Results and Discussion
Link between wax esters and natural
waxes
Thermal behavior
In Fig. 1 , the DSC heating thermograms of different natural
waxes are given. Below each thermogram, the relative frequency of
different CNs in the respective WE-fraction is depicted. They are
stacked in order of decreasing WE content from top to bottom. For
comparison, the thermogram of a pure WE (42 (22_20)) is given on top of
the figure.
The melting thermograms show different behavior for different waxes in
agreement with many other publications. The pure WE exhibits a single,
narrow endothermic peak smeared out primarily due to thermal lag as a
function of the heating rate (5 K/min). For the two most homogeneous
waxes, SFX and RBX, a similar peak shape is observed. However, the peaks
are less symmetric. The small shoulders, more pronounced in RBX, do not
only indicate less purity than for the WE C42 (22_20) but also the
presence of multiple solid phases. In contrast, the thermograms of CRX
and CLX contain broader, or rather multiple peaks with more pronounced
shoulders while BWX displays two distinct peaks. Therefrom, a detailed
interpretation with respect to the number of coexisting solid phases and
mixing behavior is practically impossible. All thermograms gathered are
in line with former observations .
For the computation of the relative CN-frequencies of WEs, analytical
data on the presence of FaOH and FA moieties in hydrolyzed WE fractions
obtained by Doan et al. by means of GC-MS were used. The data itself
provides no information on how these moieties were recombined to
constitute the WEs. Only little is known about selectivity in the
biological synthesis of WEs, combining FA and FaOH moieties. It is known
that triacylglycerol profiles of natural vegetable fats can be predicted
quite reliably based on the respective fatty acid profile and the
Coleman-assumption . Essentially, a preference for unsaturated FAs on
the sn-2 position is combined with a random distribution of FAs over the
three possible glycerol positions. Based on this, a similar statistical
approach was chosen for the recombination of the profiles of FaOH and FA
moieties given by Doan into WEs. Cumulating all symmetries at equal CN,
this resulted in relative frequencies of the WEs in a range between 34
and 56 carbon atoms. A publication by Vali et al., reports the WE
composition of five different rice bran waxes as well as the chain
lengths of the FaOH and FA residues of the saponified WEs. Computation
of the WE frequencies as outlined above resulted in good agreement
between the experimental CN-profile of the WEs and the prediction
suggested here. Thus, the WE profiles in Fig. 1 are computed
this way.
This utter arbitrary juxtaposition of the melting behavior of natural
waxes and their respective WE CN distribution depicted in Fig.
1 suggests anecdotal evidence that the WE composition relates to the
thermal properties of natural waxes. Without any doubt, the
crystallization behavior is a function of the CN-distribution of the
WEs. However, this relation rather is of thermodynamical nature and not
simply geometrical. However, the distributional width - mono- or
polydispersity or bimodality – gives indications for mixed crystal
formation and the presence of multiple solid phases. Fig. 1suggests that this even holds for waxes with low WE content such as CLX.
The fact that high melting temperatures correspond in the first place
with higher amounts of long-chained WEs is well established. This is
illustrated by comparing the thermograms of SFX and RBX and,
particularly, of CRX. For this wax, the computed WE profile is in line
with the information given elsewhere that even higher WEs are present in
the wax than possible with the data given by Doan. The fact that the
computed data underestimate the high-melting fraction is most likely due
to the challenge to quantify the long FA and FaOH moieties. Overall,
this analysis reveals that there is a need to better understand the
crystallization behavior of WEs and their mixtures in order to control
wax crystallization.
Microstructure
Micrographs of oleogels based on 10 % (w/w) of either a pure WE (C34
(16_18) or a natural wax (SFX) are shown in Fig. 2 . Herein,
the respective microstructure of the oleogels crystallized at different
cooling rates after stabilization is depicted.
The image of SFX at a cooling rate of 1 K/min (a) indicates needle-like
structures, though these are stated to be platelet-like crystals
arranged orthogonally to the objective slide . This could either be
attributed to phenomena relating to the hydrophilic glass surface and
the hydrophobic wax crystals or just be a result of heterogenous
nucleation on the glass surface. Those platelets are sintered and form a
three-dimensional network, just as repeatedly reported in the literature
. After being cooled at the same rate, the pure WE (b) forms almost
two-dimensional platelets which are oriented parallelly to the cover of
the objective slides and without discernible junctions. Hence, at equal
cooling rates of 1 K/min, the natural wax and the pure WE exhibit quite
different microstructures. As reported earlier from our lab , when
crystallized at very low cooling rates of 0.05 K/min in purified canola
oil, SFX crystallizes into well-defined, flat crystal fragments (c).
