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A Publication of Ultrasonics Society of India
A Publication of Ultrasonics Society of India
Journal of Pure and Applied
ISSN 0256-4637
Website : www.ultrasonicsindia.org
Ultrasonics
VOLUME 44 NUMBER 3-4 JULY-DECEMBER 2022
Ultrasonics
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Journal of Pure and Applied Ultrasonics
Editorial 44
S.K. Jain
Temperature dependent ultrasonic characterization of AuRE intermetallics 45
Mohd Aftab Khan, Chandreshvar Prasad Yadav, Mahendra Kumar, Dharmendra Kumar Pandey,
Devendra Nath Mishra and Renuka Arora
Acoustical and excess thermodynamic studies of binary liquid mixtures at varying temperature using 52
ultrasonic technique
C. Duraivathi, J. Jeya Priya, J. Poongodi and H. Johnson Jeyakumar
Ultrasonic study of calcium soaps (laurate and myristate) 58
Mahesh Singh Khirwar, Ashish K. Singh, Sandeep K. Singh, M.K.Rawat and Gyan Prakash
Estimation of effective Debye temperature of polymeric solutions at 303.15 K based on 64
quasi-crystalline model
Monika Dhiman, Arun Upmanyu, Pankaj Kumar, D.P. Singh and Harsh Kumar
Effect of sonication on enhancement of mechanical properties of epoxy blended rattan fibre 74
Susanta Behera, G. Nath and J.R.Mohanty
Investigation of temperature dependent mechanical, thermophysical and ultrasonic properties of 79
ScZrHf ternary alloy
Shakti Yadav, Ramanshu P. Singh, Devraj Singh and Giridhar Mishra
Ph.D. Thesis Summary : Study of intermolecular interaction in binary liquid mixtures through 86
ultrasonic speed measurement at 303.15K
Dr. Seema Agarwal
New Members 87
USI Awards 87
Author Index 88
(Authors have stated that the papers have neither been published nor have been submitted for publication elsewhere)
ISSN 0256−4637
VOLUME 44 JULY-DECEMBER 2022
NUMBER 3-4
CONTENTS
Website : www.ultrasonicsindia.org
A Publication of — Ultrasonics Society of India
EDITORIAL
Dear Colleagues,
Your next issue of JPAU has arrived almost in time easing off your wait for new developments in Ultrasonics.
In the time since my last editorial, the Elections of the Executive Council of USI have taken place in the month
of July 2022. Elections have been held within the scheduled time interval of two years.
I am pleased to inform you that Prof. Raja Ram Yadav has taken over the reins of the Society as President,
USI. The Executive Council of USI decided in its meeting held 1st July 2022 that all past presidents shall be
members of Advisory Board. Prof Krishan Lal and Prof Vikram Kumar have now been honored as members of
Advisory Board of USI.
Other office bearers of USI including Former Vice-President Dr V R Singh, and Chief Editor of JPAU, Dr
S K Jain. Former Chief Editor, by virtue of their long association with USI, have been bestowed with the honor
of Patron of USI. I) Dr Ashok Kumar and (ii) Prof. Vilas Ajabrao Tabhane, both Life Fellow of USI, have been
honored with the Life-Time Achievement Award in Ultrasonics. The new Executive Council of the Society is
published on the inside cover of every issue.
Six papers have been published in this issue which include evaluation of thermodynamic ultrasonic parameters
of calcium soaps and other binary mixtures, elastic parameters of intermetallics and evaluation of effective debye
temperature of polymeric solutions to show that they have their quasi-crystalline structures.
As you may be aware that JPAU is being exchanged with the journal "INSIGHT" which is published by
BINDT (British Institute of Non-destructive Testing). Past issues of the journal are placed in USI Office, at Force
Standards Annexeg of National Physical Laboratory. The same can be accessed for physical consultation by prior
information to the General Secretary, USI, Dr P K Dubey, or Chief Editor, of JPAU, Dr S K Jain. In recent issues
of the journal INSIGHT, the published research is focused on ultrasonic NDT. Notable among these are related
to development of digital tools for remote ultrasonic inspection and a study on concrete modified with rice husk,
brick aggregates, etc. The latter study is by R N Bhowmick and J Pal from Dept. of Civil Engg., N.I.T., Agartala.
Authors are referred to my editorial published in previous issue (JPAU, vol. 44, issue no. 1-2 (2022))
regarding JPAU's offer to submit papers related to ultrasonics, piezoelectric devices and materials published in
conferences, etc. for consideration in JPAU. JPAU reviews such papers positively with suggestions to improve
the contents of manuscripts.
— S. K. Jain
Chief Editor
Journal of Pure and Applied Ultrasonics
J. Pure Appl. Ultrason. 44 (2022) p. 44
45
KHAN ET AL.: TEMPERATURE DEPENDENT US CHARACT. OF AURE
Temperature dependent ultrasonic characterization of AuRE intermetallics
Mohd Aftab Khan1*, Chandreshvar Prasad Yadav1, Mahendra Kumar1,2,
Dharmendra Kumar Pandey1, Devendra Nath Mishra3 and Renuka Arora1
1Department of Physics, P.P.N. (P.G.) College, Kanpur, Uttar Pradesh-208 001, India
2Department of Physics, D.B.S. (P.G.) College, Kanpur, Uttar Pradesh-208 006, India
3Department of Physics, United College of Engineering & Research, Naini, Prayagraj, Uttar Pradesh-211 010, India
*E-mail: aftab1516@gmail.com
The present work incorporates computation of elastic, ultrasonic and thermal properties of B2 structured
AuRE (RE= Sm, Tb, Ho, Tm) intermetallics in temperature range 300K-900K. Initially, elastic constants are
determined under potential model approach. Later on, ultrasonic velocities are obtained in the same temperature
range for wave propagation along <100> and <111> crystallographic directions. Besides it, Debye temperatures,
thermal energy density, thermal expansion coefficient and melting temperature are also determined for chosen
intermetallics. The obtained results are compared and analyzed to explore the inherent properties the chosen
material.
Keywords: Rare-earth intermetallics; elastic properties; ultrasonic velocity; thermo-physical properties.
Introduction
Intermetallics are homogeneous and composite
materials comprising of two or more types of metal
atoms that differ in structure in comparison to their
constituent metals. Binary rare-earth intermetallic
compounds have many practical applications due to their
superior mechanical, thermo-dynamical, electrical and
magnetic properties in comparison to ordinary metals.
These materials possess high strength, ductility, melting
point, good corrosion resistance and low specific heat
which make them important for aerospace and
commercial aircraft turbines applications. Rare-earth
intermetallics exhibit B2 type CsCl structure. The rare
earth elements have electron occupancy in f-shell with
a regular change in atomic dimension and electro-
negativity on moving along rare-earth group1-6.
The study of binary rare earth intermetallics represents
an important part of the metallic phases7. The ab-initio
and First principle studies of rare earth intermetallics
monoaurides (Ln-Au) and Lanthenide-Gold (R-Au)
respectively have been done by Sardar Ahmad and his
co-workers1, 8 to examine the electronic and magnetic
properties of the materials. The thermodynamic
assessment of Au-Ho and Au-Tm binary systems have
J. Pure Appl. Ultrason. 44 (2022) pp. 45-51
been reported in literature7 using the CALPHAD (Cal-
culation of Phase Diagrams) method. The experimental
study of Au, Ag, Cu and Al based rare-earth intermetallic
compounds are reported elsewhere2, 7. The survey of
literatures indicate that, in spite of technological
importance of these rare earth intermetallic, a systematic
theoretical study of structural, elastic, ultrasonic and
thermo-physical properties of these materials, is still
needed for future applications. The present work is
focused on characterization of elastic, ultrasonic and
thermo-physical properties of AuRE (RE= Sm, Tb, Ho,
Tm) intermetallic at different temperature. The
temperature dependent second order elastic constants
(SOECs) and elastic moduli (bulk modulus-B, Young's
modulus-Y, shear modulus-G, Lame modulus-
λ
&
µ
and
Poisson's ratio-
σ
) of chosen binary intermetallics have
initially been determined using potential model approach
considering interaction up to second nearest neighbors.
Subsequently, the ultrasonic velocities for wave
propagation along two crystallographic directions
(<100> and <111>) and thermo-physical properties
(Debye temperature, thermal energy density, thermal
expansion coefficient and melting temperature) are also
determined at same physical conditions.
46 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
Theory
Theory of elastic constants
The elastic constants of B2/CsCl structured materials
have been evaluated on the basis of only two basic
parameters (lattice and non-linear parameters) using
potential model approach. Under this model, second
order elastic constants are determined by the second
order differentiation of elastic energy density with
respect to strain while elastic energy density is attained
by considering interaction among atoms defined by
Coulomb and Born-Mayer interaction potentials
assuming interaction up to second nearest neighbours
with respect to reference atom.
According to Hooke's law, stress is directly
proportional to strain under small deformation. The
generalized form of this law can be written as:
6
IIJJ
I,J = 1,...,
X= C η
∑(1)
Where, XI and
η
J are the components of stress and
strain tensor, respectively. The term CIJ is called as
coefficient of elastic constants. The elastic energy
density (F) of undeformed crystal is function of these
components of strain tensor (
η
J; J=1,….,6)9.
F = (
η
1,
η
2,
η
3,
η
4,
η
5,
η
6) (2)
Therefore, the elastic energy density (F) can be
expanded in terms of strain using Taylor's series
expansion as:
F=
(3)
The strain derivative of elastic energy density provides
the elastic constant of the material. Therefore, the second
order elastic constants are defined as:
2
IJ
or 1 2
6
IJ F
C
; I J , , ...,
ηη
∂
==
∂∂ (4)
The expansion of elastic energy density (F) in terms
of strain is defined by Eq.(3). Using this expression, the
free energy density in square terms of strain can be
written as:
2IJI
J
1
2
FC
!
ηη
=(5)
The tensor form relationship between stress and strain
gives 36 types of SOECs or stiffness constants. Under
the symmetry condition of cubic B2 structured material,
these stiffness constants (CIJ) reduce to only three types
as C11, C12 and C44 while rest becomes zero10. Therefore,
the extension of Eq. (5), under symmetry condition of
B2 structured materials takes the following form:
22 2
21111223312112211332233
222
44 12 23 31
12
2 C
F(/) [C ( ) C( )
( )]
ηηη ηηηηηη
ηηη
⎫
=+++++
⎪
⎬
+++ ⎪
⎭(6)
The elastic energy density (F) at finite temperature
(T) is sum of energy density at 0K (F0) and increase in
energy (vibrational part of energy density: FV) with
enhancement in temperature by amount T11.
0V
F=F +F
(7)
Therefore, Eq. (4) has the following form.
20 2V
IJ IJ IJ
FF
C
ηη ηη
∂∂
=+
∂∂ ∂∂ (8)
The first part in R.H.S of Eq. (8) is termed as static
part of SOECs (
0
I
J
C
) which have a constant value while
second part varies with temperature because vibrational
free energy density depends on temperature recognized
as vibrational parts of SOECs (
V
I
J
C
). Hence, the
expression of SOECs at finite temperature T becomes
as:
0
V
IJ IJ I
J
C
CC=+ (9)
The static and vibrational part of SOECs (
0
I
J
C
&
V
I
J
C
)
can be computed with the expressions given in our
previous work11-16. The estimation of second order
elastic constants requires only lattice and non-linear
parameters as an input under this potential model
approach. The second order elastic constants can be
utilized for the evaluation of bulk modulus (B), Shear
modulus (G), Young's modulus (Y), Lame modulus (
λ
,
µ
) and Poisson ratio (
σ
) as reported in literature12,17.
Theory of ultrasonic velocity
When an ultrasonic waves propagates in the medium,
there are three types of ultrasonic velocities for each
direction of propagation in cubic crystals, one is
longitudinal velocity (VL) and other two are shear
velocities (VS1 and VS2), which can be estimated with
following expressions for the wave propagation along
<100> and <111> crystallographic directions12, 18.
()()
L11 S44
L 11 12 44 S 11 12 44
a
long 100
a
long 111 2 4 3 2 3
: V C / d ; V C / d
: V C C C / d ; V C C C / d
⎫
<> = = ⎪
⎬
<> = + + = −+ ⎪
⎭
(10)
The density of CsCl/B2 structured material can be
determined with the expression given in literature19.
Debye average velocity is an important parameter in the
47
KHAN ET AL.: TEMPERATURE DEPENDENT US CHARACT. OF AURE
low temperature physics because it is related to elastic
constants through ultrasonic velocities. The Debye
average velocity (VD) is defined as given in equation18.
13
D33
LS
11 2
3
-/
V
VV
⎡⎤
⎛⎞
=+
⎢⎥
⎜⎟
⎜⎟
⎢⎥
⎝⎠
⎣⎦ (11)
On the basis of evaluated second order elastic
constants, the ultrasonic velocities of the material also
have been calculated.
Thermo-physical Properties
The temperature corresponding to maximum
frequency of phonon that can propagate through
crystalline media is termed as Debye temperature. The
Debye temperature (TD) is indirectly related to elastic
constants through Debye average velocity18.
1
3
B
2
2
Da
D
hV (6 n )
T
k
π
π
=(12)
Here, h is Planck's constant and na is atom
concentration.
The thermal expansion behavior of material can be
described on the knowledge of thermal expansion
coefficient (
α
) and melting point (MP). The variation in
density/volume of crystalline material is caused by
alteration in lattice parameter with temperature/pressure.
The reduction in density or enhancement in volume of
crystalline medium with increase in temperature leads
the following expression of linear thermal expansion
coefficient14-16.
00
0
33
VV d
d
= T V T d
α
−−
=(13)
Here, d0 and V0 are density and volume at 0K. The
thermal expansion coefficient can be determined using
Eq. (13) on the knowledge of temperature dependent
density. The melting temperature (MP) of the CsCl
structured intermetallic compounds are well related to
the thermal expansion coefficient and is equal to 0.38(α
+ 7.0 × 10–6)–1,14
Results and Discussion
The temperature dependent lattice parameters (a) of
chosen AuRE intermetallics are calculated with help of
formulation given in literature14 and using corresponding
room temperature lattice parameters1, 8. The molecular
weight of AuSm, AuTb, AuHo and AuTm intermetallics
are 347.33 gm, 355.89 gm, 361.90 gm and 365.90 gm,
respectively. The non- linearity parameters (b) of AuSm,
AuTb, AuHo and AuTm intermetallics are determined
under equilibrium condition and are found 0.798 Å,
0.721 Å , 0.663 Å and 0.381 Å, respectively. The
calculated lattice parameters and molecular weights are
utilized for the estimation of the densities at different
temperatures for the chosen intermetallics using the
expression given in literature12, 19. Both the calculated
temperature dependent lattice parameters and densities
are given in Table 1.
Table 1 – Basic parameters lattice parameters (a), non-linear parameters (b), Molecular weight (Mw) and density (d) of AuRE
compounds.
AuRE→AuSm AuTb AuHo AuTm
Parameters↓T[K]
a (Å)
300K 3.621 3.576 3.541 3.516
450K 3.633 3.588 3.553 3.526
600K 3.646 3.600 3.565 3.536
750K 3.659 3.613 3.577 3.547
900K 3.672 3.625 3.589 3.557
b (Å) 0.798 0.721 0.663 0.381
Mw (gm) 347.33 355.89 361.90 365.90
d (gm cm–3) 300K 12.14 12.92 13.53 13.97
450K 12.02 12.78 13.39 13.85
600K 11.89 12.65 13.26 13.73
750K 11.76 12.52 13.13 13.61
900K 11.64 12.39 12.96 13.49
48 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
Thermal expansion coefficient and melting
temperature of selected intermetallics are estimated with
the help of Eq. (13) and are shown in Fig. 1.
The lattice parameters of selected intermetallics have
been found to enhance with temperature and decay with
materials which are received to be maximum for AuSm
while minimum for AuTm intermetallic (Table 1). The
similar nature of lattice parameters of AgRE
intermetallics with temperature have also been reported
in literature16. The increase in lattice parameter is due
to enhancement in average inter-atomic distance caused
by increase in thermal/lattice vibration energy. The low
value of lattice parameter confirms the high inter-atomic
forces; therefore the material AuTm will have higher
inter-atomic strength than AuSm intermetallic. Since,
the selected intermetallics are B2 structured materials
whose densities vary inversely to be third power of
lattice parameter, hence the density of selected
intermetallics has been found to enhance from AuSm to
AuTm while received to decay with temperature. The
calculated density of AuTb, AuHo and AuTm are similar
to reported literature values at 300K1. The electronic
configuration of Au, Sm, Tb, Ho and Tm are
[Xe]4f145d106s1, [Xe]4f66s2, [Xe]4f116s2 and
[Xe]4f136s2, respectively. The configuration reveals that
Au posses half filled s-sub shell while the vacancy of
f-sub shell saturates from Sm to Tm. Therefore, Au with
Tm will form more stable intermetallics having strong
inter-atomic bond strength in comparison to
intermetallics formed with Sm. The maximum change
in lattice parameter of AuSm, AuTb, AuHo and AuTm
for temperature variation of 600K has been received
1.41%, 1.37%, 1.35% and 1.17% respectively. In the
similar way, the density has been found to decay by
4.12%, 4.10%, 4.21% and 3.44% respectively. The low
percentage change in lattice parameter and density with
temperature of AuTm with respect to other selected
intermetallics confirms its strong inter-atomic bond
strength.
The evaluated second order elastic constants and
elastic moduli B, G, Y,
σ
,
λ
and
µ
are evaluated using
estimated lattice parameters and is depicted in Table 2
under variation of temperature range 300-900 K for
selected intermetallics. The second order elastic
constants, bulk modulus, Young's modulus, shear
modulus, Lame module and Poisson's ratio of the AuSm
to AuTm at 300K are approximately same as given in
literature1. Thus, present second order elastic constants
and elastic moduli of chosen compounds under potential
model approach are justified. Since, our potential model
approach for evaluation of second order elastic constants
needs only lattice parameter and provides good results,
it is therefore, better than other models. Since, the
interatomic bonding strength for AuRE intermetallic
enhances as RE changes from Sm toTm, therefore elastic
moduli of AuRE intermetallics are found to increase
from AuSm to AuTm (Table 2).
The temperature dependent SOECs are very important
for understanding the mechanical strength, stability and
phase transition of material under effect of temperature.
