Hindawi Publishing Corporation Physics Research International Volume 2013, Article ID 927324, 9 pages http://dx.doi.org/10.1155/2013/927324 Research Article Applicability of Different Isothermal EOS at Nanomaterials Deepika P. Joshi 1 and Anjali Senger 2 1 Department of Physics, G. B. Pant University of Agriculture and Technology, Pantnagar, Uttarakhand 263145, India 2 Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore Correspondence should be addressed to Deepika P. Joshi; [email protected] Received 4 April 2013; Accepted 20 May 2013 Academic Editor: Ravindra R. Pandey Copyright © 2013 D. P. Joshi and A. Senger. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e present study explains the behaviour of nanomaterials such as AlN, CdSe, Ge , WC , and Ni- and Fe-filled-MWCNTs under high pressure. Among the number of isothermal EOSs available, we prefer only two parameter-based isothermal equations (i.e., Murnaghan equation, usual Tait’s equation, Suzuki equation and Shanker equation). e present work shows the theoretical study of thermo-elastic properties especially relative compression (/ 0 ), isothermal bulk modulus ( / 0 ), and compressibility ( () / 0 ) of nanomaterials. Aſter comparing all formulations with available experimental data, we conclude that pressure dependence of relative compression (/ 0 ) for the nanomaterials, are in good agreement for all the equations at lower pressure range. At higher pressure range, Suzuki and Shanker formulations show some deviation from experimental values. 1. Introduction Nanomaterials are currently in the focus of intense research due to their potential for revolutionary technological appli- cations in diverse areas [1, 2]. Nanomaterials can be metals, ceramics, polymeric materials, or composite materials. eir defining characteristic is the very small feature size in the range of 1–100 nanometers (nm). Nanomaterials are not just simply another step of minimization but an entirely different arena. At the nanomaterial level, some material properties are affected by the laws of atomic physics, rather than behaving as traditional bulk materials. Many of the mechanical properties at nanolevel are mod- ified and different from their bulk counterpart, including the hardness, elastic modulus, fracture, toughness, scratch re- sistance, and fatigue strength. High hardness has been dis- covered in many nanomaterials system. Nanosemiconductors with reduced dimensions recently have been shown to exhibit electronic and optical properties which vary with size of the particles, thus making them potential candidates for applications involving tenability of optical and/or electronic properties [3–5]. Tungsten carbide WC is an important nanocomposite, because of its high melting point and hardness; it is impor- tant materials in both industry and high-pressure research. WC finds extensive applications in industrial machinery as cutting tools and abrasives. WC is also widely used as anvil materials in multianvil high-pressure instruments and as seats in diamond anvil cells. Moreover, nano-WC with average grain sizes less than 100 nm has been the subject of active research over the past decades, primarily due to the significant roles grain-size reduction played in the enhance- ments of mechanical properties [6, 7]. ere is another class of nanosystems that are quite interesting due to their unique features, that is, nanotubes and nanowires. In particular, due to the high mechanical strength and ballistic electronic conduction, carbon nanotu- bes are beginning to find several uses such as for scanning probes, electronic transistors, field-emitting devices, and energy storage. Nanotubes can also be filled with biological molecules, raising the possibility of applications in biotech- nology. Encapsulation of various metals in multiwalled carbon nanotubes (MWCNTs) is being used to study the physical properties of nanowires and nanoparticles of these metals. Transition-metal nanowires, encapsulated inside the
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Hindawi Publishing CorporationPhysics Research InternationalVolume 2013 Article ID 927324 9 pageshttpdxdoiorg1011552013927324
Research ArticleApplicability of Different Isothermal EOS at Nanomaterials
Deepika P Joshi1 and Anjali Senger2
1 Department of Physics G B Pant University of Agriculture and Technology Pantnagar Uttarakhand 263145 India2Division of Physics and Applied Physics School of Physical and Mathematical Sciences Nanyang Technological University Singapore
Correspondence should be addressed to Deepika P Joshi deepikakandpalgmailcom
Received 4 April 2013 Accepted 20 May 2013
Academic Editor Ravindra R Pandey
Copyright copy 2013 D P Joshi and A Senger This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The present study explains the behaviour of nanomaterials such as AlN CdSe Ge WC and Ni- and Fe-filled-MWCNTs underhigh pressure Among the number of isothermal EOSs available we prefer only two parameter-based isothermal equations (ieMurnaghan equation usual Taitrsquos equation Suzuki equation and Shanker equation)The present work shows the theoretical study ofthermo-elastic properties especially relative compression (119881119881
0) isothermal bulk modulus (119870
1198751198700) and compressibility (120572
(119875)1205720)
of nanomaterials After comparing all formulations with available experimental data we conclude that pressure dependence ofrelative compression (119881119881
0) for the nanomaterials are in good agreement for all the equations at lower pressure range At higher
pressure range Suzuki and Shanker formulations show some deviation from experimental values
1 Introduction
Nanomaterials are currently in the focus of intense researchdue to their potential for revolutionary technological appli-cations in diverse areas [1 2] Nanomaterials can be metalsceramics polymeric materials or composite materials Theirdefining characteristic is the very small feature size in therange of 1ndash100 nanometers (nm) Nanomaterials are not justsimply another step of minimization but an entirely differentarena At the nanomaterial level somematerial properties areaffected by the laws of atomic physics rather than behavingas traditional bulk materials
Many of the mechanical properties at nanolevel are mod-ified and different from their bulk counterpart including thehardness elastic modulus fracture toughness scratch re-sistance and fatigue strength High hardness has been dis-covered in many nanomaterials system
Nanosemiconductors with reduced dimensions recentlyhave been shown to exhibit electronic and optical propertieswhich vary with size of the particles thus making thempotential candidates for applications involving tenability ofoptical andor electronic properties [3ndash5]
Tungsten carbide WC is an important nanocompositebecause of its high melting point and hardness it is impor-tant materials in both industry and high-pressure researchWC finds extensive applications in industrial machineryas cutting tools and abrasives WC is also widely used asanvil materials in multianvil high-pressure instruments andas seats in diamond anvil cells Moreover nano-WC withaverage grain sizes less than 100 nm has been the subject ofactive research over the past decades primarily due to thesignificant roles grain-size reduction played in the enhance-ments of mechanical properties [6 7]
There is another class of nanosystems that are quiteinteresting due to their unique features that is nanotubesand nanowires In particular due to the high mechanicalstrength and ballistic electronic conduction carbon nanotu-bes are beginning to find several uses such as for scanningprobes electronic transistors field-emitting devices andenergy storage Nanotubes can also be filled with biologicalmolecules raising the possibility of applications in biotech-nology Encapsulation of various metals in multiwalledcarbon nanotubes (MWCNTs) is being used to study thephysical properties of nanowires and nanoparticles of thesemetals Transition-metal nanowires encapsulated inside the
2 Physics Research International
multiwalled carbon nanotubes are promising materials foruse in nanodevices and in the magnetic storage industry andfor spintronicsmaterials Several interesting results have beenobtained on filled MWCNTs for example Fe- and Ni-filledMWCNTs [8]
What are the expected effects or benefits of high pressureson nanomaterials This is simply because in addition tocomposition and synthetic routes high pressure providesan additional effective driving force to produce new struc-tures and therefore new nanomaterial properties The highpressure research has truly developed an interdisciplinaryarea that has important applications in the field of scienceThe high pressure study is important not only for betterunderstanding of matter in the world around us but also tocreate entirely new formofmatter One of themost importantoutputs of high pressure experiment is the pressure-volume-temperature (119875 119881 119879) relationship termed as equation ofstate Study of equation of state for solids has been extremelyuseful in the field of geophysics and condensed matterphysics with possible application in many fields The studybased on the EOS at high pressure is of fundamental interestbecause it permits interpolation and extrapolation into theregions for which the experimental data are not availableadequately [9ndash18]
