Acoustical Estimation of Effective Debye Temperature of Polymer Solutions in the Temperature Range 303.15K -313.15K

Abstract

On the basis of the quasi-crystalline model of the liquid state, effective Debye temperature (0D) for seven binary solutions of polymers, in the temperature range of 303.15 K – 313.15 K, has been computed using sound velocity and density data. The systems investigated herein are: polyisobutylene (PIB) + benzene, Hydroxyl-terminated polybutadiene (HTPB) + chlorobenzene, polyethylene glycol (PEG) - 200 + water, PEG - 200 + benzene, polypropylene glycol (PPG) - 400 + ethanol, PPG - 400 + 1-propanol, PPG - 400 + 1-butanol. D values of all the binary solutions are also calculated on the basis of the Lorentz–Bertholet combination rule, using heat capacity data. In addition, the ideal mixing relation has been tested for the systems under study. A comparative analysis of the theoretically evaluated and standard values of effective Debye temperature has been reported. It is pointed out that the theoretically evaluated values of effective Debye temperature and their standard counterparts show a good agreement.

Full text

1. NSA-2019 RAVENSHAW
2. National Symposium on Acoustics
3. “Frontiers in Advanced Acoustical Science and
Technology”
ACOUSTICAL ESTIMATION OF EFFECTIVE DEBYE
TEMPERATURE OF POLYMER SOLUTIONS IN THE
TEMPERATURE RANGE 303.15 K-313.15 K
Monika Dhiman1, Kuljeet Singh2, D.P. Gupta1, Arun Upmanyu1* and D. P. Singh3
1Chitkara University Institute of Engineering and Technology, Chitkara University, Punjab,
India.
2Department of Physics, SGGS College, Sector-26, Chandigarh, India.
3Acoustics Research Center, 4-215, M.V. Blvd., Mississauga, ON, L5A 1Y7, Canada.
*e-mail: arun.upmanyu@chitkara.edu.in
ABSTRACT
On the basis of quasi-crystalline model of liquid state, effective Debye temperature (D) for
seven binary solutions of polymers, in the temperature range of 303.15 K – 313.15 K, has
been computed using sound velocity and density data. The systems investigated herein are:
polyisobutylene (PIB) + benzene, Hydroxyl-terminated polybutadiene (HTPB) +
chlorobenzene, polyethylene glycol (PEG) - 200 + water, PEG - 200 + benzene,
polypropylene glycol (PPG) - 400 + ethanol, PPG - 400 + 1-propanol, PPG - 400 + 1-
butanol. D values of all the binary solutions are also calculated on the basis of Lorentz–
Bertholet combination rule, using heat capacity data. In addition, ideal mixing relation has
been tested for the systems under study. A comparative analysis of the theoretically evaluated
and standard values of effective Debye temperature has been reported. It is pointed out that
the theoretically evaluated values of effective Debye temperature and their standard
counterparts show a good agreement.
1. Introduction
Debye temperature is related to many physical properties of solids, such as acoustic velocity,
specific heat capacity and thermal expansion coefficient. It is also a very useful parameter to
understand the quasi-crystalline model for multi-component solutions. In thermodynamics,
vibrational energies of liquid mixtures can be characterized by Debye like distribution, just as in
solids. The theoretical and experimental studies of thermodynamic properties of liquid mixtures
have gained much importance earlier [1-13], as liquid mixtures and their formulations are widely
used in processing and product formulation in many industrial applications. When the different
materials are mixed together, the resulting changes in physical and thermodynamics properties can
be considered as a sum of several contributions due to free volume changes, change in energy,
change in molecular orientations etc. Properties such as ultrasonic velocity, density and their
variations with temperature and composition of the liquid mixtures are useful to predict several
thermo-acoustic parameters such as effective Debye temperature etc. in such systems.
NSA-2019 Ravenshaw University, India, October 17-19, 2019 1

