Elastic properties of various poly(hydroxybutyrate) (PHB) molecular conformations were examined using molecular mechanics calculations. Force-distance functions and Young's moduli were computed by stretching PHB molecules. Unwinding the helical conformation is initially characterized by a Young's modulus of 1.8 GPa. At higher strains, the helix transforms into a highly extended twisted form. In contrast to paraffins, planar PHB is bent, attaining maximum length in straightened but twisted conformations. Single-chain moduli depend on chain extension. Computed data were used to analyze elastic response of tie molecules in the interlamellar region of semi-crystalline PHB and how their

1. Elastic Properties of Poly(hydroxybutyrate) Molecules
Zdenko Sˇpitalsky´, Toma´sˇ Bleha*
Polymer Institute, Slovak Academy of Sciences, 84236 Bratislava, Slovakia
Fax: þ421 2 54775923; E-mail: upoltble@savba.sk
Received: December 4, 2003; Revised: March 9, 2004; Accepted: March 10, 2004; DOI: 10.1002/mabi.200300118
Keywords: chain modulus; conformational analysis; molecular mechanics; semi-crystalline polymers; tie molecules
Introduction
Poly[(R)-3-hydroxybutyrate] (PHB) is linear, isotactic bio-
polymer within the class of natural polyesters called poly-
(hydroxyalkanoates).[1–3]
The high-molecular weight form
of PHB made by various microorganisms as an intracellu-
lar carbon and energy reserve is chiral and 100% isotactic.
In vivo, PHB is stored in the form of water-insoluble gra-
nules (inclusions) in the cell cytoplasm. It has been estab-
lished that the native inclusions exist in a metastable
amorphous state. Once isolated from the microorganism,
PHB forms a semi-crystalline material similar to conven-
tional thermoplastics such as polypropylene. PHB has been
intensively investigated due to its potential applications
such as biodegradable packaging material or biocompatible
medical implants. However, practical use of PHB has been
greatly limited, owing to poor mechanical performance and
narrow processing window. Besides the long-chain form of
isotactic PHB, a short-chain variety, called complexing or
cPHB, has been isolated from cell membranes.[4]
Oligomer
molecules of cPHB serve as ion carriers in channels media-
ting the ion transport through hydrophobic environment.
Furthermore,syntheticsyndiotacticPHB,synPHB,waspre-
pared by using aluminoxane catalysts.[5]
The molecular conformation of PHB has been investi-
gated using crystallographic methods. The crystal structure
of a PHB polymer determined by X-ray diffraction[6]
was found to take the form of a left-handed helix with
21 screw symmetry along the chain axis and the fiber
repeat 0.298 nm. Two antiparallel chains are packed in an
Summary: Elasticity of various poly(hydroxybutyrate)
(PHB) molecules of regular and irregular conformational
structure was examined by the molecular mechanics (MM)
calculations. Force – distance functions and the Young’s
moduli E were computed by stretching of PHB molecules.
Unwinding of the 21 helical conformation H is characterized
at small deformations by the Young’s modulus E ¼ 1.8 GPa.
The H form is transformed on stretching into the highly
extended twisted form E, similar to the b-structure observed
earlier by X-ray fiber diffraction. The computations revealed
that in contrast to paraffins, the planar all-trans structure of
undeformed PHB is bent. Hence, a PHB molecule attains the
maximum contour length in highly straightened, but slightly
twisted conformations. A dependence of the single-chain
moduli of regular and disordered conformations on the chain
extension ratio x was found. The computed data were used to
analyze elastic response of tie (bridging) molecules in the
interlamellar (IL) region of a semi-crystalline PHB. A modi-
fication of the chain length distribution function of tie
molecules t(N) due to secondary crystallization of PHB was
conjectured. The resulting narrow distribution t(N) com-
prises the taut tie molecules of higher chain moduli prone to
overstressing. The molecular model outlined is in line with
the macroscopically observed increase in the modulus and
brittleness of PHB with storage time.
