PHYSICAL REVIEW E 98, 062701 (2018)

Effect of Sm- A short-range order on the activation energies of translational and rotational
viscosities of nematic liquid crystals with highly polar molecules
M. Praveen Kumar, D. Venkata Sai, and Surajit Dhara*
School of Physics, University of Hyderabad, Hyderabad-500046, India(Received 27 August 2018; revised manuscript received 31 October 2018; published 19 December 2018)
We report experimental studies on the temperature-dependent translational and rotational viscosities of afew nematic liquid crystals made of highly polar molecules. Cyanobiphenyl (nCB) with longitudinal andcyanobicyclohexane molecules (CCN-mn) with transverse dipole moments were chosen and the measurements
were made in the nematic phase. The translational viscosity (shear viscosity) is measured by using a rheometerand the rotational viscosity is measured from the dynamics of the Freedericksz transition by measuring theoptical phase-retardation decay of the sample. The associated activation energies are obtained from the fitting of
the appropriate equation in the nematic phase. Only 8CB and CCN-47 exhibits N to Sm-A phase transition andtheir activation energies of translational and rotational viscosities are larger compared to the shortest homologs
in the respective series. Both activation energies of 8CB are significantly larger than that of CCN-47. The
temperature-dependent steady states of the director distribution under shear are comprehensibly reflected in
the rheodielectric measurements of 8CB than that of CCN-47. The results are explained based on the differencesin the Sm-A short-range orders and the critical slowing down of the order parameter fluctuations. We present
a simple physical model of molecular association that gives rise to different Sm-A short-range orders in bothclasses of compounds.
DOI: 10.1103/PhysRevE.98.062701
I. INTRODUCTION

The flow properties of nematic liquid crystals made ofelongated molecules are very complex compared to theirisotropic counterparts because in most cases the flow disturbs
the alignment and conversely the change in alignment may
induce flow in the nematic [1]. The translational (shear)viscosity due to flow depends on the orientation angle of thedirector (n̂) with respect to the shear and the velocity gradientdirections. Miesowicz first defined and measured three flow
viscosities, namely [2,3] η1 (n̂||∇v), η2 (n̂ ⊥ ∇v), and η3(n̂ ⊥ v and ∇v). Figure 1 schematically represents mutual orientation of the director, velocity (v), and the velocity gradient(∇v) directions, leading to three viscosities. These viscosities
can be expressed in terms of six Leslie coefficients αi [4–6].
There are several studies on the effect of molecular structureon the viscosities and their temperature dependencies [7–15].
The flow alignment angle of the director√ with respect to theshear direction is given by θ = tan−1 ( α3 /α2 ) [16]. In thevicinity of the nematic-isotropic (N-I) transition, usually bothare negative and the director is aligned along the flow direction
with a small angle (θ  5◦ ).
Safinya et al. reported an interesting effect of criticalslowing down of the Sm-A order fluctuations on the dynamics
of nematic director of the liquid crystals exhibiting N-Sm-A
phase transition [16,17]. They showed that near the N-Sm-A
transition, renormalized α3R becomes positive due to the divergence of correlation length ξ|| and the nematic director evolves
through a precessional motion along the neutral direction, z*

sdsp@uohyd.ernet.in

2470-0045/2018/98(6)/062701(7)(perpendicular to both the velocity and velocity gradient directions) which can be described by an equation of ellipse [16]:n2x (t )/n2x0 + n2y (t )/n2y0 = 1, where nx (t ) = nx0 cos(ω0 t ) andny (t ) = ny0 sin(ω0 t ). The precessional frequency ω0 depends
on the shear rate γ̇ and the Leslie coefficients α2 and α3R .
Consequently, several steady states with different director distributions are observed as the N-Sm-A transition is approacheddepending on the relative amplitudes of the projected director components along x and y directions, respectively.
A schematic representation of different distribution types ofn̂ is shown with shaded areas covering the unit spheres inFig. 1(e). They are denoted as am , as , a(b), and ac [16].
In am type, nx0 > ny0 ; in as type, nx0  ny0 ; in a(b) type,nx0 < ny0 [not shown in Fig. 1(e)]; and in ac type, nx0  ny0 .
The director dynamics presented in Fig. 1 has been capturedby three-dimensional nonuniform orientation simulations using Leslie-Ericksen equations by Han et al. [18,19]. The
effect of such distribution of director has also been investigated experimentally by nonequilibrium rheodielectric studies[20–22]. Rotational viscosity (γ1 ) is another important property, which is directly related to the electrooptical responsetime of the liquid crystal displays. There are several experimental studies on the measurement of rotational viscosity[14,15]. There are also some reports in which attempts weremade to correlate these two viscosities [10,11].
In this paper we report experimental studies on the temperature dependence of shear and rotational viscosities of ahomologous series of cyanobiphenyl with longitudinal andcyanobicyclohexane liquid crystals with transversely polar
molecules. We show that the activation energies in the nematic
phase of both shear and rotational viscosities of compounds
having N-Sm-A phase transition are much higher than that

