c c -cz)E=(c c -c2)E
22 33 23 l 11 33 13 2 ll 22 12 3

2

2
°11°22°33 + °11°23°31 ' <:12°33

2 2

' °23°11 ' C31°22 2'20

Later in this work expressions for the elastic strain energy
and . . . . . .
equatlons of motion W11]. be needed for comparison w1th equations

Obt -
3‘ lhed from microsc0pic considerations, in order to evaluate the

interaction force constants .

20

strain energy density U, we note that

all of which may be

Thus,

and

on
C
on

U

as

Q)
m
y—I
II
q
N

1 2 2 2
U ‘ 2(°11°l +°22°2+°33°3)
’+(°12°1°2'+°13°1°3'+°23°2°3)
+1—(c e + e + 62)
2 44 4 °55 5 C66 6
aU _ _
361 " C11°l+°12°2+°13°3 “l
3U _ _
862 " C12°1+°22°2+°23°3 " “2
aU _ _
353 ’ °13°1'*°23°2'*°33°3 ‘ “3
8U _C E _6 EU _C _a aU
3124‘ 44 4' 4' 355“ 55 5‘ 5' 356

obviously satisfy 2. 21.

To get such an expression for the elastic

2.21

satisfied if U takes the following form:

2.22

2.23

°oo°o ' “6 '

Now, the equations of motion may be written in the form '(‘12,‘ 14):

fi: aal+3€£+€i§
po 3i av az

3215

not L

$53.

21

 

. _ 8176 + 80'Z + 864
p0 8x 8y 82
80' 80' 8a
.. _ 5 4 3
p w - Tx + “8y + —8z 2.24

where po is the density. To Specialize these. equations for an

orthorhombic lattice, we must substitute 2. 4 and 2. 5 in these, the

result is:

azu a2u an2 av2 8W2

"o"i = °11 3x2 + °66a y? + °55 :2 + (°12+°66)ax_Y" av +(°13"°55)'z§“'_xaz
62v 82v av2 3“2 3va

po I ' °66ax 2+ °22 "'2' + °44 :2 + (°12+°66) 8x8y + (°23+°44)"'"" 8y82
2 2 2 2 2

__ 8 w 8 w 8 w 8 u 8 v

Pow ’ °55 a")? + C44—Tay + C33——8z2 + (°13+°55) 8x 82 I (°23 °44) av" 62'

2.25

Since the orthorhombic lattice has nine independent elastic

constants cij instead of three, like a simple cubic structure,

expressions for the Young's moduli, shear moduli, POisson's ratios,
Strain enerSY. and equations of motion are naturally much more

in"(fl-Ved.
2' 2 Intramolecular and Intermolecular Force Constants

Various molecular force constants are described in this section,
and a. procedure is developed for calculating some of them which are
not known. In Section 2. 4 the. macroscopic elastic constants c J
dis c"flesed in Section 2.1 will be obtained in terms of the above-

me -
htl oned microscopic force constants; for this purpose the two

22

following types of forces are important:
(1) Intramolecular or primary bond forces

(ii) Intermolecular or secondary bond forces

i) Intramolecular Forces
The carbon atoms constituting the backbone of a linear poly-

ethylene molecule are held together by covalent bonds formed by
sharing pairs of valence electrons. These C-C bonds feature the
primary or intramolecular forces with which we will be concerned.
The other covalent bonds are carbon-hydrogen or C-H bonds by means
of which hydrogen atoms are held to carbon atoms. The C-C bonds

have a length of 1. 54A, as against 1.10A for the C-H bonds, and they
serve different purposes insofar as their contribution to the strength
of the crystals is concerned. The strength of a polyethylene chain
depends entirely on the strength and degrees of freedom of the C-C

bond. On the other hand, because "of its geometric and steric
configuration, the C-H bond plays an important role in determining

the c rystal structure and providing the intermolecular electrokinetic

fox-c e s to be discussed later.
In a polyethylene molecule all four valencies of carbon are

8-at7i»131:'ied and its four bonds are directed in space as shown in
F o
181“are 1. 3. In general, segments of this molecule are free to rotate

a
bout the C-C bond in such a manner that any three carbon atoms always
to
nth a plane. However, as described in Section 1.3, the molecular

Cha ~
11:18 take up a planar zigzag conformation in polyethylene single

"Ya .
tals and thereby prevent any rotation about the C -C bond.

n 9
me ’I\
.ve “6

o- :t‘
I
the 0V

Tie I

23

The strength of C-C bonds for various types of deformation
has been determined by Mizushima and Shimanouchi (15), and values
for these and other geometric parameters are listed in Section 2. 5.
The force between two alternate chemical units of -CHZ- along the
chain axis is repulsive due to their being too near each other. In
the absence of any free rotation about the C-C bond, deformation of
molecular chains will take place by a process of deformation involving
stretching, bending and repulsive force constants only. This fact will

be made use of later in Section 2. 4, to obtain the interaction force

c onstants.

ii) Intermolecular Forces
As discussed in Section 1. 3, the crystal lattice of polyethylene
is such that adjacent molecular chains occupy an orthorhombic cell.
The attractive forces between these chains, which bind them together
in the solid crystalline form, are called the intermolecular or
Sec ondary valence forces.

Polyethylene is a nonpolar material for two reasons. First,
because all the valencies are satisfied and, second, because both
Garb on and hydrogen are equally electronegative. Therefore, the
attractive intermolecular force is not due to permanent dipole
moments, but rather to time varying dipole moments resulting from
diffe rent instantaneous configurations of the electrons and nuclei.
The a . . , . .

e are also called London dispersion forces, the potential govermng
them is proportional to the inverse sixth power of the distance (20).
Between any two molecules, there is also a repulsive force

due
to the interference of the electron clouds surrounding the nuclei.

24

This force is short-range compared to the London attractive force,
decreasing exponentially with distance (21, 22, 23).

At the equilibrium separation the net force is zero; and, of
course, the net energy, or the potential curve, will have a minimum
at this distance.

Different portions of the exponentially-decreasing repulsive
potential function may, for convenience, be matched by different
inverse powers of the distance (24, 25). If for polyethylene, as
suggested by Geil (6), we approximate this potential by the twelfth

power, the total potential energy function 4) may be written as

d>(r) = 315—- %, 2.26
r

1‘

where r represaits the separation distance and A and B are
constants called the Lennard-Jones potential constants. This is the
standard form of the Lennard-Jones 6-12 potential (26).
Evaluation of A and B the Lennard-J ones Constants

The shape of the general potential curve ¢(r) is as shown

below:

<I>(r)

 

    

 

 

IF(r)

Figure 2.3. Lennard-Jones 6-12 potential curve (Mr) and
force curve F(r).

gives

DI

25

The gradient of ¢(r) will give the force F(r) between two adjacent

molecules. Thus, we may write

- 92.111 2.27

 

F(r) : dr
or
F(r) = ”if; - 9-?- . 2.28
r r

The curve of F(r) is also shown in Figure 2. 3.
The force F(r) being zero at the known equilibrium distance
r0 imposes the condition that the constants A and B have a certain

definite ratio;

F(r) = 0 2.29

give 8

 

3 - T: O 2.30
O

01‘

E- r . 2.31a

wiere

I
npg ass

in; 3 u

._ ,
.gtqv

, A r
'VVvs a

' ‘ Y
P'D‘ —
~...C..

11"3'53
Lethe
2:85
.1.

...e C
i

me e:
:ccu:

3.151

26
Rewriting 2. 31a a's

A=pB, 2.311)
6

where p = i— ro is a known constant, 2. 32
and substituting for A in terms of B from 2.31b in 2.26, we

obtain the potential ¢('r) in the form

¢<r> = 3% - 135. 2.33
r

1'

which involves only one unknown constant.

The constant B can now be determined by computing the

crystal potential energy density, in a manner similar to that used by
Lennard-Jones (26) for cubic crystals, and by comparing it with the
cohesive energy density—an experimentally-determined value (16).
The crystal potential energy density, denoted by E, is defined as
the energy per mole of polyethylene,in which the individual units
occupying the lattice points are surrounded by an infinite matrix.
It is computed by summing the lattice energy of the individual units
in a mole of the crystal. The lattice energy of a unit, , denoted by
U, is the energy of the unit when in the lattice of an infinite crystal
and is the sum of the contributions ¢(r) , due to all surrounding

units, where
Mr) = 1A: - 32-
r

with r the distance of the surrounding units from the unit whose
lattice energy is being calculated. Cohesive energy density, denoted
by A , is the energy per mole of a substance that is required to

remove a unit from the matrix to a postion far from its neighbors.

o
n
I O
as
e

.1
at-

-Cu‘
10
Let:

) Ab.
Q's-o
-n

3

27

Evaluation of Lattice Energy and Crystal Potential Energy Density

 

For purposes of evaluating the crystal potential energy, we
will consider polyethylene single crystals to be made up of units of
(-C2H4-) (Figure 2. 2). This model will also be used later in
connection with the development of interaction force constants in
Section 2. 3. The lattice structure may, thus, be represented as

shown below:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b-axis
(o. 6. 0) ck e e o
‘3’ <3 ’5
(00 4! .0) c C} \( $
\ X ’\
x. \1 yr
(0: 29 0) d’ ‘3 \> D
A 2 <3
(09 Os 0) Q; A J Ax} -
(2.0.0) (4.0.0) (6.0.0) 3-.....

Figure 2. 4. Schematic representation of two-dimensional
lattice structure with lattice points occupied by.

(~CZH4-) units.

Letting

2.34

and

:here

ie 2,

ever. <
2115
leis-r.-
“will

5““ V

tzatv

The:

28

we may then write the position vector of a lattice point (i, m, n) as

A

.s _ A e .
rlmn - Zelil + mezi2 +ne3i3 , 2.35

A
where 2, m, n are integers and il, 12, i3 are unit vectors along
the a, b and c axes respectively. Therefore, the distance rlmn

is
_ [a 2 + 2 + 211/2
rlmn ‘ °1) (Inez) (“'33) . 2.36

From Figure 2. 4 it can be seen that only when 2 and m are both
even or both odd is the point occupied by a real unit. Also, all the
units for which both 2 and m are zero should be excluded; they
belong to a single chain and, hence, are permanently attached
through the C-C bonds.

It follows from 2. 26, which gives the energy of a pair of units,

that we can calculate the lattice energy U of a (0, 0, 0) unit from

1 °° '
: — Z .
U 2 l, m, n:-ao ¢£mn(r) 2 37
1 A B
=22 (_1"2" - T" 2°38
lmn rACmn
l l
=-2+[2(-‘IEZ—-—6——)]B, 2.39
rZmn rafmn

where from 2. 32

"U
II
NIH
A
H
O
V
0‘

29

and both Land m are odd or even and n equals any integer, but 1,
m,n ;/ 0.

This reduces the problem to one of calculating sums of the

 

 

type
°° l
A = Z S 2. 40
s £,m,n;-°° r1
mn
for s = 12 or 6.
Substituting for
2 _ 2 2 2
rlmn —(1e1) + (mez) + (ne3).
from 2.36, we get
A 5 1
S 1,31, n: -m [(181)2 + (mez)2 + (ne3)2] 8/2
2.41
Therefore we may write U in the form
U - is( A - ) 2 42
‘ 2 P 12 6 '
= l B A 2 43
2 12-6 '

where A12-6 = pA12 - A6 .

Crystal Potential Enggy and Cohesive Enem
A gram mole of a substance contains 6. 0249 x 1023 units,
called Avogadro's' number and denoted by N . Let M be the

molecular weight of the lattice units (-CZH4-). Therefore, M

30

grams of polyethylene will contain NA units. Thus, the crystal

potential energy per mole E is

E = NAU’ 2.44

where U is the lattice energy of a single unit, as determined in
2. 43.