The force – length plots of deformation of the helical H
form (*) and of the non-planar Tn form (&) of the PHB
hexamer.
Macromol. Biosci. 2004, 4, 601–609 DOI: 10.1002/mabi.200300118 ß 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Full Paper 601

2. orthorhombic unit cell. Thin lamellar single crystals of the
thickness of about 4–7 nm are readily formed by chain-
folding. On the other hand, the right-handed 31 helix was
found in crystals of oligomeric forms of PHB.[4]
The conformational structure of PHB in solutions, melts
and other disordered phases is still a matter of discussion. An
early study[7]
indicates that PHB forms randomly coiled or
partially helical structures, independence onthe solvent type
and temperature. Later experimental and theoretical stu-
dies[8,9]
suggested the random-coil behavior of PHB in dilute
solutions, with the mean chain dimensions of PHB in the
range typical for the flexible polyesters. Conversely, other
reports[10,11]
maintain that intrinsic propensity of PHB to
form a helix depends on the solvation strength. In good solv-
entsthe rod-like helices are preferred and ingoing fromgood
to poor solvents a sharp helix-coil transition occurs.[10]
A
presence of helical segments promotes the chain aggregation
and subsequent gelation of PHB solutions.[11]
A high degree
of local chain persistence was also reported[12]
from small-
angle neutron scattering (SANS) measurements in a melt of
PHB. According to SANS data the PHB chains prefer in melt
the rod-like structures involving helical segments. One may
hypothesize that this rod-like structure of PHB molecules is
also present in the native granules and reconstituted PHB
latex particles.[13]
The stability of PHB storage granules and
latex particles in the amorphous state can be explained on
the basis of nucleation kinetics of PHB.[2]
The information on the molecular conformation of
PHB chains in a solution and a melt provides some hints
about the chain conformations most probably encountered
in the disordered regions between crystal lamellae in a
solid semicrystalline PHB. PHB crystallizes from a melt
to form large spherulites on rapid cooling of a mould. Sub-
sequently secondary crystallization takes place on storage
of PHB samples even at room temperature.[2]
The disor-
dered interlamellar (IL) regions in the spherulitic structure
of PHB feature several types of molecules differing in their
conformations.[14,15]
The IL region involves chain loops,
which start and end in the same lamella, tails with one free
end, and bridges (tie molecules) that join up two lamellae.
Tie molecules that traverse the non-crystalline regions play
a central role in transferring stress effectively from one
lamella to the next when strained. On tensile drawing of
polymers the tie molecules straighten and entire IL region
becomes more oriented. It is often presumed that the frac-
tion of the load-carrying taut tie molecules controls the
elastic modulus of a polymer.[14]
The recent remarkable progress in techniques of single-
molecule manipulations such as AFM atomic force micro-
scopy (AFM) or optical tweezers offers direct access to
the mechanical properties of individual macromolecules.
Force-distance profiles from single-chain mechanical ex-
periments were reported for nucleic acids, proteins, poly-
saccharides and some synthetic polymers.[16–18]
In a
parallel development, the mechanical response of indivi-
dual macromolecules is investigated by atomistic model-
ing, mainly by using the molecular mechanics (MM)
and molecular dynamics (MD) methods. For example, the
MM method was used to link elasticity of the DNA frag-
ments and of the protein a-helices or b-ribbons to their
structure and energetics.[18,19]
A similar MM approach
was exploited[20–22]
to examine the deformation of highly
extended polyethylene (PE) chains involving the confor-
mational defects. The gauche-trans conformational transi-
tions induced by axial mechanical loading of the defect
PE molecules resulted in a sawtooth-like profile of the
force-length curve.[21]
It was argued[21]
that the sawtooth-
like profile is a common feature of mechanochemistry
both in highly stretched polymer chains and in biopoly-
mers[16]
where compact domains unfold on stretching.