The purpose of the present study is to access the validityof some important and widely used EOS at high pressure onnanomaterials We have done the study of the Murnaghanusual Tait Suzuki and Shanker equations under variouspressure ranges for different classes of nanomaterials andresults are compared with available experimental data Thepresent work describes the theoretical study of thermoelasticproperties especially relative compression (119881119881
0) bulk mod-
ulus (119870(119875)1198700) and compressibility (120572
(119875)1205720) of AlN (10 nm)
Ge (13 nm) CdSe (54 nm) Ni- and Fe-filled MWCNTs andWC (25 nm) nanomaterials at high pressure
2 Method of Analysis
In this work we have employed the potential-free modelwhich is developed by incorporating several important ther-modynamical relationships [9 19 20] Our main purposeof the present study is to provide a straightforward andsimple method rather than considering a potential modelbased on several approximations to analyze the thermoelasticproperties of nanomaterials We have studied the pressuredependence of volume compression bulk modulus andcompression coefficient of nanomaterials such as n-AlN n-Ge n-CdSe Ni- and Fe-filled MWCNT and n-WC using thefollowing equations of state
21 Murnaghan Equation of State The well-known andwidely used EOS [21] is the Murnaghan equation of statewhich is based on the assumption that isothermal bulkmodulus119870 is a linear function of pressure at any temperaturethat is
119870 (119875 119879) = 1198700+ 1198701015840
0119875 (1)
Using the definition of bulk modulus and integrating equa-tion (1) at constant temperature we get the Murnaghanequation of state as follows
119881
1198810
= (1 +1198701015840
0
1198700
119875)
minus11198701015840
0
(2)
Bulk modulus is written as
119870119875
1198700
= (119881
1198810
)
minus1198701015840
0
(3)
Using the well-established thermodynamic approximation[22ndash24] that is under the effect of pressure the product of 120572
(119875)
and119870(119875)
remains constant
120572119870119875= 12057201198700 (4)
where 1205720and 119870
0are the values of 120572 and 119870 at zero pressure
We get the expression
120572119875
1205720
= (119881
1198810
)
1198701015840
0
(5)
Equation (5) is a useful relation for predicting the pressuredependence of 120572
(119875)along isotherm
22 Usual Tait Equation of State The usual Tait equationis most useful for nonlinear relation of compression andpressure for different class of solids and liquids [25] Kumar[26 27] presented the derivation of this equation The usualTaitrsquos equation (UTE) can be written as follows
119881
1198810
= [1 minus1
1198701015840
0+ 1
ln1 + (1198701015840
0+ 1
1198700
)119875] (6)
Using usual Taitrsquos equation (UTE) the expression for isother-mal bulk modulus119870
(119875)is written as follows [28 29]
119870(119875)
119870(0)
=119881
1198810
exp(11987010158400+ 1) (1 minus
119881
1198810
) (7)
This is the equation for isothermal bulk modulusUsing the well-established thermodynamic approxima-
tion [22ndash24] that is under the effect of pressure the productof 120572(119875)
and119870(119875)
remains constant
120572119870119875= 12057201198700 (8)
where 1205720and 119870
0are the values of 120572 and 119870 at zero pressure
we get the expression following
120572119875
1205720
= (119881
1198810
)
minus1
expminus (11987010158400+ 1) (1 minus
119881
1198810
) (9)
Equation (9) is the relation of the pressure dependence of 120572(119875)
along isotherm
Physics Research International 3
23 Equation of State Based on Suzuki Formulation San-Miguel and Suzuki [18 19] have followed the Gruneisentheory of thermal expansion based on the Mie-Gruneisenequation of state [30]
119875119881 + 119883 (119881) = 120574119864Th (10)
where 119875 is pressure 119883(119881) = (119889Φ119889119881) Φ is potential energyas a function of volume only 120574 is the Gruneisen parameterregarded as constant and 119864Th is the thermal energy of latticevibration After applying Taylorrsquos expansion to the secondterm in (10) with respect to the second order and solving weget
119881
1198810
=
[1 + 2119896 minus (1 minus (4119896119864Th119876))12]
2119896 (11)
In the Mie-Gruneisen EOS
119875Th =120574119864Th1198810
(12)
since
119876 =11987001198810
120574 (13)
Using (12) and (13) (11) becomes
119881
1198810
=1 + 2119896 minus (1 minus (4119896119875Th11988111987001198810))
12
2119896 (14)
Taking 119896 = ((1198701015840
0minus 1)2) where 1198701015840
0is the first pressure
derivative of bulk modulus we get
119881
1198810
=
1 + (1198701015840
0minus 1) minus [1 minus 2 ((119870
1015840
0minus 1) 119870
0) 119875Th]
12
(1198701015840
0minus 1)
(15)
119881
1198810
minus 1 =
1 minus [1 minus 2 ((1198701015840
0minus 1) 119870
0) 119875Th]
12
(1198701015840
0minus 1)
(16)
where 119875Th is the thermal pressure Following the argumentsof Shanker and Kushwah [24] when 119875 is not equal to zero(16) may be rewritten as follows
119881
1198810
minus 1 =
1 minus [1 minus 2 ((1198701015840
0minus 1) 119870
0) (119875 minus 119875Th)]
12
(1198701015840
0minus 1)
(17)
Now when thermal pressure is zero (119875Th) (17) gives thefollowing simple relation
119881
1198810
minus 1 =
1 minus [1 + 2 ((1198701015840
0minus 1) 119870
0) 119875]12
(1198701015840
0minus 1)
(18)
or
119881
1198810
=
1 minus [1 + 2 ((1198701015840
0minus 1) 119870
0) 119875]12
(1198701015840
0minus 1)
+ 1 (19)
119870(119875)
1198700
= [119881
1198810
(1 + (1198701015840
0minus 1) (1 minus
119881
1198810
))] (20)
Using the identity 120572119870119875= 12057201198700where 120572
0and119870
0are the value
of 120572 and119870 at zero pressure we get the following
120572(119875)
1205720
= [119881
1198810
(1 + (1198701015840
0minus 1) (1 minus
119881
1198810
))]
minus1
(21)
Equation (21) shows the variation of 120572(119875)
with pressure atconstant temperature
24 Shanker Formulation The Gruneisen theory of thermalexpansion as formulated byBorn andHuanghas been used byShanker et al [31] These authors included higher order termfor the change in volume in the expansion of potential energyand claimed to derive a new expression for119881119881
0which reads
as follows
119881
1198810
minus 1 =
1 minus [1 + 2 ((1198701015840
0+ 1) 119870
0) 119875Th]
12
(1198701015840
0+ 1)
(22)
It has been argued by Kushwaha and Shanker that the aboveEOS may be rewritten as follows when 119875 is not equal to zero
119881
1198810
minus 1 =
1 minus [1 + 2 ((1198701015840
0+ 1) 119870
0) (119875 minus 119875Th)]
12
(1198701015840
0+ 1)
(23)
When thermal pressure is zero (119875Th = 0) (23) gives
119881
1198810
minus 1 =
1 minus [1 + 2 ((1198701015840
0+ 1) 119870
0) 119875]12
(1198701015840
0+ 1)
(24)
or
119881
1198810
=
1 minus [1 + 2 ((1198701015840
0+ 1) 119870
0) 119875]12
(1198701015840
0+ 1)
+ 1 (25)
Now bulk modulus
119870(119875)
1198700
=119881
1198810
(1 + (1198701015840
0+ 1) (1 minus
119881
1198810
)) (26)
Using the identity120572119870119875= 12057201198700with (26) we get the following
120572(119875)
1205720
= [119881
1198810
(1 + (1198701015840
0+ 1) (1 minus
119881
1198810
))]
minus1
(27)
Equations (26) and (27) are useful relations for showing thepressure dependence of119870
(119875)and 120572
(119875) respectively
3 Results
High pressure study of nanomaterials has been describedunder the Murnaghan usual Tait Suzuki and Shankerformulations because of their simple and straightforwardapplications in high pressure physics In this paper we havereported the results obtained for thermophysical propertiesof somenanomaterials that is AlN (10 nm)Ge (13 nm)CdSe(54 nm)Ni- and Fe-filledMWCNTs andWC (25 nm) underthe effect of high pressure The values of input parameters
4 Physics Research International
ExperimentalMurnaghanUTE
SuzukiShanker
093
094
095
096
097
098
099
1
101
0
251
755
105
141
Pressure (GPa)
VV
0
(a) AlN (10 nm)
ExperimentalMurnaghanUTE
SuzukiShanker
082084086088
09092094096098
1102
Pressure (GPa)
0
112
257 5
676 9
107
6
127
6
136
1
VV
0
(b) Ge (13 nm)
ExperimentalMurnaghanUTE
SuzukiShanker
082084086088
09092094096098
1102
0
085
265
365
475
575
645
Pressure (GPa)
VV
0
(c) CdSe (52 nm)
ExperimentalMurnaghanUTE
SuzukiShanker
082084086088
09092094096098
1102
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47
Pressure (GPa)
VV
0
(d) Ni-filled MWCNT
086088
09092094096098
1102
0 2
45 5 7 8 10
145
187
5
Pressure (GPa)ExperimentalMurnaghanUTE
SuzukiShanker
VV
0