The cohesive energy density has been determined by Small (16).
However, he gives values of 6, which is the square root of the
cohesive energy per unit volurre;thus, 62 determines the cohesive
energy per unit vdume. In order to convert this to a molar value we
must determine the vdime of a mole of crystalline polyethylene. If

p is the density of such material, the volume V of M grams will

be
V = 5’1- 2.45
p
Therefore, the cohesive energy per mole, A , is
A : 1:31-1- 62 ; 2. 46

and equating the values of A and E obtained in 2. 46 and 2. 44, we

arrive at:
A = E 2.47
M 2
T)- 6 .. NAU 2.48
But from 2.43
U E- B A
2 12-6 ’

which when substituted in 2. 48 gives

-9
'6

U.

31

2M62

NApA

 

12-6

The expression for A then follows from 2. 31b:

A pB

ZRM 62

2.50
NA P A12.6

 

The Lennard-Jones potential constants A and B, defined
by expressions 2. 49 and 2. 50 will be evaluated numerically later
in Section 2. 5, by substitution of the values of several physical
constants such as M, NA’ p and p, and by making use of the

series summation for A12-6 developed in Appendix I.

2. 3. Interaction Constants and Elastic Constants

In order to calculate continuum or macroscopic elastic
constants in terms of molecular force constants, it is necessary to
consider the forces of interaction that result when a lattice unit
moves relative to the units which surround it. Since the force fields
vary nearly linearly with distance for small displacements, the
slopes of the force curves at the separation distances of the surrounding
units determine the so-called interaction constants. These constants
may, therefore, be obtained in terms of the intramolecular force
constants (such as the C-C bond stretching or contraction, bending
and repulsive force constants) and the intermolecular Lennard-Jones
potential constants.

In this section a connection between the elastic constants and

the interaction constants of polyethylene single crystals is established

 

1“».
Nd.

1.
Av.
h.“

A

v

32

by following a procedure similar to von Kafrmafn's for simple cubic
crystals, as discussed in Reference (14) by Kittel. Components of
the net force acting on a unit are obtained by considering its inter-
actions with the surrounding units up to second nearest -neighbors.
These expressions, which involve finite displacements, are converted
into partial differential equations by introducing the lattice parameters
a, b and c, and by taking limits. Newton's law is then applied to
convert the force equations into equations of motion, in order to
compare these with the corresponding continuum Equations 2. 25.

A comparison of the coefficients of appropriate partial
derivatives in the two sets of equations yields the desired expressions
for the elastic constants in terms of the interaction constants. These
expressions may be modified to apply to first nearest-neighbor
interactions only simply by eliminating the terms pertaining to second
nearest-neighbors. Also, the central force assumption is applied in
a rather limited manner to polyethylene single crystals. The C-C
bonds along the chain axis have strong resistance to bending in
directions normal to the chain and, thus, make the forces between
units on the same chain non-central. The forces between units on
different chains may, however, be treated as central.

Model and Notation

 

Consider again, as in Section 2. 2, that the lattice points are
occupied by (-C2H4-) units. Figures 2. 5 and 2. 6 show both the
first and the second nearest-neighbors in one quadrant formed by
the positive x, y and z-axes. An additional axis (x') along the

diagonal of the rectangle (a, b) in the xy-plane is also shown.

33

Y
(0.1.0)
\

 

b (1',0,0)

 

 

L

C (0.0.0) a ($70.0) x
(0.0.1)

Figure 2. 5. First nearest-neighbors in the first quadrant only.

 

(0.1.1)

 

 

I
(0.0.0. 1 '0'1

 

 

 

1,0,1)

Z

Figure 2. 6. Second nearest-neighbors in the first quadrant only.

 

34

If y' is considered to be in the xy-plane, normal to x',

Figure 2. 7 then represents the x‘y'z set of axes.

 

90

 

 

 

Figure 2. 7. Rotation of x' y' z with respect to x y 2.

Similarly, by considering x" to be along the diagonal of the rectangle

(-a,b), the x"y"z set of axes is shown in Figure 2. 8.

xII y
b

 

 

 

 

-a X

YII

Figure 2. 8. Rotation of x" y" z with respect to x y z.

1
m

a
N“

C‘.)

I“w

4.“ ‘_

35

Considering a, b, c and -:—~/a2 + b2 as units of distances
along the respective axes x, y, z, x' and x", the lattice points may
be labelled by their (3, m, n) coordinates, 2, m and n being integers.
Thus, the points listed below indicate first nearest-neighbors:

(l, 0, 0), (-l, 0, 0);(0, l,0)‘,.(0,-1, 0);(0, 0,1),(0, 0, -l);(1', 0, 0), (-l ', 0, 0)

and (1",0,0),(-l",0,0) 2.51

Only four of the first ten nearest-neighbors are shown in figure 2.5.
It may be observed that nearest-neighbors, as defined here, are not
equidistant from the central unit (0, 0, 0). This is due to the geometry
of the orthorhombic cell, for which the three parameters a, b and c
are inherently unequal. An additional feature peculiar to the poly-
ethylene lattice structure is that units corresponding to (1 ‘, 0, 0) and
(1", 0, 0), along the x' and x"-axes respectively, are considered to
be first nearest-neighbors.

Similarly, the points

(0,1,1),(0, -l, 1),(0, 1, -l), (0, -l, -l); (l, 0,1),(-l, 0,1),(1, 0, -l), (-1, 0, -1);
(1', 0,1),(-1', 0,1),(1', 0, -1), (-l', 0, -l) and (1”, 0,1),(-1", 0,1),

(1", 0, -l), (-l ", 0, -l) 2. 52
are the second nearest-neighbors. These are sixteen in number;
however, only three are shown in Figure 2. 6, in addition to the four
first nearest-neighbors. It should be noted that the units corresponding
to (1,1, 0) are excluded because units corresponding to (1', 0, 0) and
(1 ", 0, 0) lie between these and the central unit (0, 0, 0).

Equations of Motion
Let Fx’ Fy and F2 be the components of the force on the

unit (0, 0,0)along the axes x, y and 2, respectively. Similarly, the

 

:-

no
‘5‘)

36

displacement components of the unit at (l, m, n) may be represented
by utmn' v‘mn and wlmn . Interaction constants will'be different
for different units and will also depend upon the direction of the
displacement. Defining kliii’nn as the force on the unit (0, 0, 0) along
the x1 -axis per unit displacement uj of the unit (I, m, n), where
i,j = 1,2,3 and x1, x2, x3 correspond to x, yand 2 while ul, u2

and 113 correspond to u, v and w, respectively, the interaction force

components Fx’ Fy and F2 are derived below:

_ k11 11
Fx ' k100(‘1100 + “-100 ' zuooo) + k010(“010 + ULo-lo ' 2‘1000)

k11
”001(“001 + u00-1 ‘4me + k1'100H‘11'00 “-1'00 ' “000)

11

+ (“1"00 + u-l"00 ' “000” + k101[ (‘1101 + “-10-1 " 211000)
+(u +u -2u )]+k13[(w +w -2w )

-101 10—1 000 101 101 10-1 000
-(w +w )]+k“[(u +u -2u )

-101 10-1 W000 011 011 0-1-1 000
+ (1.101”1 + uO-ll - 2u000)] 2.53
+k1'01{[(“l'01+“'.1'0-12‘1000) ”‘1 1'01 +111'0..1 ' zuooon
+[(“1“01 +“-1”o-12uooo) ”‘1 1"01 +ul"0-l " zuooon}
+k1?01{[(w1'01+W-1'0-1 ‘ zwooo) ' (“V—1'01 + v".1'0-1 ' zwooon

+[(w1“01 + v"--1"o.-1 ' zwooo) " (W-l”01 +W1"0-1 zwoo’om
+k k1'01{[ ("1'01”r v-1'0-1 ' 2v000) ' (V—1'01 + Y-1'0—1 ‘ zvooo”

+[(""1"01 + V-..1"0.1 ' 2v000) ‘ (V-1”01 +Vinc-.1 ' “000’” °

37

Expressing (ulmn +u-l-m-n' 2u u000)’ for example, as (ulmn ), we
may rewrite 2.53 in the form
F -k (E )+k (E )+k“(E )
x 100 100 010 010 001 001
11 - —
+ k1'00[ (“1'00) + (Elnoofl + k101[(u101) + “1.101“
101[ (W101) (‘V 101)] +k011[ (“011) + (301.1“

+k kl}01{[(u1'01) +(u .1'01):I +[(u11101) +(u _11101)]}

+ kl3'01{[(wl'01) ($.1'01 )] +[(-‘;1I101) ’ (13111101111

‘1' k11201{[(V1101) " (V _1101)] +[(V1”01)-(V_1"01)]} , 2.54

which involves the following ten interaction constants:

11 11 11 11 11 13 11 11 13 12
k100’ k010' k001' kl'OO’ k101’ k101' k011’ k1'01' k1'01 and k1'01
2,55

The component of force FY may be written as

22 —
Fy ‘ k100(V10

k22
k1'00[ (Vl'OO) + (v1..00)] + k011[ (V011) + (V01 1)]

22
o)+k

010(V010’ + 1‘

001(V001)

+
+k01l“W 011) ' (EV—01-1)]
+ kl'201{[(vl'01) +(V .1'01):l +[(v1'”bl)'+ (:1; _11101)]}
+ k1I01{[(u1101) " (11 -1101” +[(u11101) "‘ (‘3 _11101)]}

2 -— _ , _.
+k1'301{[(w1'01) (w-l'01)] +[(W1"01) -(w-l”01)] o 2.56

which involves the nine following interaction constants:

22 22 22 22 22 23 22 21 23
k100’ k010' k001’ k1'00’ k011' k011' k1'01' k1'01 and k1'01
2.57

38

And, similarly, the component of force Fz is

33 — 33 —- 33

Fz z klooIWloo) + k010IW010) + k001IW001

)
+1'00[IW1'00)+IW1“00)] +1‘101IIW101HIW101’I
k101II“101) I“ “-101” +k011IIW 011)+IW01-1)]
321IIV011) ' IV01 -1”
“‘1'301IIIW 1'01) +IW-1"01)] +[(3v'1..01) +IW-lH01)”
+k1'01IIIV1'01)'IV-.1'01)] +IIV1"01)'IV1“01)]}

+kl'01{[(u 1'01 )u-( -1'01)] +[Iqu1HOl) ’Iu_11101)l} b 2-58
involving the eleven following interaction constants:

33 33 33 33 33 31 33 32 33

100' k010' k001' k1'00' k101' k101’ k011' k011' k1'01'

32 31
k1'01 an“ k1'01

k

2. 59

Thus, the total number of interaction constants involved in all three
force equations is 10+9+11 = 30 .

Dividing through by the respective lattice distances and taking
limits, the above difference equations 2. 53-59 can be converted into

the partial differential equations given below:

F =aLk11 282u +kll b2 azu +k11 C2 82u +k11 42.4.2
x 100aL "'2' 010 “"7 001'"'2 1'00“?"