The SOECs C11 and C12 of AuRE decrease with increase
in temperature while C44 increases with enhancement
in temperature. Since, the lattice parameters are found
to enhance with temperature therefore, interaction force
among atoms will reduce with temperature. This is the
reason behind the decay of C11 and C12 with temperature.
The result is also verified by literature4. If body is
elongated in axial direction then its dimension reduces
in the transverse direction. This indicates that if
interaction forces among the atoms reduce with
temperature along axial direction then it will enhance
with temperature in the transverse direction. Due to this
reason, C44 is found to increase with temperature.
Table 2 represents that the quantities Y, B, G,
µ
,
λ
and
σ
are found to decrease with temperature in AuRE
intermetallics. The reduction in these quantities with
temperature is due to decay in SOECs with temperature.
This confirms the decay in average potential energy
and ionic nature of bond among the atoms of AuRE
intermetallics at high temperature. The reduction in
σ
Fig. 1. Thermal expansion coefficient (
α
) and melting point
vs. AuRE intermetallics
49
KHAN ET AL.: TEMPERATURE DEPENDENT US CHARACT. OF AURE
with temperature also confirms that chosen intermetallics
have comparatively large ionic bond character at high
temperature. The decay of B with temperature indicates
that compressibility of AuRE intermetallics increases
with temperature. The quantities Y, B, G,
λ
and
σ
with
temperature of AuRE intermetallics varies in similar
way as AgRE ( RE=Tb, Dy, Ho, Er, Tm) intermetallics
reported elsewhere16.
The ultrasonic velocities are evaluated with the help
of second order elastic constants and densities using
Eq. (11) for ultrasonic wave propagation along <100>
and <111> crystallographic directions. The calculated
ultrasonic velocities are shown in Table 3. Debye average
velocity (VD) are calculated using Eqs. (12) which are
also shown in Table 3. The estimated VD is utilized for
the calculation of Debye temperature (TD).
The ultrasonic velocity defines the anisotropic
characteristic of crystalline material as it is related to
the second order/stiffness constants of the material. The
estimated ultrasonic velocities of AuTb, AuHo and
AuTm are quite similar to values reported elsewhere at
300K1. Hence, our calculated velocities of chosen
intermetallics are justified. The ultrasonic velocities (VL
and VS) are found to vary not only with temperature but
also with the direction of propagation of ultrasonic wave.
The temperature dependent VL and VS of AuRE materials
for wave propagation along <100> direction are found
to resemble the variation of C11 and C44 with
temperature. At constant temperature, the variation of
ultrasonic velocities with material (AuSm to AuTm) for
<100> direction of wave propagation are found to be
governed by density. The ultrasonic velocities along
<111> direction have been found to decrease with
temperature while increase with material. The combined
effect of second order elastic constants is the dominating
factor towards the ultrasonic velocities in AuRE
intermetallics for <111> direction of wave propagation.
Therefore, the study of ultrasonic velocity in AuRE
with temperature and direction provides the information
about variation trend of elastic constants, lattice
parameter and density of the material. The longitudinal
ultrasonic velocities are obtained larger than that of
shear ultrasonic velocity for each direction of
propagation because stiffness constant C11 is found
Table 2 – Second order elastic constants (SOECs), bulk modulus (B), Young's modulus (Y), shear modulus (G), Lame modulus
(
λ
) and Poisson ratio (
σ
) of AuRE compounds at different temperatures.
AuRE ↓TC
11 C12 C44 YBG
λσ
[K] (GPa) (GPa) (GPa) (GPa) (GPa) (GPa) (GPa)
AuSm 300 60.7 42.6 42.9 56.3 48.1 21.6 33.7 0.305
450 59.8 41.8 42.3 55.7 47.4 21.3 33.1 0.304
600 59.0 41.1 41.7 55.1 46.6 21.1 32.5 0.303
750 58.2 40.4 41.1 54.5 45.9 20.9 31.9 0.302
900 57.4 39.6 40.5 53.9 45.1 20.7 31.3 0.301
AuTb 300 66.1 45.3 45.7 61.7 51.7 23.7 35.9 0.301
450 65.3 44.5 45.2 61.2 50.9 23.5 35.3 0.299
600 64.5 43.7 44.6 60.6 50.1 23.3 34.6 0.298
750 63.6 43.0 44.0 60.0 49.4 23.2 33.9 0.297
900 62.8 42.2 43.4 59.4 48.5 22.9 33.3 0.296
AuHo 300 70.9 47.6 48.2 66.6 54.9 25.7 37.8 0.298
450 70.1 46.8 47.6 66.0 54.0 25.5 37.0 0.296
600 69.3 45.9 47.0 65.5 53.2 25.3 36.3 0.295
750 68.4 45.1 46.4 64.9 56.5 25.1 35.1 0.293
900 67.6 44.3 45.9 64.4 51.5 24.9 34.9 0.292
AuTm 300 93.8 50.3 52.3 89.4 64.2 35.2 40.7 0.268
450 92.9 49.3 52.3 89.2 63.2 35.2 39.7 0.265
600 92.0 48.2 52.2 89.1 62.2 35.3 38.7 0.261
750 91.1 47.2 52.1 88.9 61.2 35.3 37.6 0.258
900 90.2 46.1 52.1 88.7 60.2 35.4 36.6 0.254
50 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
greater than C12 or C44 for all the intermetallics under
study. The estimated values of Debye temperature (TD)
of AuSm, AuTb, AuHo and AuTm are 204K, 207K, 210
K and 221 K, respectively. Since, the Debye average
velocity is maxima for AuTm among selected
intermetallics, therefore Debye temperature is found
large for AuTm intermetallic. The ultrasonic velocities
at different temperature of AuRE intermetallics along
<100> and <111> directions varies from AuSm to AuTm
as AgRE (RE=Tb, Dy, Ho, Er, Tm) intermetallics16.
The thermal expansion coefficient (
α
) is a thermo-
physical property of materials which defines the melting
point under the effect of external stimuli. Figure 1 reveals
that the thermal expansion coefficient (
α
) decreases
while melting point increases for AuRE intermetallics
as RE varies from Sm to Tm. The thermal expansion
coefficient is inversely proportional to the density (Eq.
14) which is received to increases from AuSm to AuTm,
therefore thermal expansion coefficient is obtained to
decrease respectively. Since, melting temperature has
inverse relationship with thermal expansion coefficient,
hence, its nature is found to have opposite behavior
with respect to thermal expansion coefficient under the
variation of rare earth element in AuRE intermetallics
(Fig. 1).
Conclusion
On the basis of above discussion, we conclude that
our potential model theory for evaluation of second
order elastic constants is justified for AuRE
intermetallics. The elastic moduli and density of AuRE
are found to enhance with decay in temperature and
variation of RE from Sm to Tm. This feature is found to
be evinced by change in lattice parameter with
temperature and material. The ultrasonic velocities in
chosen intermetallics are found to be quantitatively
governed by second order elastic constants, direction of
propagation and temperature. The intermetallic AuTm
is found to have high elastic constant, mechanical
strength and melting point among the chosen AuRE
intermetallics. The analysis of obtained results also
reveals that these intermetallics posses lower elastic
constant, mechanical strength, ultrasonic velocity and
melting point in comparison to intermetallics formed
Table 3 – The velocities of ultrasonic wave (103m/sec) of AuRE intermetallics for wave propagation along [100] and [111]
direction at different temperatures
Direction →T [K] <100> <111>
AuRE ↓
VLVSVDVLVSVD
AuSm 300 2.235 1.879 1.972 2.951 1.293 1.443
450 2.224 1.869 1.962 2.935 1.288 1.438
600 2.213 1.859 1.951 2.918 1.284 1.432
750 2.201 1.849 1.942 2.901 1.279 1.427
900 2.190 1.839 1.931 2.884 1.274 1.421
AuTb 300 2.262 1.882 1.980 2.960 1.310 1.461
450 2.252 1.873 1.971 2.945 1.306 1.456
600 2.241 1.864 1.962 2.930 1.302 1.452
750 2.231 1.855 1.952 2.914 1.298 1.448
900 2.220 1.846 1.943 2.898 1.295 1.443
AuHo 300 2.290 1.886 1.989 2.973 1.327 1.479
450 2.280 1.879 1.980 2.958 1.324 1.476
600 2.271 1.871 1.972 2.944 1.321 1.472
750 2.261 1.861 1.962 2.929 1.318 1.469
900 2.251 1.855 1.955 2.923 1.314 1.465
AuTm 300 2.591 1.935 2.054 3.103 1.511 1.678
450 2.583 1.937 2.055 3.096 1.515 1.681
600 2.575 1.939 2.056 3.089 1.518 1.685
750 2.567 1.942 2.057 3.082 1.522 1.688
900 2.559 1.945 2.058 3.075 1.525 1.691
51
KHAN ET AL.: TEMPERATURE DEPENDENT US CHARACT. OF AURE
by chosen rare earth element with Zn, Cu and Ag. Hence,
the obtained results provide a good understanding of
elastic, mechanical, thermal and ultrasonic properties
of chosen rare-earth intermetallics that may be used for
further investigation and also in material manufacturing
industries.
Acknowledgement
Authors express their high gratitude to Prof. R.R.
Yadav, University of Allahabad and Prof. Devraj Singh,
V.B.S. Purvanchal University, Jaunpur for their valuable
discussion and support during the course of manuscript
preparation.
References
1. Ahmad S., Shafiq M., Ahmad R., Jalali-Asadabadi S. and
Ahmad I., Strongly correlated intermetallic rare-earth
monoaurides (Ln-Au): Ab-initio study. J. Rare Earths.
36, (2018) 1106.
2. Liu L., Xiaozhi W., Weiguo L., Wang R. and Liu Q., High
temperature and pressure effects on the elastic properties
of B2 intermetallics AgRE. Open Phys. 13, (2015) 142-
150.
3. Singh R.P., Singh V.K., Singh R.K. and Rajagopalan M.,
Elastic, Acoustical and Electronic Behaviour of the RM
(R = Dy, Ho, Er; M=Cu, Zn) Compounds. American J. of
Cond. Matt. Phys. 3(5), (2013) 123-132.
4. Russell A.M., Ductility in Intermetallic Compounds. Adv.
Engg. Mat., 5, (2003) 629-639.
5. Russell A.M., Zhang Z., Lograsso T.A., Lo C. H. C.,
Pecharsky A. O., Morris J.R., Ye Y.Y., Gschneidner Jr
K.A. and Slager A. J., Mechanical properties of single
crystal YAg. Acta Mater. 52, (2004) 4033-4040.
6. Buschow K. H. J. and Vucht van J. H. N., Systemetic
Arrangement of the Binary Rare-Earth-Aluminium
SYSTEMS. Philips Res. Repts. 22, (1967) 233-245.
7. Dong H.Q., Tao X.M., Laurila T. and Paulasto-Kröcke
M., Thermodynamic assessment of Au-Ho and Au-Tm
binary systems. Calphad. 37, (2012) 87-93.
8. Ahmad S., Ahmad R., Jalali-Asadabadi S. and Ahmad I.,
First principle studies of electronic and magnetic
properties of Lanthanide-Gold (RAu) binary
intermetallics. J. Magn. Magn. Mater. 422, (2017) 458-
463.
9. Brugger K., Thermodynamics definition of Higher Order
Elastic coefficients, Phys. Rev. 133(6A), (1964) A1611.
10. Singhal R. L., "Solid state physics", Kedar Nath Ram
Nath & Co. Publishers, Meerut, India, 73, (2003).
11. Yadav R. R. and Pandey D. K., Size dependent acoustical
properties of bcc metal, Acta Phy. Pol. A 107, (2005)
933-946.
12. Yadav C .P., Pandey D .K. and Singh D., Elastic and
Ultrasonic Studies on RM (R=Tb, Dy, Ho, Tm; M=Zn,
Cu) Compounds. Z. Naturforsch. A 74, (2019) 1123-1130.
13. Khan A., Yadav C .P., Pandey D .K., Singh D. and Singh
D., Elastic and thermo-acoustic study of YM
intermetallics. J. Pure Appl. Ultrasonic. 41, (2019) 1-8.
14. Pandey D. K. and Yadav C. P., Thermophysical and
ultrasonic properties of GdCu under the effect of
temperature and pressure. Phase transitions. 93(3),
(2020) 338-349.
15. Bala J., Singh D., Pandey D. K. and Yadav C. P.,
Mechanical and thermophysical properties of ScM (M:
Ru, Rh, Pd, Ag) intermetallics. Int. J. Thermophysics.
41, (2020) 46.
16. Khan M. A., Kumar M., Yadav C. P. and Pandey D. K.,
Mechanical, thermal and ultrasonic properties of AgRE
intermetallics at different temperatures. Phase
Transitions. 95(2), (2022) 131-142.
17. Moakafi M., Khenata R., Bouhemadou A., Semari F.,
Reshak A. H. and Rabah M., Elastic, electronic and optical
properties of cubic antiperovskites SbNCa3 and BiNCa3,
Comput. Mat. Sci. 46, (2009) 1051-1057.
18. Pandey D.K. and Pandey S., in Acoustic Waves:
Ultrasonic: a technique of material characterization, Eds:
Don W. Dissanayake, Scio Publisher, Sciyo Croatia,
(2010) 397-430.
19. Pillai S.O., in Solid State physics: Crystal physics, 7th
Ed. New Age International Publisher, (2005) 100- 111.
52 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
Acoustical and excess thermodynamic studies of binary liquid
mixtures at varying temperature using ultrasonic technique
C. Duraivathi1*,4, J. Jeya Priya2,4, J. Poongodi3,4# and H. Johnson Jeyakumar4
1*,4Pope's College, Sawyerpuram-628 251, Tamil Nadu, India
2,4V.O. Chidambaram College, Thoothukudi-628 008, Tamil Nadu, India
3,4Department of Physics, Kamaraj College, Thoothukudi-628 003, Tamil Nadu, India
*E-mail: cduraivathikamali@gmail.com
The ultrasonic velocity, density and viscosity of binary liquid mixtures of lactic acid with methanol,
1-propanol and 2- propanol of various compositions at 308 K, 313 K, 318 K and 323 K were determined. These
experimental data have been used to estimate the thermodynamic parameters such as adiabatic compressibility
(
β
), free length (Lf), free volume (Vf), internal pressure (
π
i) and enthalpy (H). The excess values of the above
parameters like excess adiabatic compressibility, excess free length, excess free volume, excess internal pressure
and excess enthalpy are also evaluated in order to dispute the nature of molecular interactions present among
the constituent molecules of binary liquid mixture systems.
Keywords: Thermodynamic parameters, 1-propanol, density, excess free volume, molecular interactions.
Introduction
Using the ultrasonic properties, we can explain the
properties of unblended liquids and the mixed liquids.
The state of molecular interactions in a liquid system
can be understood from the propagation of an ultrasonic
wave in liquids1. Ultrasonic propagation in liquid
medium is very much useful in the fields such as food
industry, paper industry, pharmaceutical and biological
industries. The nature and extent of the patterns of
molecular aggregation that exist in liquid mixture
resulting from intermolecular interactions2 have been
investigated by ultrasonic technique. Ultrasonic velo-
cities have been adequately employed in understanding
the nature of molecular interaction in pure liquids3,
binary, and ternary mixtures4-6. The molecular
interaction studies in the liquids and liquid mixtures are
most effective in assessing the structural properties of
the molecules.
The thermodynamic properties of liquid matter are
crucial to the chemists and chemical engineers in regards
of their applications, such as (i) predicting the properties
of liquid mixtures of similar nature and determining the
J. Pure Appl. Ultrason. 44 (2022) pp. 52-57
applicability of theories of liquids, (ii) to extract
information about the molecular interactions among the
constituents of mixtures, (iii) reducing capital costs by
designing industrial process with better precision.
In chemical process industries, materials are normally
handled in liquid form and as a consequence, the
physical, chemical, and transport properties of liquids
and liquid mixtures assume importance. Properties like
excess free volume, excess enthalpy, excess adiabatic
compressibility etc., associated with liquids and liquid
mixtures detecting extensive application in chemical
engineering design, process stimulation, solution theory
and molecular dynamics. The energy of ultrasonic is
applied in various fields as diagnostic tool. In order to
understand the nature of molecular interactions between
the components of the liquid mixtures, it is of interest to
discuss the same in terms of excess parameters rather
than the actual values7.
Lactic acid is an over-the-counter chemical exfoliant
that comes from the fermentation of lactose - a
carbohydrate found in milk. Like glycolic and mandelic
acids, lactic acid belongs to the alpha hydroxy acid
(AHA) family. AHAs are water-soluble chemical
compounds that provide remarkable skin benefits when
# Life Member, Ultrasonics Society of India
53
DURAIVATHI ET AL.: US STUDY OF BINARY MIXTURES OF LACTIC ACID WITH ALKANOLS
used in cosmetic compositions. Lactic acid has an
additional benefit that other AHAs do not have. It not
only improves the appearance of the skin, but it also
keeps it hydrated naturally.
In the present work the effort has been made to study
the binary mixtures of
System I: lactic acid + methanol.
System II: lactic acid + 1-propanol
System III: lactic acid + isopropyl alcohol.
We have reported the variation of excess
thermoacoustical parameters such as excess adiabatic
compressibility, excess free length, excess free volume,
excess internal pressure and excess enthalpy of the
above-mentioned binary mixtures at different
temperature and the exact molecular interaction between
the binary systems have been discussed.
Materials and Methods
Experimental : In the present work, the chemicals
used are of analytical reagent grade (AR) obtained from
Loba chemicals private Ltd. with minimum assay of
99.9%. The binary liquid mixtures of different
composition were prepared by mole fraction (X) basis.
Ultrasonic velocity for the binary mixtures for all
concentrations were measured with an ultrasonic
interferometer with frequency of 2 MHz. The viscosity
of liquid mixtures was determined by using Ostwald's
viscometer. The density of the liquid samples was
measured by specific gravity bottle of 5 ml. Ultrasonic
speed, viscosity and density measurements had been
made at different temperature. Temperature of pure
liquids and their mixtures inside the chamber is
maintained with the help of temperature-controlled water
bath with minimum accuracy of ±0.2°C.