(e) Fe-filled MWCNT
ExperimentalMurnaghanUTE
SuzukiShanker
086088
09092094096098
1102
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Pressure (GPa)
VV
0
(f) WC (25 nm)
Figure 1 Variation of relative volume compression (1198811198810) with pressure (119875) for different materials
Physics Research International 5
09
095
1
105
11
115
12
0
251
755
105
141
Pressure (GPa)
K(P)K
o
(a) AlN (10 nm)
1
11
12
13
14
15
16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Pressure (GPa)
K(P)K
o
(b) Ge (13 nm)
K(P)K
o
1
15
2
25
3
35
1 2 3 4 5 6 7
Pressure (GPa)
(c) CdSe (52 nm)
1111213141516171819
0
235
853
936
123
5
155
9
185
3
229
4
252
9
Pressure (GPa)
K(P)K
o
(d) Ni-filled MWCNT
K(P)K
o
1
12
14
16
18
2
22
24
0 2
45 5 7 8 10
145
187
5
Pressure (GPa)MurnaghanUTE
SuzukiShanker
(e) Fe-filled MWCNT
1
102
104
106
108
11
112
114
116
02
5 57
5 1012
5 1517
5 2022
5 2527
5 3032
5 3537
5
Pressure (GPa)MurnaghanUTE
SuzukiShanker
K(P)K
o
(f) WC (25 nm)
Figure 2 Variation of relative Bulk modulus (119870(119875)1198700) with pressure (119875) for different materials
used to calculate the relative volume compression (1198811198810)
isothermal bulk modulus (119870(119875)1198700) and compression coef-
ficient (120572(119875)1205720) for the above nanomaterials are reported in
Table 1 with their corresponding referencesThe application of these equations of state is fully tested by
calculating the mechanical properties of nanomaterials citedpreviously The high pressure compression in the nanoma-terials is calculated by using isothermal equations (2) (6)(19) and (25) relative bulk modulus and relative isothermal
compression coefficient at different pressure is calculated byusing (3) (7) (20) (26) (4) (9) (21) and (27) respectively
The calculated results for all the nanomaterials havebeen shown in graphical form to show the validity of usedequations of state with available experimental data Figures1(a) to 1(f) show the relative compression with pressure fordifferent materials Figures 2(a) to 2(f) show the result ofrelative bulk modulus and Figures 3(a) to 3(f) representsthe result of relative compression coefficient The error bar
6 Physics Research International
075
08
085
09
095
1
105
0
251
755
105
141
Pressure (GPa)
120572(P)120572
o
(a) AlN (10 nm)
0
02
04
06
08
1
12
00
051
12 16
257
371 5
581
676
752 9 10
107
611
91
127
613
33
136
114
57
Pressure (GPa)
120572(P)120572
o
(b) Ge (13 nm)
0
02
04
06
08
1
12
0
085
265
365
475
575
645
Pressure (GPa)
120572(P)120572
o
(c) CdSe (52 nm)
0
02
04
06
08
1
12
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47
Pressure (GPa)
120572(P)120572
o
(d) Ni-filled MWCNT
0
02
04
06
08
1
12
0 2
45 5 7 8 10
145
187
5
MurnaghanUTE
SuzukiShanker
Pressure (GPa)
120572(P)120572
o
(e) Fe-filled MWCNT
082084086088
09092094096098
1102
02
5 57
5 1012
5 1517
5 2022
5 2527
5 3032
5 3537
5
MurnaghanUTE
SuzukiShanker
Pressure (GPa)
120572(P)120572
o
(f) WC (25 nm)
Figure 3 Variation of relative compression coefficient (120572(119875)1205720) with pressure (119875) for different materials
diagrams are shown in Figures 4(a) to 4(f) for the sake ofcomparison
4 Discussion
From graphical representation (Figures 1(a) to 1(f)) we seethat the 119881119881
0decreases with the increase in pressure 119875 We
have observed that all four formulations are in close agree-ment with their experimental values for all nanomaterialsat lower pressure range Error bar diagrams from Figures4(a) to 4(f) also explain this where E1 E2 E3 and E4show the deviation of theoretical values from experimentaldata which are calculated by Murnaghan UTE Suzuki andShanker EOS respectively This shows the validity of allthese equations of state in nanomaterials at lower pressure
Physics Research International 7
0
0001
0002
0003
0004
0005
00060
251
755
105
141
Erro
r
Pressure (GPa)
(a) AlN (10 nm)
0
0004
0008
0012
0016
00
051
12 16
257
371 5
581
676
752 9 10
107
611
91
127
613
33
136
114
57minus0004
Erro
r
Pressure (GPa)
(b) Ge (13 nm)
0
0005
001
0015
0
085
265
365
475
575
645
minus001
minus0005
Erro
r
Pressure (GPa)
(c) CdSe (52 nm)
0
0002
0004
0006
0008
001
0012
0014
0016
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47minus0002
Erro
r
Pressure (GPa)
(d) Ni-filled MWCNT
0
0002
0004
0006
0008
0 2
45 5 7 8 10
145
187
5
E1E2
E3E4
minus0012
minus001
minus0008
minus0006
minus0004
minus0002
Erro
r
Pressure (GPa)
(e) Fe-filled MWCNT
0
0001
0002
0003
0004
0005
0 5 10 15 20 25 30 35
minus0004
minus0003
minus0002
minus0001
Erro
r
Pressure (GPa)
E1E2
E3E4
(f) WC (25 nm)
Figure 4 Error (119884-axis) between experimental and calculated relative volume compression (1198811198810) at various pressures (119883-axis) for different
materials
8 Physics Research International
Table 1 Input parameters for different nanomaterials at referencetemperature and zero pressure Bulk modulus 119870
0and first order
pressure derivative of bulkmodulus11987010158400are taken from the references
which are given in square brackets
Material 1198700(GPa) 119870
1015840
0
AlN (10 nm) 321 [5] 4 [5]Ge (13 nm) 112 [3] 4 [3]CdSe (54 nm) 37 [4] 11 [4]Ni-filled MWCNT 1798 [8] 53 [8]Fe-filled MWCNT 167 [8] 85 [8]WC (25 nm) 416 [7] 4 [7]
range and shows that different materials are compressibleup to different extent At higher pressure range Murnaghanequation of state shows the best result however the usual TaitEOS is also near to experimental values but data obtained byShanker and Suzuki formulations show more deviation
It is clear that variations of relative isothermal bulkmodulus increase with the increase of pressure and variationsof relative compression coefficient decrease with the increaseof pressure It may be thus concluded that all these equationsexplain the compression behaviour of nanomaterials satis-factorily for lower pressure range From Figures 2(a) to 2(f)and Figures 3(a) to 3(f) we observe that at high pressureresults of Murnaghan and UTE coincide well and Shankergives somewhat closer results to Murnaghan and UTE butSuzuki deviates
The variation of relative isothermal bulk modulus andrelative isothermal compression coefficient with pressurecould not be comparedwith the experimental values since theexperimental data on these physical properties of nanomate-rials are under study and not are available so far
5 Conclusion
Thus it is clear that in all formulations there is an incrementin the ratio of isothermal bulk modulus with pressure whilerelative isothermal compression decreases with increasingpressure and different material is compressible up to differentlevels With the increase in pressures solid becomes morerigid This is well understood as at high pressure moleculescome closer and their electron clouds ripple each other Thepresent study shows that all these equations of state are validfor nanomaterials at lower pressure range It needs somemodification at high pressure range for nanomaterials
References
[1] J Z Jiang ldquoPhase transformations in nanocrystalsrdquo Journal ofMaterials Science vol 39 no 16-17 pp 5103ndash5110 2004
[2] S Ramasamy D J Smith PThangadurai et al ldquoRecent study ofnanomaterials prepared by inert gas condensation using ultrahigh vacuum chamberrdquo Pramana vol 65 no 5 pp 881ndash8912005
[3] H Wang J F Liu Y He et al ldquoHigh-pressure structuralbehaviour of nanocrystalline Gerdquo Journal of Physics CondensedMatter vol 19 no 15 Article ID 156217 10 pages 2007
[4] S H Tolbert and A P Alivisatos ldquoThe wurtzite to rock saltstructural transformation in CdSe nanocrystals under highpressurerdquo Journal of Chemical Physics vol 102 no 11 article4642 15 pages 1994
[5] Z Wang K Tait Y Zhao et al ldquoSize-induced reduction oftransition pressure and enhancement of bulk modulus of AlNnanocrystalsrdquo Journal of Physical Chemistry B vol 108 no 31pp 11506ndash11508 2004
[6] G M Amulele M H Manghnani S Marriappan et alldquoCompression behavior of WC and WC-6Co up to 50GPadetermined by synchrotron x-ray diffraction and ultrasonictechniquesrdquo Journal of Applied Physics vol 103 no 11 ArticleID 113522 6 pages 2008
[7] Z Lin L Wang J Zhang H Mao and Y Zhao ldquoNanocrys-talline tungsten carbide as incompressible as diamondrdquoAppliedPhysics Letters vol 95 no 21 Article ID 211906 3 pages 2009
[8] H K Poswal S Karmakar P K Tyagi et al ldquoHigh-pressurebehavior of Ni-filled and Fe-filled multiwalled carbon nan-otubesrdquo Physica Status Solidi (b) vol 244 no 10 pp 3612ndash36192007
[9] O L Anderson ldquoThe use of ultrasonic measurements undermodest pressure to estimate compression at high pressurerdquoJournal of Physics and Chemistry of Solids vol 27 no 3 pp 547ndash565 1966
[10] S H Tolbert and A P Alivisatos ldquoSize dependence of a firstorder solid-solid phase transition the wurtzite to rock salttransformation in CdSe nanocrystalsrdquo Science vol 265 no5170 pp 373ndash376 1994
[11] S H Tolbert and A P Alivisatos ldquoHigh-pressure struc-tural transformations in semiconductor nanocrystalsrdquo AnnualReview of Physical Chemistry vol 46 pp 595ndash626 1995