8}: By 82
82u + 8Zu +k kll (a2 + 2) 8Zu + 8Zu
Z a Z 101 ° “'2' "Z

ax12 x21 8x13 3"31

 

 

 

 

 

 

2 2 2 2
13 2 2 a w a w 11 z 2 a u a u
+k101 (a +C) a -3 +k011(b +C) $7 + E2
1"13 x31 x23 32
+k11 (34132443) 2u + 8211 + azu + azu
1'01 4 a 2 a 2 ”'8 ‘2 _ 3"" 2'
x1'3 x311 3:1,,3 "31"
13 a2+bz+4c2\ 82w 32w 32w 82w
+ k . ( , —-—— - —— + —— - ——-
1 01 4 a 2 a 2 a 2 a 2
"13 x31' x1..3 x31H
+ k12 (a2+b2+4c2\ 82v _ 32v + 2v _ 32v
1'01 4. I “—3 2 —a 2 “—3 2 "8'2
x1'3 x311 x1"3 x31H
2.60
2 2 2
__22 26v 22 28v 22 28v
“klooa ‘7“‘010" —'Z'+koo1° 2
3x 3y 6
+k22 (az+b2) 32v + 82v +k22 (132+ 2) 2v + 32v
100 4 78x "—3 2 011 “—3 2 —a 2
12 x21 "23 x32
2 2
+ k23 (b2+C2) a w _ a w
011 '“'—'3 2 —a 2
x23 "32
+k22 (a24b2+4c2) 32v + 32v + 32v + 32v
1'01 4 a 2 a 2 a Z a 2
‘ x1'3 x31' x1"3 "31'I
+ k21 (az+bz'-l-4cz\ azu 8211 + an 3211
1'01 4 ‘ ' ‘8“2' ' ”'3 2 “a 2 ' ‘73
x1'3 x31' x1..3_ x31"
23 az+b2+4c2\ 82w 82w I 82w 82w
+k ( 1 , - — + - -——-—
1'01 4 a 2 a 2 a 2 a 2
"1'3 "31' x1"3 x31"

 

8x 3y 8z
+ k33 (a2+b2) 32w + 32w + k33 (212+ 2) 82w + 82w
1'00 4 a '8'"? 101 a a 2
x12 x21 "13 x31
2 2 2 2
31 2 2 3 u 8 u 33 2 2 8 w 8 w
+k101( + ) 73 - :2) +k011(b+ )(—7-8 + —-—ax2)
x13 "31 "23 32
2 2
32 2 2 a v a v
+k011(b +c) —2-8 --——-32)
"23 x32
+ k33 a2+b2+4c2\ 32w 32w 32w 32w
, ( , + + +
1 01 4 a ‘73 a a
x1'3 x31' x1H3 ”‘31"
+ R31 3211 8211 +. 3211 8211
1'01 "“8 2 " —a 2 ‘"'2'8 ' ‘78
x1'3 "31' x1"3 "31'H
+ k32 32v _ 32v I + 2v _ 62v
1 '01 '8' 2 "a "2 “'3 2 "a '2'
"1'3 x31' "11'3 x31"

2. 62

Here, the xij diagonal axes lie I in the xixj -plane, where
i,j = l, 2, 3,1' and l", such that x1,3 is the diagonal axis along the
rectangle (%~j'az + b2, c) in the x'z-plane. These axes are shown

in Figure 2. 9 a-e below:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

v y Y
X
b
a a
a a x
x31
z Z
(a) (b)
y Y
b b
-a
a X x
C x1"3
x1'3 c
Z
x31' z x31”

(d) (e)
Figure 2. 9. Geometrical representation of axes xij
(a) xl‘2 and x21 (b) x13 and x31 (c) x23 and x32

((1) x1,3 and x31, (e) 3:1,,3 and x31” .

42

Transforming partial derivatives with respect to the x. .'8
into partial derivatives with respect to x, y and z in Equations

2.60 -62, by utilizing the transformations derived in Appendix II,

these equations bec ome:

F =k11 2 3Zu 11 2 32u 11 2 32

u
x 1002‘ 2 +k010b ' Z “‘001 C ""2
3x 3y 32
2 2 2 2
111 23u 23u 11 23.u 23u
+‘2'k1'00 a‘7+b"2 +2k101a‘7VC‘7
3x 3y 3 32
2 2 2
13 aw 11 23u 23u
+k101ac5—-5—x z +2k011 b ‘76 +c -—2-)
y 32
2 2 2
11 23u 23u 2 3n
“(1,01 a——z—+b—-2-+4c 7
3x 3y 3z
+4kl3 bc i!- +4k12 b 33-3?— 2 63
1'01 3x32 101 ° 3y32 '
_k22 a2 32v +k22 b2 32v +k22 232v
y" 100 '3 2 010 "2 001c "2’
x 3y 32
2 2 2 2
122 23v 23v 22 23v 23v
+2 kl'OO 7 +1) ——2- +2k011 b 7 +C "'7
3x 3y 3y 32
2 2 2 2
23 3w 22 23v 23v 23v
+4k011bc 3y32 +kl'01 a -—7 +b -—-2- +4c —-2-
3x 3y 3z
2 2
21 3u 23 3w
+ 4k1,01bc m + 4k1,01bc m; 2. 64

3x 32
2 2 2 2
l 33 2 3 w 2 3 w 33 2 3 w 2 3 w
+2-k1,00 a—z- +b -—-2— +2k101 a——2-+c -—-2-
3x 3y 3x 32
2 2 2 2
31 3 u 2 3 w 2 3 w 32 3 v
+ 4k101 ac m + ZkOll (b ~78}, + c .782 ) + 4ko11 bc WY 2
+ k33 a2 32 w + b2 32w + 4C2 2w + 4k31 bc 3zu
1'01 “:2 70y “‘2: 1'01 W
+ 4k3 bc 32" 2 65
1'01 532 '

By collecting coefficients of the various partial derivatives,

the above equations may be rewritten as:

_ 2 11 1 11 11 11 32u 11 1 11 11
Fx’ 3 Ik100 V2k1'00 V Zk101 V k1'01) "2 V b P‘VIV‘010 21‘1'00 V 2k011
k )32u+c2.(k11-+2k11 +Zk11 ) 31.12 +4 k13 32w
1'01 WY 2 .. . 001' 101 011V 4k1'01 “‘2 ac 1012—52
2 2
13 3w 12 3v
+4bckl,01-5—5-z— +4bckl,01W 2.66
F _a2(k22 +11(22 +1: k22 ) 3v2 +b 2(k22 1k22 +2132 +k )3zv
y” 100 2' 1'00 1'01 “'2 010 2 1'00 011 1'01 ”28),
2(k22 +21<’22 +4 kzz ) 32" +4b C(kz +)1<23 82W
+c 001 011k1'01 “—2 011V 1'01 W
2

21 3 u
+4bckl,01 W 2.67

44

2
_ 2 33 1 33 33 33 3w
Fz‘a Ik100V2k1'00V2k101Vk1'01V‘a'x'2
+1320(33 +1193 +2193 +k33 ) 32w
010 2 1100 011 1101 “'73),
2 33 33 33 33 32w
+c (1:001 + 2 k101+2 k011+4k1,01)—-2—az
2 2 2
31 3u 31 3u 32 32 3 v
V4a°k101 '“ax'Tz V4b° k1'01 —3y3"z“ V4b C Iko11 +kl'01)3y3z
2.68

Newton's law for the force components in the directions x, y

and 2 may, of course, be written as

F

_3.‘._ = {3

abc p

.5; ~

abc = p v

and
Fz O.
m- = p W Z. 69

where abc = volume of a unit cell.

The corresponding continuum equations 2. 25 based on the

generalized Hooke's law are relisted below to facilitate comparison:

" - 32“ + La“ + 32“ +( + )fi-H +C )Wazw
P u - C11“? C66 2 C55' '2 c12 C66 axay C13 55 x z
ax By 82
2 2 2 Z 2
,, _ 3 v 3 V 3_V 3 u 3 w
PV - C663? ”2287 V C44 022 + (°12+°66’5§75§VI°23 ”44V 5W
32 32w 2 3 2

.. _ w 3 w 3 u 3 v
pw -c55 73x +c44—2-ay +C33_Taz +(c13+c55)-5--5--x z +IC23+C44)3__—y3z

45

Comparing coefficients in the two sets of equations we obtain

the following expressions for the elastic constants

the interaction constants kg'nn :

a 11 1 11 11

c.. in terms of
1J

11

C11: "152' Ik100 V2 k1'00 V 2 k101 V k1'01’
C66= EPEIV‘310V2V‘1100V21‘311VkiIm) 2'70
C55 z 3%Ik31111 V 2 k131V 2 1‘31 V 4 ki'lm)

612 V C66 ._. 0

C13 V °55 =1? kigl
C66 = EVE “130 V2 kIgloo V kV3201)
C22 = ab: Ikgfo V2 k177200 V 2 1‘31} V k173201) 2' 71
C44 z 396' “(2131 V 2 kgfl V 4 ki'zm)

C12 V C66 ‘ 0

C23 V c44 = 21V Ik(21:11 V k13301)
655 = BEE Ikigo V 2 k1'300 V Z k11311 V k11301)
C44 = '33 “((3110 V '2' k1,1300 V 2 kgil V @301) 2'72
c:33 = 2913 “‘331 V Z 16131 V Z kgil V 4 k1?01)

C13 V C55 = “E k15:11

C23 V C44 '"' g “(3111 V k$01)

46

These in turn may be combined and rewritten to give:

a 11 l 11 11 11

C11: 13'; Ik100 V2 k1'00 V 2 k101 V k1'01’
(‘22 :29; ”‘ng V2 k1'200 V Z kgfl V R3201)
c33 '“' 5303531 V 2 k1:31 V 2 kgil V 4 k1?01’
C44 = 2% Ikcznzn V 2 1‘31} V 4 kf'zm)
°r C:44 = 2% Ikgio V2 k1:300 V 2 1‘31 V kfiol)
C55 = 3913' “(31111 V 2 k131 V Z k311 V 4 R1101)
or C55 2 fi— (kigo +é- kfim + 2 #1531 + kffbl) 2.73
c66 = 21% “(310 V2 k1100 V 7' k311 V ki'lm)
°r C66 '"' Ba? “(ff-10 V2 k11200 V kf'zm)
C12 ._. ' c66
c = 5- kl3 - c
13 b 101 55
= g kfln ' c55
C23 '“' g Ikgil V k1.1301) ' c44
= '3’ “3011 V k103201) “' c44

The exPressions 2. 73 may be simplified by excluding the terms
involving the second nearest neighbor interactions; this yields the

following expressions for first nearest-neighbor interactions:

47

_ __ 11 _1_ 11
C ’ Ik100V2 k1'00)

b 22 1 22

C22 = 5‘6 Ik010 V 2 k1'00)

C33 : 2% k313n

c44 : 3% kgfn‘r: 2% Ikgio V2 k'1'300) 2' 74
C55 = 5% 1‘33““ 6% Ikigo V'12 k:1'300)

__1_3_11111=_a_1_22_1_22
C66 ‘ ac Ik010 V 2 1‘1'00’0'r bc Ik100 V 2 R1100)

C12: ’C66
°13= ‘°55
C23= “C44

Constants C44, c55 and c66,-:and correspondingly C12’ on and C23
are double-valued. An appropriate single numerical value will be

selected fori'thes'e; later in Section 2. 5.

Central Force Assumption
If only central forces are allowed, the following sixteen of the

thirty interaction constants entering Equations 2. 54-59 vanish:

11 13 11 13 12- 22 23 .21: 23
k010' k101' k011' k1'01' k101' k100' k011i" k17.'01i"k1'01'
33 33 33 31 32 32 31 ,

k100' k010' k1'00' k101' k011' k1'01' k1'01 ' 2'75

while the following fourteen will still be involved:

48

kiéo’ k331' ki'loo' k11111: ki'lor k310' k331'

kf'zoo' kgfl’ ki'zov @361: 1‘31: kgil’ k11,01 2°76

It may be noted that, unlike nonpolymeric crystals, the
constants kg?” and kggl do not vanish because of the bending
resistance of the C-C bonds. This means that the central force
assumption is being applied in a limited manner.

The expressions for the elastic constants cij under this
central force assumption, including interactions up to second
nearest-neighbors, become:

C11 ' biIk130 V'12k1I00 V 2 k131V R1101)
C22 = 5% “((2)10 V l21‘12‘1200 V 2 R311 V #1201)
C33 V ab ”‘331 V Z kigl V 2k311 V “$01)
c44 = 3'1? “331 V 2 1‘31 V 4 kf'zm)
°r c44 = 3136 I2 kgil V 1615.301)
C55:;%(k(l):)l+2k1:11+4k1'101 2.77
°r C55 z Bi— IZ kigl V kfim)
C66 : 5% I2 k1100 V kiI01)
°r C66 = BEE-(1: kf'zoo V kf'zm)
C12 = ‘ C66

Magmtudes of c23, c13, c12 are equal to those of C44, c55, C66 due to
drOpping the interaction constants for the central force assumption.
This does not imply that the number of independent elastic constants

is reduced to six.