MM computations of loading of specific conformers of a
PE were used to model the elastic response of the extended
tie chains in the IL phase of semi-crystalline PE.[22]
In this paper the energy-elastic deformation of PHB
molecules of regular (helical or all-trans) and of some
disordered conformations was examined by performing
the MM calculations as a function of imposed distance
constraints. The force-length profiles and the Young’s
moduli E of individual molecular conformations of PHB
were evaluated and the stress-induced structural changes
described. Furthermore, an attempt was made to utilize the
computed elastic parameters of individual conformers of
PHB in rationalization of the changes in the structure and
mechanics of tie molecules in solid semi-crystalline PHB.
Methods
The static potential energy of molecules involving con-
formational defects was calculated by the Allinger MMþ
molecular-mechanics method[23]
by the procedure describ-
ed in previous papers.[21,22,24]
The static potential energy
of a molecule U is expressed in the method as the sum of
several contributions:
U ¼ Ur þ Uy þ Uf þ UvdW þ Ue ð1Þ
where the terms Ur and Uy represent the bond length and
bond angle deformation, respectively, Uf is the inherent
ethane-like torsional potential respecting the cosine type
periodicity of torsional angle f. The term UvdW is a sum-
mation of all non-bonded pair interactions in the molecule.
The electrostatic term Ue is calculated from interaction of
the bond dipoles assigned in a molecule in the MMþ force
field, using the dielectric constant e ¼ 1.5. The individual
energy terms in MMþ methods are expressed by simple
analytical functions involving numerous adjustable para-
meters. The MMþ method and its former variant MM2
provides reliable predictions of the structural and thermo-
dynamic data of wide group of chain molecules at ambient
temperature including polyester models.[25,26]
In few cases
we have performed parallel calculations by the other
602 Z. Sˇpitalsky´, T. Bleha
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3. force-field methods employed in conformational studies
of biomacromolecules (AMBER, BIOþ and OPLS) with
results similar to MMþ method.
The dimer unit of isotactic PHB is shown in Figure 1 with
notation of the four chain backbone torsional angles per unit.
Most calculations were performed with hexamer segment
terminated at both ends by tert-butyl groups to eliminate the
effects of end groups. By optimization the MMþ method
gives the equilibrium static energy of molecule U and the
valence geometry parameters (the torsional angles fi, bond
anglesyi andbondlengths ri),includingthe conformationsof
the CH3 side groups and terminal tert-butyl groups.
Several PHB models with either regular or irregular
starting conformation were considered. In the formergroup,
the fixed sequence of four torsional angles f, c, t, o in a
PHB unit was assumed. In the latter, the values of some
torsional angles were randomly selected. The end-to-end
length of molecule, R, is given as a distance of ether oxygen
atoms in the first and last units. The molecule was stretch-
ed by a gradual increase in the length R from the unde-
formed value Ro and the molecular strain e ¼ (R À Ro)/Ro
evaluated. The energy of a stretched molecule, almost
rigidly constrained at given R, was optimized and the equi-
librium static energy U and the valence parameters were
obtained. An implicit deformation force F is collinear with
the vector of the distance R. The force F and the Young’s
modulus E were computed as a function of the distance R
by the first and second differentiation of the static energy
U according to R. Inferring a force from the change of
potential energy obtained by minimization obviously neg-
lect any entropic contribution. Resulting asymmetrical dis-
tribution of the stored elastic energy along the chain length,
e.g. enhanced deformation in the vicinity of chain ends,
imitates the deformation pattern in real materials. An alter-
native procedure that assumes a uniform deformation of all
units along the chain and conserves the symmetry of a
helical molecule on stretching is used in analysis of chain
deformation in a crystal lattice.[25–27]
Results and Discussion
Helical Conformers
The energetics of stretching was explored for the helical
conformer of PHB, denoted as H, identified in crystallo-
graphic studies. The H conformer shown in Figure 2 was
constructed by using the torsional angles f ¼ 156.58, c ¼
À51.28, o ¼ À46.18 and t ¼ À168.28 deduced[25]
from the
X-ray diffraction data by refinement of a 21-type of heli-
cal structure of PHB (the a phase). It corresponds to the
TGGT type of helix where G is the plus or minus gauche
position of a torsional angle around 608 and 3008, respec-
tively, and Tis the trans position of a torsional angle at 1808.