A number of theoretical and experimental evidences [14-16] now are available, which support the
quasi crystalline model for liquids. Earlier workers [6-9,17-21] calculated Debye temperature of
large number of liquid mixtures as a function of pressure and temperature utilizing ultrasonic
absorption, velocity, density, compressibility of liquids. Recently, Pandey and co-workers [16] have
formulated the relation exclusively for liquid mixtures. Kandpal et al. [14] have estimated θD for a
large number of binary, ternary and quaternary liquid systems at 298.15 K. These results are in good
agreement with experimental findings. As, thermodynamic and transport properties of the polymer
solutions provide useful information about physical forces acting between the molecules of their
constituents, so in the present investigation we sought to check the validity of this formulation for
polymeric solutions in the temperature range of 303.15 K to 313.15 K.
In the present study seven polymer solutions namely polyisobutylene (PIB) + benzene, Hydroxyl-
terminated polybutadiene (HTPB) + chlorobenzene, polyethylene glycol (PEG) - 200 + water, PEG
- 200 + benzene, polypropylene glycol (PPG) - 400 + ethanol, PPG - 400 + 1-propanol, PPG - 400
+ 1-butanol, are taken for estimation of Debye temperature (θD). In addition, Lorentz - Bertholet
(L-B) combination and ideal mixing (I-M) rules [16] have been used to evaluate θD for the
polymeric solutions. A comparative analysis of these three approaches has also been reported. To
understand the intermolecular interactions and structure modification in these systems, the excess
Debye temperature values are also calculated. The obtained results are interpreted in terms of
molecular interactions prevalent among the constituents of the binary systems under study.
2. Theoretical
As reported earlier [14], effective Debye temperature for solids can be calculated using the
following expression:
3
1
33
21
4
9























tl
D
UU
V
N
k
h


(1)
where h = Plank’s constant, k = Boltzmann’s constant, N = Avogadro’s number, V = molar volume,
Ul and Ut are longitudinal and transverse sound wave velocities respectively.
The denominator part







33
21
tl UU
of above expression can be expressed in terms of density (ρ),
instantaneous adiabatic compressibility (βa) and Poisson's ratio (σ) as:
   
 
 
 


































2
3
2
3
2
3
33 213
12
2
13
121





a
tl UU
(2)
1
As reported earlier [14], Debye temperature of liquids has been computed on the basis of
approximate equivalence of isothermal compressibility (βT) and instantaneous adiabatic
compressibility (βa) just as in case of solids. Pandey et.al [26] has removed that anomaly and
modified equations (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:
NSA-2019 Ravenshaw University, Cuttack, India, October 17-19, 2019 2

 
3
1
2
3
2
3
2
3
3
4
2
1
1
4
9























































a
D
V
N
k
h
(3)
where βa can be calculated from the thermodynamic relation
 
1
2
mixmixa u

and γ is defined as
a
T




In the present investigation, Lorentz-Bertholet (L-B) combination rule [14-16] and Ideal mixing (I-
M) relation have also been used for estimating the Debye temperature for Polymeric solutions.
According to the Lorentz-Bertholet combination rule:
2211
222111
PP
DPDP
LB
DCxCx
CxCx





Pii
DiPii
LB
DCx
Cx





(4)
where CPi is the molar specific heat capacity at constant pressure of the ith component.
As per the Ideal mixing relation:
2
2
1
1DD
IM
Dxx


Dii
IM
Dx


(5)
The excess Debye temperature (θDE ) has been calculated using following relation
IM
DD
E
D


(6)
where xi is the mole fraction of ith component and θDi the Debye temperature of ith component.
The results as obtained on the basis of these formulation for the systems under study are presented
in Table 1 to Table 3, using the requisite data taken from literature [22-25].
3. Result and Discussion
In Table 1, the requisite data, for density (ρ), ultrasonic velocity (U), and molar mass (M), for pure
liquids PEG-200, PPG-400, HTPB, PIB, water, benzene, ethanol, 1-propanol, 1-butanol, and
chlorobenzene, at the temperatures 303.15 K and 313.15 K, as taken from literature [22-25], is
reported. Debye temperature (θD) values, as reported in Table 1, have been determined using
equation 3.
Table 2: Characteristic values of density (ρ), ultrasonic velocity (U), molar mass (M), Debye
temperature (θD) of pure components.
Components Type T
(K)
ρ
(kgm-3)
U
(ms-1)
M
(g mol-1)
θD (K)
[Eq-3 ]
Poly Ethylene glycol (PEG-200) [22] Polymer 303.15 1116.2 1568 200 75.72
Poly Propylene glycol (PPG-400) [ 23 ] Polymer 303.15 999.9 1350 401 50.72
Hydroxy Terminated Poly Butadiene(HTPB)[24] Polymer 303.15 1063.0 1246 2530 25.61
Hydroxy Terminated Poly Butadiene(HTPB) [24] Polymer 313.15 1054.4 1216 2530 24.78
Polyisobutylene (PIB )[ 25 ] Polymer 313.15 845.8 1361 56.10 46.13
Water [ 22 ] Liquid 303.15 995.9 1503 18.01 158.8
Benzene [ 22] Liquid 303.15 866.3 1266 78.11 78.27
NSA-2019 Ravenshaw University, India, October 17-19, 2019 3