Theory : Thermodynamical parameters derived from
ultrasonic velocity, viscosity, and density measurements
provide comprehensive information on ion-dipole
interactions, intermolecular hydrogen bonding, cohesive
and dispersive forces. Thermo-dynamical parameters
have been calculated by using following equations8-11,
Adiabatic compressibility (
β
) = 1/(U2
ρ
) (1)
Free length (Lf) = KT (
β
a)1/2 (2)
Free volume (Vf) = (MU/
η
K)3/2 (3)
Internal pressure (
π
i) = bRT (K
η
/U)1/2 (
ρ
2/3/M7/6) (4)
Enthalpy (H) =
π
iVm(5)
Excess functions for the acoustical parameters have
been computed using the relation,
AE = Aexp – [A1X1 + A2X2] (1)
X1 and X2 are the mole fraction of the component 1 and
2 respectively.
A corresponds to the thermodynamical parameters
like
β
, Lf, Vf, etc.
Results and Discussion
While mixing two liquids, the molecular interaction
between the liquids is due to the presence of dispersive
force, charge transfer, hydrogen bonding and dipole
induced dipole interactions. The acoustical parameters
which are calculated from measured ultrasonic velocity
density, and viscosity provide complete information
about interaction between ions, dipoles, the inter-
molecular hydrogen bonding and cohesive and dispersive
forces12.
In order to substantiate the presence of interaction
(either adhesive or cohesive forces) between the
molecules, it is essential to study excess parameters
like excess free volume (VE
f), excess internal pressure
(
π
E) etc., as these parameters are found to be more
sensitive towards intermolecular interactions in the
liquid mixtures13. The excess properties of the mixtures
are influenced by the following three main types of
contribution, (a) physical: due to non-specific Vander
Walls type forces (b) chemical: due to hydrogen bonding,
dipole-dipole and donor-acceptor interaction between
unlike molecules and (c) structural: due to fitting of
smaller molecules into the voids created by the bigger
molecules14.
Figure 5 summarise the excess adiabatic compressi-
bility, excess free length, excess free volume, excess
internal pressure and excess enthalpy calculated from
the experimental result of ultrasonic velocity, density
and viscosity of the binary system I (lactic acid +
methanol), system II (lactic acid + 1-propanol) and
system III (lactic acid + 2-propanol)at temperatures 308
K, 313 K, 318 K and 323 K.
The compressibility factor is a measure of
intermolecular attraction15. The negative excess
compressibility indicates the strong hetero molecular
interactions of the constituent molecules. The negative
values are due to the charge transfer, dipole-dipole and
dipole - induced dipole interaction and by the hydrogen
54 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
bonding between unlike components. Positive excess
compressibility shows the weak interactions which will
arise from the dispersive forces16,17. Further the negative
value of
β
E is associated with the structure making
tendency while a positive value is taken to indicate the
structure breaking tendency18.
In the present work, the values of excess adiabatic
compressibility found to be negative for all the three-
system considered. These values seem to be decrease at
lower concentration range and eventually increase at
higher concentration range in the system I as shown in
Fig. 1(a). Hence, the molecular interaction stronger at
lower mole fraction range and becomes weaker at the
higher mole fraction range for all the temperatures.
Whereas in system II and III, the excess adiabatic
compressibility values are negatively decrease indicate
the presence of strong molecular association shown in
Fig. 1 (b) & (c).
Fig. 1. Variation of
β
E
ad with mole fraction for (a) system I, (b) system II, (c) system III.
Fig. 2. Variation of LE
f with mole fraction for (a) system I, (b) system II, (c) system III.
From the Fig. 2(a), it is observed that the excess free
length (LE
f) values decrease at lower mole fraction range
and eventually increase at higher concentration of
solvent notably from 0.6. In the case of system II and
III, the excess free length values are negatively decrease.
In other words, the variation of excess free length values
reflects the same trend as noticed previously in excess
adiabatic compressibility. Fort and Moore19 indicated
the positive values of excess free length should be
attributed to the dispersive forces and negative excess
values should be due to charge transfer and hydrogen
bond formation. The negative values of excess
intermolecular free length are mainly due to the hydrogen
bonding between the unlike molecules20. However, our
present study shows that the increasing trend of negative
values at higher concentration range indicate the rupture
of hydrogen bonding between the components of liquid
mixtures present in system I. Whereas in system II and
III, Fig. 2 (b) & (c) shows the hydrogen bond formation
55
DURAIVATHI ET AL.: US STUDY OF BINARY MIXTURES OF LACTIC ACID WITH ALKANOLS
due to charge transfer reveals the strong molecular
interaction among the component of liquid mixtures
present in both the system.
The variation of excess free volume with the
concentration of solvent for the three systems are shown
in Fig. 3 (a), (b) and (c). For the studied mole fraction
range, the values of excess free volume (LE
f) are negative
for all the systems.In the case of system, I and II, the
values of excess free volume are negatively decrease
and show sudden increase in higher mole fraction range.
The present increasing trends of negative excess free
volume at higher mole fraction range, substantiate the
existence weak molecular interaction among the unlike
molecules. Those values get negatively decrease in
system III as shown in Fig 3 (c) implies strong
interaction.
In the study of binary liquid mixture, the variation of
excess internal pressure may give some valuable
information regarding the nature and strength of the
forces existing between the molecules. From the Fig. 4
(a), (b) in the system I and III, the values of excess
internal pressure get increases and eventually decreases
at higher mole fraction range whereas system II,
π
E
i
values get decreases as shown in Fig. 4 (c). When two
interacting molecules are having some sort of attractive
forces like that of hydrogen bonding, they result in the
increase of internal pressure. In fact, the internal pressure
is the broader concept and is a measure of the totality of
forces of the dispersion, ionic and dipolar interaction
that contribute to be the overall cohesion of the liquid
system20. The decreasing trend of excess internal
pressure indicates that, the sort range of attractive force
Fig. 3. Variation of VE
f with mole fraction for (a) system I, (b) system II, (c) system III.
Fig. 4. Variation of
π
E
i with mole fraction for (a) system I, (b) system II, (c) system III.
56 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
Fig. 5. Variation of HE with mole fraction for (a) system I, (b) system II, (c) system III.
acting among the components liquid mixtures in all the
three binary systems.
Change in HE is used to explain whether the system
under exothermic or endothermic reaction21. For all the
systems negative values are observed for excess enthalpy
as shown in Fig. 5(a), (b) and (c). Hence it is concluded
that system I, II and III are under exothermic reaction,
so that heat was released from all the three systems
considered.
Conclusion
Ultrasonic analysis on liquid mixtures reveals the clear
understanding about physico- chemical behavior as well
as existence of molecular interactions among the
component of molecules in the mixture. The
intermolecular interactions between the three binary
systems I: lactic acid + methanol, II: lactic acid + 1-
propanol and III: lactic acid + isopropyl alcohol is
depicted in this paper. In the case of system, I and II,
molecular interaction stronger at the lower mole fraction
range, it becomes weaker when increasing the
concentration solvent whereas in system III, these results
lead to the conclusion that the component molecules
are in closer proximity in the liquid mixture. This is due
to the strong dipole-dipole interactions present among
the molecules in the binary system III.
Acknowledgement
One of the authors (C. Duraivathi) would like to
express her heartfelt thanks to the Director of Collegiate
Education, Chennai - 600006 for the stipend provided
under the full time Research Scholar Scheme, since
2018.
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1. Lakshmi R.B., Tejaswini G., Ganapathi R.G. and
Chidambara K.K.N., J. pure. Appl. Ultrason., 42, (2020)
26-25.
2. Bahadur A.S., Nafeesa B.S. et al., Ind. J. Adv. Chem. Sci.,
5(3), (2017) 148-15.
3. Lakshmi K.V., Suhasini D., Reddy M.J., Ravi C., Rao
K.C. and Subha M.C.S., Ind. J. Adv. Chem. Sci., 3, (2014)
38-48.
4. Nagaraja P., Rao C.N. and Venkateswarlu P., Ind. J. Adv.
Chem. Sci., 4, (2016) 421-424.
5. Spencer J.N., Jeffrey E. and Robert C., J. Phy. Chem.,
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6. Rort R.J. and Moore W.R., Transac. Faraday Soc., 62,
(1966) 1112.
7. Apurba M., Ghosh and J.N., Ramteke., Der Chemica
Sinica, 8(2), (2017) 291-297.
8. Sumathi T., Asian J. Biochemi. Pharmaceutic. Res., 5,
(2015) 200-206.
9. Thenmozhi P A. and Krishnamurthi P., Rasayan J. Chem.,
8, (2015) 24-32.
10. Chakraborty N., Rani M. and Juglan K.C., Plant Archives,
20(2), (2020) 2825-2829.
11. Duraivathi C., Jeya Priya J., Poongodi J. and Johnson
J.H., Ultrasoni study of molecular interactions in organic
liquid with CCl4 at different temperature, Materials
57
DURAIVATHI ET AL.: US STUDY OF BINARY MIXTURES OF LACTIC ACID WITH ALKANOLS
Today: Proceedings, https://doi.org/10.1016/
j.matpr.2021.08.029.
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Ramalatha Marimuthu, Int. J. Eng. Tech., 8, (2019) 153.
13. Bala Karuna Kumr D., Rayapa Reddy K., Srinivasa Rao
G., Rama Rao G.V. and Rambabu C., J. Chem. Pharm.
Res., 3(5), (2011) 274-280.
14. Sridevi G., Int. J. Chem. Environ., 3(3), (2013) 70-80.
15. Rajesh G.M., Jayakumar S. and Dharani Rajan N. et al.,
Growth and studies on aluminium ammonium sulphate
and thiourea, J. Pure Appl. Ultrason., 42, (2020) 100-
103.
16. Fort R.J. and Moore W.R., Trans. Faraday Soc., 61,
(1965) 2102.
17. Sathyanarayana G.R., Balakaruna Kumar D., Sujatha K.,
Lakshmanarao G. and Rambabu C., J. Mol. Liq, 216,
(2016) 526-37.
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Pure Appl. Ultrason., 79(2), (2004) 26.
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21. Jeya Priya J., Duraivathi C., Poongodi J. and Amudhavalli
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58 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
Ultrasonic study of calcium soaps (laurate and myristate)
Mahesh Singh Khirwar1, Ashish K. Singh2, Sandeep K. Singh2, M.K.Rawat3 and Gyan Prakash2*
1Department of Chemistry, RBS College, Agra-282 002, India
2Department of Chemistry, V.S.S.D. College, Kanpur-208 002, India
3Department of Chemistry, Agra College Agra College, Agra-282 002, India
*E-mail: gyanprakashvssd@gmail.com
Ultrasonic studies on calcium soaps (laurate and myristate) were performed to determine the critical micelle
concentration (CMC), soap-solvent interaction, and various acoustic parameters. The values of the CMC decrease
with the increase in the chain length of the soap molecules. The ultrasonic velocity, specific acoustic impedance,
apparent molar compressibility, and relative association increase while the adiabatic compressibility, intermolecular
free length, and solvation number, decrease with increasing soap concentration. The ultrasonic velocity results
show that calcium soaps behave like a simple electrolyte in the solutions and that there is significant interaction
between the soap and solvent molecules in dilute solutions.
Keywords: Ultrasonic velocity, specific acoustic impedance.
Introduction
The term metallic soap seems to have been mentioned
specifically for the first time in 1758. Initially, the basic
need for soap was in the paint industry. Later on, a large
number of other uses1 of calcium soaps have been
reported particularly in waterproofing materials and
adjusting the hardness of organic materials.
The study of metallic soaps is becoming increasingly
important in the technology and academic field. Several
workers came forward with different methods and
techniques for studying the preparation properties and
uses of alkaline earth metal soaps2-13. The technological
applications of metals soaps are mostly based on
empirical knowledge of how and the selection of soap
is largely dependent on economic factors. The study of
molecular interaction has been a subject of an extensive
investigation by Raman14, NMR15, Infrared16 and
ultrasonic absorption17-21 measurements.
The present paper has been initiated to obtain
ultrasonic velocity measurements of calcium soaps
(laurate and myristate) in a pure 70% chloroform and
30% propylene glycol mixture.
J. Pure Appl. Ultrason. 44 (2022) pp. 58-63
Experimental
All the chemicals used were of AR grade. The calcium
soaps (laurate and myristate) were prepared by direct
metathesis of corresponding potassium soaps (laurate
and myristate) with a slight excess of the solution of the
aqueous solution of calcium nitrate at 50-60°c under
vigorous stirring. The precipitated soaps were washed
with water and then acetone and derived under reduced
pressure. The purity of the soaps was checked by
Elemental analysis and determination of their melting
points.
The density of the solutions of metal soaps was
determined by a dilatometer. The ultrasonic
measurements of the solutions of calcium soaps were
carried out with a multi frequency Ultrasonic
Interferometer (Mittal Enterprises, New Delhi) at a
frequency of 2MHz at a constant temperature (313 ±
0.05°c). The uncertainty of velocity measurements was
0.2%.
Calculation
The various acoustic parameters such as adiabatic
compressibility
β
, Intermolecular free length Lf, specific
acoustic Impendence Z, Apparent molar compressibility
59
KHIRWAR ET AL.: ULTRASONIC STUDY OF CALCIUM SOAPS
ϕ
k, Apparent molar Volume
ϕ
v, AvailableVolume Va ,
Relative association RA, and Solvation Number Sn were
evaluated by using the following relationships
β
= V–2
ρ
–1
f
L=k β
Z = V
ρ
0
k00
00
M
1000
c()
β
ϕρββ
ρρ
=−+
()
v0
00
1000 M
c
ϕρρ
ρρ
=−+
1
V
aV( V/V
)
∞
=−
1
3
0
/
0
AV
RV
ρ
ρ
⎛⎞
⎛⎞
=⎜⎟
⎜⎟
⎝⎠
⎝⎠
()
1
0
n0
00
nVβ
S= (/)
nnV β
⎛⎞
−
⎜⎟
⎜⎟
⎝⎠
Where
ρ
,
ρ
0,
β
,
β
0 and V are the density and ultrasonic
velocity of solvent and Solutions; n and n0 and M, M0 is
the number of moles and molecular weight of the solute
and solvent respectively; k and C are the temperature-
dependent Jacobson's constant and concentration in
mole –l and V
∞
equivalent to 1600 ms–1.
Results and Discussion
The density (
ρ
) of the solutions of calcium soaps
(laurate and myristate) in a mixture of 70% chloroform
and 30% propylene glycol (v/v) increases first slowly
and then rapidly with increasing soap concentration
(Table 1). The plots of density
ρ
Vs soap concentration,
C (Fig. 1) are characterized by an intersection of two
straight lines at definite soap concentration which
Table 1 – Ultrasonic velocity and other acoustic parameters of calcium soaps in 70% chloroform and 30% propylene glycol
mixture at 40 ± 0.05°C
S. Concentration Density Velocity Adiabatic Intermole- Specific Solvation Apparent Apparent Available Relative
No. C×10
3
ρ×10
–3
VCompressi- cular Acoustic Number molar com- molar Volume Association
(mol dm
–3
) (kg ml
–3
)ms
–1
bility Free Impedence, S
n
×10
–2
25 pressibility Volume V
a
×10
–3
m
3
R
A
×10
2
β×10
10
m
2
N
–1
length Z×10
–6
Kg
ϕ
k
×10
–2
M
5
N
–1
–
ϕ
v
×10
3
L
f
× 10
2
Å m
–2
S
–1
kg
–1
mol
–1
dm
3
mol
–1
Laurate
1. 4.0 1.0666 1428.0 4.59 2.63 1.523 1.523 19.033 337.58 26.87 100.03
2. 6.0 1.0684 1434.0 4.55 2.62 1.532 9.07 19.049 350.99 17.29 100.06
3. 8.0 1.0701 1441.0 4.50 2.61 1.542 9.19 19.045 346.12 12.42 100.06
4. 10.0 1.0720 1446.0 4.46 2.59 1.550 8.48 19.052 360.13 96.25 100.12
5. 12.0 1.0740 1453.0 4.41 2.58 1.560 7.71 19.048 376.21 76.55 100.15
6. 16.0 1.0750 1462.0 4.35 2.56 1.571 6.36 19.076 233.30 53.90 100.03
7. 20.0 1.0758 1470.0 4.30 2.55 1.581 5.37 19.098 138.83 40.62 99.93
8. 24.0 1.0766 1475.0 4.26 2.54 1.588 4.62 19.117 75.81 32.55 99.89
9. 28.0 1.0770 1480.0 4.23 2.53 1.593 4.05 19.135 18.45 26.78 99.81
10. 32.0 1.0776 1486.0 4.20 2.52 1.601 3.61 19.148 -19.19 22.26 99.73
Myristate
1. 4.0 1.0676 1430.0 4.58 2.62 1.527 4.18 21.467 504.46 26.56 100.08
2. 6.0 1.0696 1437.0 4.52 2.61 1.537 9.15 21.460 473.34 16.97 100.10
3. 8.0 1.0716. 1444.0 4.47 2.60 1.547 9.23 21.470 456.96 12.18 100.13
4. 10.0 1.0736 1451.0 4.42 2.58 1.558 8.52 21.475 446.48 9.31 100.15
5. 12.0 1.0760 1456.0 4.38 2.57 1.567 7.73 21.485 467.76 7.49 100.26
6. 16.0 1.0772 1464.0 4.33 2.55 1.578 6.37 21.527 299.37 5.31 100.01
7. 20.0 1.0780 1472.0 4.28 2.54 1.584 5.37 21.558 181.01 4.06 100.13
8. 24.0 1.0788 1478.0 4.24 2.53 1.592 4.62 21.579 102.07 3.22 100.07
9. 28.0 1.0796 1486.0 4.19 2.52 1.604 4.06 21.586 45.64 2.54 99.92
10. 32.0 1.0800 1491.0 4.16 2.51 1.610 3.61 21.602 -7.41 2.12 99.84
60 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
corresponds to the CMC of the soap at which there is a
sudden change in the aggregation of the soap molecules.
The CMC values decrease with an increasing chain
length of the soap molecules (Table 2).
of constants A and B. The values of these constants A
and B are recorded in (Table 2). The value of constant A
is higher than the constant B, which shows that the
solute-solvent interaction is larger than the solute-solute
interaction in soap solutions. Thus, it is now quite clear
that the soap molecules don't show appreciable
aggregation below CMC and at the definite soap
concentration i.e. at CMC, there occurs a marked
increase in the aggregation of the soap molecules.
