1. Macromol. Theory Simul. 2002, 11, 513–524 513
Energy Elasticity of Tie Molecules in Semicrystalline
Polymers
Zdenko Sˇpitalskyy, Tomµsˇ Bleha,* Peter Cifra
Polymer Institute, Slovak Academy of Sciences, 842 36 Bratislava, Slovakia
Fax: (+421) 2 5477 5923; E-mail: upoltble@savba.sk
Keywords: conformational defects; force-length relations; molecular mechanics; polyethylene (PE); polymer simulations;
Introduction
The molecular description of elasticity of semicrystalline
polymers is of considerable importance due to the techni-
cal and biological applications of a variety of materials,
ranging from thermoplastics such as polyethylene to bio-
logical fibres such as spider silk. In recent years, consid-
erable effort has been put into overcoming the complexity
in the structural and morphological hierarchy of these
materials and to getting an insight into the deformation
mechanisms of semicrystalline polymers at the molecular
level using both theory and modelling.[1–6]
The structure
of semicrystalline polymers on a molecular scale can be
approximated as consisting of two phases: (a) crystalline
regions and (b) disordered quasi-amorphous interlamellar
(IL) regions. Crystal regions typically consist of crystal
lamellae formed by regular or irregular folding of chains.
Within the IL regions four types of molecules are present
(Figure 1): (a) tails with one free end; (b) loops, which
start and end in the same lamella; (c) bridges (tie mole-
cules) which join up two lamellae and (d) floating mole-
cules which are unattached to any lamellae. Elastic prop-
erties of the crystalline phase can be predicted fairly
Full Paper: Elastic response of the disordered phase
between crystal lamellae in semicrystalline polymers is
modelled on the assumption that the stress is transferred
by bridging (tie) molecules. The deformation characteris-
tics of short poly(methylene) (PM) bridges were com-
puted by using two methods: (a) the single-molecule load-
ing by molecular mechanics (MM) calculations and (b)
the chain-ensemble averaging by lattice simulations. The
energy elastic functions ensuing from both methods differ
considerably. In MM the loading of chains containing
numerous gauche defects by an external force F yields the
sawtooth-like profile of the force (F)–length (R) functions
brought about by the stress-induced gauche–trans confor-
mational transitions. The Young's moduli E of PM chains
containing several gauche defects can be less than 1% of
the all-trans value ET; by elimination of the defects the
moduli steeply increase. In contrast, the ensemble-aver-
aging approach gives a smooth increase of the (positive)
elastic force f with chain length R and a decrease of the
(negative) energy component of the elastic force fU with
R. Both energy deformation mechanisms, single-chain
loading (by F) and statistical (by fU), are complementary
and can simultaneously be operative in the interlamellar
(IL) phase. Their proportion in the stretching process
should depend on the chain mobility and structural homo-
geneity (history) of the sample, particularly on the pre-
sence of the so-called rigid amorphous fraction in the IL
phase.
Macromol. Theory Simul. 2002, 11, No. 5 i WILEY-VCH Verlag GmbH, 69469 Weinheim 2002 1022-1344/2002/0506–0513$17.50+.50/0
A sketch of different types of chains in the interlamellar
phase: tie molecules (full lines), loops (dashed), free ends
(dotted) and floating chains (dash-dotted lines).

2. 514 Z. Sˇpitalskyy, T. Bleha, P. Cifra
well.[7]
However, the response of the IL region to
mechanical load is much less understood. Clearly,
mechanical properties of the IL phase depend on the dis-
tribution of chains into all four categories. Still, it is
believed that tie chains bridging the crystal lamellae and
threading the disordered IL, is crucial for the transfer of
stress between crystal lamellae. Tie chains may effec-
tively also include the entangled loops which permit an
indirect transmission of force between the lamellae.
Thus, the modelling of deformation of semicrystalline
polymers is in large part focused on the elastic properties
of tie chains in the disordered IL phase.
Two (complementary) approaches can be envisaged in
modelling the elasticity of tie molecules: (a) the energy-
elastic deformation of a representative set of individual
conformers of tie molecules can be examined and, by a
proper averaging of the data, the elastic properties of the
IL phase can be estimated and (b) the force–distance rela-
tionship can be calculated by statistical-mechanics aver-
aging of all allowed conformations of tie molecules in a
similar way as employed in the statistical theory of rubber
elasticity.
In the first approach, the static potential energy U of
selected tie chain models stretched by an external axial
force F is computed. The energetics of axial mechanical
loading of the highly extended poly(methylene) (PM)
chains containing kink conformational defects was exam-
ined by Kausch.[3]
He observed the interconversion of the
defect structures on stretching to more extended confor-
mations, via smooth transitions, until all conformational
defects were eliminated. Later, a similar problem was
addressed by molecular mechanics (MM) (force-field)
calculations of the deformation potential in the single-
defect chains.[8–10]
These calculations suggested that the
transition of a defect into a more extended conformation
proceeds by an abrupt change of the deformation poten-
tial U(R). A flip-flop interconversion of the torsional
angles within a defect brings about the change from the
shorter (kink) into the longer (all-trans) state[8–11]
and lib-
erates the stored elastic energy. In our recent paper[12]
the
force–length functions F(R) and Young’s moduli E of tie
chain models containing several conformational defects
were reported. Abrupt discontinuities due to gauche-to-
trans conformational transitions were established on the
force–length curves F(R) as well. The sequential annihi-
lation of the chain defects on stretching resulted in a saw-
tooth-like profile of the F(R) curve. It was suggested that
the sawtooth-like profile is a common feature of mechan-
ochemistry of bridging polymers with a restricted number
of conformations or in biopolymers where compact
domains unfold on stretching.