Three regions can be identified on the deformation
potential of the H-form in Figure 3. At first the energy U
slightly increases on stretching. The process is accompa-
nied by minor discontinuities due to abrupt changes of
some torsional angles on deformation. At strain about 0.3
the weak-force region is succeeded by a region of a steep
increase in the static energy. Then, at e about 0.55, the
non-monotonous changes of energy due to conformational
rearrangements are seen in Figure 3. A subtle local mini-
mum corresponds to a metastable structure E of a highly
Figure 1. The dimer unit of isotactic PHB with notation of
torsional angles.
Figure 2. The helical structure H of a PHB hexamer constructed
from the X-ray data[25]
shown after optimization prior stretching
(upper panel); the extended helix E constructed from X-ray
data[25]
(center) and the all-trans model TP (bottom).
Figure 3. Energetics of stretching of PHB hexamer in the helical
form H (&) and in the straightened form Tn (&). The points
(*) show the strictly all-trans form Tp and the proposed[25]
extended helical structure Eh (b form).
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4. straightened PHB molecule with the length of about
2.73 nm.
The computed deformation potential U(R) of the H
form of PHB can be related to experimental data[25]
on
cold-stretched PHB films where the X-ray fiber diffraction
patterns revealed a formation of the b-crystalline phase of
PHB. From experimental data a highly stretched helical
conformation of PHB molecules was proposed for the
b-phase by MM energy minimization.[25]
The observed
fiber repeat of the b-form, 0.460 nm was over 50% larger
than the value 0.298 nm found in the standard a-phase of
PHB. It was suggested[25]
that the b-form arises by stress-
inducedcrystallizationofthequasi-amorphousILphaseina
PHB sample.
We have used the valence geometry data of a helical
chain in the b-phase[25]
to arrange the PHB hexamer into the
corresponding extended helical form shown in Figure 2.
After energy optimization (by keeping the pre-set helical
values of torsional angles) the structure with the energy
114.4 kJ Á molÀ1
and the length 2.73 nm was found. This
helical structure, denoted as Eh in Figure 3, has almost the
same length as the non-helical structure E in the local
minimum of the potential U(R) at high strains. Hence, the
computations suggest that depending on the constraints
used in energy minimization the extended helix Eh (i.e. the
b-phase of PHB) or the non-helical twisted conformation
E can be formed on stretching. The conformational trans-
formation of the initial helix H into the disordered confor-
mation E along the energy path shown in Figure 3 should
be relevant to stretching of the non-crystalline IL phase
of PHB involving numerous partially helical segments.
On the other hand the transformation of the H form (the
a-phase) into the extended helix Eh (the b-phase) may
indicate the stress-induced solid-solid phase transition
often encountered in many types of polyesters.[25]
By differentiation of the deformation potential U(R)
the force-length curve F(R) of the H form was obtained
(Figure 4). At small strains a broad plateau is seen in
Figure 4; a small, almost constant force is needed to unwind
the helix H in this region. Then, at e about 0.3 the upturn
of force into the hard elastic region is seen. Stretching of
the helical form along a vector approximately parallel to
the helix axis proceeds by the changes in the torsional
angles in the chain backbone. The (hard) torsional angles
t and c remain essentially constant, even at high strains,
while the other two (soft) angles, f and o, change on stret-
ching (Figure 5). The variation with strain of soft angles in
two adjoining monomers slightly differs. The rather high
forces, of several hundreds of pN, needed to stretch a PHB
molecule in the vicinity of the extended E form (Figure 4)
are related to the fact that all torsional angles advance to
1808 in this region of high strains (Figure 5).