[12] J Z Jiang J S Olsen L Gerward and S Morup ldquoEnhancedbulk modulus and reduced transition pressure in 120574-Fe
2O3
nanocrystalsrdquo Europhysics Letters vol 44 no 5 article 6201998
[13] J Z Jiang L GerwardD Frost R Secco J Peyronneau and J SOlsen ldquoGrain-size effect on pressure-induced semiconductor-to-metal transition in ZnSrdquo Journal of Applied Physics vol 86no 11 pp 6608ndash6610 1999
[14] J Z Jiang and L Gerward ldquoPhase transformation and conduc-tivity in nanocrystal PbS under pressurerdquo Journal of AppliedPhysics vol 87 no 5 article 2658 3 pages 2000
[15] B Chen D Penwell L R Benedetti R Jeanloz and M BKruger ldquoParticle-size effect on the compressibility of nanocrys-talline aluminardquo Physical Review B vol 66 no 14 Article ID144101 4 pages 2002
[16] Y He J F Liu W Chen et al ldquoHigh-pressure behavior ofSnO2nanocrystalsrdquo Physical Review B vol 72 no 21 Article
ID 212102 4 pages 2005[17] Z Wang L L Daemen Y Zhao et al ldquoMorphology-tuned
wurtzite-type ZnS nanobeltsrdquoNature Materials vol 4 pp 922ndash927 2005
[18] A San-Miguel ldquoNanomaterials under high-pressurerdquoChemicalSociety Reviews vol 35 no 10 pp 876ndash889 2006
[19] I Suzuki ldquoThermal expansion of periclase and olivine and theiranharmonic propertiesrdquo Journal of Physics of the Earth vol 23no 2 pp 145ndash159 1975
[20] I Suzuki K Seya H Tokai andY Sumino ldquoThermal expansionof single-crystal manganositerdquo Journal of Physics of the Earthvol 23 no 1 pp 63ndash69 1979
Physics Research International 9
[21] F D Murnaghan ldquoThe compressibility of media under extremepressuresrdquoProceedings of theNational Academy of Sciences of theUnited States of America vol 30 no 9 pp 244ndash247 1944
[22] J L Tallon ldquoThe thermodynamics of elastic deformation-IEquation of state for solidsrdquo Journal of Physics and Chemistryof Solids vol 41 no 8 pp 837ndash850 1980
[23] F Birch ldquoEquation of state and thermodynamic parameters ofNaCl to 300 kbar in the high-temperature domainrdquo Journal ofGeophysical Research vol 91 no 5 pp 4949ndash4954 1986
[24] J Shanker and S S Kushwah ldquoAnalysis of thermodynamicproperties for high-temperature superconducting oxidesrdquoPhys-ica Status Solidi (b) vol 180 no 1 pp 183ndash188 1993
[25] H Schlosser and J Ferrante ldquoLiquid alkali metals equation ofstate and reduced-pressure bulk-modulus sound-velocity andspecific-heat functionsrdquo Physical Review B vol 40 no 9 pp6405ndash6408 1989
[26] M Kumar ldquoHigh pressure equation of state for solidsrdquo PhysicaB vol 212 no 4 pp 391ndash394 1995
[27] M Kumar ldquoThermal expansivity and equation of state upto transition pressure and melting temperature NaCl as anexamplerdquo Solid State Communications vol 92 no 5 pp 463ndash466 1994
[28] J R Macdonald ldquoSome simple isothermal equations of staterdquoReviews of Modern Physics vol 38 no 4 pp 669ndash679 1966
[29] O L Anderson andD Issak ldquoThedependence of theAnderson-Gruneisen parameter 120575
119879upon compression at extreme condi-
tionsrdquo Journal of Physics and Chemistry of Solids vol 54 no 2pp 221ndash227 1993
[30] M P Toshi ldquoCohesion of ionic solids in the born modelrdquo SolidState Physics vol 1 pp 1ndash120 1964
[31] J Shanker B Singh and S S Kushwah ldquoOn the high-pressureequation of state for solidsrdquo Physica B vol 229 no 3-4 pp 419ndash420 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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2 Physics Research International
multiwalled carbon nanotubes are promising materials foruse in nanodevices and in the magnetic storage industry andfor spintronicsmaterials Several interesting results have beenobtained on filled MWCNTs for example Fe- and Ni-filledMWCNTs [8]
What are the expected effects or benefits of high pressureson nanomaterials This is simply because in addition tocomposition and synthetic routes high pressure providesan additional effective driving force to produce new struc-tures and therefore new nanomaterial properties The highpressure research has truly developed an interdisciplinaryarea that has important applications in the field of scienceThe high pressure study is important not only for betterunderstanding of matter in the world around us but also tocreate entirely new formofmatter One of themost importantoutputs of high pressure experiment is the pressure-volume-temperature (119875 119881 119879) relationship termed as equation ofstate Study of equation of state for solids has been extremelyuseful in the field of geophysics and condensed matterphysics with possible application in many fields The studybased on the EOS at high pressure is of fundamental interestbecause it permits interpolation and extrapolation into theregions for which the experimental data are not availableadequately [9ndash18]
The purpose of the present study is to access the validityof some important and widely used EOS at high pressure onnanomaterials We have done the study of the Murnaghanusual Tait Suzuki and Shanker equations under variouspressure ranges for different classes of nanomaterials andresults are compared with available experimental data Thepresent work describes the theoretical study of thermoelasticproperties especially relative compression (119881119881
0) bulk mod-
ulus (119870(119875)1198700) and compressibility (120572
(119875)1205720) of AlN (10 nm)
Ge (13 nm) CdSe (54 nm) Ni- and Fe-filled MWCNTs andWC (25 nm) nanomaterials at high pressure
2 Method of Analysis
In this work we have employed the potential-free modelwhich is developed by incorporating several important ther-modynamical relationships [9 19 20] Our main purposeof the present study is to provide a straightforward andsimple method rather than considering a potential modelbased on several approximations to analyze the thermoelasticproperties of nanomaterials We have studied the pressuredependence of volume compression bulk modulus andcompression coefficient of nanomaterials such as n-AlN n-Ge n-CdSe Ni- and Fe-filled MWCNT and n-WC using thefollowing equations of state
21 Murnaghan Equation of State The well-known andwidely used EOS [21] is the Murnaghan equation of statewhich is based on the assumption that isothermal bulkmodulus119870 is a linear function of pressure at any temperaturethat is
119870 (119875 119879) = 1198700+ 1198701015840
0119875 (1)
Using the definition of bulk modulus and integrating equa-tion (1) at constant temperature we get the Murnaghanequation of state as follows
119881
1198810
= (1 +1198701015840
0
1198700
119875)
minus11198701015840
0
(2)
Bulk modulus is written as
119870119875
1198700
= (119881
1198810
)
minus1198701015840
0
(3)
Using the well-established thermodynamic approximation[22ndash24] that is under the effect of pressure the product of 120572
(119875)
and119870(119875)
remains constant
120572119870119875= 12057201198700 (4)
where 1205720and 119870
0are the values of 120572 and 119870 at zero pressure
We get the expression
120572119875
1205720
= (119881
1198810
)
1198701015840
0
(5)
Equation (5) is a useful relation for predicting the pressuredependence of 120572
(119875)along isotherm
22 Usual Tait Equation of State The usual Tait equationis most useful for nonlinear relation of compression andpressure for different class of solids and liquids [25] Kumar[26 27] presented the derivation of this equation The usualTaitrsquos equation (UTE) can be written as follows
119881
1198810
= [1 minus1
1198701015840
0+ 1
ln1 + (1198701015840
0+ 1
1198700
)119875] (6)
Using usual Taitrsquos equation (UTE) the expression for isother-mal bulk modulus119870
(119875)is written as follows [28 29]
119870(119875)
119870(0)
=119881
1198810
exp(11987010158400+ 1) (1 minus
119881
1198810
) (7)
This is the equation for isothermal bulk modulusUsing the well-established thermodynamic approxima-
tion [22ndash24] that is under the effect of pressure the productof 120572(119875)
and119870(119875)
remains constant
120572119870119875= 12057201198700 (8)
where 1205720and 119870
0are the values of 120572 and 119870 at zero pressure
we get the expression following
120572119875
1205720
= (119881
1198810
)
minus1
expminus (11987010158400+ 1) (1 minus
119881
1198810
) (9)
Equation (9) is the relation of the pressure dependence of 120572(119875)
along isotherm
Physics Research International 3
23 Equation of State Based on Suzuki Formulation San-Miguel and Suzuki [18 19] have followed the Gruneisentheory of thermal expansion based on the Mie-Gruneisenequation of state [30]
119875119881 + 119883 (119881) = 120574119864Th (10)
where 119875 is pressure 119883(119881) = (119889Φ119889119881) Φ is potential energyas a function of volume only 120574 is the Gruneisen parameterregarded as constant and 119864Th is the thermal energy of latticevibration After applying Taylorrsquos expansion to the secondterm in (10) with respect to the second order and solving weget
