INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,  
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ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIV, Issue XI, November 2025  
An Approximation of Energy Difference, Bayer’s Nonlinearity  
Parameter in Two Dissimilar Chain of Mesogens  
Rihana Banu  
Govt.First Grade College, Yelahanka Bangalore  
DOI: https://doi.org/10.51583/IJLTEMAS.2025.1411000054  
Received: 18 November 2025; Accepted: 27 November 2025; Published: 08 December 2025  
ABSTRACT  
In the present work density measurements are carried out on P-Cyanobenzylidene P- Nonyloxyaniline , liquid  
crystalline compound. The density is observed to drop with grow of temperature in mesogic phases expect in  
the vicity of stage transition it shows steep grow before it attains equilibrium value of the next part. Using  
density facts the various thermodynamic parameters like Moelwin Hughes parameter, Isochoric hotness  
coordinated of internal pressure, Sharma parameter Huggin’s parameter, Gruneisen parameter, etc. are assessed.  
The Bayer’s nonlinearity parameter is also estimated. All variables exhibit discontinuity at phase transformation.  
The outcome is discussed with instance to the literature facts accessible on number of samples.  
Keywords: Mesogens, density, phase transition, thermodynamic parameters, Bayer’s nonlinearity parameter.  
Preface  
The Mesogenic materials shows a different mesophases with coin of hard core, side chain and terminal groups  
[1-4]. The lengthy terminal side chains quench the high temperature nematic phase thereby manifesting the  
smectic phases. The position of oxygen atom in the molecular moiety changes the transition temperatures.  
Further the dilatometric studies involving temperature variation across different phase transformation in liquid  
crystal materials provide information regarding the nature of phase transition and pre transitional effects [5-9].  
The compound viz., P-Cyanobenzylidene P-Nonyloxyaniline differs from other compounds with electro  
negative oxygen atom on right side of the rigid core. The selected sample shows Nematic and Smectic phases.  
In the present paper the density estimation as a function of temperature are carried out and the thermal expansion  
co-efficient at phase transition are noted. By using density and thermal expansion coefficient data the number of  
thermodynamic parameters is evaluated and outcomes are discussed.  
The molecular formula of the liquid crystals used is shown below.  
C1: NCC6H4CH = NC6H4O (CH2)8CH3 (P-Cyanobenzylidene P-Nonyloxyaniline)  
The Structure of the sample as shown below:  
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15  
62.78  
10  
Cooling  
5
97.08  
0
102.94  
-5  
Heating  
-10  
-15  
20  
40  
60  
80  
100  
120  
Temperature oC  
Thermograms and Transition temperature of compound P- Cyanobenzylidene P- Nonyloxyaniline shown in  
Fig.1, Table 1.  
Table 1: Transition temperature in oC recorded by using Differential Scanning Calorimetry (DSC)  
Transition temperature in oC  
Compound  
C1  
I-N  
N-SmA SmA-Cr  
97.08 62.78  
102.94  
Experimental  
The density computation is done by using specially made pyknometer is used. The pyknometer made of  
capillaries with a radius of about 600μm and 5x10-2 to 8 x10-2 meters of length is mounted on ‘U’ shaped designed  
glass tube. The pyknometer is adjusted by measuring the molar volume of water at various temperatures. The  
above liquid crystalline compound is loaded in pyknometer and its weight is noted using the highly accurate  
chemical balance.  
The pyknometer is kept in heating block of dilatometer and the temperature of the heating block is raised 5oC  
above the clearing temperature. Then the sample is slowly cooled until the sample level reaches the marks in  
capillaries. The excess sample in the cups of the capillaries was removed by syringe. Conventional cathetometer  
was used to measure the liquid crystal level in the capillaries. The main scale and vernier scale is replaced with  
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a digital scale. A CCD camera is attached to the telescope and the levels of the capillary were observed on a  
monitor with a high magnification. By this technique the temperature variation of density and hence thermal  
expansion coefficient was measured to estimate various thermo dynamical parameters.  
Theory and expressions:  
The theory for the estimation of different thermodynamic parameters using the coefficient of thermal expansion  
(α) is reported by several authors (Ranga Reddy et al., 1999[10]; Fakruddin et al., 2010[11]). The Moelwyn-  
Hughes parameter (Moelwyn-Hughes, 1951[12]), Beyer’s nonlinearity parameter and the reduced molar volume  
˜
V
(
)
are evaluated from the following expression.  