A response of torsional angles to molecular strain shown
in Figure 5 is in harmony with the curvature of individual
torsional potentials. The torsional potentials were calcu-
lated by the MMþ method under assumption[28]
of inde-
pendent bond rotations in a PHB unit. The computed
potential barriers (Figure 6) are close to values reported
previously.[29]
The soft torsional angles exhibit lower bar-
riers, about 4 and 10 kJ Á molÀ1
for angles f and o, respec-
tively. The hard torsional angles c and t are characterized
by the higher barriers about 18 and 30 kJ Á molÀ1
, respec-
tively. However, it is important to realize that the curvature
of the deformation potential U(R), rather than the appro-
priate energy barrier, defines the torsional stiffness of bonds
and thus controls the elastic response of a macromolecule.
Correspondingly, the torsional potentials are much steeper
in Figure 6 in case of the hard torsional angles than of
soft torsional angles.
The force F1 needed to elongate a molecule by 1% is a
useful measure of chain stiffness. The values of F1 for the
molecular conformations H and E of PHB are 27 pN and
34 pN, respectively. The longitudinal Young’s modulus
E ¼ 1.8 GPa was computed at initial strains for the helical
form H by assuming the helix cross-section area A ¼
Figure 4. The force – length plots of deformation of the helical
H form (*) and of the non-planar Tn form (&) of the PHB
hexamer.
Figure 5. Variation of the torsional angles on stretching of the
helical H form of PHB hexamer.
604 Z. Sˇpitalsky´, T. Bleha
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5. 0.38 nm2
deduced from PHB crystal lattice data.[27]
The
chain modulus of the E form is 4.2 GPa when the same
cross-section area is used. A rather low value of modulus
of the PHB helix H is typical for very open helices. For
example the chain modulus 3.0 GPa and force F1 ¼
29 pN were determined[30]
in polyester poly(pivalolactone),
[CH2–C(CH3)2–COO] exhibiting a related helical molec-
ular conformation (TTGG)2 in a crystal.[27]
Similarly, the
single-chain calculations yielded[31]
a chain modulus
1.48 GPa of syndiotactic polypropylene of the same type
of molecular conformation as PHB. However, a strong
reinforcement of chains was found[32]
by introducing the
helical molecules of PHB into a crystal lattice: the cal-
culated chain moduli were in the range of 13–16 GPa,
slightly higher than the experimental crystal modulus of
PHB of about 9 GPa at À50 8C.
The All-trans Models
In molecules with paraffinic backbone such as PE the
all-trans conformation with all torsional angles set strictly
to 1808 displays the maximum span of a molecule R
achievable by variation of torsional angles only (the con-
tour length). However, the calculations show that this rule
is not valid in case of PHB and related polyesters. Here,
the fully planar all-trans structure Tp of undeformed PHB
does not show a rectilinear axis typical for paraffins. Ins-
tead, a slightly curved structure is predicted from MM
computations (Figure 2). This bending of the planar
all-trans structure of PHB is a consequence of uneven
bond lengths and bond angles in the PHB unit. It is well-
established that the differences in adjacent bond angles in
planar all-trans poly(dimethylsiloxane) or polyphosphate
chains lead to cyclic structures.[28]
Actually, the bond
angles differ considerably in the planar form Tp of PHB
(Table 1). The preference for cyclic forms is further ampli-
fied by a disparity between the lengths of CC and CO bonds
in the chain backbone (Table 1). The all-trans structure Tp
generated by the constrained optimization with torsional
angles fixed to 1808 exhibits the end-to-end length R ¼
2.45 nm and the energy about 150.8 kJ Á molÀ1
(Figure 3).
This unfavorable energy is related to the fact that the tors-
ional potentials of angles f and o do not show a minimum
at 1808 (Figure 6).