119881
1198810
=
[1 + 2119896 minus (1 minus (4119896119864Th119876))12]
2119896 (11)
In the Mie-Gruneisen EOS
119875Th =120574119864Th1198810
(12)
since
119876 =11987001198810
120574 (13)
Using (12) and (13) (11) becomes
119881
1198810
=1 + 2119896 minus (1 minus (4119896119875Th11988111987001198810))
12
2119896 (14)
Taking 119896 = ((1198701015840
0minus 1)2) where 1198701015840
0is the first pressure
derivative of bulk modulus we get
119881
1198810
=
1 + (1198701015840
0minus 1) minus [1 minus 2 ((119870
1015840
0minus 1) 119870
0) 119875Th]
12
(1198701015840
0minus 1)
(15)
119881
1198810
minus 1 =
1 minus [1 minus 2 ((1198701015840
0minus 1) 119870
0) 119875Th]
12
(1198701015840
0minus 1)
(16)
where 119875Th is the thermal pressure Following the argumentsof Shanker and Kushwah [24] when 119875 is not equal to zero(16) may be rewritten as follows
119881
1198810
minus 1 =
1 minus [1 minus 2 ((1198701015840
0minus 1) 119870
0) (119875 minus 119875Th)]
12
(1198701015840
0minus 1)
(17)
Now when thermal pressure is zero (119875Th) (17) gives thefollowing simple relation
119881
1198810
minus 1 =
1 minus [1 + 2 ((1198701015840
0minus 1) 119870
0) 119875]12
(1198701015840
0minus 1)
(18)
or
119881
1198810
=
1 minus [1 + 2 ((1198701015840
0minus 1) 119870
0) 119875]12
(1198701015840
0minus 1)
+ 1 (19)
119870(119875)
1198700
= [119881
1198810
(1 + (1198701015840
0minus 1) (1 minus
119881
1198810
))] (20)
Using the identity 120572119870119875= 12057201198700where 120572
0and119870
0are the value
of 120572 and119870 at zero pressure we get the following
120572(119875)
1205720
= [119881
1198810
(1 + (1198701015840
0minus 1) (1 minus
119881
1198810
))]
minus1
(21)
Equation (21) shows the variation of 120572(119875)
with pressure atconstant temperature
24 Shanker Formulation The Gruneisen theory of thermalexpansion as formulated byBorn andHuanghas been used byShanker et al [31] These authors included higher order termfor the change in volume in the expansion of potential energyand claimed to derive a new expression for119881119881
0which reads
as follows
119881
1198810
minus 1 =
1 minus [1 + 2 ((1198701015840
0+ 1) 119870
0) 119875Th]
12
(1198701015840
0+ 1)
(22)
It has been argued by Kushwaha and Shanker that the aboveEOS may be rewritten as follows when 119875 is not equal to zero
119881
1198810
minus 1 =
1 minus [1 + 2 ((1198701015840
0+ 1) 119870
0) (119875 minus 119875Th)]
12
(1198701015840
0+ 1)
(23)
When thermal pressure is zero (119875Th = 0) (23) gives
119881
1198810
minus 1 =
1 minus [1 + 2 ((1198701015840
0+ 1) 119870
0) 119875]12
(1198701015840
0+ 1)
(24)
or
119881
1198810
=
1 minus [1 + 2 ((1198701015840
0+ 1) 119870
0) 119875]12
(1198701015840
0+ 1)
+ 1 (25)
Now bulk modulus
119870(119875)
1198700
=119881
1198810
(1 + (1198701015840
0+ 1) (1 minus
119881
1198810
)) (26)
Using the identity120572119870119875= 12057201198700with (26) we get the following
120572(119875)
1205720
= [119881
1198810
(1 + (1198701015840
0+ 1) (1 minus
119881
1198810
))]
minus1
(27)
Equations (26) and (27) are useful relations for showing thepressure dependence of119870
(119875)and 120572
(119875) respectively
3 Results
High pressure study of nanomaterials has been describedunder the Murnaghan usual Tait Suzuki and Shankerformulations because of their simple and straightforwardapplications in high pressure physics In this paper we havereported the results obtained for thermophysical propertiesof somenanomaterials that is AlN (10 nm)Ge (13 nm)CdSe(54 nm)Ni- and Fe-filledMWCNTs andWC (25 nm) underthe effect of high pressure The values of input parameters
4 Physics Research International
ExperimentalMurnaghanUTE
SuzukiShanker
093
094
095
096
097
098
099
1
101
0
251
755
105
141
Pressure (GPa)
VV
0
(a) AlN (10 nm)
ExperimentalMurnaghanUTE
SuzukiShanker
082084086088
09092094096098
1102
Pressure (GPa)
0
112
257 5
676 9
107
6
127
6
136
1
VV
0
(b) Ge (13 nm)
ExperimentalMurnaghanUTE
SuzukiShanker
082084086088
09092094096098
1102
0
085
265
365
475
575
645
Pressure (GPa)
VV
0
(c) CdSe (52 nm)
ExperimentalMurnaghanUTE
SuzukiShanker
082084086088
09092094096098
1102
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47
Pressure (GPa)
VV
0
(d) Ni-filled MWCNT
086088
09092094096098
1102
0 2
45 5 7 8 10
145
187
5
Pressure (GPa)ExperimentalMurnaghanUTE
SuzukiShanker
VV
0
(e) Fe-filled MWCNT
ExperimentalMurnaghanUTE
SuzukiShanker
086088
09092094096098
1102
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Pressure (GPa)
VV
0
(f) WC (25 nm)
Figure 1 Variation of relative volume compression (1198811198810) with pressure (119875) for different materials
Physics Research International 5
09
095
1
105
11
115
12
0
251
755
105
141
Pressure (GPa)
K(P)K
o
(a) AlN (10 nm)
1
11
12
13
14
15
16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Pressure (GPa)
K(P)K
o
(b) Ge (13 nm)
K(P)K
o
1
15
2
25
3
35
1 2 3 4 5 6 7
Pressure (GPa)
(c) CdSe (52 nm)
1111213141516171819
0
235
853
936
123
5
155
9
185
3
229
4
252
9
Pressure (GPa)
K(P)K
o
(d) Ni-filled MWCNT
K(P)K
o
1
12
14
16
18
2
22
24
0 2
45 5 7 8 10
145
187
5
Pressure (GPa)MurnaghanUTE
SuzukiShanker
(e) Fe-filled MWCNT
1
102
104
106
108
11
112
114
116
02
5 57
5 1012
5 1517
5 2022
5 2527
5 3032
5 3537
5
Pressure (GPa)MurnaghanUTE
SuzukiShanker
K(P)K
o
(f) WC (25 nm)
Figure 2 Variation of relative Bulk modulus (119870(119875)1198700) with pressure (119875) for different materials
used to calculate the relative volume compression (1198811198810)
isothermal bulk modulus (119870(119875)1198700) and compression coef-
ficient (120572(119875)1205720) for the above nanomaterials are reported in
Table 1 with their corresponding referencesThe application of these equations of state is fully tested by
calculating the mechanical properties of nanomaterials citedpreviously The high pressure compression in the nanoma-terials is calculated by using isothermal equations (2) (6)(19) and (25) relative bulk modulus and relative isothermal
compression coefficient at different pressure is calculated byusing (3) (7) (20) (26) (4) (9) (21) and (27) respectively
The calculated results for all the nanomaterials havebeen shown in graphical form to show the validity of usedequations of state with available experimental data Figures1(a) to 1(f) show the relative compression with pressure fordifferent materials Figures 2(a) to 2(f) show the result ofrelative bulk modulus and Figures 3(a) to 3(f) representsthe result of relative compression coefficient The error bar
6 Physics Research International
075
08
085
09
095
1
105
0
251
755
105
141
Pressure (GPa)
120572(P)120572
o
(a) AlN (10 nm)
0
02
04
06
08
1
12
00
051
12 16
257
371 5
581
676
752 9 10
107
611
91
127
613
33
136
114
57
Pressure (GPa)
120572(P)120572
o
(b) Ge (13 nm)
0
02
04
06
08
1
12
0
085
265
365
475
575
645
Pressure (GPa)
120572(P)120572
o
(c) CdSe (52 nm)
0
02
04
06
08
1
12
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47
Pressure (GPa)
120572(P)120572
o
(d) Ni-filled MWCNT
0
02
04
06
08
1
12
0 2
45 5 7 8 10
145
187
5
MurnaghanUTE
SuzukiShanker
Pressure (GPa)
120572(P)120572
o
(e) Fe-filled MWCNT
082084086088
09092094096098
1102
02
5 57
5 1012
5 1517
5 2022
5 2527
5 3032
5 3537
5
MurnaghanUTE
SuzukiShanker
Pressure (GPa)
120572(P)120572
o
(f) WC (25 nm)
Figure 3 Variation of relative compression coefficient (120572(119875)1205720) with pressure (119875) for different materials
diagrams are shown in Figures 4(a) to 4(f) for the sake ofcomparison
4 Discussion
From graphical representation (Figures 1(a) to 1(f)) we seethat the 119881119881
0decreases with the increase in pressure 119875 We
have observed that all four formulations are in close agree-ment with their experimental values for all nanomaterialsat lower pressure range Error bar diagrams from Figures4(a) to 4(f) also explain this where E1 E2 E3 and E4show the deviation of theoretical values from experimentaldata which are calculated by Murnaghan UTE Suzuki andShanker EOS respectively This shows the validity of allthese equations of state in nanomaterials at lower pressure
Physics Research International 7
0
0001
0002
0003
0004
0005
00060
251
755
105
141
Erro
r
Pressure (GPa)
(a) AlN (10 nm)
0
0004
0008
0012
0016
00
051
12 16
257
371 5
581
676
752 9 10
107
611
91
127
613
33
136
114
57minus0004
Erro
r
Pressure (GPa)
(b) Ge (13 nm)
0
0005
001
0015
0
085
265
365
475
575
645
minus001
minus0005
Erro
r
Pressure (GPa)
(c) CdSe (52 nm)
0
0002
0004
0006
0008
001
0012
0014
0016
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47minus0002
Erro
r
Pressure (GPa)
(d) Ni-filled MWCNT
0
0002
0004
0006
0008
0 2
45 5 7 8 10
145
187
5
E1E2
E3E4
minus0012
minus001
minus0008
minus0006
minus0004
minus0002
Erro
r
Pressure (GPa)
(e) Fe-filled MWCNT
0
0001
0002
0003
0004
0005
0 5 10 15 20 25 30 35
minus0004
minus0003
minus0002
minus0001
Erro
r
Pressure (GPa)
E1E2
E3E4
(f) WC (25 nm)