In the second approach, using the statistical treatment of
elasticity of tie molecules, the force (f)–length (R) rela-
tions for single linear macromolecules are calculated from
the configuration free energy Ael averaged over all avail-
able conformations of tie chains. To differentiate between
the two approaches, the statistical force is denoted by f in
contrast to the force F of loading of individual conformers.
The elastic Helmholtz energy Ael is related[13–15]
to the co-
ordinate probability density function W(R) of a macromo-
lecule having one end fixed at the origin and the other end
fixed in space at the distance R
Ael = AR –Af = –kT ln W(R) (1)
where AR and Af are the mean configurational Helmholtz
energy of constrained and free chains, k is the Boltzmann
constant and T is the temperature. The force f is deter-
mined by the differentiation f = dAel/dR; the direction of
the force f is collinear with the chain vector R.
In ideal chains, the distribution W(R) is approximated by
the Gaussian function[13]
and the corresponding force fG ori-
ginates solely from the entropy factors. In real chains, the
non-Gaussian distribution functions W(R) can be deduced
from Monte Carlo (MC) simulations, by taking into
account factors such as the finite chain length, excluded
volume, solvent quality, etc.[13–16]
These real-chain fea-
tures give rise to the internal energy changes at stretching
Uel and to the non-zero energy contribution fU to the elastic
force f.[14, 17]
In experiment, the energy contribution fU can
be determined from the force–temperature (“thermoelas-
tic”) measurements on polymer networks.[13]
The magni-
tude of the term fU is an important indicator of the prevail-
ing molecular mechanism of chain stretching and thus fU
should be zero in ideal chains and ideal elastomers.
The spatial configurations of the chains in melts or in
amorphous networks are regarded as independent of
Figure 1. A sketch of different types of chains in the interla-
mellar phase: tie molecules (full lines), loops (dashed), free ends
(dotted) and floating chains (dash-dotted lines).

3. Energy Elasticity of Tie Molecules in Semicrystalline Polymers 515
intermolecular interactions and are described by the con-
formational statistics of a single chain in the theta sol-
vent. Additionally, according to one of the major assump-
tions in the molecular theory of elasticity[13]
intermolecu-
lar interaction should not depend on degree of the chain
deformation. Under these circumstances intermolecular
interactions should not contribute to the elastic energy Uel
and to the energy contribution to force fU in the amor-
phous state. In reality, the conformational statistics of tie
chains in the semicrystalline polymers may differ from
that appropriate for the ideal amorphous phase. Still, the
single-chain MC simulations provide a useful limiting
description of the mechanics of tie chains in a purely
amorphous IL phase. The quasi-static calculations of the
single-chain loading can furnish the contrasting portrayal
of this mechanics in the other limit of vanishing entropy
contribution at stretching.
The main objective of the present paper is to illuminate
the relation between the above two types of the energy-
elastic response of tie molecules, i.e., between the F–R
functions describing the individual tie chain loading and
the fU–R functions from statistical ensemble considera-
tions. These functions are calculated using the MM and
MC methods, respectively, for the chain segments rele-
vant to the disordered IL phase of semicrystalline poly-
ethylene (PE). The molecules investigated by the MM
method contain several uncorrelated gauche defects and
should model the “slack” bridges. The data for these
models supplement the results from a related study[12]
focused on the highly extended tie chains involving the
correlated defects of the kink or jog type. The tetrahedral
(diamond) lattice MC simulations of the elastic properties
of short PM chains were performed under conditions clo-
sely matching the MM computations. The variation of the
statistical force f and of its energy component fU by dis-
placement R was evaluated.
Methods and Models
MM Calculations
The potential energy of stretched molecules was calcu-
lated by the Allinger MM+ molecular-mechanics
method[18]
by the procedure described in the previous
paper.[12]
The total static potential energy of a molecule U
is expressed in the method as the sum of several contribu-
tions
U = Ur + Uh + Ub + UvdW (2)
where the terms Ur and Uh represent the bond length and
bond angle deformation, respectively, Ub is the inherent
ethane-like torsional potential respecting the cosine type
periodicity of the torsional angle b. The term UvdW is a
summation of all non-bonded pair interactions in the
molecule. The individual energy terms in MM + methods
are expressed by simple analytical functions involving
numerous adjustable parameters. The parameters, specific
for a given class of compounds, were determined[18]
by
fitting an extensive set of experimental data and opti-
mised to give the best performance of the method. For
long-chain alkanes the MM+ method provides reliable
predictions of the structural and thermodynamic data at
ambient temperature.
The PM tie molecules were modelled by n-eicosane
with 20 carbon atoms in the backbone. The distance of
terminal carbon atoms C1–C20 in n-eicosane was regarded
as the length of a molecule, R. A C20 molecule involving
the conformational defects was stretched by a gradual
increase in R. The energy of a stretched defect molecule,
almost rigidly constrained at a given R, is optimised and
the equilibrium static energy U and the geometry param-
eters (torsional angles bi, bond angles hi and bond lengths
ri) are obtained. The implicit deformation force F is colli-
near with the vector of the end-to-end distance R. A few
calculations performed for longer chains, of up to C40,
corroborate the consistency of the C20 data.