49

=-c

c13 55

C23 " C44

But for first neighbors only these may be further simplified to give:

a 11 1 11
C11“ 3? Ik100 V2 k1'00)

b 22 1 22

c=22 : E'e' Ik010 V2 k1'00)

C33 = 2% k3031

C44: 29b" kgfn °r °44=°'
:__g_:__ kll

ab 001 °r C55:0

1 .19.. k11 1 22

__ ..__§_
°66‘2 ac 1'00°r“'2 bc k1'00 2'78

c:12 “' 'C66
C13 - 'C55
C23 = '°44

These show that polyethylene single crystals possess shear
resistance even when first nearest—neighbor interactions and central
forces are assumed. It may also be remarked that identical expressions
are obtained for first nearest-neighbors from strain energy consider-

ations, as shown in Appendix III.

50

2. 4. Interaction Constants and Molecular Force Constants

The elastic constants were related to the interaction constants
in the last section, and in Section 2. 2 it was explained that interaction
forces result directly from relative motions of the lattice units in
the intramolecular and intermolecular force fields. The objective
of the present section is to obtain expressions for the interaction
constants in terms both of the intramolecular force constants, such
as those for C-C bond stretching, bending and repulsion, and the
intermolecular force constants, such as those appearing in the
Lennard-J ones potential.

It is assumed as before that the weak secondary bond forces
between units lying on different chains are central. However, the
same cannot be said of the forces between units lying along a single
chain; these are due to the strong connecting C-C bonds which provide
resistance to bending in lateral directions. For this reason the central
force assumption is limited to intermolecular forces only.

The following interaction constants,

11
100’

11 ll 11 ll _ 22 22

001' k1'00' k101’ k1'01’ k k

k 010’ 001

k

22 22 22 33 33 33 33

k k k k101' k011' k1'01

k1'00’ 011' 1'01' 001'

will be obtained in terms of the molecular force constants such as
the C-C bond stretching, bending and repulsion force constants K,
H and F, respectively, and the Lennard-J ones potential constants
A and B. The rest of the interaction constants (listed in 2. 75-)
vanish under the limited type of central forces that exist between

the lattice units, as . discussed in the preceding paragraph.

51

Constants Along the Chain
Of the fourteen interaction constants needed for interactions
up to second nearest-neighbors under the central force assumption,

three are for forces between units along the same chain, viz. ,

11 k22 and k33

k001' 001‘ 001 °

These will primarily involve the C-C bond constants. Further, if
displacements remain small and the planar zigzag conformation of
the polyethylene chain does not change, it may be assumed that no
torsion takes place; and any deformation may be accomplished merely
by stretching and bending the C-C bond. Thus, if as in Figure 2.10,
6r = change in the bond length r
60. = change in the bond angle a/Z
and

6d

ll

change in the distance Vii-between two alternate C-atoms,

 

 

 

 

r a/Z o/Z r

 

C

Figure 2.10. Geometry of two corresponding C-C bonds showing
the bond length r, bond angle a and the distance
between alternate C atoms.

52

the strain energy V becomes

In—n

V = 2[%2-K(6r)2 + 2 H (r6c1)z - :12— F(6d)2] 2.79

where K, H and F represent the stretching, bending and repulsive

force constants.

. . . ll 22
To determine the interaction constants k001’ k001 and

kggl, particular expressions for V must be derived by considering

the respective deformations in the x, y and z —directions; these are

illustrated below:

 

 

 

 

 

 

 

 

 

 

Z
K\
\\\
\\\\
6r”):
0 ‘1’ I 62
54 44' ““1
' 1
r / : o
/ 1 ‘”’ Y
0 : ”’4:\ p
l
0 K 59 5 \
41° ' / Y \
' I
4 <64, 1' x \ IV 6x
' I
p\\ 1 /I 4>
2\__-:._J’ x
I \‘\ '”6X
_’->1
.6},

(a) (b)

Figure 2.11. Deformation of C-C bonds (a) in a general direction
(b) in x-direction.only.

53

 

 

 

 

 

 

 

z e
I

. 343

54> V /

¢ VI /
\
\\ 69/
" 423:3 V 1/2
53*
/ 54°44'
X

 

(C) (d)

Figure 2.11. Deformation of C-C bonds (c) in y-direction‘only'
(d) in z-d'irec't'ion only. ’ ' ° -- 1 . .

11
001

The Constant k

By definition, kggl is the force on the unit (0, 0, 0), per unit
displacement u of the unit (0, O, l). Denoting this displacement by
6x, as inFigure 2.11 (b), the quantities Br, 60. and 6d of Equation

2. 78 may be obtained in the following way. Letting

cs
ll

N|=1
l

le

where 921 = 54044',

-410 49 .

.9.
ll
N|=I

[\-

pl

54

The bond length r is given by
r2=x2'+y2+z2 ;
and differentiating this expression we get

2r 6r 2x 6x + 2y by + 22 6z,

11
where for R001 , 6y = 6z 2 0 .

Therefore,

2r 6r 2x 6x

01'

Gr = 6x

Hlx

Now, since p = r cos ((1/2), as shown in Figure 2. 11(a),

x E cos (b r c1 0. a. . o.
— = = - cos — cos = cos — 6.05 = cos -- 811'! 4.1
r r r 2 IV) 2 (I) 2 ’7

and we have

6r = cos % sin 410 6x
= cos 54. 7° sin 41° 6x
= g 6x ,

where

g = cos 54.7o sin 410 .

Also, from Figure 2.11 (b) we have

0
69 =£9_§—riL 6x

Hlo.

5x,

55

where
d = cos 41°
and
6d = 0 .
Consequently, substitution for 6r, 66 or 60., and 6d in
2. 79 yields

V = K(g 6x)2+ H(d 6x)2 ; 2.80

and differentiating this with respect to 6x, we get an expression for

F :
x
_ dV _ 2 2
Fx— '“‘—d6): — -(2gK+2dH)6x 2.81
Therefore, kcl)31 , being the force per unit displacement, is given by
F
11 _ x _ _ 2 2

The negative sign indicates attraction for positive displacement.

22

The Constant kOOl

Following a procedure similar to the above for 1:331, but

considering only the displacement 6y, analogous expressions for

22 .
k001 may be obtained:
2 2
V = K(h6y) +I-I(e6y) , 2. 83
h = cos 54.70 cos 410
e = sin 410

56

_ dV_ 2 2
Fy—-d6y—-(2h K+2e H)6y 2.84
22 _ 2 2
1.001- 2(hK+e H) 2.85
33

The Constant k001

 

Similarly, by considering only the displacement 62, the

corresponding expressions for R301 can be derived:

v = K(iéz)?‘ + H(f6z)2 - 1:782)Z , 2.86
1 = sin 54.70
f = cos 54.70 .

.2 2
Fzz-(21K+2fH- F)6z 2.87
33 __ .2 2
1:001 —2(1K+fH-ZF) 2.88

Other. constants are essentially derived from the Lennard-Jones
6-12 potential curve <l>(r) shown in Figure 2.3. This defines the force
existing between the lattice units and its variation with separation
distance as well. If the latter is less than the equilibrium distance
r , the force will be repulsive and if it is greater than the equilibrium
distance, it will be attractive. For the small displacements with which
we are concerned, the force may be assumed to vary linearly, though
the rate of variation will evidently be different for different separation
distances. Such an assumption makes it possible to evaluate the inter-
action constants by determining the slope of the force curve at the

various lattice distances. Thus, if F(r) is given by

 

F(r) = If? .. 9% , 2.28
r r

then the derivative of F(r) with respect to r is

dF _ 156A 42B
:1"; — --;—i-Z- '1' :‘g— . 2.89

This determines the interaction constants for the first nearest-neighbors

 

8.8
11 _ dF _ _. 156A 4213
k100 ‘ " IE?) ' 14 " 8 2‘90
1‘38. a a
and
22 _ dF _ 156A 4213
k01o - ‘ (a?) - ‘74" - T ' 2°91

r=b b b

The interaction constants for the diagonal units and second nearest -
neighbors are determined from the components of the diagonal force,
or the force along the line joining the central unit with the surrounding

units . Thus,

11 __d__ a 12A 6B

 

1'00 dr ,1 . 13 7 ‘
[3.2+ b2 r r r _ 1 2 2
" 2- a + 1)

giving:
k11 _ a 156A , 4213 2 92
1'00 — 2 2 7 '- 2 2 4 °
2 2 a +b . ,a +b
3. + b 4 _ ___._4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

k22 _ b. . 156A ___4_2_8__
' — . -
V 00 2+b2 (a2+b2)7 a2+b2 4
a 4 4
k11 ___ a . 156A 428
101 az+e2 a2 + C2)? (a2 +? 4
k33 __ c . 156A 428
101 ‘ ' ' 2 2 7 ' 2 2 4
Ia2+c2 (a + c ’ (a + c )
k22 _ b . 156A 428
011 ‘ — - 2 2 7 —'—‘2“2 4
/b2+C2 (b v+c ) {b +e )
k33 = c 156A 428
011 b2+cz (b +62)? (b2+C2)4
k11 _ a 156A 428
1'01“ 2 2 2 7 " 2 2 2 4
fi2+b2+4e2 a +b +4e \ a +b +4c
4 I 4
k22 _ b 456A . 428
1'01 ‘ 2 2 2'7" ‘ 2 2 2 4
[a2+b2+4cz (a +b +4c \ (a +b +4e I
4 I 4 I
k33 _ 2e f 156A 4 428
1101’ 2 2 42 7 ' 2 2 24
J3. +102 + 4C2 (a +b4+4c ) (a +b 4+4e }

 

2. 94

2.95

2. 96

2. 97

2. 98

2. 99

 

2.100

59

Numerical values of the interaction constants will be obtained
along with the other constants in the next section (2. 5), by substituting
the values of the intermolecular force constants A and B and the

intramolecular force constants K, H and F.

2. 5. Numerical Values of Constants

Expressions for the Lennard-Jones potential constants were
derived in Section 2. 2, while in Section 2. 4 expressions for the inter-
action constants were obtained in terms of the C-C bond stretching,
bending and repulsive force constants and geometric parameters such
as bond length, bond angle, lattice distances and setting angle. The
connection between the elastic constants and the interaction constants
was established in Section 2. 3. In the present section, numerical
values for all of these constants are obtained: first the Lennard-Jones
constants, secondly, the interaction constants, and lastly, the elastic

c onstants.

Lennard-Jones Constants
In Section 2. 2, the expressions for the Lennard-Jones potential

constants A and B, 2.49 and 2.50,

 

 

B: 2M 62
NAPAIZ-é
2
2pM6
Azsz ,
N pAlZ-é

involve various quantities to which numerical values may now be
assigned. The molecular weight M of the (-CZH4-) lattice units

is:

60
M = 2x12+4 = 28 2.101

Avogadro's : number NA is:

NA = 6.0249x10Z3 2.102
The cohesive energy density 62 for polyethylene, determined
experimentally by Small (1 6), is:
62 = 62 cal/cm3 2.103
.. 9 3
— 2. 595320 x 10 erg/cm

2. 595320 x 10'15 erg/A3

The density p of crystalline polyethylene . varies from one
manufacturer to another; however, the variation is small and one
representative value, listed in the commercial bulletin of the Dow

Chemical Company, Midland, Michigan (27), is:
3 g .
p = 0.964 gram/cm 2.104

The factor A = p A - A6 has been evaluated in Appendix I;

12-6 12

its value is:

_ -3 -6 -
12-6 - 2.345833 x10 A I-35

The factor p = %- (r0)6 has also been evaluated in Appendix I:

p = 3. 897619 x103 A6 1-34

Substituting these values in the above expressions for B and A,

yields:

61

2.177300 x10"10 erg A6

B =
= 2.177300 x10.58 erg cm6 2.105
A = 8. 486677 x 10'7 erg A12
= 8.486677 x10"103 erg cm12 2.106

Interaction Constants

In Section 2. 4 the interaction constants kiifjmn have been divided
into two categories:

(a) Interaction constants for units on the same chain.