When, in contrast, the pure WE is crystallized at significantly
increased cooling rates of 30 K/min, morphologies typical for SFX,
related to upright platelets are observed (d). This dependence of the
crystal morphology on the cooling rate underlines that the crystal
structures occurring in natural waxes and pure WEs are congeneric.
Furthermore, the importance of understanding the role of the cooling
rate and the presence of polar minor components on the crystallization
is emphasized.
Pure component properties
Wax esters are composed of a long-chained FA and a long-chained FaOH
moiety. Commonly, the alkyl chain length of the FA residue ranges
between 16 and 24 C-atoms, while the FaOH residues tend to be longer,
between 18 and 32 carbon atoms .
In the solid state, aliphatic esters crystallize in unimolecular layers
with the alkyl chains oriented lengthwise, either orthogonally or tilted
with respect to the methyl end plane . The length of a saturated WE
molecule in orthogonal chain arrangement is calculated here, using eq.
(1), based on simple geometric considerations. The layer thickness\(d_{\text{ortho}}\) is build up from three contributions: The alkyl
chains (number of carbons in FA (\(n\)) and FaOH (\(m\)) residue)
multiplied by the carbon-carbon-distance in chain direction (1.26 Å ,
the ester-oxygen atom chain contribution (\(b=1.09\ A\) and the
distance between two terminal methyl end groups and thus between two
planes (\(c\)). The latter being specific for aliphatic esters (2.05 Å)
.
\(d_{\text{ort}ho}=\left(m+n\right)*a+b+c\) (Eq. 1)
For the tilted chain orientation, the thickness of a layer is
conveniently considered as the product of the sine of the inclination
angle (62.5 °) and the orthogonal layer thickness (eq. 1) , represented
by eq. 2.
\(d_{\text{tilted}}=d_{\text{ort}ho}*\sin\left(\alpha_{\text{tilt}}\right)\)(Eq. 2)
There is different rotational freedom for either the FA or FaOH residue
due to the orientation of the ester bond. Consequently, the position of
the ester bond in the WE chain molecules matters. The WE properties
depend thus on the symmetry of the molecule. The latter is expressed asCN, the difference between the CN of the FaOH and the FA
residue (eq. 3).
\(\text{CN}=CN_{\text{FaOH}}-CN_{\text{FA}}\) (Eq. 3)
Fig. 3 shows the evolution of the calculated layer thicknesses
(lines) as a function of CN. According to eq. 1 and 2, the layer
thickness increases linearly with increasing carbon number. The tilted
orientation always results in a smaller layer thickness at equal CN. The
markers represent averaged values computed from published experimental
long spacing data . The small angle XRD (SAXS) data are derived for neat
WEs of different \(CN\). The experimental long spacing data corresponds
quite well with the simply calculated layer thickness data. This holds
for both, the orthogonal and the tilted chain arrangement. A careful
look at the data reveals that only WEs with \(CN\) of \(+2\) or \(-4\)appear to crystallize in orthogonal configuration. All other WEs
crystallize with an inclination angle to the methyl end plane. At this
point, it has to be pointed out, that Kreger and Schamhart found two
long spacings for tetradecyl octadecenoate (14_18; \(CN-4\)) and
docosanyl octadecenoate (22_18; \(CN+4\)), indicating that both
configurations can be assumed by this WE . This observation hints at
possible monotropic polymorphism in WEs as known for many molecules
containing aliphatic chains.
The data gathered in Fig. 4 show a clear relation between the
caloric properties and the CN of the WEs. The data on the melting
temperature and molar heat of fusion of pure WEs are averaged data per
CN and \(CN\), accumulated after critical review of the different
sources . Most temperature data was obtained by methods like FT-IR or a
hot-plate. However, only few literature data regarding the heat of
fusion exists. Black crosses indicate onset temperatures and molar heat
of fusion data determined experimentally in this work.