MC Simulations
The simulation procedure used is described in more detail
in previous papers.[14–16]
The chains formed by N = 20–
100 beads (i.e., of the length N-1 segments) were gener-
ated on a tetrahedral lattice. The athermal model with
zero value of the reduced energy of intrachain segment–
segment contacts eS = eij /kT was assumed. The conforma-
tional parameter eg = DUtg /kT, related to the gauche–
trans energy difference DUtg in alkanes, was assigned the
value 1.5, and represented the PM-like chains at ambient
temperature T. Conformations g+
g–
and g–
g+
(the “pentane
effect”) in the chain backbone were completely sup-
pressed. The chain thickness was accounted for explicitly
by forbidding the occupation of the first neighbour sites
on a lattice. The reptation movement of chains on a lattice
and usual Metropolis energy expression were employed.
The radial distribution function of the end-to-end distance
P(R) = 4pR2
W(R) was computed by grouping the similar
chain vector lengths into a histogram with the interval of
one tetrahedral lattice spacing a. The elastic Helmholtz
energy Ael was calculated by using the Equation (1) and
by its polynomial fitting and differentiation the elastic
force f = dAel/dR was determined. The elastic internal
energy Uel of a molecule is given by the difference
between the internal energy of a chain restricted by the
condition of fixed R and of a free chain[14]
Uel = kT Dng eg (3)
where Dn = pngRP – pngP is the change on stretching in the
average number of gauche bonds between a stretched
chain with the fixed R and a free chain. The energy com-

4. 516 Z. Sˇpitalskyy, T. Bleha, P. Cifra
ponent of the force is given by the differentiation fU =
dUel/dR.
Results and Discussion
Single-Conformer Calculations
In quasi-static MM calculations the length of model
chains is conveniently expressed relative to the length of
all-trans (T form) zigzag Rzz = 2.422 nm (the contour
length). An insertion of the conformational defects into
the T form of C20 decreases the length and increases the
potential energy of a molecule. The length of a molecule
relative to the T form can be expressed either by the dif-
ference DR = Rzz–Rd, or by the ratio x = Rd/Rzz, where Rd
is the length of a defect molecule. Our previous study[12]
was focused on the elasticity of the highly extended PM
bridges involving several three-bond sequences g+
tg–
(the
kink defects) or five-bond sequences g+
tttg–
(the jog
defects). Such molecules with the correlated gauche pairs
in the defects served as models of tight PM bridges of
rather high chain extension ratio x A 0.85. In the present
study, the assortment of models is broadened to include
uncorrelated gauche defects forming more loose, “slack”
bridges with x A 0.59. The investigated defect chains are
designated as nG according to the number of gauche
bonds involved (Table 1). For comparison the data for the
relevant kink structures 1Ka and 2Ka from the previous
paper[12]
are included in Table 1.
The potential energy of a defect chain Ud represents the
static conformational energy at 298 K without the vibra-
tion contributions and is expressed relative to the energy
of the T form (51.79 kJ N mol–1
). As seen from Table 1
the energy Ud is roughly proportional to the number of
gauche bonds in a molecule.
Deformation Potential U(R) of the Defect Chains
At first, the deformation energetics of several molecules
containing two gauche bonds of the opposite sign was
compared (Figure 2). The deformation potentials U(R) of
molecules of the type 2G display the same energy discon-
tinuities as observed[12]
for a kink or a jog: at the critical
strain ec = (R–Rd)/Rd, the defect chains undergo the tran-
sition into the T form and an abrupt decrease of the
energy is observed (Figure 2b). At the critical strain ec
the length of stretched defect molecules becomes com-
parable to the length of the T form. The transition occurs
from a hypersurface of the deformed defect into a hyper-
surface of the T form and the accumulated elastic energy
DUaccu is set free. Thus, when the energy DUaccu =
24.5 kJ N mol–1
, almost entirely due to bond stretching
and bond angle bending, is released at annihilation of the
2Ga defect in Figure 2. In contrast to the kink defect, all
other 2G defects undergo transitions ahead of reaching
the equilibrium length Rzz of the zigzag (corresponding to
Ud = 0). In all 2G defects two gauche angles are changed
in a concerted way: at first their values continuously
change by about 308 from the gauche equilibrium values,
then in the vicinity of the transition point they abruptly
change into 1808. The fully extended zigzag undergoes
the energy-elastic deformation (through ri and hi vari-
Table 1. Deformation parameters of the PM chains with the gauche defects.
Defect Sequence DR
nm
x P2 DUd
kJ N molÀ1
DUlib
aÞ
kJ N molÀ1
F1
bÞ
10À10 N
e = 1%
E
GPa
E/ET
T – – 1.000 0.927 0.00c)
– 3.16 175.42 1.00
1Ga ttttttttg-tttttttt 0.269 0.889 0.683 3.708 17.313 0.16 8.696 0.05
1Gb tg+ttttttttttttttt 0.795 0.967 0.881 3.840 14.572 1.47 81.852 0.46
1Ka tttttttg+
tg–
ttttttt 0.782 0.968 0.864 8.11 41.65 2.18 121.1 0.69
2Ga ttttttg+tttg-tttttt 0.166 0.931 0.764 7.447 31.986 0.87 48.134 0.27
2Gb tttg+tttttttttg+ttt 0.258 0.893 0.644 7.519 35.024 0.40 21.946 0.12
2Gc ttttg+tttttttg-tttt 0.254 0.895 0.657 7.426 29.039 0.42 23.338 0.13
2Gd ttg+tttttttttttg-tt 0.255 0.895 0.699 7.481 29.869 0.43 23.792 0.13
2Ge g+tttttttttttttttg- 0.169 0.930 0.885 7.351 31.387 0.95 52.843 0.30
3Ga tg+ttttttg-ttttttg+t 0.585 0.759 0.421 11.340 22.509 0.05 2.932 0.02
3Gb tttg+ttttg-ttttg+ttt 0.919 0.621 0.269 11.104 18.930 0.02 1.269 0.007
3Gc tg+tttttg+tttg+ttttt 0.278 0.885 0.676 11.255 47.265 0.25 13.815 0.07
3Gd tt(g+tg-)tttttttg+tttt 0.256 0.894 0.712 11.831 19.495 0.20 11.305 0.06
2Ka ttg+
tg–
tttttttg+
tg–
tt 1.643 0.932 0.817 16.22 71.51 1.58 88.0 0.50
4Ga g+tttg-tttttttg+tttg- 0.258 0.893 0.724 14.832 45.895 0.89 49.699 0.28
4Gb tg+tttttg+tttg+ttg-tt 0.565 0.766 0.502 14.909 18.217 0.05 2.898 0.02
5G tg+ttg-ttg+tttg+ttg-tt 0.989 0.591 0.103 18.446 17.403 0.01 0.697 0.004
a)
The energy liberated at the first conformational transition.