Ethanol [23] Liquid 303.15 783.7 1152 46.07 85.30
1-Propanol [23] Liquid 303.15 795.7 1190 60.09 80.50
1-Butanol [23] Liquid 303.15 801.9 1226 74.12 77.79
Chlorobenzene Liquid 303.15 112.56
Chlorobenzene Liquid 313.15 112.56
Using equations (3), (4) and (5), the theoretically obtained values of θD for the seven polymer
solutions, are reported in Table 2 and Table 3. Table 2 presents the computed values of θD and θDE for
six binary solutions i.e. PEG-200 + water, PEG-200 + benzene, PPG-400 + ethanol, PPG-400 + 1-
propanol, PPG-400 + 1-butanol, HTPB + chlorobenzene at 303.15 K. Table 3 presents the computed
values of θD and θDE for two binary solutions i.e. PIB + benzene, and HTPB + chlorobenzene at
313.15 K. The average percentage deviation (APD), of the results obtained from equations (4) and
(5), as compared to θD values as obtained by using equation (3), is also calculated and reported for
all these binary solutions, in Table 2 and Table 3.
Table 3: Effective Debye temperature of binary polymeric solution at 303.15 K
x1
Mole fraction
of solute
ρ
(kgm-3)
U
(ms-1)
θD (K)
[Eq-3 ]
θD (K)
[Eq-4]
θD (K)
[Eq-5]
θDE (K)
[Eq-6]
PEG - 200 + WATER
0.0246
1028.8
1602
158.04
148.59
156.73
1.31
0.0631
1062.2
1663
150.68
136.22
153.54
-2.86
0.0917
1076.2
1702
146.41
128.99
151.16
-4.75
0.1315
1091.6
1705
138.04
120.89
147.85
-9.81
0.2877
1109.7
1757
113.28
101.38
134.88
-21.6
APD
7.20
-4.36
PEG - 200 + BENZENE
0.1117
910.4
1284
76.11
78.09
78.21
-2.1
0.2512
972.1
1347
76.70
77.84
78.11
-1.41
0.3348
1001.1
1387
77.27
77.67
78.04
-0.77
0.4301
1030.2
1431
77.84
77.47
77.94
-0.1
0.6681
1080.9
1512
77.97
76.86
77.54
0.43
APD
-0.39
-0.74
PPG - 400 + ETHANOL
0.1
872.5
1226
76.47
70.74
81.84
-5.37
0.2
916.8
1273
71.01
63.83
78.39
-7.38
0.3
940.9
1299
66.64
59.81
74.93
-8.29
0.4
961.0
1321
63.44
57.17
71.47
-8.03
0.5
966.9
1326
60.17
55.30
68.01
-7.84
0.6
974.7
1335
57.76
53.92
64.55
-6.79
0.7
980.3
1342
55.69
52.84
61.09
-5.4
0.8
984.5
1344
53.73
51.99
57.63
-3.9
0.9
988.6
1347
52.09
51.30
54.18
-2.09
APD
5.67
-8.02
PPG - 400 + 1-PROPANOL
0.1
936.3
1295
77.75
69.73
77.52
0.23
0.2
957.6
1308
71.19
63.82
74.55
-3.36
0.3
979.9
1319
66.67
60.07
71.57
-4.9
NSA-2019 Ravenshaw University, Cuttack, India, October 17-19, 2019 4