The ultrasonic velocity (v) of calcium soaps (laurate
and myristate,) solutions increases with increasing
Concentration and chain length of the soap. (Table 1).
The variation of ultrasonic velocities with soap
concentration depends on the concentration derivative
of density
ρ
and adiabatic compressibility
β
.
dv /dc = –v /2 [ 1/
ρ
(d
ρ
/dc)+1/
β
(d
β
/dC)]
The results show that with the increase of soap
concentration, the density increases while the adiabatic
compressibility decreases and so the quantity (dv/dc) is
positive, while (d
β
/dC) is negative. For the soap
solutions, the value of 1/
β
(d
β
/dC) is larger than the
values of 1/
ρ
(d
ρ
/dc). Hence, the concentration derivative
of velocity (dv/dc) is positive, i.e. as the soap
concentration increases the ultrasonic velocity increases.
These results are in agreement with the results reported
for electrolytic solutions, which show that calcium soaps
behave as simple electrolytes in solutions of mixed
organic solvents.
A break is observed at a definite soap concentration
for the plots of ultrasonic velocity v/s soap concentration
(Fig. 2) for solutions of calcium soaps in a mixture of
organic solvents. This definite soap concentration
Table 2 – Values of CMC Constant (A & B) obtained from
density measurements at 40 ± 0.05°C
Calcium Soaps CMC ( mol dm
–3
)A B
Laurate 0.0137 0.82 –8.18
Myristate 0.0125 1.14 –8.42
Fig. 1. Concentration Vs Density
Fig. 2. Concentration Vs Velocity
The plots of density
ρ
Vs soap concentration, C below
the CMC are extrapolated to zero soap concentration
and the extra polated values of density,
ρ
0 are in harmony
with the experimental value of the density of the pure
solvent mixture. It is concluded that the soap molecules
do not show appreciable aggregation below the CMC
whereas at this definite soap Concentration i.e. at CMC
there is a marked change in the aggregation of the soap
molecules.
The density results have been explained in terms of
Root's equation:
ρ
=
ρ
0 + AC – BC1/2
Where
ρ
and
ρ
0 are the densities of the solution and
solvent, respectively and C is the soap concentration
(mol dm–3). The constants A and B represent the solute-
solvent and solute-solute interaction, respectively.
The plots of (
ρ
–
ρ
0)/C Vs C1/2 indicate a break at a
definite soap concentration, which corresponds to the
CMC of the soap. The intercept and slope of the plots of
(
ρ
–
ρ
0)/C Vs C1/2 below the CMC represents the values
61
KHIRWAR ET AL.: ULTRASONIC STUDY OF CALCIUM SOAPS
corresponds to the CMC of these soaps. As the chain
length of the soap increases, the CMC decreases (Table
3). The cause of the micellization process is the energy
change arising from the polar head group of the soap
molecules.
The plots of ultrasonic velocity v/s soap concentration
have been extrapolated values of velocity, V0 (1416.0
and 1418.0) are in close agreement with the experimental
values V (1416.0 and 1416.0) in the organic solvent
mixture.
The solutions having concentration below the CMC,
the variation in ultrasonic velocity, V with soap
concentration follows the relationship:
V = V0 + GC
Where, G = Garnsey,s22 Constant.
The slop of the plots of V v/s C gives the values of the
Garnsey constant for calcium soaps in an organic solvent
(v/v) mixture. These values of Garnsey constant increase
with an increasing chain length of soap molecules. The
decrease in the adiabatic compressibility of calcium
soaps, with the increase in soap concentration chain
length of the soap, is observed (Table. 1).
The decrease in the adiabatic compressibility
β
of
these soap solutions decreases with increasing soap
concentration (Table 1). The decrease in adiabatic
compressibility,
β
is because, the soap molecules in
dilute solution are considerably ionized into cation and
fatty acid anion RCoo– (where R is C11H23 and C13H27,
Laurate and Myristate) and these ions are Surrounded
by a layer of solvent molecules firmly bound and oriented
towards the ions. The influence of the electrostatic field
of the ions is responsible for the orientation of solvent
molecules around the ions and as a result, The internal
pressure increases which lower the compressibility of
soap solutions i.e., solution becomes harder to compress.
The plots of adiabatic Compressibility,
β
v/s Soap
concentration, C (Fig. 3) show a break at a definite soap
concentration corresponding to the CMC of calcium
Soaps. These plots are extrapolated to Zero soap
concentration and the extrapolated values of
β
0(4.68
×1010m2N–1 and 4.65×1010m2N–1 are in close agreement
with the experimental values (4.69 ×1010m2N–1 and
4.69×1010m2N–1) of the adiabatic compressibility of
the organic solvent mixture.
The Results of adiabatic compressibility have been
explained in terms of Bachem, S Equation
B =
β
0 + AC + BC3/2
where A and B are constants, C is the concentration of
the soap in mole/dm–3. The values of the constants A
and B have been determined from the slope and intercept
of (
β
–
β
0)/C Vs C1/2 and are recorded in (Table 3).
The inter molecular free length, Lf decreases while
the acoustic impedance, Z increases with the increase in
concentration and chain length of soap (Table 1). This
is a clear indication of a significant interaction between
the soap and solvent molecule. This interaction
considerably affects the structural arrangement. The
increase in the value of Z with increasing soap
concentration can be explained based on the lyophobic
interaction between soap & solvent molecules which
increases the intermolecular distance making relatively
wider gaps between the molecule and becoming the
main cause of impediment in the propagation of
ultrasonic waves. The plot of Lf vs. C (Fig. 4) show a
break at a definite soap concentration which corresponds
to the CMC of these soaps.
Table 3 – Values of allied parameters obtained from ultrasonic
measurements at 40 ± 0.05°C
Calcium Laurate Calcium Myristate
CMC(mol dm
–3
) 13.75×10
–3
13.00×10
–3
G3000.0 3111.1
-A × 10
10
26.4 29.0
B × 10
10
60.0 70.0
ϕ
k0
× 10
–2
19.021 21.432
S
k
×10
–2
0.529 0.800
Fig. 3. Concentration Vs adiabatic compressibility
62 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
Fig. 4. Concentration Vs intermolecular free length
The values of apparent molar compressibility,
ϕ
k
increases with increasing soap concentration (Table 1).
The
ϕ
k is related to the molar concentration, C by the
relationship:
ϕ
k =
ϕ
k0 + Sk C1/2
Where
ϕ
k0 and Sk are the limiting apparent molar
compressibility and a constant respectively. The values
of
ϕ
k0 and Sk have been obtained from the intercept and
slope of the plots of
ϕ
k and C½ below the CMC and are
recorded in (Table 3).
The solvation numbers of the solutions of calcium
soap (laurate and myristate) increases and relative
association, RA decreases with increasing with increasing
soap concentration (C). The plot of RA v/s C are
characterised by a break at the CMC. The decrease in
value of RA with increasing soap concentration has been
attributed to the decreased association of soap molecules
at higher soap concentration.
The ultrasonic velocity results shows that the calcium
soaps (laurate and myristate) behaves as a simple
electrolyte in the solutions .This is the confirmation of
the fact that there is a significant interaction between
the soap and solvent molecules in dilute solutions and
soap molecules do not aggregate appreciably below the
CMC. The CMC values are in close agreement with the
values obtained from other parameters.
References
1. Wilfred Gallay Ira Euddington, The Recrystalisation of
calcium soaps in Mineral oils, Canadian Journal of
Research, 22(b)6, (1994) 76-89.
2. Varma R.P. and Jindal R., Studies of Infrared Spectra of
Cerium (IV) Soaps and their Solubilities in Benzene and
Benzene-Methanol Mixtures, Tenside detergents, 22,
(1982) 193.
3. Mehrotra K.N., Gahlaut A.S. and Meera Sharma,
Conductometric investigations on lanthanum soap
solutions Conductometric investigations on lanthanum
soap solutions, Journal of the American Oil Chemists
Society, 63, (1986) 1571-1575.
4. Mehrotra K.N., Mithlesh Chauhan and Shukla R.K.,
Influence of alkanols on the micellar behavior of
samarium soaps, Surfactants & detergents, Journal of the
American oil Chemists, 67, (1990) 446-450.
5. Mehrotra K.N. and Rawat M.K., Ultrasonic and
Viscometric Studies on Magnisium Dicaprate in Organic
Solvants, Indian Journal of Pure and Applied Physics,
29, (1991) 131-133.
6. Shukla R.K., Mithlesh Chauhan and Vikas Mishra, Effect
of electrolytes and non-electrolyte on micellar behaviour
of lanthanum soaps in non-aqueous solvents, Physics and
Chemistry of Liquids an International Journal, 45, (2007)
345-349.
7. Rawat M.K. and Singh N., Ultrasonic Velocity and other
Allied Parameters of Fe(III) Soaps in Non Aqueous
Medium, Journal of Material Science Research India,
3(1), (2006) 72-77.
8. Rawat M.K., Ultrasonic Measurements and other Allied
Parameters of Alkaline Earthmetal soaps in Mixed organic
Solvents, Journal of Chemtracs Bodhgaya, 9, (2007)
9. Rawat M.K. and Sharma G., Ultrasonic velocity and allied
properties of manganese, cobalt and copper soaps in non-
aqueous medium. Journal of the Indian Chemical Society,
84, (2007) 46-49.
10. Rawat M.K., Sharma Y. and Kumari S., Molecular
interaction and compressibility behaviour of beryllium
soaps in non-aqueous medium, Asian Journal of
Chemistry, 20(2), (2008) 1464-1472.
11. Rawat M.K., Ultrasonic study of molecular interaction
and Compressibility Behavior of Magnisium
Carboxylates, Journal of Applicable Chemistry, 3(2),
(2013) 168-175.
12. Sangeeta and M.K. Rawat, ultrasonic velocity and other
allied Parameters of Dysprosium Laurate and Myristate,
Journal of applicable chemistry, 3(1), (2014) 354-359.
13. Rawat M.K. and Sangeeta, Ultrasonic study of molecular
interactions and compressibility behaviour of strontium
soaps in chloroform-propylene glycol mixture, Journal
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of Pure and Applied Physics, 46, (2008), 187-192.
14. Francesca Caterina Izzo, Matilde Kratter, AustinNevin
and Elisabetta Zendri, A Critical Review on the Analysis
of Metal Soaps in Oil Paintings, Chemistry Open, 10,
(2021) 904-921.
15. Wei-Chuwan Lin and Shyr-Jin Tsay, Nuclear magnetic
resonance studies on the intermolecular association in
some binary mixtures. I. Chloroform and proton-acceptor
solvents, The Journal of Physical Chemistry, 74(5),
(1970) 1037-1041 .
16. Coburn W.C. Jr. and Ernest Grunwald, Infrared
Measurements of the Association of Ethanol in Carbon
Tetrachloride, Journal of American Chemical Society,
80(6), (1958) 1318-1322.
17. Ryszard Zielinski, Shoichi lkeda, Hiroyasu Nonura and
Shigeokato, Effect temperature on micelle formation in
Aqueous solution of Alkyl tri methyl ammonium bromide,
Journal of Collide and Interface Science. 129, (1989)
175-184.
18. Subramanyam Naidu P. and Ravindra Prasad K.,
Ultrasonic velocity and allied parameters in solutions of
cypermethrin with xylene and ethanol, Indian J. Pure &
Appl. Phys., 42, (2004) 512.
19. Furniss A.S., Ilannaford A.J., Rogers V., Smith P.W.G.
and Tachell A.R., Vogel's Text Book of Practical Organic
Chemistry, 4th Edition, (1980). Longmann.
20. Subramanyam Naidu P. and Ravindra Prasad K.,
Molecular interactions in binary mixtures -an ultrasonic
study, Indian J. Pure & Appl. Phys., 40, (2002) 264.
21. Jacobson B., Anderson W.A. and Arnold J.T., A proton
magnetic resonance study of the hydration of
deoxyribonucleic acid. Nature, 173, (1954) 772-773.
22. Garnsey R., Boe R.J., Mohoney R. and Litovitz T.A., J.
Chem. Phys., 50, 5222 (1969).
64 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
Estimation of effective Debye temperature of polymeric solutions at
303.15 K based on quasi-crystalline model
Monika Dhiman1, Arun Upmanyu1*, Pankaj Kumar1, D.P. Singh2* and Harsh Kumar3#
1Chitkara University Institute of Engineering and Technology, Chitkara University, Punjab-140 401, India
2Acoustics Research Center, Mississauga, ON, L5A 1Y7, Canada
3Department of Chemistry, Dr. B.R. Ambedkar National Institute of Technology, Jalandhar-144 011, India
*E-mail: arun.upmanyu@chitkara.edu.in
Based on the quasi-crystalline model of liquids, effective Debye temperature (
θ
D) of six binary polymeric
solutions has been computed at 303.15 K using ultrasound velocity and density data. The binary systems studied
herein are polypropylene glycol (PPG) 400 + ethanol, PPG-400 + 1-propanol, and PPG-400 + 1-butanol,
polyethylene glycol (PEG) 200 + methyl acrylate, PEG 200 + ethyl acrylate, and PEG-200 + n-butyl acrylate.
To understand the anharmonic and quasi crystalline behaviour of these systems, several other parameters such
as excess Debye temperature, diffusion constant, latent heat of melting, Debye frequency, pseudo-Grüneisen
parameter (
Γ
), Bayer's parameter (B/A) and Lennard-Jones potential repulsive term exponent have also been
determined. Excess Debye temperature values are fitted with the Redlich-Kister polynomial equation. From the
knowledge of R-K coefficients, the physico-chemical characteristics of the binary mixtures are analyzed and
discussed in terms of molecular interactions leading to the existence of quasi-crystalline structures in the systems
under study. The effect of the variation of the alkyl chain of alkanols on the properties of PPG-400 + n-alkanol
binary systems has been investigated. In addition, the effect of alkyl chain variation of acrylates on the properties
of PEG-200 + alkyl acrylate binary systems has also been studied. The analysis of the obtained results points
out that the effective Debye temperature and other related parameters can be successfully estimated for these
binary polymeric solutions using quasi-crystalline model. The present investigation provides significant information
about the existence of quasi crystalline structures and the related structural changes occurring in these polymer
solutions with variation of the type and concentration of a solute.
Keywords: Effective Debye temperature, polymers, quasi-crystalline state, excess parameters, binary polymer
solutions.
Introduction
The famous neutron scattering experiment1 established
the quasi-crystalline model for liquids. Based upon this
model, researcher2,3 put forward the concept of effective
Debye temperature (
θ
D) for liquids. Subsequently, the
original formulation was simplified4 and later
successfully used by various workers5-7 to determine
the effective Debye temperature (
θ
D) of binary, ternary,
and quaternary liquid mixtures. Recently, Pandey et al.8
reported the effective Debye temperature for pure ionic
liquids by assuming the existence of quasi crystalline
structure of the ionic liquids. Shrivastava et al.9
estimated the Debye temperature for binary mixtures of
J. Pure Appl. Ultrason. 44 (2022) pp. 64-73
ionic liquids. Insert Kandpal et al.10 computed for four
quaternary mixtures over the entire range of composition
at 298.15 K Sirhoi et al.11 extended same approach to
determine the Debye temperature of ionic liquids at
elevated pressure. It is reported12,13 that quasi-crystalline
liquid mixtures possess unique acoustical,
thermodynamical, and viscous properties owing to their
non-stick and scratch-resistant characteristics. These
properties find extensive applications in paint industry,
the synthesis of super fluids and improving camouflaging
materials. Recently, the study of thermoacoustic and
thermodynamical properties of polymer solutions
attracted the intention of researchers owing to their
application in chemical and pharmaceutical industry14,15.
The physical properties of polymeric solutions depend
# Life Member, Ultrasonics Society of India
65
DHIMAN ET AL.: ESTIMATION OF EFFECTIVE DEBYE TEMPERATURE OF POLYMERIC SOLUTIONS
directly on the nature of the spatial configuration.
Bonded by the intermolecular interactions, the spatial
structure of polymer solutions has the main
characteristics of short-range order. The direct
correlation between spatial arrangement and quasi-
crystalline state for polymer solutions was the main
motivation behind the research work of present study.
Therefore, the determination of Debye temperature and
other thermodynamical parameters such as diffusion
constant and latent heat of melting etc., for polymeric
solutions has been carried out based on the quasi-
crystalline model of the liquids.
Polypropylene glycol (PPG) is extensively used in
industry in the formulation of polyurethanes, as a
rheology modifier, in foams, membranes and the
manufacture of paint balls16,17. Three alkanols viz;
ethanol, 1-propanol, and n-butanol, chosen to dissolve
the PPG-400, are used as solvents in varnishes, perfumes,
drugs, plastics, lacquers, and plasticizers. Polyethylene
glycol (PEG) is also considered a unique class of
chemical substances. It is widely used as a defoaming
agent, lubricant, and viscosity modifier in many
products. It is also used as a coating for fresh fruit, as a
binder and modifier in latex paints18-20. The three
acrylates viz; methyl acrylate, ethyl acrylate and n-butyl
acrylate, chosen to dissolve the PEG-200, are extensively
used as processing agents, moulding resins, adhesives,
paints, coatings, and emulsions.
Keeping in view the applications of PPG-400, PEG-
200, n-alkanols and alkyl acrylates, eight pure systems
and their six binary solutions at 303.15 K have been
chosen for present study. The pure systems studied here
are polyethylene glycol (PEG)-200, polypropylene
glycol (PPG)-400, methyl acrylate (MA), ethyl acrylate
(EA), n-butyl acrylate (BA), ethanol, 1-propanol, and
1-butanol. The binary systems studied herein are PPG-
400 + ethanol, PPG-400 + 1-propanol, PPG -400 + 1-
butanol; and PEG-200 + MA, PEG-200 + EA, PEG-200
+ BA.
In this investigation, effective Debye temperature is
calculated assuming the quasicrystalline nature of
polymer solutions. Besides that, the effects of the
variation of the alkyl chains of alkanols and acrylates
on the polymer solutions' properties are also investigated.