b)
The force needed to bring about a 1% elongation.
c)
Static potential energy U = 51.79 kJ N mol–1
.

5. Energy Elasticity of Tie Molecules in Semicrystalline Polymers 517
ables) also above the contour length Rzz, up to the chain
fracture. The energy liberated at the transitions Ulib, given
as the sum of the accumulated elastic energy DUaccu and
the difference of the defect static energies DUd, is listed
in Table 1.
For a given number of gauche bonds, their location
within a molecule and the sign of gauche rotation affect
to a great deal the chain length and the overall shape of
the molecule. This observation is illustrated in Figure 3a
for three different defects of the 3G type. The deforma-
tion potentials U(R) in Figure 3b show that the energy
changes on stretching become more gradual for coiled
defect chains of smaller chain extension ratio x. Interest-
ingly, at the first transition, the 3Ga and 3Gc defects
transform to the 1G structure while the 3Gb defect trans-
forms to the 2G structure. On the second transition the T
form is produced from these structures.
The deformation response of the defect chains should
be related to the mutual orientation of the force F and the
defect bonds. In looking for such a correlation the orien-
tation function P2 defined as
P2 = (3pcos2
hPav – 1)/2 (4)
was used, where cosh = ai N b is given by the dot product
of vectors. The vectors ai are specified by co-ordinates of
the first and fourth carbon atoms in the chain backbone
(xj+3–xj; yj+3–yj; zj+3–zj) for each torsional angles bi. The
chain-end vector b(x20–x1; y20–y1; z20–z1) defines the
orientation of force F. In a C20 molecule an averaging
over 17 backbone torsional angles bi is carried out. P2 = 1
Figure 2. Molecules with two gauche defects: the projection of
different structures (a), the deformation potentials (b) and
force–length functions (c).
Figure 3. Molecules with three gauche defects: the projection
of different structures and their orientation factor P2 (a) and the
deformation potentials involving two interconversions (b).

6. 518 Z. Sˇpitalskyy, T. Bleha, P. Cifra
at full parallel or antiparallel alignment of vectors. The
connection between the stretching energy data and the
orientation function P2 is readily seen for the 3G type
defects (Figure 3). In a loop-forming structure 3Gb the
axes of two (of the three) gauche bonds are oriented
nearly perpendicularly to the chain-end vector b and thus
the P2 value is low. In contrast, in a fairly extended 3Gc
structure, the gauche bond axes and vector b are more
aligned and a higher P2 value applies.
Stretching of molecules containing multiple gauche
bonds results in the U(R) potentials with numerous
gauche–trans transitions. For example, the deformation
of a 5G defect chain proceeds by a “reaction” path
sketched in Figure 4a. The corresponding deformation
potential (Figure 4b) displays individual transitions by
which the gauche defects are sequentially annihilated.
Force–Length Relationships
The force–length functions F(R), obtained by differentia-
tion according to R of the fitted deformation potentials
U(R), feature an unusual, discontinuous quality (Figure
2c and 4c). The F(R) curves are separated into several
portions by the sudden drops in force. The portions of
F(R)curves become steeper after each elimination of a
defect. In the 2G chains the individual portions of the
F(R) curves are nearly linear (Figure 2c), whereas in the
5G chain (Figure 4c) the deviations from linearity are
evident. The force at the transition points, Fc, represents
the maximum load, which a chain can bear prior to
“yielding” by a conformational interconversion.
Superposition of F–R functions of several gauche
defect chains shows that stretching of tie molecules fol-
low a common pattern with two distinct regions of elastic
response (Figure 5). The minor forces, up to about
0.2 nN, produce a fairly sizeable stretching of slack
bridges in the region of moderate chain extension ratio
0.6 a x a 0.85. The high extension region, 0.85 a x a 1
corresponds to the tight bridges. In this region the maxi-
mum force Fc, which the chain can bear before a flip-flop
jumps occur, gradually increases from about 0.6 to about
0.9 nN. The resulting shape of the F(R) function in Figure
5 bears a resemblance to the sawtooth-like pattern found
for the tight kink bridges.[12]
The qualitative differences
in elastic response of slack and tight bridges are apparent
also from the force F1 needed to extend a defect chain by
1% (Table 1). The force F1 is of the order of several pN
for slack bridges and in the range of tenths and hundreds
pN for tight bridges.