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M. PRAVEEN KUMAR, D. VENKATA SAI, AND SURAJIT DHARA

FIG. 1. [(a)–(c)] Schematic diagrams showing three different director orientations with respect to shear and corresponding Miesowicz viscosities. Schematic representation of different types of distribution of n̂ on the unit sphere [16]. The first sphere from theright-hand side represents the “b” orientation corresponding to theviscosity η2 . The shaded regions on the subsequent unit spheres show
the excursion of the director under shear in different distribution
types.

of compounds having no N-Sm-A phase transition in thehomologous series. We discuss the effect of Sm-A short-rangeorder on the rotational viscosity and the critical slowing downof the Sm-A order parameter fluctuations on the shear viscosity of cyanobicyclohexane and cyanobiphenyl compounds
exhibiting N-Sm-A phase transition. Both the effects are muchstronger in liquid crystals with longitudinal dipole moments
than that of transversely polar molecules.
II. EXPERIMENTAL

We studied a few compounds from the homologous seriesof cyanobiphenyl (nCB) and cyanobicyclohexane (CCN-mn)
liquid crystals. The cyano group (-CN) is oriented along thelongitudinal and transverse directions for nCB and CCN-mn
molecules, respectively. The phase-transition temperaturesand molecular structures of the compounds are shown in
Table I and in Fig. 2, respectively. The CCN-mn compounds
have bicyclohexane core hence exhibits very low birefringence (n  0.03) [23]. CCN-35 exhibits only nematicisotropic (N-I) phase transition. CCN-47 exhibits both N-I and
N-Sm-A phase transitions. CCN-73 and CCN-55 exhibits bothN-I and N-Sm-B phase transitions. The dielectric anisotropyof the compounds are largely negative ( < 0) due to thepresence of transverse dipole moments (see Fig. 2) [23–26].
The nCB compounds are well known. They exhibit large
positive dielectric anisotropy and large birefringence.
TABLE I. Phase-transition temperatures of CCN-35, CCN-73,
CCN-47, and CCN-55 compounds.
SamplePhase transitions (◦ C)

Nematic range (◦ C)

CCN35
CCN73
CCN47
CCN55

Cry 38.4 N 49.3 I
Cry 38.6 Sm-B 38 N 50.2 I
Cry 25.6 Sm-A 28.2 N 57.3 I
Cry 25 Sm-B 30 N 66.4 I

10.9
12.2
29.1
30.4

PHYSICAL REVIEW E 98, 062701 (2018)

FIG. 2. Chemical structure of cyanobiphenyl (nCB) and cyano
bicyclohexane (CCN-mn) compounds. The red arrows show thedirections of permanent dipole moments of -C≡N group.