Fig. 4 shows that for a single CN, distinct different data
points are found. The WEs experimentally studied for this manuscript are
selected such that at equal CN either low values of\(\text{CN}\ \in[-2;0;\ +2]\) or higher values of\(\text{CN}\ \in[-8;\ +8]\) are studied. This data
clearly shows that substantial asymmetry,\(CN\in[-8;\ +8]\), causes lower caloric values
compared to more symmetric references. Because of the limited
possibility to study more complete rows of symmetry, the data does not
suggest other insights. The experimental data gathered are, however, in
general in agreement with the literature values (grey circles). The
broad spreading at equal CN displayed is due to either different
symmetry and also the difference in data acquisition, method and purity
of the WEs. Overall, the melting temperature shows a non-linear increase
with increasing CN. For n-alkanes and monoacid TAGs, Chickos and Nichols
successfully modeled the melting temperatures using a hyperbolic
equation . In contrast, the increase of the molar heat of fusion with CN
could be considered linear. Such a behavior of the thermal properties
has been previously reported for other aliphatic long-chained molecules
like n-alkanes , FA or FaOH .
In Fig. 5 , the same averaged data on the melting temperatures
of pure WEs of different CNs and symmetries is depicted as a function of\(CN\) . Again, black crosses indicate the onset temperatures of the WEs
investigated in this manuscript by means of DSC.
Symmetric WEs (\(\text{CN}=0\)) exhibit the highest melting
temperatures for a given CN. With increasing asymmetry, thus increasing
absolute value of CN, the melting temperatures decrease. This
effect is almost independent from the type of asymmetry, being caused by
either a longer FA or FaOH residue. Astonishingly, WEs with aCN of \(+2\) systematically yield higher melting
temperatures (supported by bold lines) than their isomers with aCN of \(-2\). This may correspond to the data shown inFig. 3 , revealing that WEs with a CN of \(+2\)crystallize in orthogonal chain arrangement. Esters with a value of\(\text{CN}=-4\) were found to crystallize in the orthogonal
configuration as well (see Fig. 3 ). However, their melting
temperatures do not differ considerably from the smooth evolution of
melting temperatures over CN as for \(+2\).
Considering the crystallographic and thermal properties data of pure
WEs, it is underlined that next to the CN also the symmetry, thus the
position of the ester bond in the alkyl chain is an important
characteristic that needs to be taken into account. In a recent
publication , it was already established that symmetric and asymmetric
WEs crystallize differently and that these differences also propagate
into WE-based oleogels.
Mono-Ester oleogels
Thermal properties
The DSC analysis was performed for the oleogels composed of MCT-oil and
10 % (w/w) of a single WE (mono-ester gels). In line with the data
displayed in Fig. 1 , the thermograms for oleogels formed with
MCT-oil and either SFX (gray) or pure WE C42 (22_20) (black), dosed at
10 % (w/w) respectively, are shown in Fig. 6 . The data reveals
a more accentuated peak of the gel-sol transition for the WE-gel. The
WE-gel undergoes this transition at lower temperatures, being attributed
to the longer average CN of the WEs in SFX. The fact that the WE-gel has
a more accentuated peak is accounted for by the single crystalline phase
in this gel compared to multiple solid phases in the SFX-based oleogel.
The latter is indicated by the shape of the peak, showing a more
pronounced hump. In line with the thermogram shown in Fig. 6 ,
all mono-ester gel systems revealed a single melting peak without
indicating any polymorphic transitions under the present conditions.
This was independent of the CN and \(CN\). With increasing CN, the
gel-sol transitions occur at higher temperatures.
In Fig. 7 , the dissolution temperatures and the molar heat of
fusion as a function of CN are gathered for pure WEs (gray) and the
oleogels consisting of MCT-oil and a single WE (10 % (w/w)) (black).
Also for the gels, a systematic increase of the caloric properties with
increasing CN was found. Obviously, the gel-sol transition temperatures
of the mono-ester gels are lower than the melting temperatures of the
respective neat structurants.