The computed F–R data of defect chains should be
linked to the observation[19]
that force-driven conforma-
tional transitions are controlled by a mechanical torque
generated. The torque is defined as the cross vector pro-
duct of the force vector F and the lever arm vector h.
Vector h specifies the distance from the axis of rotation
i.e., from the interconverting gauche bond in our case. In
structures where the orientation of the force and lever
arm vectors is similar, such as in the kink structures 1Ka
or 2Ka, the torque is small and thus rather high forces, Fc
(up to 1.1 nN), are needed to achieve the transition.[12]
In
Figure 4. Deformation of a chain with five gauche bonds:
sketch of the reaction path (a), the deformation potential (b) and
force–length functions (c). An analogy (without averaging) to
the variation of the conformational energy Uel with R (b) and to
the variation of the term fU with R (c) is drawn by solid lines.

7. Energy Elasticity of Tie Molecules in Semicrystalline Polymers 519
contrast, in some molecules with the uncorrelated gauche
defects, the vectors F and h are oriented almost perpendi-
cularly, producing thus the large torque. Consequently,
the force Fc required to achieve the transition is smaller
in such a case. The orientation function P2 defined by
Equation (4) partially accounts for the variations of tor-
que in defect molecules.
Discontinuous stress–strain functions r(e) of the defect
chains can be computed from the F(R) functions by
assuming A = 0.18 nm2
for the defect chain cross-section
in the axial chain stress r = F/A. This value of A corre-
sponds to the zigzag PM chains packed in an orthorhom-
bic crystal.The cross-section A may be slightly higher in
molecules containing numerous gauche defects, however,
we used the above mentioned value of A to facilitate the
comparison with previous results.[12]
The initial slopes of
r(e) curves at e = 0 define the longitudinal Young’s mod-
uli E of bridging PM chains (Table 1).
The modulus E is proportional to the second derivative
of the deformation potential U(R), i.e., it depends on the
steepness of “walls” in the U(R) potential. Thus, the elas-
tic response of macromolecules is controlled by the tor-
sional stiffness of the gauche bonds rather than by the
height of appropriate energy barriers. In the multidefect
chains the deformation potentials U(R) in Figure 2–4
become steeper on stretching, i.e., the torsional stiffness
of interconverting bonds increases.
Influence of a defect on the chain stiffness is conveni-
ently expressed by the ratio E/ET of the defect chain mod-
ulus to the modulus ET of the reference T form. A marked
reduction of the relative Young’s modulus E/ET with the
concentration of the gauche defects in molecules is evi-
dent from the Table 1. The relative stiffness of defect
chains given by the ratio E/ET iscorrelated in Figure 6
with the orientation parameter P2. Besides the data in
Table 1, the results[12]
for the kink and jog structures are
also included in this plot. The defect molecules of the
small moduli, rather insensitive to the P2 values, are
located in the region of low P2 (P2 a 0.5). On the other
hand, in the region of high P2 (A0.75), the relative stiff-
ness of defect molecules steeply increases. Evidently, the
stiffness of a PM chain can be significantly reduced by an
incorporation of uncorrelated gauche defects. This reduc-
tion is much higher than the diminution of stiffness due
to presence of the kink defects.[12]
Evidently, the kinks
represent the mechanically stiffer elements than uncorre-
lated gauche defects. Usually the kink defect chains are
presumed to prevail in the more ordered IL phase while
the loose ties with uncorrelated gauche defects should be
more typical for the disordered regions of the IL phase.
Ensemble-Averaged Simulations
In order to match the MM treatment the conformational
energy DUtg = 3.7 kJ N mol–1
(eg = 1.5 at 298 K) and lat-
tice spacing a = 0.153 nm was assumed in MC simula-
tions. Using these parameters, the self-avoiding walks
generated on a tetrahedral lattice supply the ensemble-
averaged properties of short PM chains at 298 K. Some
calculated dimensional and conformational characteris-
tics of the PM-like chains are listed in Table 2. Here, the
chain extension ratio x = R/Rzz and the contour length Rzz
is the ultimate length a chain (of given N) can achieve,
since no energy deformation of the zigzag structure is
allowed in MC simulations. Similarly, the extension ratio
of a random coil of mean dimensions pR2
P1/2
can be
defined by xc = pR2
P1/2
/Rzz. A gradual development of
chain-coiling with an increase in N is readily seen in
Table 2. As expected, in the short chains N = 20 the coil
dimensions are represented by a large fraction of the con-
tour length (71.5%). The pngP values give the average
number of the gauche bonds in the free chains of dimen-
sions pR2
P1/2
. For N = 20 it means that about four gauche
(defect) bonds should be present on average in the defect
chain structures which have the extension ratio x around
0.715.
Figure 5. The force–length functions F(R) of two PM chains
involving several defects.
Figure 6. Variation of the defect chain moduli with the orienta-
tion factor P2 as defined by Equation (4).

8. 520 Z. Sˇpitalskyy, T. Bleha, P. Cifra
The elastic Helmholtz energy Ael and the elastic inter-
nal energy Uel were calculated by using the Equation (1)
and (3), respectively, for chains of the length N = 20–
100 as a function of the chain extension ratio x (Figure
7). Due to the focus on the fairly and highly extended
bridging molecules, the region x A 0.7 is of primary inter-
est. The energy term Uel results solely from conforma-
tional isomerisation on stretching. Since the number of
the low-energy trans bonds in PM chains increases on
stretching, the conformational term Uel diminishes with
increasing x.[14]
The term Uel is positive in the region of
compressed coils below xc = pR2
P1/2
/Rzz in Figure 7; above
this value of xc the energy Uel becomes negative. By sub-
tracting Uel from the Ael term the entropy term –TSel can
be evaluated. Again, the entropy Sel is defined, similarly
to other thermodynamic functions, as the difference of
entropy of stretched and free chains, SR–Sf. The entropy
factor prevails over the energy factor in Figure 7 and the
Helmholtz energy penalty Ael increases with x. Only in
very short chains N = 20, a shallow minimum is present
on the Ael curve, as noted previously.[20]
By an increase of
the chain length, the onset of upturns in the plots of Ael,
and downturns in the plots of Uel, shifts to smaller x.