Several acoustical and thermodynamical parameters such
as excess Debye temperature (
θ
DE), diffusion constant
(Di), Latent heat of melting (
∆
Hm) and Debye frequency
(
ν
m), pseudo-Grüneisen parameter (
Γ
), Bayer's
parameter (B/A) and Lennard-Jones potential repulsive
exponent term (n) have also been determined. The values
of the excess Debye temperature (
θ
DE) are fitted to a
Redlich-Kister polynomial equation. The experimental
data required for the computation of various parameters
are taken from literature17,21. To the best of our
knowledge, these acoustical and thermodynamical
parameters for PPG-400 + n-alkanols, and PEG-200 +
alkyl acrylates using quasi-crystalline model of liquids
and their effect on structural changes with change in
composition of these solutions has not been reported
earlier.
Theoretical
π
θ
⎡⎤
⎢⎥
⎢⎥
=⎢⎥
⎛⎞
⎢⎥
+
⎜⎟
⎢⎥
⎜⎟
⎝⎠
⎣⎦
D
B
lt
N
hV
k
UU
1
3
33
9
4
12 (1)
Debye temperature for solids can be calculated using
standard relations where h = Planck's constant, kB =
Boltzmann's constant, N = Avogadro's number, V = molar
volume, Ul and Ut are longitudinal and transverse sound
wave velocities respectively.
On the basis of the quasi-crystalline model for liquids,
the denominator part 33
12
⎛⎞
+
⎜⎟
⎜⎟
⎝⎠
lt
UU
of above expression
can be expressed in terms of density (
ρ
), instantaneous
adiabatic compressibility (
β
a) and Poisson's ratio (
σ
)
as:
()
()
()
()
()
33
22
32
33
121
12 2
31 31 2
⎡⎤
⎛⎞ ⎧⎫⎧ ⎫
++
⎪⎪⎪ ⎪
⎢⎥
+= +
⎜⎟ ⎨⎬⎨ ⎬
⎜⎟ ⎢⎥
−−
⎪⎪⎪ ⎪
⎩⎭⎩ ⎭
⎝⎠ ⎢⎥
⎣⎦
σσ
ρβ σσ
a
lt
UU (2)
Effective Debye temperature of liquids has been
computed on the basis of approximate equivalence of
isothermal compressibility (
β
T) and instantaneous
adiabatic compressibility (
β
a) just as in case of solids.
Pandey et al.3 has removed that anomaly and modified
Eq. (1) for liquid mixtures with the assumption that
coefficient of thermal expansion (
α
) has considerable
values in liquids and not in solids, so γ ≠ 1. The modified
equation for liquid mixtures is given below:
()
1
3
33
22
32
9
4
14
2
13
⎡⎤
⎢⎥
⎛⎞
⎢⎥
⎜⎟
⎝⎠
⎢⎥
=⎢⎥
⎧⎫
⎛⎞ ⎛⎞
⎪⎪
⎢⎥
+
⎨⎬
⎜⎟ ⎜⎟
⎢⎥
+
⎝⎠ ⎝⎠
⎪⎪
⎢⎥
⎩⎭
⎣⎦
π
θ
ρβ γγ
D
a
N
hV
k(3)
66 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
where
β
a can be calculated from the thermodynamic
relation
()
1
2−
=
βρ
amixmix
u
and γ is defined as :
=
β
γβ
T
a
The Debye frequency (
ν
m), Diffusion constant (Di)
and Latent heat of melting (
∆
Hm) can be obtained from
the following equation reported in literature21,22
B
mD
k
h
νθ
=× (4)
2
96
BoD
i
kd
Dh
θ
=(5)
2
9
128 B
mDo
k
M
Hd
h
∆θ
⎛⎞
=⎜⎟
⎝⎠ (6)
where d0 is the molecular diameter.
Lennard-Jones potential repulsive term exponent (n)
can be obtained from the Molar volume (Vm) and
available volume (Va) of the mixtures using equation
given below23.
61
3
⎛⎞
=−
⎜⎟
⎝⎠
m
a
V
nV(7)
The excess Debye temperature (
∆
DE) has been
calculated using well established relation mentioned
below.
where Xi is the mole fraction of ith component (in present
study i = 2) and
θ
Di the Debye temperature of ith
component.
=−
θθθ
EIM
DDD (8)
=∑
θθ
IM
DiD
i
X(9)
The values of
θ
DE were fitted with Redlich-Kister17
type polynomial equation given as:
()
12 1
0
21
=
=−
∑
nK
EK
K
Y
XX A X (10)
where YE is
θ
DE and, AK, is a fitting parameter obtained
using the least-squares method and n (=5) is the degree
of the polynomic expansion.
The R-square value (R2) are also calculated by using
the following equation.
2sum squared regression erro
r
R= 1– sum squared total error (11)
Results and Discussion
The requisite data for density (
ρ
), ultrasonic velocity
(U), molar mass (M), effective Debye temperature (
θ
D)
for pure polymers and liquids, viz; PEG-200, PPG-400,
methyl acrylate, ethyl acrylate, n-Butyl acrylate, ethanol,
1-propanol, and 1-butanol at temperature 303.15 K are
taken from literature16,20 and reported in Table 1.
Besides, the values of Debye frequency (
ν
m), Diffusion
constant (Di), and latent heat of melting (
∆
Hm) for pure
polymer and liquids are also presented in Table 1. All
the parameters are reported in SI units.
For systems PPG-400 + Ethanol, 1-Proponol and 1-
Butanol: The theoretically determined effective Debye
temperature (
θ
D) [calculated using Eq. (3) and (9)] and
its excess (
θ
DE) values for the three binary polymer
solutions viz; PPG-400 + ethanol, PPG-400 + 1-propanol,
PPG-400 + 1-butanol at various mole fraction (X1) of
PPG-400, and at 303.15 K are reported in Table 2. The
values of
θ
D obtained using ideal mixing rule [Eq. 9]
are follow the same trend of
θ
D values obtained using
Eq. 3. The close approximation and similar trend in the
Table 1 – Characteristic values of density (
ρ
), ultrasonic velocity (U), molar mass (M), Debye temperature (
θ
D), and other
parameters of pure components at 303.15 K
Components
ρ
UM
ρ
D(K)
υ
mDi×10–9
∆
HmReference
(kg m–3) (ms–1) (g mol–1) [Eqn(3)] (THz) (m2s–1) (KJ/mol)
PEG-200 1116.4 1596.2 200 77.09 1.605 6.240 19.13 [20]
PPG-400 999.9 1350 400 50.72 1.056 6.916 27.43 [16]
Methyl Acrylate 941.6 1162.2 86.09 72.15 1.502 3.51 4.202 [20]
Ethyl Acrylate 910.1 1144.1 100.12 67.14 1.398 3.671 4.741 [20]
n-Butyl Acrylate 889.4 1184.6 128.17 63.19 1.315 4.190 6.581 [20]
Ethanol 783.7 1152 46.07 85.30 85.30 2.922 2.14 [16]
1-Propanol 795.7 1190 60.09 80.50 80.50 3.316 3.02 [16]
1-Butanol 801.9 1226 74.12 77.79 77.79 3.68 3.98 [16]
67
DHIMAN ET AL.: ESTIMATION OF EFFECTIVE DEBYE TEMPERATURE OF POLYMERIC SOLUTIONS
θ
D values obtained using Eq. (3) and (9) ensure the
validity of ideal mixing rule for these binary polymeric
solutions. Using
θ
D values [obtained from Eq. (3)],
diffusion constant (Di) and latent heat of melting (
∆
Hm)
are calculated and reported in Table 3. Debye frequency
(
ν
m), pseudo-Gurneisen parameter (
Γ
), Bayer's number
(B/A) and Lennard-Jones potential repulsive exponent
term (n) are also calculated for these binary systems
and presented in Table 4. For better understanding,
variation of the excess value of effective Debye
temperature with mole fractions for the three polymers
solutions are presented in Fig. 1. The excess Debye
temperature values are fitted with the Redlich-Kister
(R-K) polynomial equation and reported the R-K
coefficients are reported in Table 5. A perusal of Table 2
indicates that
θ
D values show a decrease in magnitude
with increase in mole fraction (X1) of PPG-400 in all
the three binary systems under investigations. The
gradual decrease in the values of
θ
D suggests that the
quasi-crystalline structure for each binary liquid mixture
decreases at higher mole fractions of the polymer. The
reasonable values of
θ
D (~50 K - 86 K) are the strong
evidence of existence of effective Debye temperature
and quasi-crystalline state in these binary polymer
solutions and found in good agreement with the earlier
published report5. Excess parameters play a vital role in
assessing the compactness of liquid mixture24.
Inspection of Fig. 1 reveals that the excess effective
Table 2 – Effective Debye temperature (
θ
D) using Eq. (3) and by using Ideal mixing rule (
θ
DIM) from Eq. (9) and excess Debye
temperature (
θ
DE) using Eq. (10), at various mole fraction (X1) of PPG - 400 (i) in ethanol, (ii) in 1-propanol, and
(iii) in 1-butanol at 303.15 K
X1
θ
D(K)
θ
DIM(K)
θ
DE(K)
θ
D(K)
θ
DIM(K)
θ
DE(K)
θ
D(K)
θ
DIM(K)
θ
DE(K)
PPG-400 + ethanol PPG-400 + 1- propanol PPG-400 + 1-butanol
0.0 85.30 85.30 0 80.50 80.50 0 77.79 77.79 0
0.1 76.47 81.84 -5.37 77.75 77.52 0.23 76.11 75.08 1.03
0.2 71.01 78.39 -7.38 71.19 74.55 -3.36 69.90 72.37 -2.47
0.3 66.64 74.93 -8.29 66.67 71.57 -4.9 65.70 69.67 -3.97
0.4 63.44 71.47 -8.03 63.05 68.59 -5.54 62.27 66.96 -4.69
0.5 60.17 68.01 -7.84 59.96 65.61 -5.65 59.65 64.25 -4.6
0.6 57.76 64.55 -6.79 57.60 62.63 -5.03 57.42 61.55 -4.13
0.7 55.69 61.09 -5.4 55.59 59.65 -4.06 55.45 58.84 -3.39
0.8 53.73 57.63 -3.9 53.73 56.67 -2.94 53.68 56.13 -2.45
0.9 52.09 54.18 -2.09 52.15 53.70 -1.55 52.13 53.42 -1.29
1.0 50.72 50.72 0 50.72 50.72 0 50.72 50.72 0
Table 3 – Values of diffusion constant (Di) and latent heat of melting (
∆
Hm) of three polymer solutions, at various mole fraction
(X1) of PPG-400 (i) in ethanol, (ii) in 1-propanol, and (iii) in 1-butanol at 303.15 K
PPG - 400 + ethanol PPG - 400 +1-propanol PPG - 400 +1-butanol
X1Di×10–9 ∆HmDi×10–9 ∆HmDi×10–9 ∆Hm
(m2/s) (kJ/mol) (m2/s) (kJ/mol) (m2/s) (kJ/mol)
0.0 2.992 2.14 3.316 3.02 3.68 3.98
0.1 3.720 4.42 4.063 5.75 4.26 6.72
0.2 4.339 6.93 4.551 8.06 4.68 8.87
0.3 4.829 9.47 4.955 10.44 5.08 11.17
0.4 5.257 12.14 5.330 12.85 5.40 13.41
0.5 5.597 14.60 5.645 15.19 5.72 15.78
0.6 5.919 17.20 5.952 17.67 6.01 18.16
0.7 6.213 19.80 6.234 20.16 6.26 20.49
0.8 6.465 22.30 6.465 22.54 6.49 22.80
0.9 6.706 24.85 6.699 25.00 6.71 25.13
1.0 6.916 27.43 6.916 27.43 6.916 27.43
68 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
Debye temperature (
θ
DE) values are mostly negative for
all systems. The negative magnitude of
θ
DE values
indicates the existence of only weak interactions between
the component molecules. The polar nature of these
three alkanols supplements this conclusion. The close
look on Fig. 1 indicates that the order of the magnitude
of negative values of
θ
DE for three binary solutions of
PPG are PPG-400 + ethanol >PPG-400 +1-propanol >
and PPG-400 + 1-butanol.
Smaller magnitude of
θ
DE values for PPG-400 +
Ethanol is due the strong bonds percolation in the
interstitial spaces of the polymeric chains by the small
size ethanol molecules. Whereas, in case of PPG-400 +
1-butanol polymer system, the presence of greater steric
hindrance to the large sized 1-butanol molecules causes
weakest bonding in the component molecules and hence
maximum negative values of
θ
DE. The intermediate
values of
θ
DE for the system PPG-400 +1-propanol are
as expected. The excess Debye temperature (
θ
DE) values
are fitted to a Redlich-Kister (R-K) polynomial equation.
The Redlich-Kister polynomial coefficient (AK) with
R2-values for PPG-400 + ethanol, PPG-400 + 1-
propanol, PPG-400 + 1-butanol at 303.15 K are
presented in Table 5. The R2-values of all fits indicate
that the RK equation fits the experimental data very
well. An analysis of the obtained results points to the
existence of solvent-solute interactions of short-range
order among the constituents of the binary polymeric
mixtures. This conclusion is supported by Redlich-Kister
coefficients with their high R2-values for the three
functional chemical combinations (PPG-400 and
alkanols)24,26. Furthermore, the interactions become
weak with the addition of alkyl group in the chain of
these alkanols. The inspection of Table 3 reveals very
little variation in values of diffusion constant (Di) with
change in length of alkanol chain which confirms the
prevalence of only weak interactions in the binary
systems [Table 3]. Whereas the rise in the values of
diffusion constant (Di) and latent heat of melting
θ
DE)
with rise in mole fraction of the polymer, confirms the
preponderance of polymeric nature of the binary mixture
at higher concentrations of the PPG-400 [Table 3].
The Debye frequency (
ν
m) is the theoretical maximum
frequency that atoms or molecules in the liquid can
Table 4 – Values of Debye frequency (
ν
m), pseudo-Gruneisen parameter (
Γ
), Bayer's number (B/A) and Lennard-Jones potential
repulsive term exponent (n) of three polymer solutions, at various mole fraction (X1) of PPG-400 (i) in ethanol, (ii) in
1-propanol, and (iii) in 1-butanol at 303.15 K
PPG - 400 + ethanol PPG - 400 +1-propanol PPG - 400 +1-butanol
X1
ν
m (THz)
Γ
B/A n
ν
m (THz)
Γ
B/A n
ν
m (THz)
Γ
B/A n
0.0 1.776 1.195 6.427 10.281 1.683 1.201 6.473 10.418 1.619 1.212 6.510 10.531
0.1 1.592 1.136 6.572 10.716 1.618 1.100 6.693 11.079 1.584 1.075 6.735 11.205
0.2 1.478 1.111 6.654 10.962 1.482 1.083 6.724 11.171 1.455 1.067 6.746 11.237
0.3 1.387 1.097 6.700 11.099 1.388 1.065 6.753 11.260 1.368 1.067 6.758 11.275
0.4 1.320 1.085 6.738 11.213 1.312 1.066 6.765 11.295 1.296 1.061 6.768 11.305
0.5 1.253 1.081 6.748 11.243 1.248 1.063 6.771 11.312 1.242 1.063 6.777 11.332
0.6 1.202 1.077 6.763 11.289 1.199 1.064 6.779 11.338 1.195 1.063 6.785 11.355
0.7 1.159 1.074 6.774 11.323 1.157 1.065 6.786 11.359 1.154 1.059 6.792 11.377
0.8 1.118 1.071 6.780 11.340 1.118 1.059 6.793 11.378 1.117 1.058 6.796 11.387
0.9 1.084 1.068 6.786 11.359 1.085 1.058 6.797 11.392 1.085 1.058 6.799 11.396
1.0 1.056 1.057 6.799 11.397 1.056 1.057 6.800 11.401 1.056 1.057 6.800 11.401
Fig. 1.Variation of the excess value of Debye temperature
(
θ
DE) with mole fractions (X1) of PPG-400
69
DHIMAN ET AL.: ESTIMATION OF EFFECTIVE DEBYE TEMPERATURE OF POLYMERIC SOLUTIONS
oscillate at and can be derived from the speed of sound
in the liquid. It shows a decreasing trend with rise in X1
values for all the systems [Table 4]. The trend of variation
of vm with increase in the chain length of the alkanol is
also like
θ
D. The main physical meaning of the Grüneisen
approach and using the Grüneisen parameter lies in
quantifying phonon anharmonicity. Here, the Grüneisen
parameter for individual mode frequencies is defined as
the logarithmic derivative of phonon frequencies with
respect to volume change. Hence the larger the Grüneisen
parameter, the larger is the anharmonicity. The trend of
variation of pseudo-Gruneisen parameter is same as that
of Debye frequency, with rise in X1 values as well as
change in chain length of the alkanols, for all the systems
[Table 4]. The opposite trends of variation in the values
of Bayer's parameter (B/A) and Lennard-Jones potential
repulsive term exponent (n) with the increase in the
values of X1 and chain length of alkanols, supplements
the results arrived at earlier. The value of n within the
range 10.4-11.5 confirms the existence of short-range
forces in the systems, leading to existence of quasi
crystalline structures in them.
For systems PEG-200 + Methyl Acralytes, Ethyl
Acralytes and n-Butyl Acralytes: The theoretically
calculated values of effective Debye temperature (
θ
D)
[calculated using Eq. (3) and (9)] and its excess (
∆
DE)
values for the three binary polymer solutions viz; PEG-
200 + MA, PEG-200 + EA, and PEG-200 + n-BA
polymer solutions at various mole fraction (X1) of PEG-
200, and at 303.15 K are reported in Table 6. Using
θ
D
values [obtained from Eq. (3)], diffusion constant (Di)
and latent heat of melting (
∆
Hm) are calculated and
reported in Table 7. Debye frequency (
ν
m), pseudo-
Gurneisen parameter (
Γ
), Bayer's number (B/A) and
Lennard-Jones potential repulsive exponent term (n)
are also calculated for these binary systems and
presented in Table 9. The results presented in Table 6
indicate that
θ
D values [determined by using Eq. (3)]
gradually increase with increase in mole fraction (X1)
Table 5 – Redlich-Kister polynomial coefficients with R-square (R2) value for PPG-400 + ethanol, PPG-400 + 1-propanol,
PPG-400 + 1-butanol for excess Debye temperature (
θ
DE) at 303. 15 K
Binary Polymer A0A1A2A3A4A5R2
PPG-400 + Ethanol -30.930 14.135 -6.845 11.663 -14.950 2.483 0.999
PPG-400 + 1-Propanol -22.539 4.488 -7.826 19.744 45.450 -70.131 0.997
PPG-400 + 1-Butanol -18.055 4.045 -9.481 12.508 53.702 -68.369 0.997
Table 6 – Effective Debye temperature (θD) using Eq. (3) and by using ideal mixing rule (
θ
DIM) from Eq. (9) and excess Debye
temperature (
θ
DE) using Eq. (10) of three polymer solutions, at various mole fraction (X1) of PEG-200
(i) MA, (ii) in EA and (iii) in n-BA at 303.15 K.