Stretching of the conformer Tp restrained in the planar
form brings a gradual straightening of the bent structure via
deformation of the bond angles and bond lengths but energy
expenses are extreme. On the other hand, an unconstrained
optimization of the Tp form results in a non-planar structure
with slight deviations of the torsional angles from 1808. The
non-planar form of PHB, denoted as Tn (Figure 3), shows a
lower energy U ¼ 126.5 kJ Á molÀ1
and is less curved (R ¼
2.54 nm) than the Tp conformation. Subsequent deforma-
tion of the bent form Tn is described by the monotonously
raising potential U(R) shown in Figure 3. The stretching
of the Tn molecule is a low-force process in initial stages
(Figure 4) with the force F1 ¼ 12 pN and the modulus
E ¼ 2.4 GPa provided the mentioned cross-section area
A ¼ 0.38 nm2
per chain is assumed. It should be noted
that although the helical conformer H is much stable than
the Tn form (by 46.4 kJ Á molÀ1
) their elastic parameters are
similar. Again, this comparison underlines that steepness
of the deformation potential U(R) controls the elastic res-
ponse of a molecule and not its energy at minimum or the
barrier height. At later stages, the stretching of the non-
planar Tn form requires considerable forces of several
hundreds pN (Figure 4).
In analogy with PE stretching of helical molecules of
PHB and related polyesters is commonly presumed to result
in the planar zigzag structure. In reality, maximum contour
length of a PHB chain is accomplished in slightly twisted
conformations where torsional angles differ a little from
1808, such as in the forms E and Eh discussed above. Such
category of highly straightened non-planar forms should
play the same reference role in PHB as the zig-zag struc-
ture in PE-like polymers. For example, these twisted
Table 1. The valence geometry parameters in the planar form Tp
of PHB molecules calculated by the MMþ method.
Bond lengths Bond angles
nm 8
OCa 0.1418 CcOCa 117.25
CaCb 0.1547 OCaCb 108.65
CbCc 0.1522 CaCbCc 113.27
CcO 0.1349 CbCcO 111.49
Figure 6. The torsional potentials of four bonds in a PHB unit
calculated in the approximation of independent bond rotations.[28]
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6. conformations can be presumed to occur in the crystalline
inclusion complexes of PHB[33]
where PHB chains are
straightened by squeezing into narrow channels formed
by cyclodextrin molecules. Similarly, straightened non-
planar conformations should be the ultimate forms of
PHB chains in oriented fibers of PHB prepared by cold
drawing.[34]
Conformationally Disordered Molecules
A number of PHB models with disordered conformation
have been explored which presumably represent chain con-
formations in the disordered states of PHB. The chain
models were constructed rather arbitrarily by random intro-
duction of the torsional ‘‘defects’’ into the helix H or the
planar form Tp. The defect torsional angles were initially
set to the gauche state or to the respective minima in the
torsional potentials in Figure 6. The computed force-length
curves of disordered models generally follow the trend
observed in stretching of the H or T forms. Exceptionally,
some breaks were detected on the F-R curves as in the
case conformation denoted M, characterized by the set of
torsional angles f ¼ 1548, c ¼ 1788, o ¼ 618 and t ¼
À1768 (Figure 7). The breaks around R ¼ 2.8 nm are
associated with abrupt transitions into more extended
forms of a molecule. Similar but more pronounced dis-
continuities due to gauche-trans conformational transitions
were observed in highly extended PE molecules.[21,22]
Generally, smooth stress-induced transitions to longer
forms of molecules are preferred in PHB since (a) the tor-
sional potentials in PHB are much less steep than in PE
and (b) the C–O–C bond angles in the PHB chain back-
bone are much softer in deformation than C–C–C bond
angles in PE and thus much less elastic energy can be
accumulated on stretching in a PHB chain relative to PE.
It was previously outlined[21,22,24]
that the energy-elasticity
method used here is suited primarily for the straightened
chain fragments where the entropy loss suffered during
extension can be neglected. This approach can also be
applied to coiled conformations occurring under conditions
of the reduced mobility and restricted thermal equilibration.
In the case of disordered coiled conformations under full
thermal randomization the origin of elastic force should be
both energetic and entropic. These two force components
were for example evaluated from the free energy changes on
intercalation of macromolecules into narrow channels.[35]
Here, molecules were required to stretch from their
multitude of randomly coiled conformers to the relatively
small numbers of straightened, channel-bound forms. The
chain fragments whose conformations are constrained
solely by stretching were found to store considerable
amounts of elastic force, whose origin is both energetic and
entropic. In case of randomly-coiled conformations of PHB
this approach[35]
of comparing both the energies and
entropies of unstretched and stretched coiled polymer
chains can provide the single chain forces and moduli of
similar magnitudes as those obtained from a comparison of
energies between conformationally restricted and extended
chains in Figure 4 and 7.