Figure 4 Error (119884-axis) between experimental and calculated relative volume compression (1198811198810) at various pressures (119883-axis) for different
materials
8 Physics Research International
Table 1 Input parameters for different nanomaterials at referencetemperature and zero pressure Bulk modulus 119870
0and first order
pressure derivative of bulkmodulus11987010158400are taken from the references
which are given in square brackets
Material 1198700(GPa) 119870
1015840
0
AlN (10 nm) 321 [5] 4 [5]Ge (13 nm) 112 [3] 4 [3]CdSe (54 nm) 37 [4] 11 [4]Ni-filled MWCNT 1798 [8] 53 [8]Fe-filled MWCNT 167 [8] 85 [8]WC (25 nm) 416 [7] 4 [7]
range and shows that different materials are compressibleup to different extent At higher pressure range Murnaghanequation of state shows the best result however the usual TaitEOS is also near to experimental values but data obtained byShanker and Suzuki formulations show more deviation
It is clear that variations of relative isothermal bulkmodulus increase with the increase of pressure and variationsof relative compression coefficient decrease with the increaseof pressure It may be thus concluded that all these equationsexplain the compression behaviour of nanomaterials satis-factorily for lower pressure range From Figures 2(a) to 2(f)and Figures 3(a) to 3(f) we observe that at high pressureresults of Murnaghan and UTE coincide well and Shankergives somewhat closer results to Murnaghan and UTE butSuzuki deviates
The variation of relative isothermal bulk modulus andrelative isothermal compression coefficient with pressurecould not be comparedwith the experimental values since theexperimental data on these physical properties of nanomate-rials are under study and not are available so far
5 Conclusion
Thus it is clear that in all formulations there is an incrementin the ratio of isothermal bulk modulus with pressure whilerelative isothermal compression decreases with increasingpressure and different material is compressible up to differentlevels With the increase in pressures solid becomes morerigid This is well understood as at high pressure moleculescome closer and their electron clouds ripple each other Thepresent study shows that all these equations of state are validfor nanomaterials at lower pressure range It needs somemodification at high pressure range for nanomaterials
References
[1] J Z Jiang ldquoPhase transformations in nanocrystalsrdquo Journal ofMaterials Science vol 39 no 16-17 pp 5103ndash5110 2004
[2] S Ramasamy D J Smith PThangadurai et al ldquoRecent study ofnanomaterials prepared by inert gas condensation using ultrahigh vacuum chamberrdquo Pramana vol 65 no 5 pp 881ndash8912005
[3] H Wang J F Liu Y He et al ldquoHigh-pressure structuralbehaviour of nanocrystalline Gerdquo Journal of Physics CondensedMatter vol 19 no 15 Article ID 156217 10 pages 2007
[4] S H Tolbert and A P Alivisatos ldquoThe wurtzite to rock saltstructural transformation in CdSe nanocrystals under highpressurerdquo Journal of Chemical Physics vol 102 no 11 article4642 15 pages 1994
[5] Z Wang K Tait Y Zhao et al ldquoSize-induced reduction oftransition pressure and enhancement of bulk modulus of AlNnanocrystalsrdquo Journal of Physical Chemistry B vol 108 no 31pp 11506ndash11508 2004
[6] G M Amulele M H Manghnani S Marriappan et alldquoCompression behavior of WC and WC-6Co up to 50GPadetermined by synchrotron x-ray diffraction and ultrasonictechniquesrdquo Journal of Applied Physics vol 103 no 11 ArticleID 113522 6 pages 2008
[7] Z Lin L Wang J Zhang H Mao and Y Zhao ldquoNanocrys-talline tungsten carbide as incompressible as diamondrdquoAppliedPhysics Letters vol 95 no 21 Article ID 211906 3 pages 2009
[8] H K Poswal S Karmakar P K Tyagi et al ldquoHigh-pressurebehavior of Ni-filled and Fe-filled multiwalled carbon nan-otubesrdquo Physica Status Solidi (b) vol 244 no 10 pp 3612ndash36192007
[9] O L Anderson ldquoThe use of ultrasonic measurements undermodest pressure to estimate compression at high pressurerdquoJournal of Physics and Chemistry of Solids vol 27 no 3 pp 547ndash565 1966
[10] S H Tolbert and A P Alivisatos ldquoSize dependence of a firstorder solid-solid phase transition the wurtzite to rock salttransformation in CdSe nanocrystalsrdquo Science vol 265 no5170 pp 373ndash376 1994
[11] S H Tolbert and A P Alivisatos ldquoHigh-pressure struc-tural transformations in semiconductor nanocrystalsrdquo AnnualReview of Physical Chemistry vol 46 pp 595ndash626 1995
[12] J Z Jiang J S Olsen L Gerward and S Morup ldquoEnhancedbulk modulus and reduced transition pressure in 120574-Fe
2O3
nanocrystalsrdquo Europhysics Letters vol 44 no 5 article 6201998
[13] J Z Jiang L GerwardD Frost R Secco J Peyronneau and J SOlsen ldquoGrain-size effect on pressure-induced semiconductor-to-metal transition in ZnSrdquo Journal of Applied Physics vol 86no 11 pp 6608ndash6610 1999
[14] J Z Jiang and L Gerward ldquoPhase transformation and conduc-tivity in nanocrystal PbS under pressurerdquo Journal of AppliedPhysics vol 87 no 5 article 2658 3 pages 2000
[15] B Chen D Penwell L R Benedetti R Jeanloz and M BKruger ldquoParticle-size effect on the compressibility of nanocrys-talline aluminardquo Physical Review B vol 66 no 14 Article ID144101 4 pages 2002
[16] Y He J F Liu W Chen et al ldquoHigh-pressure behavior ofSnO2nanocrystalsrdquo Physical Review B vol 72 no 21 Article
ID 212102 4 pages 2005[17] Z Wang L L Daemen Y Zhao et al ldquoMorphology-tuned
wurtzite-type ZnS nanobeltsrdquoNature Materials vol 4 pp 922ndash927 2005
[18] A San-Miguel ldquoNanomaterials under high-pressurerdquoChemicalSociety Reviews vol 35 no 10 pp 876ndash889 2006
[19] I Suzuki ldquoThermal expansion of periclase and olivine and theiranharmonic propertiesrdquo Journal of Physics of the Earth vol 23no 2 pp 145ndash159 1975
[20] I Suzuki K Seya H Tokai andY Sumino ldquoThermal expansionof single-crystal manganositerdquo Journal of Physics of the Earthvol 23 no 1 pp 63ndash69 1979
Physics Research International 9
[21] F D Murnaghan ldquoThe compressibility of media under extremepressuresrdquoProceedings of theNational Academy of Sciences of theUnited States of America vol 30 no 9 pp 244ndash247 1944
[22] J L Tallon ldquoThe thermodynamics of elastic deformation-IEquation of state for solidsrdquo Journal of Physics and Chemistryof Solids vol 41 no 8 pp 837ndash850 1980
[23] F Birch ldquoEquation of state and thermodynamic parameters ofNaCl to 300 kbar in the high-temperature domainrdquo Journal ofGeophysical Research vol 91 no 5 pp 4949ndash4954 1986
[24] J Shanker and S S Kushwah ldquoAnalysis of thermodynamicproperties for high-temperature superconducting oxidesrdquoPhys-ica Status Solidi (b) vol 180 no 1 pp 183ndash188 1993
[25] H Schlosser and J Ferrante ldquoLiquid alkali metals equation ofstate and reduced-pressure bulk-modulus sound-velocity andspecific-heat functionsrdquo Physical Review B vol 40 no 9 pp6405ndash6408 1989
[26] M Kumar ldquoHigh pressure equation of state for solidsrdquo PhysicaB vol 212 no 4 pp 391ndash394 1995
[27] M Kumar ldquoThermal expansivity and equation of state upto transition pressure and melting temperature NaCl as anexamplerdquo Solid State Communications vol 92 no 5 pp 463ndash466 1994
[28] J R Macdonald ldquoSome simple isothermal equations of staterdquoReviews of Modern Physics vol 38 no 4 pp 669ndash679 1966
[29] O L Anderson andD Issak ldquoThedependence of theAnderson-Gruneisen parameter 120575
119879upon compression at extreme condi-
tionsrdquo Journal of Physics and Chemistry of Solids vol 54 no 2pp 221ndash227 1993
[30] M P Toshi ldquoCohesion of ionic solids in the born modelrdquo SolidState Physics vol 1 pp 1ndash120 1964
[31] J Shanker B Singh and S S Kushwah ldquoOn the high-pressureequation of state for solidsrdquo Physica B vol 229 no 3-4 pp 419ndash420 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Advances in Condensed Matter Physics
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Superconductivity
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Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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ThermodynamicsJournal of
Physics Research International 3
23 Equation of State Based on Suzuki Formulation San-Miguel and Suzuki [18 19] have followed the Gruneisentheory of thermal expansion based on the Mie-Gruneisenequation of state [30]
119875119881 + 119883 (119881) = 120574119864Th (10)
where 119875 is pressure 119883(119881) = (119889Φ119889119881) Φ is potential energyas a function of volume only 120574 is the Gruneisen parameterregarded as constant and 119864Th is the thermal energy of latticevibration After applying Taylorrsquos expansion to the secondterm in (10) with respect to the second order and solving weget
119881
1198810
=
[1 + 2119896 minus (1 minus (4119896119864Th119876))12]
2119896 (11)
In the Mie-Gruneisen EOS
119875Th =120574119864Th1198810
(12)
since