X1θD(K) θDIM(K) θDE(K) X1θD(K) θDIM(K) θDE(K) X1θD(K) θDIM(K) θDE(K)
PEG - 200 + MA PEG - 200 + EA PEG - 200 + n-BA
0.000 72.15 72.15 0 0.000 67.14 67.14 0 0.000 63.19 63.19 0
0.077 72.64 72.53 0.10 0.0725 67.45 67.86 -0.41 0.0730 63.39 64.21 -0.82
0.1539 73.42 72.91 0.50 0.145 67.86 68.59 -0.73 0.1459 63.72 65.22 -1.51
0.2254 74.09 73.27 0.82 0.2185 67.97 69.32 -1.35 0.2131 64.14 66.16 -2.02
0.2968 74.70 73.62 1.08 0.292 68.92 70.05 -1.13 0.2802 64.67 67.09 -2.42
0.3797 75.33 74.03 1.30 0.3660 69.58 70.79 -1.21 0.3538 65.36 68.11 -2.75
0.4626 75.87 74.44 1.43 0.4400 70.27 71.52 -1.25 0.4274 66.19 69.13 -2.95
0.5259 76.22 74.75 1.47 0.5095 70.98 72.21 -1.24 0.4963 67.09 70.09 -3.00
0.5891 76.53 75.06 1.47 0.5790 71.73 72.90 -1.17 0.5651 68.11 71.05 -2.94
0.6589 76.81 75.41 1.40 0.6485 72.51 73.60 -1.09 0.6381 69.31 72.06 -2.76
0.7287 77.03 75.75 1.28 0.7180 73.37 74.29 -0.92 0.711 70.62 73.08 -2.46
0.7956 77.18 76.08 1.10 0.784 74.19 74.94 -0.75 0.7823 72.02 74.07 -2.05
0.8625 77.29 76.41 0.87 0.8500 75.01 75.60 -0.59 0.8535 73.55 75.06 -1.51
0.9313 77.34 76.75 0.59 0.925 76.07 76.35 -0.28 0.9268 75.25 76.08 -0.82
1.000 77.09 77.09 0 1.000 77.09 77.09 0 1.000 77. 09 77.09 0
70 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
of PEG-200 in all the three binary mixtures of PEG200
+ MA, PEG200 + EA and PEG200 + n-BA. The gradual
increase in the values of
θ
D points out that the quasi-
crystalline state is more probable at higher mole fraction
(X1) of PEG-200 in these polymer solutions. The
reasonable values of
θ
D (~63 K -77 K) are the strong
evidence of existence of quasicrystalline state for these
polymer solutions5. The increase in the value of
θ
D
with increase in mole fraction (X1) of PEG-200 in
polymer solutions suggests that solute-solvent
interactions play an important role in the determination
of
θ
D for all the three binary mixtures. PEG is a relatively
non-polar molecule that interacts favourably with the
exposed non-polar surfaces when a macromolecule
unfolds. The non-polar nature of these three acrylates
supplements this conclusion. The value of effective
Debye temperature calculated using ideal mixing rule
[determined by using Eq. (4)] is comparable and follow
the same trend as
θ
D values calculated by using the
standard relation [Eq. (3)]. These results ascertain the
validity of ideal mixing rule for these binary solutions.
The values of diffusion constant (Di) and latent heat of
melting (
∆
Hm) for these binary solutions are reported in
Table 7. Magnitude of these parameters increase with
rise in value of X1 of PEG-200. This points out to the
existence of solute-solvent type of interactions leading
to quasi crystalline behaviour of the system. The Debye
frequency (
ν
m) shows an increasing trend with rise in
X1 values for all the systems (Table 9). It is interesting
to observe that the trend of magnitude of
ν
m decreases
with the increase in the chain length of the acrylates
(MA>EA>n-BA). The trend of variation of pseudo-
Gruneisen parameter is opposite to that of Debye
frequency, with rise in X1 values of PEG-200 in solution
as well as with change in chain length of the acrylates,
for all the systems (Table 9). The trends of variation in
the values of Bayer's parameter (B/A) and Lennard-
Jones potential repulsive termexponent (n) with the
increase in the values of X1 or chain length of acrylates
is opposite to the variation in pseudo-Gruneisen
parameter with such a change. The values of n within
the range 10.5-12.5 confirms the existence of short-
range forces in the systems, leading to existence of
quasi crystalline structures within these three systems
of binary solutions of PEG -200 + MA, PEG-200 + EA,
PEG-200 + n-BA. Therefore, the concept of effective
Debye temperature can be successfully applied to
understand the quasicrystalline structure of these binary
polymer solutions. For better understanding, variation
of the excess value of effective Debye temperature (
θ
DE)
with mole fractions (X1) for the three polymers solutions
are presented in Fig. 2. The excess Debye temperature
(
θ
DE) values are fitted with the Redlich-Kister (R-K)
polynomial equation and the R-K coefficients in given
in Table 8.
Table 7 – Values of diffusion constant (Di) and latent heat of melting (
∆
Hm) at various mole fraction (X1) of PEG-200 in PEG -
200 + MA, PEG - 200 + EA and PEG - 200 + n-BA at 303.15 K
X1Di×10–9
∆
HmX1Di×10–9
∆
HmX1Di×10–9
∆
Hm
PEG - 200 + MA PEG - 200 + EA PEG - 200 + n-BA
0.000 3.510 4.202 0.000 3.671 4.741 0.000 4.19 6.581
0.077 3.734 4.99 0.0725 3.836 5.37 0.073 4.280 7.02
0.1539 3.975 5.89 0.145 4.003 6.06 0.1459 4.378 7.53
0.2254 4.194 6.78 0.2185 4.144 6.70 0.2131 4.476 8.04
0.2968 4.409 7.74 0.292 4.338 7.56 0.2802 4.580 8.60
0.3797 4.654 8.91 0.3660 4.527 8.48 0.3538 4.702 9.27
0.4626 4.892 10.14 0.4400 4.720 9.47 0.4274 4.834 10.01
0.5259 5.070 11.13 0.5095 4.888 10.38 0.4963 4.966 10.77
0.5891 5.243 12.14 0.5790 5.062 11.35 0.5651 5.107 11.60
0.6589 5.430 13.30 0.6485 5.245 12.44 0.6381 5.267 12.57
0.7287 5.611 14.48 0.7180 5.436 13.62 0.711 5.437 13.64
0.7956 5.780 15.63 0.784 5.620 14.81 0.7823 5.615 14.79
0.8625 5.943 16.81 0.8500 5.806 16.06 0.8535 5.805 16.07
0.9313 6.105 18.03 0.925 6.020 17.56 0.9268 6.012 17.52
1.000 6.240 19.13 1.000 6.240 19.13 1.000 6.240 19.13
71
DHIMAN ET AL.: ESTIMATION OF EFFECTIVE DEBYE TEMPERATURE OF POLYMERIC SOLUTIONS
Table 8 – Redlich-Kister polynomial coefficients with R-square (R2) value for PEG-200 + MA, PEG-200 + EA, and PEG-200 +
n-BA for excess Debye temperature (
θ
DE) at 303. 15 K
Binary polymer
Mixture A0A1A2A3A4A5R2
PEG-200+MA 5.8394 1.4527 -0.2284 -1.4679 -1.0345 7.6160 0.9998
PEG-200+EA 5.8882 1.1895 -0.4867 -1.6535 -0.8549 7.3650 0.9995
PEG-200+n-BA 5.9064 0.7229 -0.2860 -0.3889 -1.1301 6.7980 0.9998
Table 9 – Values of Debye frequency (
υ
m), pseudo-Grüneisen parameter (
Γ
), Bayer's number (B/A) and Lennard-Jones potential
repulsive term exponent (n) of three polymer solutions, at various mole fraction (X1) of PEG-200 (i) in methyl acrylate,
(ii) in ethyl acrylate, and (iii) in n-butyl acrylate at 303.15 K
X1
υ
m (THz)
Γ
B/A n X1
υ
m (THz)
Γ
B/A n X1
υ
m (THz)
Γ
B/A n
PEG-200 +MA PEG-200 +EA PEG-200 +n-BA
0.000 1.502 1.037 6.570 10.710 0.000 1.398 1.059 6.527 10.582 0.000 1.315 1.099 6.549 10.646
0.077 1.512 1.029 6.630 10.891 0.0725 1.404 1.048 6.572 10.717 0.073 1.319 1.086 6.581 10.742
0.1539 1.528 1.026 6.692 11.076 0.145 1.412 1.039 6.618 10.853 0.1459 1.326 1.074 6.614 10.841
0.2254 1.542 1.024 6.746 11.238 0.2185 1.415 1.029 6.656 10.968 0.2131 1.335 1.064 6.646 10.937
0.2968 1.555 1.022 6.796 11.389 0.292 1.434 1.026 6.708 11.124 0.2802 1.346 1.056 6.679 11.036
0.3797 1.568 1.021 6.851 11.553 0.3660 1.448 1.022 6.753 11.260 0.3538 1.360 1.048 6.717 11.150
0.4626 1.579 1.021 6.901 11.703 0.4400 1.463 1.019 6.798 11.395 0.4274 1.378 1.041 6.757 11.270
0.5259 1.587 1.022 6.937 11.811 0.5095 1.477 1.017 6.840 11.521 0.4963 1.396 1.036 6.795 11.386
0.5891 1.593 1.022 6.971 11.912 0.5790 1.493 1.017 6.882 11.647 0.5651 1.418 1.032 6.836 11.509
0.6589 1.599 1.023 7.005 12.016 0.6485 1.509 1.017 6.924 11.772 0.6381 1.443 1.029 6.881 11.644
0.7287 1.603 1.024 7.038 12.113 0.7180 1.527 1.018 6.967 11.900 0.711 1.470 1.026 6.928 11.785
0.7956 1.607 1.026 7.066 12.199 0.784 1.544 1.019 7.007 12.020 0.7823 1.499 1.025 6.977 11.930
0.8625 1.609 1.027 7.093 12.280 0.8500 1.561 1.021 7.046 12.138 0.8535 1.531 1.025 7.027 12.081
0.9313 1.610 1.028 7.119 12.357 0.925 1.583 1.024 7.092 12.277 0.9268 1.566 1.025 7.081 12.243
1.000 1.605 1.027 7.138 12.413 1.000 1.605 1.027 7.138 12.413 1.000 1.605 1.027 7.138 12.413
Fig. 2.Variation of the excess value of Debye temperature
(
θ
DE) with mole fractions (X1) of PEG-200
MA and negative for PEG-200 + EA and PEG-200 + n-
BA polymer solutions. Positive value of (
θ
DE) for PEG-
200 + MA polymer solution indicate the enhancement
of short-range order of the molecules as compared to
the pure components whereas the negative value of (
θ
DE)
indicate the decrease in the short-range order of the
molecules as compared to its pure counterpart24.
Inspection of Fig. 2 further reveals a slight irregular
behaviour in the value of (
θ
DE) in polymer solution of
PEG-200 + EA. It suggests the deviation from the quasi-
crystalline state of this binary system. The Redlich-
Kister (R-K) polynominal fitting procedure was
implemented on these polymer solutions in a similar
way as previously. The R2 values for PEG-200+ MA,
PEG-200 + EA, and PEG-200 + n-BA polymer solutions
indicate R-K equations fits the data very well [Table
8]. R-K coefficient and R2 values in these polymeric
solutions also support the conclusion that short range
order exists in these systems25,26.
The perusal of Fig. 2 indicates that excess effective
Debye temperature (
∆
DE) is positive for PEG-200 +
72 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
Conclusion
This paper reports the theoretically determined values
of effective Debye temperature (
θ
D), for six binary
solutions at 303.15 K. Based on the quasi-crystalline
model approach, the evaluated results for the effective
Debye temperature of these binary solutions show a
reasonable value, at all compositions. The trends of
variations of other acoustical and thermodynamical
parameters supplements the conclusions about effective
Debye temperature. The effect of the variation of the
alkyl chain of alkanols on the properties of PPG400 +
alkanol binary systems and the effect of alkyl chain
variation of acrylates on the properties of PEG200 +
alkyl acrylate binary systems has been explained in terms
of solute-solvent intermolecular interactions. Redlich-
Kister coefficients with their R2 values in functional
chemical combinations (PPG-400 and alkanols; PEG-
200 and acrylates) support the conclusion that short
range order exists in these systems. Moreover, the
molecular interactions among the constituent molecules
become weak with the addition of alkyl group in the
chains of the alkanols and acrylates. The obtained results
also concluded that these polymeric solutions show quasi
crystalline behaviour due to solute-solvent interactions
prevalent in them and the effective Debye temperature
as well as other thermodynamic parameters for these
systems can be successfully determined using the applied
methodology.
References
1. Hughe, D. J., Palevsky H., Kley W. and Tunkelo E.,
Atomic motions in water by scattering of cold neutrons,
Phys. Rev., 119(3), (1960) 872.
2. Kor S. K. and Tripathi D. N., Temperature and pressure
dependence of effective Debye temperature in associated
liquids based on quasi crystalline model, J. Phys. Soc.
Jpn., 36(2), (1974) 552-554.
3. Pandey J. D., Sanguri V., Mishra R. K. and Singh A. K.,
Acoustic method for the estimation of effective Debye
temperature of binary and ternary liquid mixtures, J. Pure
Appl. Ultrason., 26(1), (2004) 18-29.
4. Nori T. S., Srinivasu C., and Fakruddin Babavali S.,
Computational study of debye temperature for liquid
mixtures-thermal energy variations, Phys. Chem. Res.,
8(1), (2020) 167-173.
5. Vyas V., Ultrasonic investigation of effective Debye
temperature in multicomponent liquid systems at 298.15
K, Phys. Chem. Liq., 42(3), (2004) 229-36.
6. Shukla R. K., Shukla S. K., Pandey V. K. and Awasthi P.,
Sound velocity, effective Debye temperature and pseudo-
Grüneisen parameters of Pb-Sn mixtures at elevated
temperatures, Phys. Chem. Liq., 45(2), (2007) 169-180.
7. Gopal A. M., Raj A. M. and Poongodi J., Estimation of
Debye temperature for binary liquid mixtures using
ultrasonic techniques, J. Pure Appl. Ultrason., 41, (2019)
90-93.
8. Pandey J. D., Shukla A. K., Singh N. and Sanguri V.,
Estimation of thermodynamic properties of ionic liquids,
J. Mol. Liq., 315, (2020) 113585.
9. Shrivastava S. C., Sanguri V., Srivastava S. and Pandey
J. D., Effective Debye temperature of ionic liquids and
their binary mixtures, J. Mol. Liq. 347, (2022) 118382.
10. Kandpal C., Singh, A.K., Dey R., Singh V.K. and Singh
D., Estimation of effective Debye temperature of
multicomponent liquid mixtures at 298.15 K, J. Pure Appl.
Ultrason. 41, (2019) 19-23.
11. Sirohi A., Dogra A., Singh D. P. and Upmanyu A., Quasi-
Crystalline behavior of imidazolium based pure ionic
liquids over the extended pressure range, ECS Trans.,
107, (2022) 8583.
12. Dubois J.M., New prospects from potential applications
of quasi crystalline materials, Mater. Sci. Eng. A., 294,
(2000) 4-9.
13. Huttunen-Saarivirta E., Microstructure, fabrication and
properties of quasi crystalline Al-Cu-Fe alloys: a review,
J. Alloys Compd., 363, (2004) 154-78.
14. Deb P. K., Kokaz S. F., Abed S. N., Paradkar A. and
Tekade R. K., Pharmaceutical and biomedical
applications of polymers, Basic fundamentals of drug
delivery. Academic Press New York (2019) 203-267.
15. Sannaningannavar F.M, Navati B.S and Ayachit N.H.,
Studies on thermo-acoustic parameters of poly (ethylene
glycol)-400 at different temperatures, J. Therm. Anal.
Calorim., 112, (2013) 1573-1578.
16. Dhiman M., Singh K., Kaushal J., Upmanyu, A. and Singh
D. P., Ultrasonic study of molecular interactions in
polymeric solution of polypropylene glycol-400 and
ethanol at 303 K, Acta Acust united Ac. 105, (2019) 743-
752.
17. Raju K., Karpagavalli K. and Krishnamurthi P., Ultrasonic
studies of molecular interactions in the solutions of poly
(propylene glycol) 400 in N-alkanols. Euro. J. Appl. Eng.
Sci. Res., 1, (2012) 216-219.
18. Upmanyu A., Dhiman M., Singh D. P. and Kumar H.,
73
DHIMAN ET AL.: ESTIMATION OF EFFECTIVE DEBYE TEMPERATURE OF POLYMERIC SOLUTIONS
Thermo-viscous investigations of molecular interactions
for the binary mixtures of polyethylene glycol-400 and
polyethylene glycol-600 with dimethyl sulfoxide and
water at different temperatures, J. Mol. Liq., 334, (2021)
115939.
19. Ottani S., Vitalini D., Comelli F. and Castellari C.,
Densities, viscosities, and refractive indices of poly
(ethylene glycol) 200 and 400+ cyclic ethers at 303.15
K, J. Chem. Eng. Data., 47, (2002) 1197-1204.
20. Han, F., Zhang J., Chen G. and Wei X., Density, viscosity,
and excess properties for aqueous poly (ethylene glycol)
solutions from (298.15 to 323.15) K, J. Chem. Eng. Data.,
53, (2008) 2598-2601.
21. Chaudhary N. and Nain A. K., Densities, speeds of sound,
refractive indices, excess and partial molar properties of
polyethylene glycol 200+ methyl acrylate or ethyl acrylate
or n-butyl acrylate binary mixtures at temperatures from
293.15 to 318.15 K, J. Mol. Liq., 271, (2018) 501-13.