The computed Young’s moduli of regular and disordered
structures are plotted as a function of the chain extension
ratio x in Figure 8. The ratio x quantifies a straightening of
undeformed chains: the initial chain length of individual
structures Ro is expressed relative to the reference length
Ro
ref
of the extended helical form E, x ¼ Ro/Ro
ref
. The scatter
of points in Figure 8 is considerable, but still an approxi-
mate correlation can be detected between the relative
straightening of undeformed molecular conformations and
their moduli. The disordered and helical forms possessing
the ratio x less than about 0.9 give the low and medium
Figure 7. Force – length curve of the disordered conformation
M characterized by the torsional angles specified in the text.
Figure 8. Relation between the chain extension ratio x of indi-
vidual PHB conformations and their chain modulus Erel (both
values normalized to the helical form E).
606 Z. Sˇpitalsky´, T. Bleha
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7. values of the modulus E (Figure 8). At higher x an upturn to
the high moduli of non-planar twisted forms takes place.
Such type of correlation of the modulus E versus the ratio
x was used previously in analysis of tie molecules in semi-
crystalline PE.[22]
Elasticity of Tie Molecules in the IL Regions
The computed energy-elastic parameters of individual con-
formers of PHB are potentially useful in rationalization of
the changes in the structure and mechanics of tie molecules
in solid semi-crystalline PHB. Theory and modelling of
mechanical properties of semi-crystalline polymers[36–38]
is a daunting task because of complexity of their structural
and morphological hierarchy. The elastic properties of
the crystalline phase consisting offolded chain lamellae can
be predicted fairly well. In non-crystalline interlamellar (IL)
regions the molecular modeling was focused[15,20–22,24]
on tie (bridging) molecules effectively transferring stress
from one lamella to the next when strained. The fraction of
the load-carrying taut tie molecules controls the elastic
modulus of a polymer. On tensile drawing of polymers
the molecular bridges straighten and entire IL region
becomes more oriented. The spherulite-lamellar morpho-
logy of semi-crystalline polymers is transformed in this
way into an oriented fibrous structure.[14,15]
Some information on chain modulus of tie molecules in a
solid semi-crystalline PHB can be inferred from the com-
puted elasticity of individual PHB conformational forms.
The average modulus of the IL phase should depend on
the chain moduli of tie molecules differing in their length
i.e. in the number of monomers N. The distribution of
tie molecules according to their length is specified by the
function t(N) which enumerate the population of slack
(coiled) and tight bridges.[38]
The distribution function
t(N) can be converted[22]
to an analogous distribution of
the chain moduli as a function of N, x(N). Such approach
was applied to literature data about various distributions
t(N) appropriate for the PE samples differing in the struc-
ture and orientation.[22]
For PHB no information on the distribution function
t(N) is available. One can suppose that freshly processed
solid PHB exhibits the distribution t(N) covering a wide
range of length of tie molecules with a maximum at the
extension ratio x in the region of coiled bridges. A sketch of
such distribution is shown in Figure 9 where a Gaussian
form of the t(N) function was supposed. A rather broad
distribution assumed is compatible with a moderate modu-
lus and toughness measured for fresh PHB samples. The
PHB bridges should straighten on stretching and the distri-
bution t(N) should move towards the right in Figure 9.