Shear viscosity was measured by using a rheometer (Anton
Paar MCR-501) in cone-plate geometry with a plate diameterof 25 mm and cone angle of 1◦ . For measuring the dielectric
constant under shear, we used a parallel plate configuration
with a gap of 75 μm. The dielectric constant was measured asa function of temperature with the help of an LCR meter (Agilent E4980A) at an applied voltage of 0.5 V and frequency
4.11 kHz. The details of the rheodielectric setup is presented
in our previous reports [21,22,27–30]. The temperature of thesample was controlled with an accuracy of 0.1◦ C. All themeasurements were made on cooling the samples from theisotropic phase.
Rotational viscosity of the CCN-mn compounds was measured in homeotropic cells by using a phase-decay timemeasurement technique [31]. The experimental setup consists of two Glan-Thompson polarizers, a He-Ne laser, anda photomultiplier tube (Hamamatsu). The nematic director
was deformed by applying a small voltage just above theFreedericksz threshold value. The director relaxes when thevoltage is switched off at t = 0. The corresponding decay inthe phase retardation is approximately expressed as δ(t ) =δ0 exp(−2t/τ0 ), where δ0 is the total phase change of the celland τ0 is the relaxation time. The rotational viscosity is givenby γ1 = τ0 K33 π 2 /d 2 , where d is the cell gap and K33 is thebend elastic constant. In case δ0 is close to nπ , the retardation
decays as δ(t ) = δ0 exp(−4t/τ0 ). τ0 is obtained from the slopeof the linear plot of ln[δ0 /δ(t )] with time, t [13,31].
III. RESULTS AND DISCUSSIONFirst we performed rheodielectric measurements of nCB
liquid crystals at a fixed shear rate (γ̇ = 100 s−1 ) and theresults are shown in Fig. 3(a). The effective shear viscosity(ηeff ) of both 5CB and 7CB decreases discontinuously below
the N-I phase transition and then continue to increase as
the temperature is reduced. The corresponding rheodielectric
data suggest that in the nematic phase εeff  ε⊥ . Hence thedirector gets “b” orientation in the nematic phase as describedin Fig. 1(b) and, consequently, ηeff  η2 . The rheodielectric

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PHYSICAL REVIEW E 98, 062701 (2018)

FIG. 3. Temperature dependence of effective viscosity (ηeff ) andthe dielectric constant (εeff ) as a function of temperature measured
at a fixed shear rate γ̇ = 100 s−1 for (a) 5CB and 7CB and for(b) 8CB. Solid and open symbols represent the viscosity and dielectric constant, respectively. The steady states with director distribution
types “a,” “a − b” (intermediate regime of “a” and “b”), “am ,” “as ,”and “ac ” are labeled in correlation to the shear induced distributions
reported by Safinya et al. [16]. Red arrow in the bottom figure
indicates N-Sm-A phase-transition temperature.

data of 8CB looks very different in comparison to 5CB and
7CB, especially near the N-Sm-A phase transition [Fig. 3(b)].
A small discontinuity around TNI − T = 2.2◦ C indicates achange of director orientation from “b” to an intermediate
regime, labeled “a − b.” Corresponding change is also reflected in the temperature variation of εeff . Below TNI − T =5◦ C, the average director orientation is changed along theneutral direction (z direction). The temperature variation ofdielectric constant reflects the effective orientation of thedirector with respect to the applied electric field under shear.
While decreasing the temperature from the isotropic phaseat a few temperatures both the values and slopes of ηeff andeff simultaneously changes indicating changes in the director
distribution. These temperatures are marked by verticallydotted line in Fig. 3(b). Both ηeff and eff increase discontinuously at the N-Sm-A phase transition (TNI − T  7.5◦ C),
which is marked by an upward arrow in Fig. 3(b). This isbecause of the weak first-order N-Sm-A phase transition of
8CB. The steady states with different director distributions,
namely am , ac , and as , are identified from the change ofslope in εeff and in ηeff and they are consistent with theresults of synchrotron x-ray studies reported by Safinya et al.
[16]. They showed that there is an interplay of frictional andflow-induced fluctuation forces on the nematic director as
the N-Sm-A transition is approached. Consequently, several
steady states as mentioned emerge due to the critical slowing