4
−
1
(
)
(
)
( )  
1
= 13  
+ 4  
1
3
3
( ) = 1 − 1  
(2)  
= [1 +  
3
]
(
)
3 1 +  
(3)  
Using the coefficient of thermal expansion Haward and Parker(Haward and Parker, 1968[13]) obtained an  
expression for the isochoric temperature coefficient of internal pressure (X) as  
−
2
(
1
+
2
)
=
(4)  
1
The Sharma parameter (So) (Sharma, 1983[14]; Reddy et al., 2007[15]) is given by the  
expression  
−
(
)
=
3 + 4  
(5)  
The isothermal microscopic Gruneisen parameter (Γ) is a measure of volume dependence of the harmonicity of  
related to F and S0 as  
the normal mode frequency (ν) of a molecular vibrations of a polymer and is  
2
Γ
=
(
3
2
+
+
4
)
+ (  
2
)
(6)  
The fractional free volume (f) is a measure of disorder due to increasing mobility of molecules in a polymer and  
can be expressed in terms of the isothermal microscopic Gruneisen parameter (Γ) as  
1
= ( ) = (  
)
(7)  
г
+
1
Where Va is the available volume of a liquid crystal.  
Thermal parameter (A*), is a dimensionless parameter which shows that at low temperatures, a liquid crystal  
tends to be ordered exhibiting a small thermal expansion and small fractional free volume, thereby making A*  
equal to unity.  
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2
∗
1
+
= ( ) = 1 + (  
)
(8)  
1
−
(
)
The Gruneisen parameter  
= (21 ) + ( ) + 2  
for liquid crystals can be found from  
P
(9)  
3
2
The isothermal, isobaric and isochoric Gruneisen parameters are identical to the corresponding acoustical  
parameters so one can write  
Γ
=
Γ
+
Γ
(10)  
i
c
h
i
t
h
i
b
a
The isochoric Gruneisen parameter Γich could be evaluated using the following equation  
E − F  
Γich = −  
Where (11)  
−
1
C1  
[
(
)−1]  
F E = − 2 + αT  
[
2
α
(
V
̃
)
] and F=-2α  
(12)  
RESULTS AND DISCUSSIONS  
The heat variation of ( ) and thermal expansion coefficient (α) for the liquid crystalline compounds is measured  
and illustrated in Fig. 3 and Fig.4. The density is found to decrease with increase of temperature in LC  
phases but in the vicinity of phase transition there is a steep jump in density. The density data is used to estimate  
the number of thermodynamic parameter viz, Moelwyn - Hughes parameter(C1), reduced molar volume(Ṽ),  
(f),  
Isochoric Temperature Co-efficient of Internal Pressure(X), Sharma parameter(So), fractional free volume  
Thermal parameter(A*), Gruneisen parameter(ΓP), Huggin’s parameter(F) isothermal microscopic Gruneisen  
parameter(Γ), isothermal Gruneisen parameter(Γith), isochoric Gruneisen parameter(Γich), isobaric Gruneisen  
parameter(Γiba) and Bayer’s nonlinearity parameter(B/A)  
and shown in table 2 and table 3.  
From our studies it is reported that the reduced molar volume and fractional free volume is found to increase  
with increase of temperature and there is a sudden increase in this values at phase transition. This is due to the  
fact that at phase transition the molecules in a compound align in a particular direction hence critical volume  
decreases, due to this reduced molar volume and fractional free volume increases. The percentage of increase in  
reduced molar volume in isotropic to nematic phase is 41% and Smectic to nematic phase is 24% in P-  
Cyanobenzylidene P-Nonyloxyaniline. temperature. Similar results are observed in fractional free volume also.  
The Moelwyn - Hughes parameter (C1) and Bayer’s nonlinearity parameter (B/A) falls with decreases of heat at  
phase transitions. There is a steep decrease in these values because these two parameters depend on  
compressibility. At phase transition the molecules in a compound are aligned in a particular direction and hence  
compressibility decreases therefore C1 and B/A values decreases. During isotropic- nematic transition the  
decreases in these values are very high when compared to smectic-nematic transitions. This is due to the fact  
that at higher temperatures compressibility is low; in liquid crystal research isotropic-nematic transition  
temperature is more than smectic - nematic transition temperature. Similarly, the values of the parameters viz,  
parameter(So), Thermal parameter(A*),  
Isochoric Temperature Co-efficient of Internal Pressure(X), Sharma  
Gruneisen parameter(ΓP), Huggin’s parameter(F) isothermal microscopic Gruneisen parameter(Γ), isothermal  
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Gruneisen parameter(Γith), isochoric Gruneisen parameter(Γich), isobaric Gruneisen parameter(Γiba) decreases at  
phases transitions.  