The statistical force f and its energy component fU are
defined by the differentiation according to R of the Helm-
holtz energy Ael and the conformational energy Uel,
respectively. The remaining part of the force fS = f–fU is
entropic in nature. In this manner in Figure 8, the respec-
tive force–length relations are computed from the curves
in Figure 7. The reduced total elastic force fRzz/kT (and its
energy part fURzz /kT) smoothly increases (decreases) with
x to their limiting values at x = 1. A very low representa-
tion of the highly extended forms among the chains gen-
erated and the resulting poor statistics disallowed to reach
the full extension limit x = 1 in the case of N = 100 in
computations.
The restoring force f in Figure 8 shows a higher-than-
linear rise at higher chain extensions, typical for the non-
Gaussian chain models.[13–17]
A positive force f corre-
sponds to the chains in tension. Negative force calculated
in Figure 8 in the zone below xc indicates that the mole-
cules are in compression. Rigorously, the statistical force
f in Figure 8 pertains to the PM chains constrained at their
end-points whereas in the IL phase tie molecules are con-
strained by the lamellar planes. The difference in force f
between these two types of constraints[20]
diminishes with
increasing x and can be neglected at higher extensions.
The energy term fU due to conformational isomerisation
of PM chains plotted in Figure 8 is negative and
diminishes with chain stretching. The absolute value of fU
is similar to f in the whole range of the extension ratio x.
Therefore, the entropy component of the force fS (not
shown) is given approximately by a two-fold of the force
Table 2. Dimensional and conformational characteristics of the
PM-like chains from MC simulations.
N Rzz
nm
pR2
P1=2
nm
xc = pR2
P1/2
/Rzz pngP
20 2.373 1.696 0.715 4.28
40 4.872 2.772 0.569 9.22
60 7.372 3.582 0.486 14.15
100 12.365 4.858 0.393 23.97
Figure 7. Plots of the elastic Helmholtz energy Ael and the elas-
tic internal energy Uel as a function of the chain extension ratio x
for variable chain length N.
Figure 8. Plots of the reduced total elastic force fRzz/kT and its
energy part fURzz/kT, (upper and lower set of curves, respec-
tively), as a function of the chain extension ratio x for variable
chain length N = 20 (solid lines), N = 40 (dashed lines), N = 60
(dash-dot lines) and N = 100 (dotted lines).

9. Energy Elasticity of Tie Molecules in Semicrystalline Polymers 521
f. The energy contribution to force for a single chain or a
polymer network is related to the variation of the force
with temperature:
fU = f – T(df/dT)R (5)
This equation, based on purely thermodynamic argu-
ments, is employed in the force–temperature (thermo-
elastic) measurements[13, 21]
of polymer networks. These
experiments provide information on the fraction fU /f of
the total force in networks which is of energetic origin.
From Equation (5) a simple relation between f and fU
and the temperature coefficient of the mean chain dimen-
sions follows for the Gaussian chains
fU = f T (dlnpR2
P/dT) (6)
When, as an approximation, the data on f and fU from
Figure 8 for the non-Gaussian chains are placed into the
latter equation, the negative coefficient d ln pR2
P/dT
results in a whole range of x. A decrease of the dimen-
sions of PM chains with temperature is well established
and is explained as follows: upon increase in temperature
the population is enhanced of the high-energy gauche
states which are more compact than the trans ones. The
negative values of the ratio fU /f were determined from
thermoelastic measurements of crosslinked polyethyl-
ene.[13]
To facilitate the comparison with the MM data, the
absolute values of statistical force f and of the term fU are
plotted in Figure 9 for PM chains of the length N = 20 at
298 K. The internal retractive force f is positive for coils
extended above xc = 0.715 and rises to a limiting value of
about 120 pN at the full extension. The positive force in
the case of molecular bridges pulls the walls together.
Hence, the statistical force f shows an orientation oppo-
site to the loading force F. The energy contribution fU is
negative in the whole range of x and it reads about
–105 pN at x = 1. Thus the force term fU pushes the walls
apart and acts in the same direction as the loading force
F. The corresponding statistical moduli are also plotted in
Figure 9 using the value of A = 0.18 nm2
for the PM
chain cross-section. The statistical Young’s modulus Eel
attains the value of several GPa in highly extended tie
molecules. The “energy modulus” (Eel)U, the conforma-
tional energy contribution to the statistical modulus,
attains similar absolute values but is negative in PM-like
molecules. The data in Figure 9 suggest that the statistical
approach predicts the mixed energy–entropy mechanism
in deformation of short PM tie chains. The contribution
of the energy changes can be important especially in the
highly extended chains.
The Relation between Energy Elasticity Measures
At low chain extensions statistical force f is an indicator
of the prevailing entropic nature of deformation mechan-
ism in flexible macromolecules. At high extensions the
conformational space that a polymer bridge can sample is
increasingly reduced and entropy effects become sup-
pressed. The deformation of highly stretched tie chains
(of x approaching 1) and of the ultimate zigzag form is
accounted for mainly by the energy effects. The energy-
elasticity functions of PM chains provided by the single-
conformer MM computations (the F(R) functions) and by
the ensemble-averaged MC results (the fU (R) functions)
differs. The energy force F is positive, increases with R
and shows sudden drops in force. The energy contribution
to force fU is negative and smoothly decreases with R.