(b) Interaction constants for units on different chains.
These are evaluated below.

(a) Expressions for constants in category (a) are derived in
Section 2. 4. These relations (2. 82, 85 and 88) involve the C-C bond
stretching, bending and repulsive force constants K, H and F which

are given by Shimanouchi, et al (17):

K 4. 0 3:10-3 dyne/A

H 0.113e10"3 dyne/A 2.107

0. 96 x10"3 dyne/A

The geometric factors g, h, i, d, e and f, are defined in Section

2. 4 in terms of the following (10,11):

C-C bond length r = 1.54-A
c-c bond angle a = 109° 28' 2.108

setting angle 8 = 41°

Substituting these values of r, 0., (3,. we obtain:

62

g2 = 0.143724
b‘2 = 0.190197
i2 = 0.666084 2.109
d2 = 0.569587
e2 = 0.430414
f2 = 0.333922

The interaction constants in category (a) thus turn out to be:

k“ = 1.275102 x 10"3 dyne/A

001

22 -3

kOOl = 1.616266x10 dyne/A 2.110
13531 = 3.482134x10"3 dyne/A

(b) The expressions for the constants in category (b) are given
in Section 2. 4 (2. 90-2.100). Their numerical values may be obtained
by substituting the values of the Lennard-J ones constants A and B
from 2.105 and 2.106 and the lattice parameters a, b and c d 7. 41A,

4. 94A and 2. 55A. The result is:

kito = - 9.180603 x 10‘8 dyne/A
kzz = - 1.371359 x 10"8 dyne/A
010
k11 _ 4 -6
1,00- ..219081x10 dyne/A
1.12:200 = 2. 809002 x 10"6 dyne/A
11 -8
k101 = - 6. 075525 x10 dyne/A
kg?” = - 2. 090739 x10"8 dyne/A
1.22 - 4. 538807 x 10'7 dyne/A

011

63

33
011

11
1'01 -
22
1'01 -

33
1'01 _

2. 342901 x10-7‘dyne/A

W
1

2.843103 x 10“7 dyne/A

W
I
I

1. 895375 x10"7 dyne/A

W
I
I

1.956759 x 10'7 dyne/A

W
I
I

2.111

Elastic Constants

The expressions 2. 77 for the elastic constants cij in terms
of the interaction constants kIVI-nn and the lattice parameters a, b
and c under the central force assumption as derived in Section 2. 4,
on substitution of the numerical values from above, yield the following

values, including second nearest-neighbor interactions:

ell = 0.948127 11:10-6 dyne/Az
-6 2
€22 = 0.288779x10 dyne/A
-4 2
e33 = 2.422665x10 dyne/A
— 1 123761 10'4d /14.2 - -
C44 - o X Yne - C23
- 0 886595x10“48 e/AZ = -c
C55 " ° Y“ 13
- - 6 515515x10“7d ’/A‘Z =-e -
c:66 ‘ ' V” 12

2.112

The corresponding expressions 2.73 for first neighbors only yield:

e11 = 1.186831x10“6 dyne/AZ
-6 2
e22 = 0.363552x10 dyne/A
-4 2
e = 2.423565x10 dyne/A

33

64

1.124921 x 10'4 dyne/AZ = - c

C44 = 23
- 0 887471 10"4 d /A2 —

c55 - . x yne - - cl3

c _ 7 006943 3410-7 d ne/AZ = - c

66 ° Y 12

2.113

As observed earlier, the constants C44, CSS, (:66, C12’ €23
and C13 are doubled-valued. However, the constants C44 and C55
should have much higher values than the constant C66, because the
former two involve movements of units belonging to the same chain.
Thus, only the higher values of C44 and c55 are used; but in the
case of C66, for which the two values are of the same order of
magnitude, the average value is taken.

Substituting the values of cij 2. 17-19 yields:

E1 = 0. 377123 1410-6 dyne/Az
-6 2
E2 = 0.266161 x 10 dyne/A
-4 2
E3 = 2.388100 x 10 dyne/A

2. 114
All the numerical values are given to six decimal places as a
matter of calculational convenience only. These may be rounded off

to three decimal places for future use without any loss of accuracy.

III. DISCUSSIONS OF RESULTS

3.1. Anisotropy of Polyethylene Single Crystals

Anisotropy of polyethylene single crystals is a compound effect
depending on the inherent nature of the lattice structure and the
directional molecular forces of different strengths that exist along
the three lattice axes. The complex orthorhombic lattice of poly-
ethylene (Figure l. 6), consisting of (-C2H4-) as the lattice units.
has been converted into a simple orthorhombic lattice (Figure 2.1)
by choosing. as a basis, the pair consisting of dains at the mid-point
and the corner of the rectangle (a, b). An orthorhombic lattice
structure has nine independent elastic constants cij’ whereas cubic
crystals have only three such constants. Thus, an orthorhombic
lattice, by itself, is anisotropic in a manner which is more complex
than the cubic lattice; and the situation is further complicated by the
directional molecular forces that exist between the units themselves.
However, this complexity has been reduced by approaching the
problem from the continuum and the discontinuum points of view
independently, then relating the results.
3. 2. Continuum Theory of the Orthorhombic Lattice

Crystals having an orthorhombic lattice, irrespective of what
the molecular forces are, would be expected to have different elastic
moduli along the “three lattice axes because of the inequality of the
lattice parameters a, b and c. Thus, there are three Young's moduli
and three shear moduli for such crystals; however, Poisson's ratios
are six in number,because

e.

._ _l _
vij — ' e. ’4
J

II
V

65

66

A uniaxial tensile stress along the a-axis would cause a certain
contraction along the b or c-axis, and this would be different from
the one caused along the a-axis by a uniaxial stress in the b or c-
directions.

Expressions 2. 17-19, for E E2 and E in terms of Cij’

1’ 3
have a common numerator but different denominators. Hence the
expression 2. 20, which relates all three Young's moduli, can be derived.
The shear moduli (323, G13 and (312 are equal to C44, c55, C66
respectively (2.11) due to the definition of the shear strains (2. 5).
Expressions 2.16, 2.16a for the six Poisson's ratios must be
obtained in pairs by considering uniaxial stresses along the three
axes each time, and by solving the resulting equations simultaneously.
The equations of motion, 2. 25, and the expression for the
strain energy, 2. 22, are slightly more involved than the corresponding
equations and expression for cubic crystals. The strain energy
relation is only an approximation, because higher-order terms
involving rotational or torsional and coupled deformations are
neglected. This is the case for cubic crystals too.
It should be pointed out again that the constants cij differ,
not only due to the inequality of the lattice parameters, but also due
to the inequality of the molecular forces in various directions. This
is discussed in more detail later when their relationship with the

interaction constants is explained.

67

3. 3. Molecular Forces

Both intramolecular and intermolecular forces are highly
directional. The intramolecular or primary bond forces which exist
along the molecular chain axis are due to the covalent C-C bonds.
These are several times stronger than the intermolecular or the
secondary bond forces existing between the chains. The latter are a
net result of the London dispersion forces of attraction and the
repulsive forces caused by the overlap of the electron clouds surround-
ing the nuclei. In fact, the intramolecular forces are so strong that
to some extent they dominate the inherent anisotropy of the orthor-
hombic lattice.

Molecular chains take a planar zigzag configuration, when in a
lattice; thereby preventing free rotation of the segments of the molecules
about the C-C bonds. This gives the chains a definite resistance to
deformation along the c-axis, which is also the chain axis, and to
bending in the lateral directions a and b. The strength of the C-C
bond for various types of deformations is fairly well known and the
values of these constants are listed in Section 2. 4.

The strength of the secondary bond forces is known only as a
measure of the cohesive energy or the sublimation energy. These
intermolecular forces have been assumed to be determined by a 6-12
Lennard-J ones potential, which involves two unknown constants. It
should be noted that the value of the sublimation energy. as determined
experimentally by Muller (28), is more than twice the value of the
cohesive energy as determined experimentally by Small (16). However,

the value of the latter quantity is more reliable; first, because it is the

68

more recent of the two and, secondly, because it is verified theoret-
ically by the latter author. Accordingly, Small's cohesive energy
data is employed here.

Values of the Lennard-J ones potential constants A and B
determined in this manner have the same order of mangitude as for
solid argon. This is to be expected because the lattice energy is of
the same order of magnitude (23).

The intermolecular forces of polyethylene arise from the
interaction between the hydrogen atoms, which are attached to the
carbon atoms of the molecular chain by covalent bonds formed by
sharing a pair of electrons. Thus, the valence electron of hydrogen
spends most of the time in the region between the carbon and hydrogen
atoms and very little time outside this region. The result is that the
dispersion type of van der Waals forces, which arise from the time -
varying instantaneous electron configurations, are very weak-ma
condition existing in rare gases too, though for a different reason,

It has been emphasized that polyethylene single crystals have
an entangled-rectangular lattice structure in which the planes of the
chains have two orientations. This fact is not considered in the
computation of the lattice energy of a unit in an infinite matrix. How-
ever, as illustrated in Appendix IV, the orientation of the chains
becomes insignificant if one considers their interaction in more detail.
The chains are situated in space in such a manner that for any two
neighboring chains the hydrogen atoms are equidistant from each other.
This determines the Lennard-Jones potential between chains; the net

or effective potential for the two units has been obtained in Appendix IV.

69

A slightly different method than the usual low density gas
approach (23) has been used to determine the two Lennard-Jones
potential constants. By taking the equilibrium separation distance
between the two nearest chains to determine the minimum in the
potential curve, the ratio of the two constants is fixed. This leaves
only one unknown constant, which is then determined by computing
the crystal potential energy density and comparing it with the
cohesive energy density. In the case of gases, the crystal potential
energy or the lattice energy is first calculated from a general
potential. Its value is then minimized to determine the value of the
equilibrium separation in terms of the lattice energy and one unknown
constant, which in turn is obtained from data for the second virial
coefficient. However, this is not applicable to solids where the
equilibrium separation and the separation for minimum crystal
potential energy are identical.

The triple inverse power series involved in computing the
crystal potential energy converge very rapidly. Their values have
been obtained by splitting them into component single, double, and
triple series. For evaluation of the single series, standard formulae
are given in reference (29) by Knopp. To compute the sums . of the
double and triple series, they were terminated at a point beyond which
the contribution of the terms is less than 0. 000015 for the sixth power
and less than 0. 00000000024 for the twelfth power.

As the units of distance along the three axes a/Z, b/Z, and c
have been used; these lead to the above -mentioned series. However,

it is only for points for which the integers along the a and b -axes

70

are even, that a real unit exists. This point has been taken into
account in computing the sums: of the series.

In order to compare the crystal potential energy with the
cohesive energy, all the units belonging to the central chains are
ignored. These units are attached to each other permanently and
remain so even when the chains are removed a considerable
distance from each other, as required in determining the cohesive

energy density.

3. 4. Interaction Constants
To establish a relationship between the continuum elastic
constants and the molecular force constants, first and second nearest-
neighbor interactions have been considered. The expressions for
the force components on a central unit due to its motion relative to
its neighbors, involve thirty constants for interactions up to second
nearest-neighbors, while for cubic crystals there are only five such
constants. If only first neighbor interactions are considered, the
expressions contain eleven constants for an orthorhombic crystal,
whereas only two constants are involved in the case of cubic crystals.
In view of the inequality of the three lattice parameters a, b
and c, all the first neighbors are not equidistant from the central
unit. Such is the case for the second neighbors too. The first nearest-
neighbors are the units which lie nearest to the central unit along the
axes x, y, z, x' and x" in either positive or negative directions. The
second nearest-neighbors are the units that form rectangles with the
first neighbors and the central unit. In this manner the four units

corresponding to (l, l, O) are eliminated from the family of second

71

nearest-neighbors, but the eight units corresponding to (l', 0, l) are
included instead.