119876 =11987001198810
120574 (13)
Using (12) and (13) (11) becomes
119881
1198810
=1 + 2119896 minus (1 minus (4119896119875Th11988111987001198810))
12
2119896 (14)
Taking 119896 = ((1198701015840
0minus 1)2) where 1198701015840
0is the first pressure
derivative of bulk modulus we get
119881
1198810
=
1 + (1198701015840
0minus 1) minus [1 minus 2 ((119870
1015840
0minus 1) 119870
0) 119875Th]
12
(1198701015840
0minus 1)
(15)
119881
1198810
minus 1 =
1 minus [1 minus 2 ((1198701015840
0minus 1) 119870
0) 119875Th]
12
(1198701015840
0minus 1)
(16)
where 119875Th is the thermal pressure Following the argumentsof Shanker and Kushwah [24] when 119875 is not equal to zero(16) may be rewritten as follows
119881
1198810
minus 1 =
1 minus [1 minus 2 ((1198701015840
0minus 1) 119870
0) (119875 minus 119875Th)]
12
(1198701015840
0minus 1)
(17)
Now when thermal pressure is zero (119875Th) (17) gives thefollowing simple relation
119881
1198810
minus 1 =
1 minus [1 + 2 ((1198701015840
0minus 1) 119870
0) 119875]12
(1198701015840
0minus 1)
(18)
or
119881
1198810
=
1 minus [1 + 2 ((1198701015840
0minus 1) 119870
0) 119875]12
(1198701015840
0minus 1)
+ 1 (19)
119870(119875)
1198700
= [119881
1198810
(1 + (1198701015840
0minus 1) (1 minus
119881
1198810
))] (20)
Using the identity 120572119870119875= 12057201198700where 120572
0and119870
0are the value
of 120572 and119870 at zero pressure we get the following
120572(119875)
1205720
= [119881
1198810
(1 + (1198701015840
0minus 1) (1 minus
119881
1198810
))]
minus1
(21)
Equation (21) shows the variation of 120572(119875)
with pressure atconstant temperature
24 Shanker Formulation The Gruneisen theory of thermalexpansion as formulated byBorn andHuanghas been used byShanker et al [31] These authors included higher order termfor the change in volume in the expansion of potential energyand claimed to derive a new expression for119881119881
0which reads
as follows
119881
1198810
minus 1 =
1 minus [1 + 2 ((1198701015840
0+ 1) 119870
0) 119875Th]
12
(1198701015840
0+ 1)
(22)
It has been argued by Kushwaha and Shanker that the aboveEOS may be rewritten as follows when 119875 is not equal to zero
119881
1198810
minus 1 =
1 minus [1 + 2 ((1198701015840
0+ 1) 119870
0) (119875 minus 119875Th)]
12
(1198701015840
0+ 1)
(23)
When thermal pressure is zero (119875Th = 0) (23) gives
119881
1198810
minus 1 =
1 minus [1 + 2 ((1198701015840
0+ 1) 119870
0) 119875]12
(1198701015840
0+ 1)
(24)
or
119881
1198810
=
1 minus [1 + 2 ((1198701015840
0+ 1) 119870
0) 119875]12
(1198701015840
0+ 1)
+ 1 (25)
Now bulk modulus
119870(119875)
1198700
=119881
1198810
(1 + (1198701015840
0+ 1) (1 minus
119881
1198810
)) (26)
Using the identity120572119870119875= 12057201198700with (26) we get the following
120572(119875)
1205720
= [119881
1198810
(1 + (1198701015840
0+ 1) (1 minus
119881
1198810
))]
minus1
(27)
Equations (26) and (27) are useful relations for showing thepressure dependence of119870
(119875)and 120572
(119875) respectively
3 Results
High pressure study of nanomaterials has been describedunder the Murnaghan usual Tait Suzuki and Shankerformulations because of their simple and straightforwardapplications in high pressure physics In this paper we havereported the results obtained for thermophysical propertiesof somenanomaterials that is AlN (10 nm)Ge (13 nm)CdSe(54 nm)Ni- and Fe-filledMWCNTs andWC (25 nm) underthe effect of high pressure The values of input parameters
4 Physics Research International
ExperimentalMurnaghanUTE
SuzukiShanker
093
094
095
096
097
098
099
1
101
0
251
755
105
141
Pressure (GPa)
VV
0
(a) AlN (10 nm)
ExperimentalMurnaghanUTE
SuzukiShanker
082084086088
09092094096098
1102
Pressure (GPa)
0
112
257 5
676 9
107
6
127
6
136
1
VV
0
(b) Ge (13 nm)
ExperimentalMurnaghanUTE
SuzukiShanker
082084086088
09092094096098
1102
0
085
265
365
475
575
645
Pressure (GPa)
VV
0
(c) CdSe (52 nm)
ExperimentalMurnaghanUTE
SuzukiShanker
082084086088
09092094096098
1102
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47
Pressure (GPa)
VV
0
(d) Ni-filled MWCNT
086088
09092094096098
1102
0 2
45 5 7 8 10
145
187
5
Pressure (GPa)ExperimentalMurnaghanUTE
SuzukiShanker
VV
0
(e) Fe-filled MWCNT
ExperimentalMurnaghanUTE
SuzukiShanker
086088
09092094096098
1102
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Pressure (GPa)
VV
0
(f) WC (25 nm)
Figure 1 Variation of relative volume compression (1198811198810) with pressure (119875) for different materials
Physics Research International 5
09
095
1
105
11
115
12
0
251
755
105
141
Pressure (GPa)
K(P)K
o
(a) AlN (10 nm)
1
11
12
13
14
15
16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Pressure (GPa)
K(P)K
o
(b) Ge (13 nm)
K(P)K
o
1
15
2
25
3
35
1 2 3 4 5 6 7
Pressure (GPa)
(c) CdSe (52 nm)
1111213141516171819
0
235
853
936
123
5
155
9
185
3
229
4
252
9
Pressure (GPa)
K(P)K
o
(d) Ni-filled MWCNT
K(P)K
o
1
12
14
16
18
2
22
24
0 2
45 5 7 8 10
145
187
5
Pressure (GPa)MurnaghanUTE
SuzukiShanker
(e) Fe-filled MWCNT
1
102
104
106
108
11
112
114
116
02
5 57
5 1012
5 1517
5 2022
5 2527
5 3032
5 3537
5
Pressure (GPa)MurnaghanUTE
SuzukiShanker
K(P)K
o
(f) WC (25 nm)
Figure 2 Variation of relative Bulk modulus (119870(119875)1198700) with pressure (119875) for different materials
used to calculate the relative volume compression (1198811198810)
isothermal bulk modulus (119870(119875)1198700) and compression coef-
ficient (120572(119875)1205720) for the above nanomaterials are reported in
Table 1 with their corresponding referencesThe application of these equations of state is fully tested by
calculating the mechanical properties of nanomaterials citedpreviously The high pressure compression in the nanoma-terials is calculated by using isothermal equations (2) (6)(19) and (25) relative bulk modulus and relative isothermal
compression coefficient at different pressure is calculated byusing (3) (7) (20) (26) (4) (9) (21) and (27) respectively
The calculated results for all the nanomaterials havebeen shown in graphical form to show the validity of usedequations of state with available experimental data Figures1(a) to 1(f) show the relative compression with pressure fordifferent materials Figures 2(a) to 2(f) show the result ofrelative bulk modulus and Figures 3(a) to 3(f) representsthe result of relative compression coefficient The error bar
6 Physics Research International
075
08
085
09
095
1
105
0
251
755
105
141
Pressure (GPa)
120572(P)120572
o
(a) AlN (10 nm)
0
02
04
06
08
1
12
00
051
12 16
257
371 5
581
676
752 9 10
107
611
91
127
613
33
136
114
57
Pressure (GPa)
120572(P)120572
o
(b) Ge (13 nm)
0
02
04
06
08
1
12
0
085
265
365
475
575
645
Pressure (GPa)
120572(P)120572
o
(c) CdSe (52 nm)
0
02
04
06
08
1
12
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47
Pressure (GPa)
120572(P)120572
o
(d) Ni-filled MWCNT
0
02
04
06
08
1
12
0 2
45 5 7 8 10
145
187
5
MurnaghanUTE
SuzukiShanker
Pressure (GPa)
120572(P)120572
o
(e) Fe-filled MWCNT
082084086088
09092094096098
1102
02
5 57
5 1012
5 1517
5 2022
5 2527
5 3032
5 3537
5
MurnaghanUTE
SuzukiShanker
Pressure (GPa)
120572(P)120572
o
(f) WC (25 nm)
Figure 3 Variation of relative compression coefficient (120572(119875)1205720) with pressure (119875) for different materials
diagrams are shown in Figures 4(a) to 4(f) for the sake ofcomparison
4 Discussion
From graphical representation (Figures 1(a) to 1(f)) we seethat the 119881119881
0decreases with the increase in pressure 119875 We
have observed that all four formulations are in close agree-ment with their experimental values for all nanomaterialsat lower pressure range Error bar diagrams from Figures4(a) to 4(f) also explain this where E1 E2 E3 and E4show the deviation of theoretical values from experimentaldata which are calculated by Murnaghan UTE Suzuki andShanker EOS respectively This shows the validity of allthese equations of state in nanomaterials at lower pressure
Physics Research International 7
0
0001
0002
0003
0004
0005
00060
251
755
105
141
Erro
r
Pressure (GPa)
(a) AlN (10 nm)
0
0004
0008
0012
0016
00
051
12 16
257
371 5
581
676
752 9 10
107
611
91
127
613
33
136
114
57minus0004
Erro
r
Pressure (GPa)
(b) Ge (13 nm)
0
0005
001
0015
0
085
265
365
475
575
645
minus001
minus0005
Erro
r
Pressure (GPa)
(c) CdSe (52 nm)
0
0002
0004
0006
0008
001
0012
0014
0016
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47minus0002
Erro
r
Pressure (GPa)
(d) Ni-filled MWCNT
0
0002
0004
0006
0008
0 2
45 5 7 8 10
145
187
5
E1E2
E3E4
minus0012
minus001
minus0008
minus0006
minus0004
minus0002
Erro
r
Pressure (GPa)
(e) Fe-filled MWCNT
0
0001
0002
0003
0004
0005
0 5 10 15 20 25 30 35
minus0004
minus0003
minus0002
minus0001
Erro
r
Pressure (GPa)
E1E2
E3E4
(f) WC (25 nm)
Figure 4 Error (119884-axis) between experimental and calculated relative volume compression (1198811198810) at various pressures (119883-axis) for different
materials
8 Physics Research International