Secondary crystallization, a unique feature of PHB
thermoplastics, should bring notable changes into the tie
molecule distribution t(N) with ageing of sample. Tie
molecules become shorter (their x increases) by secondary
crystallization as their fragments and/or end-portions are
gradually built into the crystal lamellae consisting of 21
helices (the a-phase). Thus, after secondary crystallization,
a rather narrow function t(N) displaying a maximum at
highly straightened bridges can be presumed (Figure 9) in
the IL phase. The correlation E versus x from Figure 8
suggests that modification of the distribution t(N) sketched
in Figure 9 should result in an increase in the average
modulus of tie chains: the low-modulus molecular confor-
mations are transformed by secondary crystallization into
the highly extended high-modulus forms. By subsequent
stretching of aged PHB the taut tie molecules located in
the narrow distribution t(N) close to x ¼ 1 are overstressed
(x > 1) and subsequently they broke at x >> 1. Macro-
scopically, the samples of bacterial PHB show a rapid in-
crease in brittleness (decrease of elongation at break) with
the storage time.[1–3]
In other words, an absence of coiled
tie molecules in the narrow distribution t(N) precludes the
elongation of aged PHB samples. This phenomenon of
physical ageing represents the most serious obstacle of an
extensive PHB application.
The change in elasticity of tie molecules outlined in
Figure 9 is indirectly supported by various experimental
techniques indicating much reduced mobility of PHB
chains in the IL phase relative to a melt or solutions. The
recent measurements[39,40]
suggest that the non-crystalline
region of PHB consists of two fractions, a quasi-amorphous
‘‘soft‘‘ fraction and a newly identified ‘‘rigid’’ amorphous
fraction. In this way a traditional two-component model of
semi-crystalline polymers is extended into the three-
component model. This rigid fraction comprises over 60%
of non-crystalline phase in PHB.[39]
The brittleness of
PHB is attributed to the rigid fraction composed of the
relatively immobile, tightly packed chains on interface
between the crystalline and amorphous layers.[40]
Since the
rigid fraction most probably consists of taut tie molecules,
these data lend support to the modification of distribution
t(N) outlined above. A restricted dynamics and limited
Figure 9. A sketch of presumed modification of the length
distribution t(N) of tie molecules induced by secondary crystal-
lization in the solid PHB.
Elastic Properties of Poly(hydroxybutyrate) Molecules 607
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8. thermal randomization of conformers result in the vitrifica-
tion of the rigid amorphous fraction of the IL phase. The
strain energy cannot be fully dissipated through conforma-
tional motions and is stored in the tie chains. In such
situation the deformation mechanism based on the energy
elasticity of individual molecular conformations of PHB
becomes essential.
Apparently, potential improvements in the mechanical
performance of PHB thermoplastics can be accomplished
by changes in the structure and chain packing of IL phase,
particularly in the ratio of its soft and rigid fractions. The
polymer must be flexible on the molecular level in order
the material is tough on the macroscopic level. In molec-
ular terms it means that any modification of PHB should
secure enhanced flexibility and mobility of the IL region
through a sufficiently wide chain length distribution of
tie molecules t(N).
Conclusion
The deformation response of several regular and irregular
conformations of a PHB hexamer was examined by the MM
calculations. The static energy of PHB molecules and the
elastic force was calculated as a function of the chain length
R. It was found that elongation of the helical form H
results in a highly extended twisted form E that is similar to
the b-structure observed in cold-drawn PHB. The planar
all-trans structure of undeformed PHB shows an inherent
tendency of closing upon itself in a cycle. Thus, instead
of the planar zig-zag form, the highly straightened but
slightly twisted conformations of PHB chains exhibit the
maximum contour length and ultimately should occur on
drawing of PHB fibers or on confining the PHB molecules
into narrow channels.
The computed elastic parameters of individual PHB
conformers were employed to illuminate the changes in the
structure and mechanics of tie molecules in solid semi-
crystalline PHB. Qualitative changes in the length distribu-
tion of tie molecules t(N) due to secondary crystallization
and stretching of solid PHB were suggested. The presum-
ed modification of the function t(N) from a broad distri-
bution of slack tie molecules into a narrow distribution of
highly extended chains can explain the observed increase
in the brittleness of PHB samples on storage.
Acknowledgement: The research was supported in part by the
Grant Agency for Science (VEGA), Grant 2/3012/23.
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