A limited type of central force assumption has been applied to
reduce the number of constants to fourteen for second nearest-neighbor
interactions and seven for first nearest-neighbor interactions. The
forces between the lattice units are not quite central, due to the units
being rather unsymmetrical in shape. The major attractive forces
arise from the hydrogen atoms which are located off the lattice points.
However, because the forces between units belonging to different chains
(secondary bond forces) are much smaller than the forces between
units belonging to the same chain (primary bond forces), the former
may be considered to be nearly central. This is what has been termed
a limited type of central force assumption.

The interaction constants have, therefore, been classified in two
categories:

(a) those for units belong to the same chain

(b) those for units belonging to different chains

The interaction constants of category (a) are obtained from the C-C
bond stretching or contraction, bending and repulsive force constants,
while those belonging to category (b) are obtained from the Lennard-

J'one s 6 - 12 potential.

. ll 22 ‘ 33
All three constants of the first category, k001' k001 and kOOl'
are positive and approximately 103 times stronger than kI'IOO and

kfg'oo, the only two positive constants of the second category. Other

constants of the second category are negative, and their magnitudes

are 10-100 times lower than these two.

72

- 11 22
As mentioned above, the constants kl'OO and k1,00 are

11 22 11 33 22 33

p081tive, while the constants k100’ k010’ k101’ k101’ k011, k011,
11 22 33 . . .
kl'Ol’ kl'Ol and k1,01 are negative. The pOSitive constants are

for units whose separation distance is smaller than that to the point
where the F(r) force curve has a zero slope. This occurs at a
distance of r = 4. 493 A -- slightly larger than the distance r0 =

4. 450A, at which the ¢(r) potential curve has a zero sloPe. Because
the interaction constants are negative for units having a separation
distance greater than r = 4.493A. it will be seen later that the
inclusion of second nearest-neighbor interactions lowers the values

of the elastic constants instead of raising them.

11 22

33 .
001, k001 and kOOl involve

deformations of the C-C bonds and, thus, are obtained from the strain

The interaction constants k

energy expressions in terms of the particular type of deformation
required. Identical results would be obtained if a general expression
for the strain energy involving all types of C-C bond deformation were
obtained. The constants could then be obtained by taking the appr0priate
partial derivative, :though the expressions would be slightly more

involved than the one used here.

. . ll 22 ll 22 11
The interaction constants k100' 1:010, kl'OO’ kl'OO’ km”,
33 22 33 ll 22 33 .

from the sloPes of the force curve at the appropriate separation
distances. Of course, this amounts to approximating the force
curve by straight line segments in the neighborhood of the location
of the units. However, for the infinitesimal displacements we are

concerned with,this assumption is well justified.

73

3. 5. Elastic Constants

Expressions for the nine elastic constants cij in terms of the
interaction constants kijmn have been derived in Section 2. 3. These
are obtained by comparing the coefficients of the appropriate partial
derivatives in the equations of motion, derived from continuum theory,
and the force components equations, derived from discontinuum theory.
However, some of the partial derivatives appear twice in the equations
of motion. This leads to redindant expressions for the constants 044,
C55 and C66 and, correspondingly, for the constants 023, 013 and
612 also. Selection of the appropriate expressions was not made until
their numerical values were obtained and a comparison could be'.‘
made.

The expressions for the elastic constants have been simplified
by employing the central force assumption discussed above to
eliminate some of the constants, and these have been further simplified
for first nearest neighbor interactions. It may be pointed out that, for
fir st neighbor interactions and the central force assumption,identical
expressions are obtained from strain energy considerations ianppendix
IV.

In order to decide upon one expression for the constants C44,
055 and C66, all the expressions were first evaluated by substituting
in the values of interaction constants. The constants 0'44, -c‘-55 and
c'66 are identified as the shear moduli. The first two involve move-
ments of the units belonging to the same chain whereas the constant
C66 involves movements of units on different chains. Thus, the

values of (:44 and c55 should be much higher than that of 666'

74

However, 066 has two numerical values, both of the same order of
magnitude; and in the absence of any criterion for making a selection,
it was decided to use an average of the two values as the true value
for this constant. The two lower values of C44 and 055 compare
well with this value of 066; however, their higher values are about
103 times larger. This suggests that the higher values of the constants
are the right ones.

As mentioned earlier, because the interaction constants are
negative for all units whose separation distance is greater than
4. 493A, the numerical values of the elastic constants for second
neighbor interactions are smaller than those for first neighbor inter-
actions. Thus, it appears that the crystal gets weaker as one includes
higher neighbor interactions, but the fact is that,as one includes inter-
actions of all the surrounding units in an infinite matrix, the actual
value of the constant is obtained. However, the contribution of
neighbors higher than second neighbors is negligible :because as one
goes to larger distance, the force curve levels off and its slope
rapidly approaches a zero value. Thus, the value of the constants
obtained by includingi'nteractions up to second neighbors is very
close to their true value. .

If only first neighbor interactions are considered, cubic
crystals have no resistance to shear; it is only when second nearest-
neighbor interactions are included that shear resistance is introduced.
In the case of orthorhombic crystals, however, shear resistance in
all directions is present even when only first neighbor interactions

are considered and forces are assumed to be central. This is due to

75

the existence of a unit at the mid-point of the orthorhombic -lattice
rectangle (a, b), and the restrictions imposed on the central force
assumption in the direction of the chain axis.

The values of the constants C11’ C22. and C33 are calculated to

6, 0.288729 x 10'6 and 2. 422.665 x 1074 dyne/AZ.

be 0. 948127 x 10'
The numerical value of C33 is about 100-1000 times the value of C11
and c22 which is compatible with the ratio of the order of 50 for the
C-C bond dissociation energy of 83 kcal/mole (13) and the cohesive
energy of 1.736 kcal/mole (16). Also the value of E3 = 2.388100 x
10-4 dyne/A2 calculated from oil. (2.10) is in good agreement with
the observed value of E = 2. 6 x 10"4 dyne/A2 for oriented poly-
ethylene, obtained recently by Sakarda et al. (17). Shimanouchi

et a1. (17) have also calculated a value of E = 3. 4 x 10'4 dyne/Az
for infinitely-long oriented polyethylene molecules, which is much

higher than the experimentally-determined value.

The constants C44, CS5 and C66, which are measures of the

4 4

shear moduli, have numerical values of 1.123761 x 10' , 0. 886595 x 10-
and 6. 51551 5 x 10"7 dyne/AZ. No experimental value is available with
which these may be compared to draw any useful conclusion. However,
the values of the constants C44 and c55 are about 100 -200 times the
value of the constant C66. This is reasonable because the former two
involve movements of the units belonging to the same chain, and are
thus connected by stronger C-C bonds. Further, the values of these
moduli are lower than the corresponding E1, E2 and E3 values

which is true in general for shear and Young's moduli of bulk poly-

crystalline materials.

IV. C ONC LUSIONS

1. The constants A and B in the 6-12 Lennard-Jones

potential,

 

¢(r)= 12 -—%.
r

I‘

for forces existing between polyethylene chains in the crystal lattice
are found to be

A

8.48667 x10-7 erg/A12

and
1 O

B 2.177360 x10- erg/A6 .
They are of the same order of magnitude as those for argon, whose
lattice energy is of the same order as the cohesive energy of poly-
ethylene.

2. The ratio of the dissociation energy of the primary C-C
bonds and the cohesive energy of the secondary bonds is found to be
approximately fifty. Thus, the primary bonds, or intramolecular
forces, are about fifty times as strong as the secondary bonds, or
intermolecular forces, for polyethylene single crystals.

3. For second nearest-neighbor interactions, thirty interaction
constants have been found for polyethylene single crystals, as compared
to only five such constants for crystals having simple cubic symmetry.
For first neighbor interactions only, these constants become eleven
in number, as against two for cubic crystals. On application of a

limited central force assumption to the forces between units belonging

to different chains, the numbers of constants for second nearest-

76

77

neighbors have been reduced to fourteen for polyethylene single
crystals and two for cubic crystals. In the latter case, however,
no limitation is imposed on the central force assumption. The
corresponding numbers for the first neighbors become seven and
one.

4. Interaction constants for units belonging to the same chain
turn out to be 103 -105 times higher in value than those for units
belonging to different chains, the former being of the order of 1-3 x
10..3 dyne/A, whereas the latter have a magnitude ranging from
3 x10“6 to 9 x10.8 dyne/A.

5. Interaction constants for units having a separation distance
less than 4. 493 A are positive, while, for others having higher
separation distances they are negative. The magnitudes of the
positive constants are found to be approximately 300 times the
magnitudes of the negative constants for separations up to the most
distant second neighbor. For higher order neighbors this factor will
be still higher; but the true values of the elastic constants are
approached very rapidly. In fact, second nearest neighbor inter-
actions give values of the constants which are quite close to their
acutal values.

6. The value of the elastic constant c is found to be about

33
200-1000 times the values of the constants C11 and C22' These
constants are measures of the Young's moduli along their respective

axes. The exact value of the former is

C33 = 2. 42266 5 x10”4 dyne/AZ

78

which yields a value of Young's modulus of E3 = 2. 388100 x 10-4

dyne/Az that agrees with the observed value of Young's modulus for

4 dyne/A2 and is lower than the

oriented polyethylene of 2. 6 x 10-
value of 3. 4 x 10.4 dyne/AZ calculated by Shimanouchi (17).

Values of the constants c11 and c22 are found to be:

6 dyne/A2

c 0.948127x10-

11

6 dyne/ A2

C22 0. 288779 x10

These yield values of the Young's moduli E1 and E2 as 0. 377123 x 10"6

dyne/A2 and 0. 266161 x 10-6 dyne/A‘Z and, thus, seem quite reasonable
since the value of Young's modulus of bulk polyethylene is known to

be about 10"5 dyne/AZ. This fall squarely between the minimum value

6

of 0. 266161 x 10' dyne/Az calculated for E2 and the maximum value

of 2. 388100 x 10'4 dyne/AZ calculated for E3.

_ 7. The constants C44, C55 and C66 have been calculated to

be:
-4 2
644 = 1.123761 x10 dyne/A
-4 2
C55 _ 0.866595 x 10 dyne/A
-7 2
C66 _ 6.515515 x 10 dyne/A

Thus, polyethylene single crystals are found to have shear resistance
even when only first nearest -neighbor interactions are considered
and the central force assumption is applied. This is in direct
contrast to cubic crystals, for which shear resistance is introduced
by considering the second neighbor interactions. The shear moduli

C44 and css, which involve movements of units belonging to the same

79

chain, are found to be about 100-200 times the shear modulus C66,

which involves movements of units belonging to different chains.

81

Due to the lattice units being of unsymmetrical shape, the
forces between them are not quite central. It would be of considerable
interest to determine the effect of these forces on the elastic constants
of polyethylene single crystals. The interaction constants in such a
case may be evaluated by taking their geometry and location into
account.

Bulk polyethylene is made up of spherulites which consist of
randomly-oriented lamellae of single crystals. A future study could
be directed towards finding how the elastic properties of spherulites
are related to those of single crystals. This information may in turn
be related to the elastic properties of bulk polyethylene.

Investigations identical to the present work could be extended
to include polymers having lattice structures such as tetragonal,
hexagonal, monoclinic, triclinic, and rhombohedral. A comparison
of the inherent anisotropies of these materials due to their lattice
structures would then be possible.