down of the Sm-A order parameter fluctuations. This effectis clearly revealed in the rheodielectric data. Although Negita
et al. reported rheodielectric measurements on 8CB [20] using
a homemade concentric double cylinder–type viscometer, ourrheodielectric measurements using a cone-plate and parallel
plate geometries of MCR 501 rheometer are able to capture
the fine structure of director distributions associated withdifferent steady states. It may be mentioned that the transition
temperatures of different steady states weakly depend on theapplied shear rates. To avoid this, we kept the low shear, i.e.,γ̇ = 100 s−1 , in our experiments.
There are several experimental studies on the anchoringtransition of CCN-47 [32–34]. It was proposed that there
exists a smectic short-range order in the nematic phase[35–37]. Inspired by this, we studied the rheodielectric properties of CCN-47 and also a few compounds of the homologous series. Figure 4 shows the temperature dependence ofeffective shear viscosity (ηeff ) and effective dielectric constant(εeff ). For all the compounds, ηeff decreases below the NI phase transition and then increases as the temperature is
CCN−47increases more rapidly
decreased. It is observed that ηeffin the nematic phase compared to the other homologs as the
N-Sm-A transition is approached. For example, there is asmall slope change at a temperature of TNI − T = 7◦ C, and
we identify this as a change of orientation from steady state“b” to an intermediate regime “a − b” type [Fig. 4(a)]. It
has been shown that there is a Sm-A short-range order in
the nematic phase of CCN-47 liquid crystal [35]. Hence itis legitimate to expect the occurrence of steady states dueto the critical slowing down of the Sm-A order parameter
fluctuations similar to that of 8CB. However, we do not see
any distinct variation of effective viscosity (like we see in caseof 8CB) as the N-Sm-A transition is approached except another small slope change in the ηeff at TNI − T = 24◦ C. Thischange is an indication of the change of director orientation
and attributed to the onset of appearance of other steady states“am ,” “as ,” “a(b),” and “ac .” Unlike 8CB, the steady states are
not distinguishable in CCN-47.
The effective dielectric constant εeff measured at the sameshear rate are shown in Fig. 4(b). εeff increases discontinuously for all the compounds below the N-I phase-transition
temperature. This confirms that the steady state form is “b”
type since the dielectric anisotropy of all the compounds
is negative. Above TNI − T = 24◦ C, εeff tends to increase
as indicated by a small vertical arrow in Fig. 4(b). This isconsistent with the viscosity data and is associated to theonset of subsequent steady states. However, they cannot be
distinguished as clearly as we see in case of 8CB (see Fig. 3).
This means that the Sm-A order parameter fluctuations in
CCN-47 are not as strong as that of 8CB. In other words,
ξ|| is much larger in 8CB than that of CCN-47, i.e., ξ||8CB >ξ||CCN−47 . A schematic diagram of the fluctuating Sm-A clusterwith correlation length ξ|| is shown in Fig. 5 when the Deborah
number D = γ̇ τ  1, where γ̇ is the shear rate and τ is thelongest characteristic structural relaxation time [17]. Theseclusters are anisotropic and tend to get distorted due to shear
flow and this is energetically costly.
The effect of Sm-A short-range order fluctuations can beseen on the temperature dependence of ξ|| and hence welooked at the critical exponents in both these compounds,

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PHYSICAL REVIEW E 98, 062701 (2018)

FIG. 5. Schematics of a cluster with the nematic director along
the horizontal direction (a) at rest and (b) under shear flow. The
dotted ellipse indicates Sm-A-type order with correlation length ξ||
along the director.

FIG. 4. (a) Temperature variation of effective shear viscosity(ηeff ) and (b) effective dielectric constant (εeff ) of CCN-mn compounds measured under a fixed shear rate γ̇ = 100 s−1 . The verticalsolid lines and upward arrows in the viscosity data in the top figure
indicates the viscosity increases discontinuously at the N-Sm-A or
N-Sm-B phase transitions. The temperature range of steady states,“b” and “a − b,” “am ,” “as ”, “a(b),” and “ac ” are labeled underthe viscosity data of CCN-47 with dotted arrows. The vertical solidlines and downward arrows in the viscosity data in the bottom figure
indicates the dielectric constant decreases discontinuously at the
N-Sm-A or N-Sm-B phase transitions. The upward arrow below
the dielectric data of CCN-47 indicates a small slope change dueto the occurrence of steady states.

which can be obtained easily from the temperature dependence of bend elastic constant K33 . According to the meanfield theory, the renormalized bend elastic constant varies asR∝ ξ|| (T ). The correlation length diverges as ξ|| (T ) ∼ t −ν ,K33
where t = (T − TN−SmA )/TN−SmA is the reduced temperatureand ν is the critical exponent [17,38]. From the temperaturedependent measurement of bend elastic constant near the NSm-A phase transition it is found that in case of 8CB, ν  1[39], whereas for CCN-47, ν  0.3, which is much smaller
than that of 8CB [40]. The lower critical exponent of CCN-47