0.4
983.3
1328
63.05
57.49
68.59
-5.54
0.5
986.7
1331
59.96
55.61
65.61
-5.65
0.6
988.8
1338
57.60
54.17
62.63
-5.03
0.7
990.6
1344
55.59
53.03
59.65
-4.06
0.8
996.7
1345
53.73
52.11
56.67
-2.94
0.9
998.6
1348
52.15
51.36
53.70
-1.55
APD
6.86
-4.95
PPG - 400 + 1-BUTANOL
0.1
967.5
1311
76.11
69.25
75.08
1.03
0.2
976.3
1314
69.90
64.01
72.37
-2.47
0.3
980.2
1324
65.70
60.46
69.67
-3.97
0.4
987.4
1328
62.27
57.91
66.96
-4.69
0.5
989.0
1336
59.65
55.98
64.25
-4.6
0.6
991.0
1342
57.42
54.47
61.55
-4.13
0.7
996.3
1345
55.45
53.25
58.84
-3.39
0.8
998.1
1347
53.68
52.26
56.13
-2.45
0.9
999.2
1349
52.13
51.43
53.42
-1.29
APD
4.70
-3.98
HTPB + CHLOROBENZENE
0.000089
1065.6
1242
72.02
72.92
73.05
-1.03
0.0001786
1063.9
1243
72.01
72.78
73.05
-1.04
0.0002684
1063.5
1245
72.07
72.63
73.04
-0.97
0.0003586
1063.4
1244
71.97
72.34
73.04
-1.07
APD
-0.60
-0.95
A perusal of Table 2 indicates that density (ρ) density and ultrasonic velocity (U) values increase in
first five binary systems e.g. PEG-200 + water, PEG-200 + benzene, PPG-400 + ethanol, PPG-400
+ 1-propanol, PPG-400 + 1-butanol, with rise in mole fraction (x1) of the solute polymer. The θD
values as determined by using equations (3), (4) and (5), show an decrease in magnitude with such a
change. For the binary solutions, containing hydrogen bonded solvents such as water (dipole
moment=1.85 D), ethanol (dipole moment=1.69 D), 1-propanol (dipole moment=1.68D), 1-butanol
(dipole moment=1.66 D), the L-B combination rule (equation 4) under estimate the θD values where
as I-M rule (equation 5) overestimate these, as compared to the standard values of θD, determined by
using equation 3. This can be understood in terms of the existence of strong solute-solvent
interactions (e.g. dipole-dipole interactions, dipole-induced dipole interactions) prevalent in the
systems. The non-linear behaviour of θDE values with rise in mole fraction (x1) of the solute
polymer, along with large negative magnitude of θDE confirms this inference.
The binary solution, PEG-200 + benzene, contains a non-polar solvent e.g. benzene (dipole
moment=0D). The long chained polar molecules of PEG-200 have limited feasibility for molecular
association with non-polar solvent molecules. Only, weak associative solute-solvent interactions are
expected in this system. The θD values determined by using L-B combination rule and I-M rule are
in good agreement with the standard values of θD, determined by using equation 3, in this case. This
fact is confirmed by the APD values which are ~ - 1%. Whereas this variation is much higher (≤ ±
8%) for other binary solutions as reported in Table 2.
The density (ρ) density and ultrasonic velocity (U) values remain almost constant for the sixth
binary system e.g. HTPB + chlorobenzene, with rise in mole fraction (x 1) of the solute polymer. For
the binary solution, contains a polar solvent e.g. chlorobenzene (dipole moment= 1.57D). The steric
effect of much bigger sized, long chained HTPB molecule, leaves a little chance for dipole-induced
dipole interactions between solvent-solute molecules. Thus the existence of only weak molecular
interactions in this system are expected. This conclusion is confirmed by the good agreement of the
NSA-2019 Ravenshaw University, India, October 17-19, 2019 5