22. Duc N. B., Hieu H. K., Hanh P. T., Hai T. T., Tuyen, N. V.
and Ha, T. T., Investigation of melting point, Debye
frequency and temperature of iron at high pressure, Eur.
Phys. J. B., 93, (2020) 1-7.
23. Upmanyu A. and Singh D.P., Ultrasonic studies of
molecular Interactions in Polymer Solution of the
polyisobutylene (PIB) and benzene, Acta Acust united
Ac., 100, (2014) 434-439.
24. Dash A. K. and Paikaray R., Acoustical study on ternary
mixture of dimethyl acetamide (DMAC) in diethyl ether
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Chem. Liq., 51, (2013) 749-763.
25. Sastry S. S., Babu S., Vishwam T. and Tiong H.S., Excess
parameters for binary mixtures of alkyl benzoates with
2-propanol at different temperatures, J. Therm. Anal.
Calorim., 116, (2014) 923-935.
26. Das D., Messaâdi A., Barhoumi Z. and Ouerfelli N., The
relative reduced Redlich-Kister equations for correlating
excess properties of N, N-dimethylacetamide + water
binary mixtures at temperatures from 298.15 K to 318.15
K, J. Solution Chem. 41, (2012) 1555-1574.
27. Gayathri A., Venugopal T. and Venkatramanan K.,
Redlich-Kister coefficients on the analysis of physico-
chemical characteristics of functional polymers, Mater.
Today: Proc., 17, (2019) 2083-2087.
74 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
Effect of sonication on enhancement of mechanical
properties of epoxy blended rattan fibre
Susanta Behera1, G. Nath2*# and J.R.Mohanty1
1Department of Mechanical Engineering, Veer Surendra Sai University of Technology, Sambalpur-768 018, Odisha
2Department of Physics, Veer Surendra Sai University of Technology, Sambalpur-768 018, Odisha
*E-mail: ganeswar.nath@gmail.com
Ultrasound sonication is one of the promising techniques to disperse the fiber particles into polymeric
matrix thoroughly. The present study encompasseseffect of ultrasonic treatment on mechanical property of rattan/
epoxy composite (REC). Rattan fibers were initially treated with NaOH using ultrasonic bath sonicator for
proper absorption of chemical into the fibre surface. Treated rattan fibers were blended with epoxy matrix
material by ultrasound during the winding process to enhance the adhesion. The ultrasonic cavitation, improves
the wetting between aramid fibers and resins. According to the ultrasonic treatment the interfacial properties of
the composite has been greatly improved. Various mechanical properties such as tensile strength, Young's modulus,
flexural strength, flexural modulus, impact strengths and hardness of the fabricated composite have been calculated
and analyzed with different weight percentage of fiber matrix composition. Scanning electron microscopy (SEM)
has been used for the characterization of fabricated composite. The NaOH treated composite shows better tensile,
flexural and impact strength at 45.5 MPa, 121.89 MPa and 39.445 J/m–1 respectively. The mechanical properties
of the treated REC were better than the untreated REC. This shows that composite with ultrasonic treatment has
good mechanical properties and can be used for wide range of engineering applications.
Keywords: Ultrasonication, rattan fiber, surface treatment, mechanical strength.
Introduction
Polymer matrix natural fibre composites have received
a lot of attention in the lightweight fields due to their
high particular stiffness and specific strength. Presently,
climate change is a major concern because of global
warming. Several measures are being taken worldwide
to safeguard life on the earth by making eco-friendly
bio-degradable products. Natural fibres are gaining
popularity among researchers and academics for use in
polymer composites due to their eco-friendliness and
sustainability, as well as their superior qualities when
compared to synthetic fibres1. As a result, researchers
focused more on developing lightweight biodegradable
composites. Researchers were eager to discover various
types of chemical treatment procedures with superior
properties. As a result, researchers found ultrasonic
treatment. Recently, ultrasonic treatment is used for
chemical modification of natural fibers which increases
J. Pure Appl. Ultrason. 44 (2022) pp. 74-78
surface roughness of fiber and causes increased bonding
adhesion with polymer because of the ultrasonic
cavitation effect. This would disperse the existed
agglomeration consisting of microparticles2. Among
polymer matrixes, a great amount of study has been
spent on enhancing the properties of epoxy from both a
scientific and an industrial standpoint3. Rattan is one of
the many natural fibres that have been explored so far.
It is a member of the "Calamoideae" subfamily and is
used to make chairs, clothing, medicines, handicrafts,
and artwork. Additionally, its fruit is edible. According
to the findings, natural fibres like rattan are becoming
more and more competitive with glass fibres in
composites due to their affordability and apparent
ecological superiority. Because of its superior
mechanical and thermal properties, rattan fibre to be a
suitable choice for composites4.
The goal of the current study is to create an
environmentally friendly composite material with epoxy
# Life Member, Ultrasonics Society of India
75
BEHERA ET AL.: EFFECT OF SONICATION ON MECHANICAL PROPERTIES OF EPOXY BLENDED FIBRES
resin serving as the matrix and rattan fibre as
reinforcement. The project shows how to treat the surface
of rattan fibre using non-destructive methods like
ultrasonic processing. NaOH mixed surfactants are used
to modify the surface structure. Rattan fiber's surface
becomes roughened as a result of surfactant action,
which improves the fiber's interface bonding with epoxy.
Different mechanical properties tensile strength, flexural
strength, impact strength and hardness have been
evaluated to determine its ability to resist deformation
under load. Water absoption capability of REC was
determined by water uptake test. The interactions
between polymers and fibers are characterized using
scanning electron microscopy (SEM) and Fourier
transform infrared (FTIR) spectroscopy.
Materials and Methodology
Prepreparation of Rattan Fibers: Rattan plant stem
were collected from local area, immersed in water for
30 days to allow microbial degradation which separates
the cellular tissue and pectin's surrounds the fiber
bundles. Then long fibres are extracted from RPS and
allowed to dry in an oven for 24 hr at 100°C to remove
moistures present on fiber surface. Aqueous NaOH
solution was created by impregnating a 125 kHz high
frequency ultrasonic pulse. In order to create a uniform
combination of surfactants, 100 ml of water with NaOH
pellets was dissolved in it5.
Mechanism of Ultrasonication in Modifying the
Surface of Rattan Fibre: The alkaline solution was
prepared by ultrasonic sonication technique which was
used for surface treatment of short UF. The UF are
placed in a beaker containing NaOH solvent within a
sonicator operating at 60 W and 125 kHz for well
dispersion and bleaching of the surface of UF for 60
minutes which are termed as treated rattan fiber (TRF).
This ultrasonic process separates the fibers from the
cell wall by employing cavitation in an alkaline media.
Owing to cavitation effect, ultrasound in liquid medium
can result in rise in pressure, temperature and shear
force. This leads to formation of micro bubbles6. During
the compression cycle these micro bubbles get collapsed
and networks of lignin, cellulose and xylan are homo-
lyzed. Additionally, the bursting of the asymmetrical
bubbles on the fiber surface generates a strong microjet
that collides heavily with the fiber, causing cracking
and shrinkage on the lignocellulosic fiber7.
Fabrication of composite: Extracted fibers are mixed
with epoxy resin at different weight fractions (0, 5, 10,
15, 20 and 25%) for the manufacture of the composites,
and the mixture were poured into the mould.
Compression moulding technique was used for the
synthesis of REC.
Characterization of Rattan Fiber and Composite:
SEM Quanta FEG 650 instruments operating at a 10 kV
voltage have been used to examine the surface
morphology of ruptured REC. Fourier transform infrared
(FTIR) spectra of TRF/epoxy composite was carried
out in the range of 4000-600 cm–1 using thermo scientific
Spectrophotometer (Nicolet 6700) to the different
functional groups present on the composite6.
Water Uptake Test: Water absorption test of REC
were carried out as per ASTM-D570 standard speci-
fication. The composite specimens were first cut into
10 mm in length, 10 mm width and 3 mm in thickness
then dried in an oven for 24 hr at 100°C. After cooling
samples to room temperature, the initial weight of all
the samples were observed to a precision of 0.0001 g.
For different time durations i.e., 6, 12, 24, 48 hr test was
conducted. The results are presented by applying the
following equation7.
Water uptake percentage
10
0
(Final weght – Initial weght)
Initial weght
=×
(1)
Tests for Mechanical Properties
Tensile test: An INSTRON 3382 instrument with a
load cell of 5kN was used to perform tensile testing on
the constructed rattan composite. The test specimens
were machined in accordance with ASTM D 638
standard. At a cross head speed of 2 mm/min, five
samples from each fibre loading have been evaluated.
Flexural test: Flexural testing using the instrument
(INSTRON 3382) has been implemented using a three-
point bending technique. The ASTM D790-99 standard
was followed in the preparation of the testing specimen,
which had the dimensions 127 mm, 12.7 mm, and 3.2
mm.
Impact test: Izod Impact testing equipment of the
pendulum type was used to conduct impact tests in
accordance with ASTM D 256. Five reliable test results
were selected for each fibre loading, and the average
values were recorded.
Hardness test: Vickers micro-hardness tester was used
76 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
to determine the hardness of the composite (VH3300
Buehler, USA). Five samples were evaluated at each
fibre loading.
Results and Discussion
The scanning electron microscopy (SEM) image of
sodium hydroxide treated rattan fibre and REC is shown
in Fig.1(a)
3386 cm–1. In both cases the band is similarly stretched
indicating that there is no much deformation in -OH
group. But nearly around 3859 cm–1 the composite shows
a higher band length than epoxy. This may be due to the
removal of hydroxyl group of the cellulose due to
chemical treatment if rattan fiber9. The deserted bands
from 1503 cm–1 to 1030 cm–1 in rattan fiber are of short
band length than the raw epoxy due to the more ring
side vibration of C-O-H groups of lignin and C-O-R
(glycoside) groups of hemicellulose10. Thus, the action
of surfactants on the surface of rattan fiber suitably
modifies the functional group of the lignocellusic
components for enhancement of sound absorption.
Figure 3 shows water uptake % of neat epoxy and
REC at various fiber content as a function of time in
hours. The presence of natural fiber increases the water
absorption due to its hygroscopic characteristics. The
result shows that the water absorption increases with
the increment of fiber percentage, due to number of
voids exists in the composites also increases. It can be
seen that water absorption increases rapidly at initial
stage and after 48hr it reaches saturation level without
any further increment11. Figure 4(a) depicts the tensile
strength of REC at various fibre weight percentages.
The maximum strength at 20% loading indicates that
the matrix and reinforcement have stiffened and can
withstand greater stress at the same strain level. The
decrease in strength at higher fibre loading is due to the
fact that when fibre content increases, the fibres function
as fissures and cracks form, resulting in stress
concentration zones that diminish composite stiffness12.
The fluctuation in flexural strength of REC with varied
fibre loadings is shown in Fig. 4(b). As per the research,
the flexural value of the composite appears to be similar
to the tensile strength of the composite. Flexural strength
Fig. 1.SEM of (a) NaOH Treated fiber (b) REC
Fig. 2.FTIR of neat epoxy and REC Fig. 3. Water up take (%) at different fiber loding
Figure 1(a) shows the SEM of treated fibre is distinct
and debundled with fibre measurement on a 100 m scale.
while, the SEM of REC Fig. 1(b) shows the
homogeneous circulation of rattan filaments in an epoxy
matrix. According to the SEM of the composite, there is
good bonding between the rattan fibre and the matrix,
and the fibres are coated by epoxy resin8.
FTIR spectroscopy has been performed to observe
the microscopic changes in different functional group
which constitutes the rattan fiber composite. The
deformations occurred at functional groups cellulose
are shown in Fig. 2 with respect to transmittance % and
the wave number. From the IR spectrum band, it is
clearly observed that the occurrence of broad band at
77
BEHERA ET AL.: EFFECT OF SONICATION ON MECHANICAL PROPERTIES OF EPOXY BLENDED FIBRES
and the fiber. Fiber-fiber stickiness occurs at higher
fibre loadings, resulting in non-uniform fibre dispersion
in the matrix and a reduction of flexural strength12. The
variation in impact strength of the composite with fibre
loadings is depicted in Fig. 4(c).
According to the findings, impact strength exhibits
the same behaviour as flexural strength. It increases
with fibre loadings and then declines after a specific
value. The maximal impact strength is at 20% fibre
loading because the fibre matrix adhesion is stronger at
this fibre loading. At this point, the fibre crowding is
adequate and serves as a stress exchange route. The
voids and stress concentration sites expand as the fibre
content increases, and fracture initiation occurs at those
locations during impact. When a result, as the
concentration of fibre increases, the impact strength
decreases. In Fig. 4(d), the hardness of REC increases
with fibre loadings and decreases with increasing fibre
loading12.
Conclusion
The use of ultrasonic mercerization on lignocellulosic
biomass such as rattan fibre improves the strength of
the green composite material greatly. In this study, the
mechanical properties of REC were studied. Due to
increased voids, the water uptake percent of REC
increases with fibre weight fractions. Surface treatment
with NaOH is found to improve fibre strength. The
tensile strength of REC grows and then drops after
showing the highest value. At varying weight
percentages, other mechanical properties such as
flexural, impact and hardness have been investigated.
Tensile, flexural, impact, and hardness properties of
20% weight fraction REC increased to 46.94 MPa,
113.11 MPa, 30.67 J/m–1, and 26 Hv, respectively. At
this fibre loading, the explanation could be due to greater
bonding between the matrix and the fibre. The SEM
micrographs and FTIR results support this. As a result,
it is reasonable to conclude that rattan fibre can be
efficiently used to build composites for the production
of value-added commodities.
References
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resin-bonded recycled denim composite as an efficient
sound-absorbing material, App. Acoustics. 173, (2021)
107710.
Fig. 4.Mechanical properties of REC (a) tensile strength (b)
flexural strength (c) impact strength (d) hardness No.
initially increases with increased fibre loadings because
appropriate stress transfer takes place between the matrix
78 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
2. Gholampour A. and Ozbakkaloglu T., A review of natural
fiber composites: properties, modification and processing
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natural fibers for sound absorption applications, Buil.
Environ. 94 (2015) 840.
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thermo acoustic material from residual organic wastes,
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and Roy T., Utilization and regeneration of waste
sugarcane bagasse as a novel robust aerogel as an effective
thermal, acoustic insulator, and oil adsorbent, J. Clean
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6. Zhou B., Wang L., Ma G., Zhao X. and Zhao X.,
Preparation and properties of bio-geopolymer composites
with waste cotton stalk materials, J. Clean. Produ. 245,
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7. Nayak S. and Mohanty J.R., Influence of chemical
treatment on tensile strength, water absorption, surface
morphology, and thermal analysis of areca sheath fibers.
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8. Sahoo S.K., Mohanty J.R., Nayak S. and Behera B.,
Chemical treatment on rattan fibers: durability,
mechanical, thermal and morphological properties, J.
Natural Fib. 18, (2021) 1762.
9. Jain J. and Sinha S., Potential of pineapple leaf fibers
and their modifications for development of tile
composites, J. Natural Fib., (2021) 1.
10. Nayak S. and Mohanty J.R., Influence of chemical
treatment on tensile strength, water absorption, surface
morphology, and thermal analysis of areca sheath fibers,
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11. Cao L., Fu Q., Si Y., Ding B. and Yu J., Porous materials
for sound absorption, Compo Commun. 10, (2018) 25.
12. Arenas J. and Crocker M., Recent trends in porous sound-
absorbing materials, Sound Vib. 44, (2010) 12.
13. Fan M., Dai D. and Huang B., Fourier transform infrared
spectroscopy for natural fibres, Intech Open. 3, (2012)
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14. Khan M.Z.H., Sarkar M.A.R., Al Imam F.I., Khan M.Z.H.
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79
YADAV ET AL.: TEMPERATURE DEPENDENT US PROPERTIES OF ScZrHf TERNARY ALLOY
Investigation of temperature dependent mechanical, thermophysical
and ultrasonic properties of ScZrHf ternary alloy
Shakti Yadav, Ramanshu P. Singh, Devraj Singh# and Giridhar Mishra*##
Department of Physics, Prof. Rajendra Singh (Rajju Bhaiya) Institute of Physical Sciences for Study and Research,
Veer Bahadur Singh Purvanchal University, Jaunpur-222001, Uttar Pradesh India
*E-mail: giridharmishra@rediffmail.com
In this paper, we present theoretically evaluated values of temperature mechanical, thermophysical and
ultrasonic properties of hexagonal close-packed structured medium entropy alloy ScZrHf in temperature range
of 0-900 K. By utilizing the Lennard-Jones potential model, we have computed the second order and third order
elastic constants (SOECs and TOECs) with the help of lattice parameters. While all of the SOECs have been
found to be decreasing with increase in temperature, the TOECs increases with temperature. SOECs and TOECs
have been used to compute the elastic moduli such as: bulk modulus, shear modulus, Young's modulus and
Poisson's ratio, and ultrasonic velocities at different angle along unique axis. Further, the thermal properties such
as Debye temperature, Debye heat capacity, energy density of ScZrHf in temperature range of 0-900 K and
lattice thermal conductivity of ScZrHf in temperature range of 300-900K have been estimated. The lattice thermal
conductivity decreases with increase in temperature. Finally, the ultrasonic attenuation due to phonon - phonon
interaction in both longitudinal and shear modes and themoelastic relaxation mechanism have been computed
for ScZrHf ternary alloy in the temperature range of 300-900 K and it has been found that the attenuation due
to phonon-phonon interaction is much higher than that due to thermoelastic relaxation mechanism.
Keywords: Refractory medium-entropy alloys, hexagonal closed-packed, ultrasonic behaviour, rare-earth,
transition metal.
Introduction
The high entropy alloys (HEAs) have attracted
remarkable attention in recent decades due to being a
new design concept of alloy materials1. HEAs are
defined as a solid solution of five or more metal elements
in single phase mixed in equal proportion. These alloys
are assumed to be stabilized by a high configurational
entropy of mixing2. Alloys with three or four elements
are considered as medium entropy alloys3. The refractory
high-entropy alloys (RHEAs) are composed of refractory
metals such as Ti, Zr, Hf, Nb, W, Mo and Ta4. These
RHEAs have been extensively studied due to their
excellent mechanical and thermal properties and
widespread applications5,6.