These seemingly conflicting outcomes correspond to two
contrasting assumptions on the structure of the IL phase
in semicrystalline polymers and its mechanical response.
The statistical approach to the elasticity of flexible
macromolecules in solutions or in the melt is based on
the averaging of a large number of conformational states
accessible to macromolecules. The conformational states
correspond to minima in the torsional potentials similar
to the U(R) potential. Thus, the conformational energy
Uel at a given chain length R is averaged over all confor-
mational states available at the given R. It is assumed that
intermolecular interactions do not affect the force f.
Stretching of molecules results in a translation of their
segments into new conformational states since adequate
local volume and/or sufficient mobility of neighbouring
chains is supposed. The average conformational energy
Uel is modified due to the change of the distribution of
Figure 9. Plots of statistical force f and of the term fU (upper
and lower solid curves, respectively) and of the statistical modu-
lus Eel and its energy component (Eel)U (dashed lines) for PM
chains of the length N = 20 at 298 K.

10. 522 Z. Sˇpitalskyy, T. Bleha, P. Cifra
conformational states by stretching. Individual conforma-
tional states are not deformed by stretching (stored elastic
energy is zero). An analogy (that neglects the averaging)
of the variation of the conformational energy Uel with R
can be constructed by connecting by a line the minima in
the potential U(R) in Figure 4b. The slope of this line, the
rate of the decrease of the energy of minima with R,
shown in Figure 4c, is evocative of the variation of the
term fU with R. The energy deformation mechanism is
especially significant in highly extended molecules. For
example, the conformational energy force fU at x = 0.9 in
Figure 9, of about –58 pN, is in absolute value around
15% higher than the corresponding value of force f. One
can also envisage an opposite case, of high positive
values of fU, i.e., an increase of the average conforma-
tional energy Uel on stretching. Such a situation should
occur in polyoxymethylene (POM), and related molecules
with an acetal segments at high extension ratio x. Here,
even though the trans conformation is of higher energy,
stretching of a POM chain favours the longer trans con-
formations over the shorter gauche conformations. In this
case, both f and fU exhibit orientation opposite to the load-
ing force F.
The conformational statistics of fairly extended tie
chains in the IL phase of semicrystalline polymers may,
however, markedly differ from the assumptions appropri-
ate for the purely amorphous state. For example, the
gauche–trans conformational energy difference in PM
can be significantly enhanced by the crystal lamellae con-
straints and by other effects of semicrystalline morphol-
ogy. The energy DUtg = 6.4 kJ N mol–1
found in a recent
NMR study of solid PE[22]
is almost the double of the
value 3.7 kJ N mol–1
used in our simulations. Such
enhanced conformational energy would result in an
increase in the parameter eg (in the stiffness of simulated
chains) and the energy term fU would become even more
negative.[14]
Additionally, for tie molecules hardly a full
thermal randomisation of chains can be assumed. The
conformational motion during the tie chain deformation
can be partially suppressed by neighbouring macromole-
cules and relaxation into a new equilibrium can be
blocked. Since the strain energy cannot be fully dissi-
pated through conformational motions, it should be at
least partially stored in the internal structural parameters
of individual conformers. These are conditions sharing
many features with the deformation of glassy polymers
and in this situation the energy-elastic deformation of the
individual defect conformers becomes essential.
The individual defect chains investigated by MM in
this and the previous paper[12]
represent the loose and
tight interlamellar bridges. Consideration of isolated
chains in MM calculations is in harmony with MC simu-
lations which also neglect intermolecular interactions.
Both types of results provide the useful limiting descrip-
tion of the mechanics of tie molecules unaffected by the
neighbouring chains in the melt-like IL phase. Possible
deviations from an ideal amorphous disorder in the IL
phase can be covered by an inclusion of the constraining
forces of embedding chains in modelling by MC and MM
methods. In the MM method intermolecular interactions
can be accounted for implicitly, by an additional con-
straining potential at the minimisation of the energy of
molecule[11]
or explicitly, by examining a multichain sys-
tem representing the bulk polymer.[23–25]
The elastic force
F and the elastic modulus E are defined as derivatives of
the curvature of the U(R) potential of stretched chains.
However, the effect of local packing on the U(R) poten-
tial can be quite complex; the neighbouring chains may
not simply rise the steepness of the U(R) potential. For
example, in a related problem of the rotational barrier of
the methyl group in polymers,[25]
some barriers were
higher in the solid state than in an isolated chain, but
some barriers were lower. The primary effect of the pack-
ing interactions was to broaden the range of the barrier
heights. A simple model of such a behaviour has been
proposed.[26]
Axial stretching of defect chains up to the all-trans
form proceeds by displacements of atom groups in the
chain skeleton from their (quasi)-equilibrium positions,
which are particularly striking at abrupt interconforma-
tional transitions. The single-conformer loading of defect
chains results in flip-flop jumps in force and characteris-
tic sawtooth-like profiles of F(R) curves. In multichain
systems an averaging of single-molecule jumps may take
place which would result in the smoothing of the F(R)
function. Nevertheless, the discontinuities in stress–strain
curves were also detected in the computer modelling of
glassy polymers.[23, 24]
In biopolymers, the sawtooth-like
forms of the F(R) functions were observed[27]
in single-
macromolecule manipulations by the atomic force micro-
scopy technique. Here, the jumps in force result from
transitions between compact and unfolded domains of a
biopolymer on stretching.