In many crystalline polymers hydrogen bonding provides the
intermolecular forces, which are much stronger than the London
dispersion-type of van der Waals forces. An investigation of the
influence of hydrogen bonding on the elastic properties and, hence,

the anisotropy of such polymers would be of considerable interest.

81

Due to the lattice units being of unsymmetrical shape, the
forces between them are not quite central. It would be of considerable
interest to determine the effect of these forces on the elastic constants
of polyethylene single crystals. The interaction constants in such a
case may be evaluated by taking their geometry and location into
account.

Bulk polyethylene is made up of spherulites which consist of
randomly-oriented lamellae of single crystals. A future study could
be directed towards finding how the elastic properties of spherulites
are related to those of single crystals. This information may in turn
be related to the elastic properties of bulk polyethylene.

Investigations identical to the present work could be extended
to include polymers having lattice structures such as tetragonal,
hexagonal, monoclinic, triclinic, and rhombohedral. A comparison
of the inherent anisotropies of these materials due to their lattice
structures would then be possible.

In many crystalline polymers hydrogen bonding provides the
intermolecular forces, which are much stronger than the London
dispersion-type of van der Waals forces. An investigation of the
influence of hydrogen bonding on the elastic pr0perties and, hence,

the anisotropy of such polymers would be of considerable interest.

APPENDIX I

EVALUATION OF SERIES
In order to determine the Lennard-Jones 6-12 potential
constants A and B (Section 2. 2), it is required to evaluate the
triple series
so

AS : 2, m2: n: _m [(Zel)2 + (me2)2 + (ne3)z] '8/2 I-l

for s = 6 or 12, and where Am both even or both odd, but )4 0

and n = any integer.
As defined earlier ,
_ _3; _
el - 2 — 3.705 A
_ E _
e2 — 2 — 2. 470 A
_ E ._.
e3 — 2 2. 550 A
Dividing and multiplying by ef, I-l may be written as
_ e 2 e 2 —s/2
A =esz[22+mZ(—Z—) +n2(—:1)] , l-Z
s 1 e e
1 1
Substituting
e Z 2
2 _ 2.470 _
(131-) — (3.705) — 0.444
e 2 Z
3 2.555 _
(ET) " '3.'7"05') ‘ 0'4”
in I-2, we get
AS : e1"S 23 (22 + 0.444 m2 + 0.474 n'2)'-S/2 I-3
= efs A' a I-4
s

82

83

where

- 2
A; = 2 (£2 +0.444 m2 +0.474 n2) S/

with the same conditions on 2, m, n as in I-l.

This triple series of I-5 is separated into single, double and triple
series to find its value. These component series are evaluated as
follows:

Si_ngl£ Series

For m, n = 0, we have

so 00 _
2 2'S=2223=2a , 1-6
‘:—CD [:2 S
where
°° -s
a : E 2 for .8: even only
3 2:2
I-7
or
'00 l 21/
_ Z) __ _ -
aZU — n=1 (Zn for n — any integer and

s=2v=even .

Similarly for 2, n = 0, we have

°° 2)-s/2

gm (0.444 m 2 (0.444)'S/2 aS . 1-8

m

An exact method of finding the value of this single series is

given by Knopp (29), by which

00 21/
l _ V-l 1211'?
1:1 nzu ‘”'") 2.(2v): Bzu ' 1’9

where n is any integer, and B's are Bernoullian numbers. The

first few of these numbers are

 

_ _1 _1 _ 1 __1_ _ 691
36‘1'31‘2'32‘6'34‘ ”30' B6‘42' 1312 '2730 °
1-10
and
B3=B5= °=B2V+1:O for 2V+l:3.
Expression I-9 for odd terms only is
°° 1 v-l Q6)“ -11'2V
2 _fi = (‘1) 2 (2u)' 1'11
n=1 (Zn-l) ' '

This is obtained from I-9 by substracting the series for even terms

12 2v

_ 0° 1/ _ V -1 1r
as — Z (2;) — (-1) m B21! I-lZ
n=1
Setting 3 = 21/ = 6 in I-12 yields:
a - 3% l -( 1)2 "6 B I 13
_ _ - t -
6 n=1 (2n)6 2.(6). 6
Substituting 36 = 217 from 1-10 in 1-13, we get
a6 = 0. 015895916 . I-14
Similarly for s = 12, we have:
m 12
l 5 11'
a = Z —-— = (-1) —""'— B 1'15
12 n=1 (2n)12 2.(12)1 12
. . _ 691 .
Substituting B12 — - _2730 from I-10 in I-15, we get
a = 0.000244198 . I-16

12

85

Double Series

 

For n = 0, we have

so 2 2 -s/Z
E (2 + 0.444 m ) e
I, m: -°°
cn 2 Z -s/2
4 2(1 +0.444m) = 4bs. 1-17
I, m=2
where
00 -s/2
b = 23 (£2 + 0. 444 mg) for l, m = even only.
S I, m=2
I-18
Similarly for m = 0, we have
°° 2 2 '5/2
>3 (2 +0.474n) :4b' , 1-19
2, n: -°° s
where
°° 2 2 .8/2 for t = even only and
b' = z (2 +0.474n)
S 2:2, n=1 n 2 any integer;
I-20
and for l = 0, we have
°° 2 2 '5/2
E (0.444m +0.474n) 24b", I-21
m, n: -°° S
where
co 2 2 -s/Z
b" = 2 (0.44m +0.474n)
s
m=2,n=l
for m = even and
n = any integer.

I-22

86

The double series bs’ b's and by are evaluated by direct
summation, and to do this with reasonable accuracy all the terms

whose c ontributi on is

3
1
< — z =
(40) 0.000015 for s 6,
and is
1 6
< '46) z 0.00000000024 for s :12,

are neglected. Such a termination of these sums is justifiable because
of the fast convergence of the series. Though the number of terms
increases rapidly with the increasing values of l, m and n, the order
of error involved would still be small because, first, the size of the
polyethylene crystals is small (being finite) and, secondly, the
contribution of only the first few terms is significant as compared to
higher order terms.

Thus:

b6 = 0. 34911017055

0‘
ll

0.101079817615962

12

b'6 = 0.01782841530 1-23
bl2 2 0.0001156995141015

b" = 0.1201500514

b'l'Z = 0. 008143632999903

Trixie Series

We have

as -s/Z
2 (£2 + o. 444 m2 + o. 474 n2)
to ma 1'1: -0)

8c , I-24

where

Evaluating the sums cS

double series, we have:

(1‘2 + 0. 444 m

87

2 -s/2

2+0.474n)

for i, m

even and

11 any integer.

I-25

with the same order of accuracy as for

C6 = 0.20449870704
I-Z6
clz = 0.0209123979253897
Rewriting the A'S of I-5 as;
- 2
°° 2 2 2 S/
I _
AS _ ,m,n:_m(2 + 0.444m +0.474n)
00 ‘30 -s/Z
= 2 z: 2'5 + 2 2 (0.444m2)
2:2 m=2
co 2 Z -s/Z co 2 2 -S/Z
+4 23 (2 +0.444m) +4 2 (2 +0.474n)
f, m=2 .12, 11:1
°° 2 2 ‘8/2
+4 2: (0.444m +0.474n)
m=2,n=l
°° 2 2 2 '3/2
+8 2 (2 +0.444m +0.474n) . 1-27
,m=2,n=1
we have
-s/2
A'=2(l+0.444 )a +4(b +b'+b")+8c .
S S S S S S
I-28
Substituting
0.444“3 = 0.087528384 for s—

6

88

and

6

0.444" = 0.0076612 for 5 =12

in I-28, we have

A' = 2.175056768a

6 +4(b6+b'

6+bé‘)+8c6

6

and

._ I
A' - 2.0153224a +4(b12+b +b'1'2)+8c

12 12’

I-29

12

while substituting the numerical values of as, bs’ b's, b's' , cS in the
above, yields:

Aé = 3. 97935343732

1-30

'12 = 0. 6673191079229836

To obtain numerical values of the sums AS , these numerical values

of A; must be substituted. Thus, for s = 12 and 6

-12

A12 2 61 A12
and 1-31
._ '6 1
A6 — 61 A6 3
where
e1 = 3.705A
such that
el'l‘2 = 2, 587.95131"Z 14'12
and I-32
e'6 = 2, 587.95131'1 A'6,

89

Substituting I-32 in I-31, gives

A12 = 9.96372 x10.8 14'”

and I-33

A6 = 1.537646x10“3 A’6

Now, from 2. 43 ,

A12.6 : I"”‘12 'A6 '
where
p =-:- (r0)6 =—:— (4. 45)6 =% (7, 795.23947455)
= 3, 897.6197 A6 ; 1-34
therefore,
__ -3 -6
A _ -l.149298x10 A . 1-35

12-6

APPENDIX II

TRANSFORMATION OF PARTIAL DERIVATIVES
Let ci. be the direction cosines of the axes x', y', z', with

reapect to the axes x, y, z, as illustrated in the figure given bdow showing

only the x' axis.

 

 

 

 

 

Figure II-l. Rotation of axes.

Thus,
C11:C°S 0., c12=cos (3, c13=cosy II-l
such that
._. l l I
x—cllx +c21y _+c3lz
y = ch x' + c22 y' + C32 2' II-Z
_ i I l
z—cI3x +c23y +c33z

Differentiating II-2 with respect to x, y, 2, respectively, yields

3x _ 8 _ 32 _ .
a? " C11' . 35 “ °21’ ’63? " C31 ° H'3

90

91

If a displacement function q is given as

q = q(x, y, z) II-4
then
231 z 23 2:; 23 :31 9.4 23 -
and ex the + By 3x' + 8x the ° H 5
Substituting from II-3 into II-5 results in
9.9. = 9.9. 23 29. - -
the C11 3x “:21 8y +C31 az ' H 6
and differentiating II-6 again, we get
2 2 2 2
a 9 = _8__ 5’51 8 a. 3 a
,2 3x' (3):") ' c11 “11 + C21 axay " C31 3x3z)
3x 3x
2 2 2
a a a
+ C21 “11 8x3 + C21 3y l c31 ayaz)
829 829 82
+ c31 (C11 axaz + c21 ayaz + ('31 32%)
2 2 2 2
_ 2 a g 2 a 2 a g a 9
" c11 2 +821 ‘2' + c31 2 +2C11°218x8y
3x By 32
2 2
. . 3 L9.
+2 °11¢31 xaz + 2 °21°31 ayaz '

II-7

It is required to transform second partial derivatives of the

displacements with respect to the axes x12, x21; x13, x31; x23, x32;

x1 ,3, 1:31,; x1,,3, x31“ into partial derivatives with respect to x, y, 2.
In order to do so, the direction cosines of these axes are needed. They

are listed below along with the corresponding axes.