is an indicator of weaker Sm-A short-range order. Anothersubstantial difference in the physical properties of these twocategories of compounds is that the bend elastic constantof CCN-47 is much smaller than splay elastic constant, i.e.,
K33 < K11 , whereas in case of 8CB, K33 > K11 [23,40].
The temperature-dependent linear part of the shear viscosity in the N phase of all the compounds are fitted to
the Arrhenius equation, ηeff (T ) = η0 exp(ES /kT ), to get theactivation energy of the translational viscosity. The fittings
for CCN-mn compounds are shown in Fig. 6. For nCB thefittings are performed on the data shown in Fig. 3 but notshown here. The activation energies of shear viscosity in thenematic phase (ES ) of all the compounds are summarized inTable II. This table brings out some interesting features of
the activation energies of the nematic liquid crystals made ofhighly polar molecules with transverse and longitudinal dipole
orientations. It is noticed that ES of CCN-47 which exhibits
N-Sm-A transition is much larger than that of other homologs.
For example, ES of CCN-47 is about 66% larger than thatof CCN-35 (shortest homolog). Similarly in nCB compounds,
8CB has much larger ES than 5CB or 7CB. For example, ES
of 8CB is 185% larger than that of the shortest homolog 5CB.
Hence the activation energy of translational viscosity in the

FIG. 6. Estimation of activation energy of shear viscosity fromthe Arrhenius equation, ηeff (T ) = η0 exp(ES /kT ). Variation ofln(ηeff ) with 1000/T . Solid lines represent the least square fits tothe data. Due to shorter temperature range, CCN-35 and CCN-73 areshown separately in the inset.

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PHYSICAL REVIEW E 98, 062701 (2018)

TABLE II. ES →Activation energy of shear viscosity in thenematic phase. ER → Activation energy of the rotational viscosity
in the nematic phase. ER of 5CB, 7CB, and 8CB were obtained from
Ref. [13], Ref. [42], and Ref. [41], respectively.
SamplesES (N-phase, kJ/mole]

ER (N-phase, kJ/mole]

CCN-35
CCN-73
CCN-47
CCN-55
5CB
7CB
8CB

27
26
45
31
20
30
57

42
39
48
39
27 [13]37 [42]
71 [41]

nematic phase is significantly larger in polar compounds withlongitudinal and transverse dipole moments which also havea Sm-A phase.
We measured the rotational viscosity (γ1 ) of the CCNmn compounds as a function of temperature and they arepresented in Fig. 7(a). We notice that except for very close to
N-I, the γ1 of CCN-47 is relatively larger than other homologs

FIG. 7. (a) Temperature dependence of rotational viscosity (γ1 )of CCN-mn compounds. (Inset) Birefringence (n) of the samples
adapted from Ref. [23]. (b) Inverse temperature dependence ofln(γ1 /n). Solid line represents the least-squares fits to the data.

and it increases rapidly as the N-Sm-A phase transition isapproached. This is expected, as the smectic correlation√ lengthincreases with decreasing temperature, i.e., γ1 ∝ ξ , whereξ ∼ [T /(T − TNA )]0.33 [43]. The rotational viscosity of thenematic is related to the Leslie coefficients and is given by
γ1 = α3 − α2 [10,44]. The renormalized α3R due to the Sm-A
short-range order fluctuations is proportional to η2 (α3R ∝η2 ). Thus Sm-A short-range order is reflected in both thetranslational and rotational viscosities. The temperature dependence of rotational viscosity can be expressed as γ1 (T ) ∼Sexp(ER /kT ) [15], where S is the orientational order parameter and is proportional to the birefringence n, and ER is theactivation energy of the rotational viscosity. The temperature
variation of n of the compounds are shown in the inset of
Fig. 7(a). In the nematic phase it is almost the same for all thecompounds. Figure 7(b) shows a linear variation of ln(γ1 /n)with 1/T . The activation energies obtained from the fitting arealso presented in Table II. The activation energies of rotational
viscosities of nCB compounds were obtained from several
references [13,41,42]. ER of CCN-47, which has Sm-A phase,is also comparatively larger than that of other homologs. For
example, ER of CCN-47 is about 15% larger than the shortest
homolog CCN-35. Similarly, in the nCB homologous series,
8CB has a much larger ER than 5CB or 7CB. For example, theER of 8CB is 163% larger compared to the shortest homolog,
5CB. Hence the activation energy of rotational viscosity in thenematic phase is significantly larger in longitudinally polar
compounds which have a Sm-A phase. This clearly suggeststhat both CCN-47 and 8CB have Sm-A short-range order but
it is comparatively weaker in CCN-47.
We present a simple physical model of molecular association that accounts for the smaller Sm-A short-range orderand activation energies in the nematic phase of CCN-47 thanthat of 8CB. It is known that liquid crystal molecules withstrong permanent dipole moments exhibit antiparallel association between pairs of molecules. The interaction becomesstrongest when the dispersion energy between the aromaticcores having high polarizabilities [45,46]. The induced dipole
moment by the cyano group of 8CB of a neighbor is parallel tothe permanent dipole of the cyano group of the given dipole;consequently, the effective dipole moment of each molecule
is enhanced as shown in Fig. 8(a). The chain-chain dispersion
interaction energy is negligible in this case. Compounds withsuch a strong antiparallel correlation are found to exhibit
strong Sm-A short-range order even if there is no long-range
Sm-A order such as in 5CB or 7CB. In fact, in the case of nCB
the effective molecular length is enhanced due to antiparallelcorrelation. CCN-47 has a strong transverse dipole moment
and a possible molecular association is shown in Fig. 8(b),although in this case the effective transverse dipole moment
is enhanced but the interaction is not strong as the core ismade of bicyclohexane, which has very low polarizability[n  0.03 for CCN-47, see inset of Fig. 7(a)] comparedto the biphenyl moiety. The effective molecular length isalso not enhanced. In this configuration the Sm-A correlation
length and hence the short-range order is expected to bemuch smaller than that of 8CB. The antiparallel correlationof dipoles is usually reflected in the dielectric data across theN-I phase transition. It is known that the average dielectric
constant ε [= (ε|| + 2ε⊥ )/3] of nCB is reduced in the nematic