θD values, determined by using L-B combination rule and I-M rule with its standard values. This
conclusion is also in agreement with the existence of only weak solute-solvent interactions in the
binary system of PEG-200 + Benzene, which show a similar trend for θD values. On the basis of θDE
values, the binary systems reported in Table 2, can be arranged in the order of their decreasing
strength of solute-solvent interactions as, PEG-200 + water > PPG-400 + ethanol >PPG-400 + 1-
propanol, >PPG-400 + 1-butanol, > HTPB + chlorobenzene >PEG-200 + benzene .
Using equations (3), (4), (5) and (6), the theoretically obtained values of θD and θDE for two polymer
solutions i.e. PIB + benzene, and HTPB + chlorobenzene, at 313.15 K, are reported in Table 3. The
average percentage deviation (APD), of the results obtained from equations (4) and (5), as
compared to θD values as obtained by using equation (3), is also calculated and reported for these
two binary solutions as well.
Table 4: Effective Debye temperature of binary polymeric solution at 313.15 K
x1
ρ
(kgm-3)
U
(ms-1)
θD (K)
[Eq-3 ]
θD (K)
[Eq-4 ]
θD (K)
[Eq-5]
θDE (K)
[Eq-6]
PIB + BENZENE
0.0082
855.5
1228
76.33
77.19
77.09
-0.76
0.0171
855.3
1228
75.19
77.01
76.82
-1.63
0.0268
852.7
1230
74.10
76.81
76.51
-2.41
0.0376
851.9
1231
72.93
76.59
76.18
-3.25
0.0627
848.9
1238
70.74
76.07
75.39
-4.65
0.0775
847.0
1242
69.58
75.75
74.93
-5.35
0.0942
845.9
1247
68.43
75.40
74.41
-5.98
APD
-4.09
-3.59
HTPB + CHLOROBENZENE
0.000089
1055.5
1212
69.68
70.51
70.64
-0.96
0.0001786
1055.0
1212
69.63
70.37
70.63
-1
0.0002684
1054.7
1214
69.70
70.22
70.63
-0.93
0.0003586
1054.5
1214
69.65
69.93
70.62
-0.97
APD
-0.57
-0.92
A perusal of Table 3 indicates that density (ρ) density, slowly decrease but ultrasonic velocity (U)
values increase in the first binary system e.g. PIB + benzene, with rise in mole fraction (x1) of the
solute polymer. The θD values as determined by using equations (3), (4) and (5), show an decrease
in magnitude with such a change. As the binary solution, both the components PIB (dipole
moment=0 D) and benzene (dipole moment=0 D) are non-polar. So only weak solute-solvent
interactions (e.g. van der Waal type) are expected in the binary solution. In this case, both, the L-B
combination rule as well as I-M rule, only slightly overestimate θD as compared to its standard
values determined by using equation 3. This fact is confirmed by the APD values which is ~ - 4 % in
both cases. This can be understood in terms of the existence of weak dispersive forces present in the
systems, due to the steric effect of non-polar PIB molecules. The non-linear behaviour of θDE values
with rise in mole fraction (x1) of the solute polymer, along with large negative magnitude of θDE
confirms this inference.
For the second binary solution e.g. HTPB + chlorobenzene, at 313.15K, the density (ρ) density and
ultrasonic velocity (U) values remain almost constant with rise in mole fraction (x1) of the solute
polymer (Table 3). For the binary solution, contains a polar solvent e.g. chlorobenzene (dipole
moment= 1.57D). The steric effect of much bigger sized, long chained HTPB molecule, leaves a
little chance for dipole-induced dipole interactions between solvent-solute molecules. Thus the
NSA-2019 Ravenshaw University, Cuttack, India, October 17-19, 2019 6

existence of only weak molecular interactions in this system are expected. This conclusion is
confirmed by the good agreement of the θD values, determined by using L-B combination rule and I-
M rule with its standard values, determined by using equation 3. This fact is confirmed by the APD
values which is ~ - 1% in both cases. Another point of interest, is that these θD values are smaller
than their corresponding values, for the system, at 303.15 K. This is as expected. The results
obtained, in present investigation, by using above technique (e.g. using equations (3), (4) and (5)),
tally fairly well with the results reported in literature [14] for similar systems.
4. Conclusion
The present investigation reports the theoretically determined values of Debye temperature (θD), for
seven polymeric solutions, at 303.15 K and 313.15 K. The evaluated results for the Debye
temperature of the binary solutions studied here, show a reasonable agreement, at all compositions.
Thereby, it logical to conclude that polymeric solutions like any other pure liquid, liquid mixtures
and liquid metals etc., retain their solid behaviour. So, it is rational to investigate binary polymer
solutions on the basis of quasi crystalline model. This approach can be useful for studying many
other thermo-physical properties of binary solutions. Also, the trends of variation of density,
ultrasonic velocity, Debye temperature, excess Debye temperature, and average percentage deviation,
with change in mole fraction of the solute, are good indicators of the nature, type and strength of the solute-
solvent interactions going on in the bulk of the binary systems under study.
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