In the MM approach, instead of the conformational
equilibrium, the dynamic interconversion processes (the
reaction path) are crucial. The elastic parameters of the
defect molecule do not include the entropy term even
though the appropriate averaging over the set of defect
molecules is feasible. For example, the mean elastic mod-
ulus of the IL phase in solid PE can be calculated from
the data[28]
on the distributions of the number of mono-
mers and number of defects in tie molecules. The positive
elastic force F raises up to the value Fc of several hun-
dreds pN at the yielding point. The elastic modulus E (the
second derivative of the U(R) potential) is defined in a
pointed contrast to the concept of the statistical modulus
Eel. Interestingly, the values of statistical moduli Eel
shown in Figure 9 are in the same range of values as the
chain moduli E of the coiled defect structures of low
orientation factor P2 (Table 1).

11. Energy Elasticity of Tie Molecules in Semicrystalline Polymers 523
One can compare the energy expenditures on chain
stretching calculated by the MC and MM methods. The
Helmholtz energy change DAel on stretching of a chain of
N = 20 (Figure 7) from equilibrium coil dimensions to the
full contour length (that is from xc to x = 1) is
15.36 kJ N mol–1
at a temperature 298 K or about
0.54 kJ N mol–1
per change of x by 0.01 in this region. The
corresponding value for the term DUel is –20.08 kJ N mol–1
or –0.70 kJ N mol–1
per Dx = 0.01. The MM energy expen-
ditures on the defect chain loading differ considerably in
the regions below and above x = 1. For example the exten-
sion of the defect 3Ga by Dx = 0.166 to the first transition
point requires 14.86 kJ N mol–1
(Figure 3b), that is about
0.90 kJ N mol–1
per Dx = 0.01. The stretching of the zigzag
form by 3% requires 24.11 kJ N mol–1
, i.e., about
7.53 kJ N mol–1
per Dx = 0.01. These energy considerations
are relevant to the IL phase since one can suppose that
while most of the tie chains remain in the range of x a 1, a
fraction of tie molecules is stretched beyond the contour
length Rzz to the values of x A 1.
In most cases both energy deformation mechanisms,
statistical (by fU) and defect chain loading (by F), should
be functioning in the IL phase of semicrystalline poly-
mers. In the highly extended bridges of restricted mobi-
lity and far from thermal equilibrium, the loading energy
factors should even dominate over statistical ones at
deformation. The proportion of both energy mechanisms
in the stretching process may depend on the history and
structural homogeneity of the investigated sample. Tie
chain mechanics can be particularly sensitive to the pre-
sence of zones in the IL phase differing in the structural
order and chain packing. Such discernible structural non-
uniformities are frequently alleged for the IL phase of PE
and two non-crystalline phases, amorphous (I) and rigid
amorphous (II), were actually detected there.[29]
The rigid
phase II, midway between the amorphous and crystalline
phase, is non-crystalisable, but does not participate in the
glass transition. Hence, in the structural domains in the IL
phase of PE with prevailing content of the phase I or
phase II, either the ensemble-averaging or the defect
chain loading mechanisms of the energy-elasticity may
respectively dominate.
Conclusions
The deformation of tie molecules transferring the stress
through the disordered phase in semicrystalline polymers
was examined by two computational methods, molecular
mechanics and lattice MC simulations. Two types of the
response of tie molecules in short PM chains resulted
from these methods: (a) the F–R functions describing the
loading of defect tie chains by an external force F and (b)
the function fU–R for the energy component of the elastic
force fU from statistical ensemble considerations.
The model chains investigated by the MM method con-
tain several uncorrelated gauche conformational defects
and represent the “slack” bridges. The stretching of defect
chains yields the sawtooth-like profile of the force (F)–
length (R) functions due to the stress-induced gauche-
trans conformational transitions. The chain Young’s
moduli are considerably reduced by incorporation of the
defects into the chains. The chain stiffness correlates with
the orientation factor P2 and the torque generated. The
description of the energy elasticity by the defect chain
approach is appropriate under conditions of restricted
thermal equilibriation in tie molecules (similar to the
glassy state).
The lattice simulations of the elastic properties of short
PM chains were performed under conditions closely
matching the MM computations. In the ensemble-aver-
aging approach used, the stretching of PM chains is
accompanied by a re-equilibriation of the conformer dis-
tribution only, no barriers or stored elastic energy are pre-
sent. The smooth increase of the (positive) elastic force f
and a decrease of the (negative) energy term fU on chain
stretching was obtained. In mixed energy–entropy
mechanism of deformation found for short PM molecules,
the contribution of the energy changes is substantial,
especially in the case of highly stretched molecules.The
elastic parameters computed by the statistical approach
are appropriate for the tie molecules under full thermal
randomisation, similar to coils in solutions or melts.
The differences between two sets of the energy elastic
parameters are related to the contrasting assumptions on
the chain mobility in the IL phase employed in the two
approaches used. Both energy deformation mechanisms,
the single-chain loading (F) and statistical redistribution
of conformers (fU), are complementary and can simulta-
neously be operative in the IL phase. Their share in
stretching process may depend on the structural homoge-
neity and the history of the sample investigated.
Acknowledgement: The research was supported in part by the
Grant Agency for Science (VEGA), grant 2/7056/20.
Received: December 20, 2001
Revised: March 21, 2002
Accepted: March 26, 2002
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