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M. PRAVEEN KUMAR, D. VENKATA SAI, AND SURAJIT DHARA

FIG. 8. Schematic diagram showing the antiparallel configuration of (a) two 8CB molecules and (b) two CCN-47 molecules. The
arrow with the solid red line represents the direction of permanent
dipole moment and the one with dotted line the induced one. The
polar cyano group (-C≡N) is parallel and perpendicular to the longaxes of the 8CB and CCN-47 molecules, respectively.

phase below the N-I phase transition in highly polar (longitudinal) compounds due to the strong antiparallel correlationof dipoles [46,47]. For CCN-47 and its homologous series,
it has been found that ε remains unchanged across the N-I
transition [24,48]. This confirms that antiparallel associationof molecules and short-range order is smaller in CCN-47 than
8CB.

PHYSICAL REVIEW E 98, 062701 (2018)

and transverse dipole moments in compared to the otherhomologs in the respective series. The enhancement is significant in the case of 8CB than CCN-47. Although both 8CB
and CCN-47 compounds have the same dipolar group, theactivation energies of the rotational and translational viscosities of 8CB are much larger than those of CCN-47 due tostronger Sm-A short-range order in the nematic phase. Usinga simple physical model we showed that the Sm-A short-rangeorder due to strong antiparallel correlation of longitudinal
dipole moments of 8CB molecules which enhances both theactivation energies is expected to be larger in 8CB than in
CCN-47. The director excursion due to the critical slowing
down of the Sm-A order parameter fluctuations in CCN-47
is smaller compared to the 8CB liquid crystal; consequently,
the steady states are not observed. Computer simulations on
nematic liquid crystals made of highly polar molecules in theframework of Leslie-Ericksen formalism can shed more lighton the proposed mechanism. We emphasize that, as far as weknow, ours is the first among a few rheodielectric experiments
to probe the effect of smectic short-range order and criticalslowing down of the Sm-A order parameter fluctuation as the
N-Sm-A transition is approached in highly polar compoundswith both longitudinal and transverse dipole moments. Suchstudies also have important relevancy on our understanding of
the phase transitions in liquid crystals away from equilibrium.
ACKNOWLEDGMENTS

The short-range smectic order in the nematic phase enhances the activation energies of translational and rotational
viscosities of highly polar compounds with both longitudinal

M.P.K. acknowledges UGC for CSIR-UGC fellowship(Ref No: 21/06/2015(i)EU-V). S.D. gratefully acknowledges
the support from the DST (DST/SJF/PSA-02/2014-2015). We
thank N. V. Madhusudana from Raman Research Institute,Bangalore, for useful discussion.

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IV. CONCLUSION

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