The display of data reconfirms that the symmetry of the WE alkyl chain
is of significance to define both, pure component (Fig. 4 andFig. 5 ) and gel properties, see Fig. 7 . The effect on
the gel properties, thus molar heat of fusion (right) and gel-sol
transition temperatures (left), is less clearly documented than for the
pure components (see Fig. 5 for comparison). However, the
higher stability of WE C34(18_16) over WE C34 (16_18) is found in the
molar heat of fusion as much as the effect of asymmetry for the WE with
a CN of 36. Considering the right graph showing the molar heat of
fusion, the pure WE data (gray symbols) are shown for comparison. The
data reveal that the differences between different WE are magnified in
the gels, see CN 34 and CN 36. It needs to be pointed out here that a
proper accounting for the solubility of WEs in MCT oil should have let
to identical values of the heat of fusion for the WEs either neat or in
the gel. Data on the solubility are however currently not available.
Actually, the differences in enthalpy of fusion illustrated inFig. 7 allow to calculate the solubilities of the WE in
MCT-oil. With that in mind, the fact that the distance between the
curves drawn in Fig. 7 becomes smaller with increasing CN
properly illustrates the reduced solubility of larger molecules within a
homologous series.
Overall, Fig. 7 illustrates that the behavior of the pure WEs
propagates to their solidification in oleogels.
Viscoelastic behavior
The amplitude test revealed that the amplitude of 0.005 % was within
the linear-viscoelastic range (LVR) for all oleogels formed by 10
% (w/w) of a single WE in MCT-oil. Further, as all samples exhibited
storage moduli (G’) larger than the loss moduli (G”) in the LVR, they
behaved as gels at 5 °C. This is not surprising since the chosen
concentration is high compared to the critical gelling concentration
known for natural waxes and higher than the concentration used by
Avendaño-Vásquez et al. to investigate the behavior of pure WEs in oil .
Whilst holding the samples at low oscillatory stress for 30 min, no
significant effect of hardening due to further sintering of the primary
crystals or softening due to a weakening influence of small oscillatory
shear stress was detected.
During the temperature tests, the phase transitions (sol-gel or gel-sol)
occur within a small temperature range. This is illustrated inFig. 8 for the oleogel formed with MCT-oil and 10 % (w/w) of
WE C42 (22_20). In logarithmic scaling of the ordinate, this results in
a characteristic, almost sigmoidal shape. The complex shear modulus G*
does not change significantly outside the temperature range in which
phase transitions occur (gray highlighting). For both, gel-sol and
sol-gel transition, good agreement between DSC and rheological data was
found. This indicates that the shear applied at the level used here does
not influence the crystallization behavior as such. The data, however,
do not offer any insights with respect to the effect of shear on the gel
network structure.
The maximum G* values of the mono-ester gels after the holding time,
measured at 5 °C within the LVR, are depicted as a function of CN inFig. 9 . The complex shear modulus is the length of the
resulting vector of the viscous (G”) and the elastic (G’) component and
therefore reflects the viscoelastic behavior of the sample. For
predominantly elastic samples, it is considered as a measure for the
rigidity of the sample, not to be mistaken for the gel hardness . The
highest G* values
(G*C30 = 7.54x106 Pa;
G*C34(18_16) = 7.49x106 Pa) are
observed for gels formed by the shortest WEs with a CN of 30 and 34. As
the CNs increase, the G* of the mono-ester gels decreases
(G*C46 = 2.2x106 Pa). Basically the
same observation was made by Avendaño-Vàsquez for the storage moduli
(G’) of 3 % (w/w) solutions of pure WEs in safflower oil . Also in the
rheological characterization, the two WEs with a CN of 34 differ
significantly. WE C34 (18_16) has a significantly higher maximum G*
than WE C34 (16_18). Within the group of WEs with a CN of 36, it is
remarkable that the WE C36(14_22) yields a higher G*
(G*C36(14_22) = 6.2x106 Pa) than
both, the other WEs with asymmetric, C36(22_14), or symmetric
configuration, C36 (18_18). This is again in line with Avendaño-Vàsquez
as well .
At first sight, it seems to be counterintuitive that at same inclusion
levels, WEs with higher CN and lower solubilities yield gels with
decreased G*. A line of thought to be possibly explored is considering
the dosage of WE in the gel. In this work, dosage is done per mass,
hence, yielding less WE molecules in the sample at higher CN.
Avendaño-Vàsquez et al. observed larger crystals for WEs with higher
CNs, relating this to changes in the crystal-crystal interaction and
elasticity of the oleogels . Unravelling the link between
crystallization behavior, crystal size and viscoelastic behavior of the
gels certainly necessitates more attention.