 

x31'

x1"3

x3111

 

 

 

 

 

 

 

 

 

 

 

( a . -—P-— . 0 )
Va2+b2 .1542
( ' a 9 b i 0 )

Vaz+bz «I 2+bZ
( a . o . C )
N/a2+c N/az +cz
( ' a a 0 9 C )

Va2+cz N/a
( o . b . ° )

N/bz+c2 Vbz Z
( O , _ b c )
N/bz+c2 «I b

( a b c )
Va2+bz+4c2 Klaz +b 2+4c2 Va2+bz+4cz
( _ a. _ b C )

N/a2+b2 +4c2 Vaz+bz +4cZ N/a2+b2 +4c2
(- a b 2C )

Va2+b2+4c2 Vaz+bz+4cz ~la2 +b z+4eZ
( a. - b 2C )

v 2 Z 2 ’ ’
a “D +4° ~laz+bz +4e2 Va2+b2+4c2

II-8

93

Thus, for differentiation with respect to the x12 axis, we have

.33 -23-_2_§_i+_‘2__251

3x - ax' '-
12 N/az+bz Vaz+bz

and differentiating again

 

 

 

 

2
9.9..=_L(§_9_)._.__a__(_2__9_3+____
8x2 ax' 3’“ 2 2 2 2 8x2

12 «la +b N/a +b
b a all
+ (~—
2 2 2 2 3""
N/a +b N/a +b
01'
2 2 32 2
—§-3 = a2 —‘216 +Zab-g—9— + b2 __g_6 2 )
3x12 a 2+b 8x 8y
Similarly for the ax1s x21
2 2 2 2
a _ 1 2 a a 2 a
+_ 22 (a—§-2abaa+b —29-)
£3le a +b 3x 3y
Adding II 9 and II 10 yields
029 + 629 _ 1 2 82 +21)2 82 )
2 " 2 2
6x12 8x21 a +b 3x By

and subtracting II-lO from II-9 gives

2 2
89 _ 3 _ 4 ab 6
2 2c _ 2 2 5x5‘y '
3x12 8x21 a +b

Repeating the above operation for the other sets, we have

2 Z Z Z
3 3 g 2 3 Z 3 9
3x 3x a 2+b 3x 82

8x 3y

b 32
2 2 5‘33’)
Va +b
b 829 )
2 a 7‘
a +b y
11-9
. 11-10
, 11-11
11-12
11-13

 

94

 

8 8 _ 4ac 8 _
f2" ' “—13 2 - 7:2 82 ' ”'4
x13 x31 3
2 2 2 2
- 2
§_§_+§_g_=-%——2(2b923 ,+2c a ), 11-15
8x23 8x32 b +c " ' 8y 82
2 2 2
1%.. .. 9.1L = .2422. _3_9_ ; II-16
8 8 2 b +2 8y82
"23 x32 C
2 2 2 2 2
8x13 8x31, +b +4e 8x Y 8y
2 32
+ 8 —9-2 ), 1117
82
82 32 1 82 82
firm—im (8a°8719“+8b°8'%’)' 11""
a a +b +4 x z Y z
Xl,3 X31, a C
2 2 2 2 2
a +——9—a =———————2 12 2(2a2__382 -8ab-g;%—+2bza
8x1”3 8x231” a +b +4c 8x y 8y
2
+8cz a—-g-) , II-l9
82
829 829 132
a2 " 2 zTT‘T"8a°8xaz ”“8“???" “'20
1:1,,3 8x31” a +b +4c
2 a2 a2 1 2 a2
3% +_2q_ + _.9._ +§_9_ = (4a
a a a 2 a 2+b2+4 Z a
x1'3 x31' x1H3 x31H a C Y
2 2
+4bZ9—% +1667“? ).
8y 82
11-21
2 2 2 2 2 ’
a _89 + 39 ___9._3 =——————' (16bc——‘1-a ).
2 2 2 2 2 2 8y8z
8x1,3 8x31, 8x1”3 8x31” a +b +4c

II-22

- 1 . .‘...u_\?-"'

 

 

95

Equations II-9 through II-22 are the transformation. relations utilized

in deriving the expressions 2.63 -65 from 2.60-62-

 

_‘ .-. r1'.n:5.."
b
.

I] 7..

APPENDIX III

STRAIN ENERGY CONNECTION BETWEEN THE ELASTIC AND
INTERACTION CONSTANTS

In Section 2. 3, the elastic constants were obtained in terms
of the interaction constants from a comparison of the equations of
motion derived from continuum and disc ontinuum theories. Second
nearest-neighbor interactions were considered; and the expressions !
thus obtained were simplified for first nearest-neighbors only and a ,
limited type of central force assumption. Expressions identical to }
these (2. 78) may also be obtained from strain energy considerations.
In this appendix an expression for the strain energy is derived by
considering motions of the first nearest-neighbor lattice units relative
to the central unit. This is then compared with the corresponding
continuum Expressions 2. 22: of the Section 2.1.

First nearest-neighbors surrounding a central unit (0, 0, 0)
are shown in Figure 2. 5 in Section 2. 3. Using the same notations
as were used for the secondnearest-neighbors, the strain energy
per lattice unit uI in terms of displacements u, v, w and the

interaction constants, under the central force assumption becomes:

u = -'—k '1 (u -u )2 +3.1. (v 1"
I 2 k100 100 000 2' 0k10 "010 “000
1 k11 1 k33
+‘2'k001(“001'“000) +2'k001'"001"’000' +2 k001‘w001'w000)
+-1-1<1 [(u1 )2 + (u -u ) ]
2 k1100 1100 u000 11100 000
1 k22 2
"2" 1.100 [("1100""000)Z + ("11100'"000' ]
111-1

96

97

By dividing each term of the above expression by the appropriate lattice

distance and using the well known finite difference relations we obtain:

1 2 k11 u2 1 2 k22 1 2
“1:2" k100(8x) "'2'" "010‘8YZ' *2 ° ["001('5?)
k22
“L" 001 (az'z + "001 (T) 2]
a+z 8v 2
+ ___:{k1100[('g:l_' )2+(ax212) ] +k1|zoo[(Taav 2)2+(‘5;;) ]}

III-2

Once again,by transforming differentiations with respect to x12 and

x21 into those with respect to x, y, 2, III -2 may be written as:

“1:2'a2"l00'ax'z"b2"121l0'a;2) +“210(111'522)
+ k001(82 v2) +"001(6):; ”)2”
+-l-{kl [a —-:—+b + b 8“2]
8 "1100("%’ 8')2 “it" y)
. 2
”(lizoo[(a%—b%1)2+(a%§'b%yi)]}

III-3

which may rewritten by collecting coefficients of the respective

partial derivatives as:

___ k11 1k 2 11 (:2 k11
“13‘2"100+4"1100"ax‘y) +71?" "1100 (a) "'2' "001%?2
1 2 22 k22 1 1 c2 k22
+4" "1100('5;t')Z “’2 (2' "010"? 1100) (15"'2c "8001(2‘3'

1 2 k33

+ 2 C "001 "67’2111'4

 

98

To obtain an expression for the strain energy density, let us multiply
both sides of the above by the number of units per cell and divide by

the volume of the unit cell. From Figure 2. 2, we have

8013-) + 2 Q) = 2 111—5

number of lattice units per cell

volume of the unit cell = abc .

Thus, the strain energy density U in terms of uI may be written as:

2 uI
abc HI - 6

 

U :
The corresponding expression from continuum theory is:

_ 1 2 r 2 2
U ‘ '2' (“1151 + c22'52 + c3363) + (“126162 + c13‘51'53 + °23€2‘3)

2

1 2 2
+ '2' (C4464 + C5565 + c6666)

which, on substitution for the values of Ej from 2. 5 and 6, yields:
_ 1 8u 2 8v 2 8w 7-
U - 2[°11 (8;) 1“sz5:," ”3318;”

+[c12 ($1 63%) + cl3 (3%) (133) + c23 (3%) ($21)]

8u2

2
l 8v 8w 8w 8v 8u2
”'2' °44'8‘2 "8;" + c55 Wilt)?" + C66 ('5;+§7) ] 111--7

Collecting coefficients, this may be rewritten as:

l 1 Bu 2 1
U = “___

8u2 8u2 1 av?-
2' C1195) +2°66 "2°55'52') +2°66"5§'

l 8 l 8v)2
2

2
v
+2"sz ('83?) *2 C44 (“8"

1 8w 2 1 8w 2 1 8w 2
"2655(7)? +2°441871 +2°33 (8'2"
213 '21 9.13 2111.. 9.1.1 211.11
+ c12 (8x) (By) J” C13 (axl (82) + <323 (8y) (82)

8v 8w 8w 8u 8v 8u

 

99

Comparing III-4 and III-8 by using III-6, we obtain the following

11

expressions for c.. in terms of k!
13 mm

11 ll

 

611 = 15% ("100 +'i’"1100'
C22 '"‘ 2": ("0:0 + 2":1200)
C33 = {CH- "001 a
(:44 = 2913 "301 °r “44 = 0
c55 = 3215 "(1101 °r (‘55 = 0

 

C 1.21.11 ._.1.
66 ' 2 ac 1'00" 2 bc 1'00

C12 = 0
‘213 -= 0
C23 = 0 0 111-9

Except for C12’ C13 and c23 these are identical to the Expression
2.78. Even redundancy in expressions for C44, c55 and C66 is same
as before. This provides a check on the two procedures. Such an
agreement will not, however, be obtained if non-central forces are
considered because different higher order terms are neglected and

included in the two cases.

APPENDIX IV

A NOTE ON THE LENNAR Dl-JONES POTENTIAL
Intermolecular forces are approximated by a 6-12 Lennard-

Jones potential,

 

 

(Mr) = l2 - —%— 2.26
1‘ 1‘
01'
1' . r .
¢(r) = e[:2( Tm)” wig—13f] . 1v-1

Both of these forms are listed by Hirschfelder, et a1 (21). The first

is used herein, and this is also the form employed by Born and

Huang (23). However, 'Peterlin, et a1 (30), McMahon and McCllOugh‘Gl)
use variations of the second. Further, they employ one potential along
the b-axis and another along the diagonal (a, b) axis, thus calculating
different values of rmin along these directions. This is probably done
because of the inherent properties of the orthorhombic lattice of poly-
ethylene:

(i) There is an additional unit at the midpoint of the rectangle

(a, b).

(ii) This unit is at a distance of -lz--1\/a2 + b2 , which is not
equal to b.

(iii) The unit has an orientation different from the corner
units.

Nevertheless, a closer study of the lattice structure of poly-
ethylene reveals that such difficulties may be resolved without using

different potential forms in different directions.

100

 

 

101

 

 

 

 

 

 

 

 

b
a
A
/
/ \
//r4=2.63A ‘\ '
/ \rl
/ \
l // b\\ x
/ r3 =./2.£3 r' \ \\\ \
/ [I \\ 171:2.63 \\~
/ \\ \
___- —r' - \ ._. -
\. \ r 2 63A
H \\2— °
1'3 \‘ '\ l
\\ \
V

 

Figure IV-l. Detailed positions of the hydrogen atoms in the
(a, b) plane.

 

102

Figure IV-l is a detailed drawing of a single cell of polyethylene
in the plane (a,b), showing the positions of the hydrogen atoms in
space relative to each other. It may be observed from this figure
that the separation distances r1, r2, r3 and r4 of the hydrogen
atoms are all equal. This suggests that:

(i) The geometry of the polyethylene molecules and their
location in space determine the lattice structure.

(ii) The London dispersion forces between hydrogen atoms
determine the Lennard-J ones potential such that the hydrogen atoms

are at equilibrium separation.

 

Thus, the net force between any two lattice units in a plane is a result
of the interaction of their hydrogen atoms. The potential between the

two nearest hydrogen atoms can be written as

¢'(r) = if? -

1'

IV-2

"04"?

where the ratio of A' and B' is determined by their equilibrium
distance r2) . The net potential 4) between the two units at the lattice

points is:
49(1') = 2[¢'(r1)+ ¢'(r2)+ 9'.(-r3) + ¢'(r4)l
+ 2 [ ¢'(ri) + ¢'(r'2) + 1l>'(r'3) + 11>'(1r11)]

= 8 [ 4111-10) + ¢1(r1)] Iv-3

10.

ll.

12.

13.

14.

15.

’ VI. BIBLIOGRAPHY

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31111111231

H. Geil, "Growth and Perfection of Crystals, " Editors: R. H.

Doremus, B. W. Roberts and D. Turnbull, Wiley, New York,

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H. B. Huntington, "The Elastic Constants of Crystals, " Academic
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103

 

16.
17.

18.
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20.

21.

22.

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29.

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104

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J. O. Hirschfelder, C. F. Curtiss, R. B. Bird, ”Molecular Theory
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C. K. Ingold, "Introduction to Structure in Organic Chemistry, "

 

G. Bell and Sons Ltd., London, 1956.
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2089 (1964).

 

IllllllfllflHflH

 

1804