Zr and Hf are two of the excellent refractory metals
which shows extraordinary mechanical and thermal
J. Pure Appl. Ultrason. 44 (2022) pp. 79-85
properties such as high tensile strength and resistance
to wear and tear and high temperature, Sc is the lightest
transition metal with good thermal conductivity and
tensile strength. Considering the individual properties
of these metals, ScZrHf becomes a potential candidate
for widespread applications where high tensile strength
and excellent resistant to high temperature is required.
Hexagonal close-packed Ti, Zr, Hf based RHEAs are
being synthesized for industrial applications. Therefore,
ScZrHf is a promising alloy for high pressure and high
temperature applications.
Huang et al.6 have studied effect of Sc and Y addition
on properties of HCP structured TiZrHf alloy and found
improved the strength and ductility in TiZrHfSc alloy
as compared to the TiZrHf alloy. Another study by Huang
et al.7 have examined the thermoelastic behaviour of
ScTiZr, ScTiHf, ScZrHf, and ScTiZrHf and evaluated a
number of properties of these alloys.
#Life Fellow, Ultrasonics Society of India
## Life Member, Ultrasonics Society of India
80 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
There have been a few other experimental and
theoretical studies focused on HCP structured high
entropy alloys for their mechanical and thermal
properties8-11.
Despite these studies, temperature dependent second
order elastic constants (SOECs, CIJ) third order elastic
constants (TOECs, CIJK), thermal properties such as
Debye temperature, heat capacity, thermal energy
density, thermal conductivity and ultrasonic properties
such as ultrasonic velocities and attenuation for HCP
structured ScZrHf alloy are yet to be investigated.
Therefore, this paper presents the determination of
temperature dependent SOECs, TOECs, elastic moduli,
Poisson's ratio, ultrasonic velocities in different modes
of vibration along different directions in the alloy crystal,
Debye average velocity, Debye temperature, thermal
energy density, heat capacity and ultrasonic attenuation
for HCP structured ScZrHf alloy utilizing a theoretical
approach.
Theory
Lennard-Jones potential method has been utilized to
compute the SOECs and TOECs at different temperature
for ScZrHf. The lattice parameters for computing SOECs
and TOECs were found in the literature7. The
formulation for calculating the six independent SOECs
and ten independent TOECs have been taken from
literature12-14. The bulk modulus (B), shear modulus
(G), Young's modulus (Y) and Poisson's ratio have been
computed by Voigt-Reuss-Hill method15,16 for hexagonal
crystals.
The ultrasonic velocities are important parameters
while estimating the mechanical properties of materials,
and for computing the ultrasonic velocities in
longitudinal (VL), quasi-shear (VS1) and shear (VS2)
modes along different angles with unique axis (c-axis)
of HCP crystal, Debye average velocity (VD) and Debye
temperature (
θ
D) we have use formulation from
literature8,12-14.
As the medium-entropy alloys are potential materials
for application in high pressure and temperature
conditions, it becomes crucial to study thermal properties
such as heat capacity, thermal energy density and thermal
conductivity at different temperature. The heat capacity
(CV) and thermal energy density (E0) have been evaluated
using the Debye model for heat capacity12,17,18.
Morelli and Slack19 have described theoretical
formulation for computing the lattice thermal
conductivity κ, which is given by equation (1).
s
aD
22/s
n
Mθδ
κ=A γT(1)
Where A is a proportionality constant (with very slightly
dependence on
γ
),
δ
(in Å) is the cube root of volume
per atom, n is the number of atoms per unit cell, Ma (in
amu) is average atomic mass, T is the temperature (in
K), γ is Grüneisen constant which can be calculated by
ρ
α
γ
V
B
=C where
α
is the volume thermal expansion
coefficient, B is bulk modulus, CV is heat capacity and
ρ
is density of the material.
The mechanical properties such as elastic moduli,
mechanical stability, thermal conductivity, heat capacity
of solid and liquids is directly correlated to the ultrasonic
attenuation. The major causes for the ultrasonic
attenuation in materials at high temperature are phonon-
phonon interaction (Akhiezer loss)14,20 and loss due to
thermo-elastic relaxation mechanism8,21. The
formulation for evaluating the ultrasonic absorption
coefficient (
α
) over frequency (f) squared (
α
/f2) due to
phonon-phonon interaction for longitudinal and shear
modes in terms of the acoustic coupling constant (D)
was developed by Mason and Bateman22 and is given
as follows:
2
2th 0
3
4πτ ED
(
α/f )= 2ρV(2)
j
i
j2
i
() –
#
γ
γ
V
0
32CT
D=9 E(3)
Where V is ultrasonic velocity in longitudinal and shear
modes, j2
i
()
γ
and
2
j
i
γ
are square average and
average square Grüneisen numbers, respectively for
longitudinal and shear modes and V
th 2
D
3
τCV
=
κ
is thermal
relaxation time.
The ultrasonic attenuation due to thermoelastic
relaxation mechanism have been computed by:
⎛⎞
⎜⎟
⎜⎟
⎝⎠
2
2j
i
2th S
L
4
2
πγ κ
T
α
ρV
=
f(4)
81
YADAV ET AL.: TEMPERATURE DEPENDENT US PROPERTIES OF ScZrHf TERNARY ALLOY
Results and Discussion
The computed values of SOECs and TOECs for
ScZrHf at different temperatures in range of 0-900 K
are presented in Fig. 1 and Fig. 2.
It is clear from Figs. 1 and 2 that the SOECs show a
decrease with increase in temperature while TOECs
exhibit opposite behaviour than that of SOECs i.e., an
increase with increase in temperature. The SOECs also
follow the mechanical stability criteria23 which is given
as C44>0, C11>|C12|, (C11+2C12)>2C13 for hexagonal
structured crystals. This confirms that the ScZrHf alloy
maintains a high mechanical stability over the
temperature range 0-900 K.
The temperature dependence of bulk modulus (B),
shear modulus (G) and Young's modulus (Y) are
presented in Fig. 3. It is evident from the Fig. 3 that all
of the elastic moduli i.e., B, G and Y decrease with
increase in temperature. The values of B, G and Y at 300
K are found to be 80.27 GPa, 51.35 GPa and 126.99
GPa respectively which are in good agreement with the
value for similar materials available in literature7,11.
The Poisson's ratio is found to be varying from 0.2364
to 0.2362 which is comparable to the Poisson's ratio of
ScZrHf and similar alloys available in literature7,8,11,24.
The longitudinal (VL) quasi-shear (VS1), shear (VS2),
and Debye average (VD) velocities at different angles
with the unique axis in the temperature range 0-900 K
have been evaluated using the SOECs and density of
the alloy and are plotted in Fig. 4.
The longitudinal wave velocity VL decreases with
increase in temperature but decreases with increase in
angle
θ
up to 45° with the unique axis and start increasing
again from 45-90°. The maximum value of VL is
5.10×103 ms–1 at angle 90° with the unique axis at
temperature 0 K. The quasi-shear wave velocity VS1
also decreases with temperature but have a maximum
value of 3.15×103 ms–1 at angle 45°. The shear velocity
VS2 monotonically decreases with temperature while
increases with angle
θ
. The Debye average velocity VD
shows similar nature to quasi-shear wave velocity with
maximum value of 3.30×103 m/s at 0 K temperature
and angle
θ
= 55°. We could not find the values of
ultrasonic velocities of ScZrHf HCP alloy but on
comparison to similar materials8,25, a good agreement
in dependence with temperature and angle has been
found.
The Debye temperature
θ
D decreases from 303.2 K to
292.8 K in temperature range of 0-900 K. This shows
that the Debye temperature does depend on temperature
but the dependence is not very significant.
The heat capacity CV and thermal energy density E0
have been evaluated in temperature range of 0-900 K
Fig. 1. Temperature (T) dependence of SOECs of ScZrHf
Fig. 2. Temperature (T) dependence of TOECs of ScZrHf
Fig. 3. Temperature (T) dependence of elastic moduli: bulk (B),
shear (G) and Young's (Y) modulus of ScZrHf
82 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
Fig. 4.Ultrasonic (a) longitudinal (VL), (b) quasi-shear (VS1)
and (c) shear (VS2) and (d) Debye average (VD) velocities
of ScZrHf at different temperatures.
by employing the Debye model and have been plotted
in Fig. 5.
Figure 5 depicts that CV increases with temperature
but the dependence becomes less and less significant as
the temperature increases making the plot with
temperature plateau at higher temperature. The thermal
energy density (E0) shows an almost linear increment
with increases in temperature.
The thermal conductivity is a key factor in material
characterization and to evaluate it for the alloy, we
used Eq. 1. The Grüneisen numbers required for the
computation of thermal conductivity have been
calculated with the help of the thermal expansion
coefficient from the literature7. The temperature
Fig. 5. Temperature dependence of heat capacity (CV) and
thermal energy density (E0)
Fig. 6.Temperature dependence of thermal conductivity of
ScZrHf
83
YADAV ET AL.: TEMPERATURE DEPENDENT US PROPERTIES OF ScZrHf TERNARY ALLOY
dependence of thermal conductivity is plotted in the
Fig. 6.
The thermal conductivity is 11.52 W/mK at 300 K
and decreases with increase in temperature from 300 K-
900 K.
The major goal of the present investigation is to
examine the ultrasonic behaviour the metal alloy ScZrHf
due to the fact that the ultrasonic attenuation is directly
correlated to thermoelastic properties of the material.
As the ultrasonic attenuation due to phonon-phonon
interaction known as Akhiezer loss and due to
thermoelastic relaxation are of great interest due to their
significance over other losses in a perfect crystal at
high (>100 K) temperature, these attenuation have been
evaluated using Eqs. (2)-(4) in the temperature range of
300-900K and are presented in Fig. 7.
The ultrasonic attenuation in longitudinal mode
(
α
/f2)L shows highest value of 474.41×10–17 Np s2/m at
temperature 900 K and angle (
θ
) 45° and lowest value
of 218.97×10–17 Np s2/m at 300 K of temperature and
90° angle (
θ
) with unique axis. Akhiezer20 already
suggested that the attenuation due to phonon-phonon
interaction dominates over other forms of attenuations
including thermo-elastic relaxation mechanism and is
plotted in Fig. 7 clearly exhibit this. The ultrasonic
attenuation due to thermo-relaxation mechanism (
α
/f2)th
is in range of 6.87×10–24 – 17.12×10–24 Np s2/m in the
temperature range of 300-900 K.
The total attenuation (
α
/f2)Total is sum of ultrasonic
attenuation due to phonon-phonon interactions
(longitudinal and shear modes) and due to thermo-
relaxation mechanism. As attenuation in longitudinal
mode dominates over shear mode and attenuation due
to thermo-relaxation mechanism, the total attenuation
shows similar nature with temperature and angle with
unique axis.
Conclusion
Based on the obtained results and discussion, the
following conclusions have been drawn:
(i) The results obtained for SOECs and TOECs of
ScZrHf are in good agreements with other studies
available in literature. This confirms the
significance and successful application of the
Lennard-Jones potential approach.
(ii) The alloy shows strong mechanical stability.
Fig. 7. Direction and temperature dependence of (a)
longitudinal, (b) shear, (c) due to thermos-relaxation
mechanism and (d) total ultrasonic attenuation of ScZrHf
84 J. PURE APPL. ULTRASON., VOL. 44, NO. 3-4 (2022)
(iii) The heat capacity followsDulong-Petit law and
the plot with temperature becomes plateau at
high temperature.
(iv) The lattice thermal conductivity decreases with
temperature. It suggests that the Electronic
thermal conductivity dominates at high
temperature in the alloy.
(v) The ultrasonic attenuation in longitudinal mode
is predominant over shear mode and
thermoelastic attenuation.
Acknowledgement
One of us (SY) is thankful to CSIR-HRDG for
providing financial support in the form of CSIR-Senior
Research Fellowship (09/1014(0012)/2019-EMR-I).
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86
J. Pure Appl. Ultrason. 44 (2022) p. 86
Ph.D. Thesis Summary
Study of intermolecular interaction in binary liquid mixtures through ultrasonic
speed measurement at 303.15K
(Ph.D. Degree Awarded to Dr. Seema Agarwal by Bundelkhand University, Jhansi (U.P.), India in 2022)
In recent years the ultrasonic study of properties of
liquid mixtures and salvation number find direct application
in chemical and biochemical industry. Thermodynamic
and transport properties of liquid of liquid mixtures have
been extensively used to study the departure of a real
liquid mixture behavior from ideality. The measurements
of ultrasonic velocity, viscosity, refractive index and density
have been adequately employed in understanding the
molecular interactions in liquid mixtures. The Ethyl acetate
and 1, 4-dioxane is selected as a solvent in the present
work since it finds a variety of applications. Alcohols
play an important role in many chemical reactions due to
the ability to undergo self-association with internal
structures.
In Literature Reviewa critical review of equilibrium and
transport properties of liquid and their theoretical aspects
has been presented by many workers. Under study of
ultrasonic velocity the thesis deals with the sound velocity
of liquid and liquid mixtures. It gives information about
the extent of interactions and some other parameters such
as adiabatic compressibility, intermolecular free length,
free volume etc. In view of the above, it was thought
worthwhile to perform ultrasonic velocity measurement
on binary mixtures of ethyl acetate with 1-alkanols and
1, 4-dioxane with 1-alkanols.
Viscosity measurements can also be used for the study
of molecular interactions between components of binary
liquid mixtures. Grunberg explained on the basis of
viscosity measurement, the role of molecular shape of the
interacting molecules in molecular interactions. However,
binary mixtures having ethyl acetate, 1, 4-dioxane as one
of the component have really been studied. Thus it was
though important to perform viscosity measurements for
the study of molecular interactions in binary mixtures
containing ethyl acetate, 1, 4-dioxane as one of the
component.
The refractive index measurement can also be used for
the study of molecular interaction between components of
binary liquid mixtures. In this chapter the study of
molecular interactions in the binary mixtures by refractive
index measurement in presented. Refractive index
measurement for the binary mixtures ethyl acetate with
1-alkanols and 1, 4-dioxane with 1-alkanols at 303.15 K
have been made. Deviations in molar refraction have been
calculated and the studies on model calculations are useful
to understand the mixing behaviour of liquids in term of
molecular interactions and orientational order-disorder
effect. Excess molar volume of mixing has served as
valuable check different theories of solutions.
A study on the thermodynamic parameter free volume
has been made. Several relations have been proposed for
the evaluation of this parameter; however the concept of
free volume varies with the model chosen. Researcher
evaluated free volume in case of a number of pure liquids
using varies methods and in the case of twelve binary
liquid mixtures. In the present investigation the negative
excess free volume (VfE) for binary mixtures may be
attributed to hydrogen bond formation through dipole-
dipole interaction.
Researcher has evaluated internal pressure of a number
of pure liquid and liquid mixtures. It is useful in
understanding the molecular interactions, internal structure
and clustering phenomena of the liquids. The excess
internal pressure (PiE) is another important parameter
through which molecular interactions can be explained. In
the present investigation for the six binary systems it is
observed that, the values of PiE are almost negative and
gradually decrease and move towards the positive values
by the increase with increase in x1. When the concentration
of ethyl acetate increased, the corresponding decrease in
concentration of ethyl acetate leads to specific interactions
i.e., the interactions move from weak to strong which
supports the above arguments is case of other parameters.
The present investigation show that greater molecular
interaction exists in the mixtures which may be due to
hydrogen bond formation. Also the weak molecular
interaction that exists which may be due to the dominance
of dispersion forces and dipolar interactions between the
unlike molecules. In this manner all the parameters which
have been evaluated from the measured values supports
one another to give a conclusion that the interactions
becoming strong starting from weak interaction among
the component molecules of the mixtures.
87
New Members - 2022
J. Pure Appl. Ultrason. 44 (2022) p. 87
Patron:
P-4
Dr Ved Ram Singh
Retired as Scientist, 'G',
CSIR-National Physical Laboratory (NPL),
New Delhi-110012
P-5
Dr S K Jain
Retired as Chief Scientist,
CSIR - National Physical Laboratory (CSIR-NPL),
New Delhi-110012
LF-111
Dr K Sakthipandi
Associate Professor
SRM TRP Engineering College,
Tiruchirappalli 621 105
Tamil Nadu, India
LM-381
Dr Eaglekumar Tarpara
Researcher
Electronic Detector Group (EDG)
Brookhaven National Laboratory (BNL), NY, USA
AM-2022-1
Ms Kalpna Yadav
Researcher
Ultrasonics Section
CSIR National Physical Laboratory
New Delhi-110012
USI Awards
Life Time Achievement Award presented at USI Foundation Day 5th November at Bundelkhand
University Jhansi to
1. Dr Ashok Kumar, Former Chief Scientist, CSIR-NPL New Delhi and
2. Dr V. A. Tabhane, Professor Emeritus, Deptt of Physics, University of Pune, Pune.
88
Authors Index
J. Pure Appl. Ultrason. 44 (2022) p. 88
Agarwal, Seema 28, 86
Amireddy, K. K. 37
Arora, Renuka 17, 44
Balasubramaniam, Krishnan 37
Behera, Susanta 72
Dhiman, M. 62
Duraivathi, C. 50
Jeyakumar, H.J. 50
Khan, M.D. 44
Khirwar, M.S. 56
Kriti 42
Kumar, Harsh 62
Kumar, Mahendra 17, 44
Kumar, Pankaj 62
Mishra, D.N. 44
Mishra, Giridhar 77
Mohanty, J.R. 72
Narayanan, M. M. 41
Nath, G. 72
Pandey, D. K. 17, 44
Pandey, Ekta 28
Poongodi, J. 50
Prakash, Gyan 56
Priya, J.J. 50
Rajagopal, Prabhu 37
Rawat, M.K. 56
Sharma, D. K. 28
Singh, A.K. 56
Singh, D. P. 62
Singh, Devraj 77
Singh, Dhananjay 17
Singh, Kanika 3
Singh, R.P. 77
Singh, S.K. 56
Singh, V. R. 3
Upmanyu, A. 62
Yadav, C. P. 17, 44
Yadav, Shakti 77
R.N. 39355/81
Dr. Sanjeev Kumar Shrivastava
Associate Professor & Convener
Department of Physics, Bundelkhand University, Jhansi
Mobile: 9415055565; E-mail: sksphys7@gmail.com, sanjeev@bujhansi.ac.in