Mixed oleogels
Thermal properties
Building on the data of the mono-WE gels, also gels using binary
mixtures of WEs as structuring agent were investigated. Four binary
structurant systems were applied at inclusion levels of 10 % (w/w) in
MCT-oil. Within the two WEs, three mixing ratios (1:2; 1:1; 2:1) were
used. The constituting WEs differed in terms of CN and CN.
The heating thermograms of the resulting mixed oleogels are gathered inFig. 10 (black lines). For completeness, the thermograms of the
related mono-ester gels are depicted are shown as well (dashed lines).
Prior to discussing the data, it is meaningful to emphasize that these
data are on oleogels and not on binary mixtures of pure WEs. This
designates the systems, depending on the interpretation of the MCT-oil,
to be at least a ternary system. As dictated by the Gibbs phase, this
has some consequences regarding the degrees of freedom , e.g. the points
relating to three coexisting phases under isobaric conditions do not
need to follow an isothermal line.
Mixing WEs with a large difference in CN, 16 (Fig. 10 (a)) or
12 carbon atoms (Fig. 10 (b)) leads to two separate peaks with
clearly distinguished transition temperatures. It should be noted that
the sharp peaks obtained for the pure WEs (see Fig. 1 ) –
isothermal as equilibrium thermodynamics dictates – are smeared out due
to continuous dissolution of the WEs as the temperature increases. This
is in line with the remark made above. This is also valid for the
mixtures of WEs. For the mixed structurant systems with a high
difference in CN, two thermal events are observed. The evolution of the
peaks is very regular. Starting from the mono-WE gel with the higher
gel-sol transition temperature, the peak temperatures develop
corresponding to the reduced concentration of this WE. Starting from the
WE with the lower gel-sol transition temperature, a similar behavior is
observed. The thermal distance between the two individual peaks observed
in the mixtures increases with higher inclusion levels of the
higher-melting WE. This is in line with the underlying principles of
solubility due to differences in the heat of fusion of the WEs. In
general, the evolution of the peaks implies individual crystallization
of the WEs, suggesting a eutectic behavior in the binary mixtures of
WEs. This involves that the peaks at lower temperatures in the mixtures
relate to three phase (solid-solid-liquid) equilibria. For the system
containing a 1:1 WE mixture of C30 + C46, the high temperature peak is
accompanied by another high temperature shoulder. Currently, the data
available do not allow to derive any explanation, such as the formation
of a compound or alike. One might be tempted to relate this to the
discrepancy in the lower transition temperatures comparing the two
systems, relating in both systems to C30 (16_14). This is, however,
directly related to the fact that a fixed mass fraction C42 (22_20) has
a higher molar dilution effect than the one of C46 (22_24).
The systems containing WEs with smaller difference in CN (two or zero
carbon atoms) behave quite differently (Fig. 10 (c) and (d)).
In Fig. 10 (c), the CNs of the oil structuring WEs differ only
in two carbon atoms. Considering the information shown in Fig.
3 , the actual difference between the two constituting WEs is bigger
than it appears at first sight. The shorter WE C36(18_18) has aCN of \(0\) and crystallizes in orthogonal chain arrangement,
whereas the longer WE C38(20_18) with a CN of \(+2\)crystallizes in the orthogonal chain arrangement. In the first place,
the evolution of the peaks – only a single thermal event in the
mixtures – suggests that the two WEs mix ideally in the solid phase of
the oleogels. Nonetheless, the changes of the peak shape and the
relation to the mono-ester gels indicate a different behavior. However,
in an early contribution by Kreger and Schamhart, the spacings of binary
mixtures of WEs (CN difference of two carbon atoms) have been
investigated by means of XRD . They found that independent of the mixing
proportion, mixtures of WEs crystallizing individually in tilted chain
arrangement, never crystallized in orthogonal chain arrangement. Even if
a WE individually crystallizing in orthogonal chain arrangement is added
to a WE individually crystallizing in tilted chain arrangement at equal
mass ratio (1:1), the mixture still accepts the tilted chain
arrangement. Still awaiting further confirmation, it appears likely that
all mixtures assume the tilted configuration. Additionally, the
different shape of the peak observed in the oleogel formed by the 2:1
mixture recommends further investigations of this system.