Fig. 3: Temperature versus density and thermal expansion coefficient graph of compound P-Cyanobenzylidene  
P-Nonyloxyaniline  
Table 2: Thermodynamic parameters of liquid crystalline compound (C1) P-Cyanobenzylidene P-  
Nonyloxyaniline  
̃
T(K)  
Α
V
X
f
*
C
1
S
o
Γ
p
A
369.5  
370  
0.00033 12.69326 1.112668  
0.00033 12.6824 1.112808  
0.0012 7.170531 1.34042  
0.00027 14.44625 1.093841  
0.00027 14.433 1.093959  
-0.64162 1.118897 0.074523 1.006001 6.181835  
-0.6415 1.118908 0.074593 1.006013 6.176404  
-0.4623 1.104525 0.146692 1.025218 3.421006  
-0.65704 1.117194 0.064816 1.004492 7.058294  
-0.65694 1.117206 0.064879 1.004501 7.051666  
-0.65685 1.117218 0.064943 1.004511 7.045056  
370.5  
371  
371.5  
372  
0.00027 14.41978 1.094078  
0.00027 14.40659 1.094196  
0.00027 14.39344 1.094314  
0.00027 14.38033 1.094433  
0.00027 14.36725 1.094551  
0.00027 14.35421 1.094669  
0.00027 14.34121 1.094787  
0.00031 13.07551 1.107934  
0.00196 6.667083 1.487115  
372.5  
373  
-0.65675 1.117229 0.065007  
1.00452 7.038464  
7.03189  
-0.65665 1.117241 0.065071 1.004529  
373.5  
374  
-0.65655 1.117253 0.065134 1.004538 7.025334  
-0.65646 1.117265 0.065198 1.004547 7.018796  
-0.65636 1.117276 0.065261 1.004556 7.012275  
-0.65626 1.117288 0.065325 1.004566 7.005772  
-0.64548 1.118503 0.072162 1.005612 6.372951  
374.5  
375  
375.5  
376  
-0.35106 1.044021 0.167921 1.033888  
3.16977  
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376.5  
377  
0.00021 17.08298 1.075076  
0.00021 17.06634 1.075171  
0.00021 17.04975 1.075265  
-0.67253 1.115146 0.054236  
1.00311 8.376621  
-0.67245 1.115157 0.054291 1.003117 8.368304  
-0.67238 1.115169 0.054347 1.003123 8.360009  
377.5  
378  
0.00021 17.03321  
1.07536  
-0.6723  
1.11518 0.054403  
1.00313 8.351736  
378.5  
0.00021 17.01671 1.075454  
-0.67222 1.115191 0.054459 1.003137 8.343485  
Table 2(continued)  
T(K)  
F
Г
B/A  
Γ
Γ
Γ
iba  
ith  
ich  
369.5  
370  
1.479037  
1.478702  
0.869499  
1.523481  
1.523204  
1.522927  
1.522649  
1.522372  
1.522095  
1.521818  
1.52154  
12.41861  
12.40612  
5.816981  
14.42836  
14.41319  
14.39807  
14.38299  
14.36794  
14.35294  
14.33798  
14.32306  
14.30818  
12.85773  
4.955182  
17.43807  
17.41912  
17.40021  
17.38136  
17.36255  
5.846632  
5.841201  
3.085265  
6.723127  
6.716499  
6.709889  
6.703297  
6.696722  
6.690166  
6.683627  
6.677106  
6.670603  
6.037757  
2.833541  
8.041489  
8.033172  
8.024877  
8.016604  
8.008353  
3.072015  
3.069195  
1.574987  
3.520581  
3.517164  
3.513757  
3.510358  
3.506969  
3.503589  
3.500218  
3.496856  
3.493504  
3.169746  
1.352442  
4.191147  
4.186884  
4.182633  
4.178393  
4.174165  
2.774617  
2.772005  
1.510278  
3.202546  
3.199335  
3.196132  
3.192938  
3.189753  
3.186577  
3.183409  
3.18025  
11.69326  
11.6824  
6.170531  
13.44625  
13.433  
370.5  
371  
371.5  
372  
13.41978  
13.40659  
13.39344  
13.38033  
13.36725  
13.35421  
13.34121  
12.07551  
5.667083  
16.08298  
16.06634  
16.04975  
16.03321  
16.01671  
372.5  
373  
373.5  
374  
374.5  
375  
1.521263  
1.490278  
0.368445  
1.567112  
1.566894  
1.566676  
1.566457  
1.566239  
3.177099  
2.868012  
1.481099  
3.850342  
3.846287  
3.842243  
3.83821  
375.5  
376  
376.5  
377  
377.5  
378  
378.5  
3.834188  
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Fig. 4: The variation of Beyer’s parameter of nonlinearity (B/A) Isochoric Temperature Co-efficient of Internal  
Pressure (X) with temperature in P-Cyanobenzylidene P-Nonyloxyaniline compound  
˜
V
Fig. 5: The variation of reduced volume  
(
), Fractional free volume (f) with temperature in P-Cyanobenzylidene  
P-Nonyloxyaniline compound  
Fig. 6: The variation of Moelwyn- Hughes parameter (C1), Beyer’s nonlinearity parameter (B/A) with  
temperature in P-Cyanobenzylidene P-Nonyloxyaniline compound  
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CONCLUSION  
From these studies it is noted that the sudden change in density and discontinuity in thermal expansion co-  
efficient attributes to the sudden change from ordered liquid crystal phases disordered isotropic phase.  
These phases differs mainly in the degree molecular orientation due to this the thermodynamic parameters also  
changes at phase transitions.  
ACKNOWLEDGMENT  
The author is thankful, GFGC Yelahanka Bangalore for providing laboratory facilities to carry out this work.  
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