The thermograms for oleogels structured by binary mixtures of two WEs
with identical CN 36 but reverse symmetry, either with CN of\(-8\) or \(+8\), are shown in Fig. 10 (d). Except for the
thermogram of the 1:1 mixing ratio, the data suggest ideal mixing in the
solid phase with almost indistinguishable peaks, just as seen inFig. 10 (c). This is also supported by practically invariable
heat of fusion data compared to the mono-ester gels. Similar to the data
displayed in Fig. 10 (a), the mixing ratio of 1:1, here also
equal mole fractions, behaves differently. Again, a small second
dissolution peak at elevated temperatures is discernible. This is
accompanied by a heat of fusion (sum of both peaks), lower than for the
mono-ester gels. Taking into account that the second peak, particularly
the shoulder, has the highest transition temperature, the observation is
possibly related to the interplay of equilibrium and kinetics. The
system might tend to form a 1:1 compound crystal, highest melting point,
which cannot be formed under the given process conditions due to the
complication of the growth regime at equal mole fractions. This may
yield a suboptimal non-equilibrium solid state. The mixtures of two WEs
with identic CN but reverse symmetry and a \(CN\) of either \(-6\) or\(+6\) – C36(14_22) and C36(22_14) – showed single dissolution
peaks at the mixing ratios of 1:2 and 2:1. Their heat of fusion was
comparable to the mono-ester gels. At a mixing ratio of 1:1, thus equal
mole fractions of both WEs, a second dissolution peak at higher
temperatures is discernible. The heat of fusion of the overall thermal
event is lower compared to the mono-ester dissolution peaks.
Interestingly, the second peak occurs at higher temperatures than the
dissolution of the single components what is quite interesting. Until
now, no explanation for this behavior is possible and requires further
investigations.
Without further detailed analysis and crystallographic data, the chain
arrangement of these mixtures remains unresolved.
Viscoelastic behavior
The mixed oleogels studied were subjected to rheological
characterization of the viscoelastic behavior. The amplitude-strain test
confirmed that a strain of 0.005 % lies within the LVR. During the
temperature tests, a two-step phase transition behavior was observed for
the mixtures with a large CN difference of the constituting WEs,
illustrated by Fig. 11 as an example for the systems of WE C30
(16_14) and C42 (22_20). The DSC data of this system are depicted inFig. 10 (b). The systematic development of G* on temperature
increase almost perfectly matches the DSC data, confirming a two-step
dissolution process for the mixed systems with high CN difference. The
temperature test data also confirmed a two-step gel-sol transition
process for the 1:1 mixture of C36(14_22) and C36(22_14) (data not
shown).
In Fig. 12 , the maximum G* values of the different mixed
oleogel systems are depicted. The mixing ratios of the WEs are again
2:1, 1:1 or 1:2. The values were determined after the samples were
stabilized for 30 min at 5 °C.
The two systems containing mixtures of WEs with a large chain length
difference, i. e. C30 + C46 (black diamonds) and C30 + C42 (empty
circles), show clearly higher G* values than the gel containing solely
the long-chained WE. The values for the 1:1 mixture of C30+C46
(G*30+46(1:1) = 1.17x107 Pa) are also
substantially higher than those for the short-chained WE, e.g.,
C30(14_16) (G*C30 = 7.54x106 Pa. For
the systems containing WEs C30+C42, the picture follows a similar trend,
but less pronounced. The G* level of the mixtures remains in the range
of the short chain reference. Thus, if the magnitude of the G* values is
taken as a measure for the rigidity of the samples, the mixture of two
WEs with a large difference in CN leads to a more rigid oleogel network.
The observation of lipid network structures developing higher resistance
to deformation if sequential crystallization events occur is well
established for fat crystal networks . This mechanism is usually
designated as sintering. Accordingly, it is presumable that the lower
melting WEs crystallize on the original scaffolding of the higher
melting component. This type of observations in the field of non-TAG
oil-structuring was already made by other authors .
Regarding the G* values of the systems containing mixtures of WEs with a
small chain length difference, i.e., C36 + C38 (black triangles) and C36
+ C36 (empty squares) do not show a positive synergy upon mixing. In
contrast, at a mixing ratio of 1:1, lowest G* values were yielded
(G*C36+38 (1:1) = 4.17x105 Pa) and the
C36+C36 (G*C36+38
(1:1) = 1.13x106 Pa).