Table 1 Input parameters for different nanomaterials at referencetemperature and zero pressure Bulk modulus 119870
0and first order
pressure derivative of bulkmodulus11987010158400are taken from the references
which are given in square brackets
Material 1198700(GPa) 119870
1015840
0
AlN (10 nm) 321 [5] 4 [5]Ge (13 nm) 112 [3] 4 [3]CdSe (54 nm) 37 [4] 11 [4]Ni-filled MWCNT 1798 [8] 53 [8]Fe-filled MWCNT 167 [8] 85 [8]WC (25 nm) 416 [7] 4 [7]
range and shows that different materials are compressibleup to different extent At higher pressure range Murnaghanequation of state shows the best result however the usual TaitEOS is also near to experimental values but data obtained byShanker and Suzuki formulations show more deviation
It is clear that variations of relative isothermal bulkmodulus increase with the increase of pressure and variationsof relative compression coefficient decrease with the increaseof pressure It may be thus concluded that all these equationsexplain the compression behaviour of nanomaterials satis-factorily for lower pressure range From Figures 2(a) to 2(f)and Figures 3(a) to 3(f) we observe that at high pressureresults of Murnaghan and UTE coincide well and Shankergives somewhat closer results to Murnaghan and UTE butSuzuki deviates
The variation of relative isothermal bulk modulus andrelative isothermal compression coefficient with pressurecould not be comparedwith the experimental values since theexperimental data on these physical properties of nanomate-rials are under study and not are available so far
5 Conclusion
Thus it is clear that in all formulations there is an incrementin the ratio of isothermal bulk modulus with pressure whilerelative isothermal compression decreases with increasingpressure and different material is compressible up to differentlevels With the increase in pressures solid becomes morerigid This is well understood as at high pressure moleculescome closer and their electron clouds ripple each other Thepresent study shows that all these equations of state are validfor nanomaterials at lower pressure range It needs somemodification at high pressure range for nanomaterials
References
[1] J Z Jiang ldquoPhase transformations in nanocrystalsrdquo Journal ofMaterials Science vol 39 no 16-17 pp 5103ndash5110 2004
[2] S Ramasamy D J Smith PThangadurai et al ldquoRecent study ofnanomaterials prepared by inert gas condensation using ultrahigh vacuum chamberrdquo Pramana vol 65 no 5 pp 881ndash8912005
[3] H Wang J F Liu Y He et al ldquoHigh-pressure structuralbehaviour of nanocrystalline Gerdquo Journal of Physics CondensedMatter vol 19 no 15 Article ID 156217 10 pages 2007
[4] S H Tolbert and A P Alivisatos ldquoThe wurtzite to rock saltstructural transformation in CdSe nanocrystals under highpressurerdquo Journal of Chemical Physics vol 102 no 11 article4642 15 pages 1994
[5] Z Wang K Tait Y Zhao et al ldquoSize-induced reduction oftransition pressure and enhancement of bulk modulus of AlNnanocrystalsrdquo Journal of Physical Chemistry B vol 108 no 31pp 11506ndash11508 2004
[6] G M Amulele M H Manghnani S Marriappan et alldquoCompression behavior of WC and WC-6Co up to 50GPadetermined by synchrotron x-ray diffraction and ultrasonictechniquesrdquo Journal of Applied Physics vol 103 no 11 ArticleID 113522 6 pages 2008
[7] Z Lin L Wang J Zhang H Mao and Y Zhao ldquoNanocrys-talline tungsten carbide as incompressible as diamondrdquoAppliedPhysics Letters vol 95 no 21 Article ID 211906 3 pages 2009
[8] H K Poswal S Karmakar P K Tyagi et al ldquoHigh-pressurebehavior of Ni-filled and Fe-filled multiwalled carbon nan-otubesrdquo Physica Status Solidi (b) vol 244 no 10 pp 3612ndash36192007
[9] O L Anderson ldquoThe use of ultrasonic measurements undermodest pressure to estimate compression at high pressurerdquoJournal of Physics and Chemistry of Solids vol 27 no 3 pp 547ndash565 1966
[10] S H Tolbert and A P Alivisatos ldquoSize dependence of a firstorder solid-solid phase transition the wurtzite to rock salttransformation in CdSe nanocrystalsrdquo Science vol 265 no5170 pp 373ndash376 1994
[11] S H Tolbert and A P Alivisatos ldquoHigh-pressure struc-tural transformations in semiconductor nanocrystalsrdquo AnnualReview of Physical Chemistry vol 46 pp 595ndash626 1995
[12] J Z Jiang J S Olsen L Gerward and S Morup ldquoEnhancedbulk modulus and reduced transition pressure in 120574-Fe
2O3
nanocrystalsrdquo Europhysics Letters vol 44 no 5 article 6201998
[13] J Z Jiang L GerwardD Frost R Secco J Peyronneau and J SOlsen ldquoGrain-size effect on pressure-induced semiconductor-to-metal transition in ZnSrdquo Journal of Applied Physics vol 86no 11 pp 6608ndash6610 1999
[14] J Z Jiang and L Gerward ldquoPhase transformation and conduc-tivity in nanocrystal PbS under pressurerdquo Journal of AppliedPhysics vol 87 no 5 article 2658 3 pages 2000
[15] B Chen D Penwell L R Benedetti R Jeanloz and M BKruger ldquoParticle-size effect on the compressibility of nanocrys-talline aluminardquo Physical Review B vol 66 no 14 Article ID144101 4 pages 2002
[16] Y He J F Liu W Chen et al ldquoHigh-pressure behavior ofSnO2nanocrystalsrdquo Physical Review B vol 72 no 21 Article
ID 212102 4 pages 2005[17] Z Wang L L Daemen Y Zhao et al ldquoMorphology-tuned
wurtzite-type ZnS nanobeltsrdquoNature Materials vol 4 pp 922ndash927 2005
[18] A San-Miguel ldquoNanomaterials under high-pressurerdquoChemicalSociety Reviews vol 35 no 10 pp 876ndash889 2006
[19] I Suzuki ldquoThermal expansion of periclase and olivine and theiranharmonic propertiesrdquo Journal of Physics of the Earth vol 23no 2 pp 145ndash159 1975
[20] I Suzuki K Seya H Tokai andY Sumino ldquoThermal expansionof single-crystal manganositerdquo Journal of Physics of the Earthvol 23 no 1 pp 63ndash69 1979
Physics Research International 9
[21] F D Murnaghan ldquoThe compressibility of media under extremepressuresrdquoProceedings of theNational Academy of Sciences of theUnited States of America vol 30 no 9 pp 244ndash247 1944
[22] J L Tallon ldquoThe thermodynamics of elastic deformation-IEquation of state for solidsrdquo Journal of Physics and Chemistryof Solids vol 41 no 8 pp 837ndash850 1980
[23] F Birch ldquoEquation of state and thermodynamic parameters ofNaCl to 300 kbar in the high-temperature domainrdquo Journal ofGeophysical Research vol 91 no 5 pp 4949ndash4954 1986
[24] J Shanker and S S Kushwah ldquoAnalysis of thermodynamicproperties for high-temperature superconducting oxidesrdquoPhys-ica Status Solidi (b) vol 180 no 1 pp 183ndash188 1993
[25] H Schlosser and J Ferrante ldquoLiquid alkali metals equation ofstate and reduced-pressure bulk-modulus sound-velocity andspecific-heat functionsrdquo Physical Review B vol 40 no 9 pp6405ndash6408 1989
[26] M Kumar ldquoHigh pressure equation of state for solidsrdquo PhysicaB vol 212 no 4 pp 391ndash394 1995
[27] M Kumar ldquoThermal expansivity and equation of state upto transition pressure and melting temperature NaCl as anexamplerdquo Solid State Communications vol 92 no 5 pp 463ndash466 1994
[28] J R Macdonald ldquoSome simple isothermal equations of staterdquoReviews of Modern Physics vol 38 no 4 pp 669ndash679 1966
[29] O L Anderson andD Issak ldquoThedependence of theAnderson-Gruneisen parameter 120575
119879upon compression at extreme condi-
tionsrdquo Journal of Physics and Chemistry of Solids vol 54 no 2pp 221ndash227 1993
[30] M P Toshi ldquoCohesion of ionic solids in the born modelrdquo SolidState Physics vol 1 pp 1ndash120 1964
[31] J Shanker B Singh and S S Kushwah ldquoOn the high-pressureequation of state for solidsrdquo Physica B vol 229 no 3-4 pp 419ndash420 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
4 Physics Research International
ExperimentalMurnaghanUTE
SuzukiShanker
093
094
095
096
097
098
099
1
101
0
251
755
105
141
Pressure (GPa)
VV
0
(a) AlN (10 nm)
ExperimentalMurnaghanUTE
SuzukiShanker
082084086088
09092094096098
1102
Pressure (GPa)
0
112
257 5
676 9
107
6
127
6
136
1
VV
0
(b) Ge (13 nm)
ExperimentalMurnaghanUTE
SuzukiShanker
082084086088
09092094096098
1102
0
085
265
365
475
575
645
Pressure (GPa)
VV
0
(c) CdSe (52 nm)
ExperimentalMurnaghanUTE
SuzukiShanker
082084086088
09092094096098
1102
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47
Pressure (GPa)
VV
0
(d) Ni-filled MWCNT
086088
09092094096098
1102
0 2
45 5 7 8 10
145
187
5
Pressure (GPa)ExperimentalMurnaghanUTE
SuzukiShanker
VV
0
(e) Fe-filled MWCNT
ExperimentalMurnaghanUTE
SuzukiShanker
086088
09092094096098
1102
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Pressure (GPa)
VV
0
(f) WC (25 nm)
Figure 1 Variation of relative volume compression (1198811198810) with pressure (119875) for different materials
Physics Research International 5
09
095
1
105
11
115
12
0
251
755
105
141
Pressure (GPa)
K(P)K
o
(a) AlN (10 nm)
1
11
12
13
14
15
16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Pressure (GPa)
K(P)K
o
(b) Ge (13 nm)
K(P)K
o
1
15
2
25
3
35
1 2 3 4 5 6 7
Pressure (GPa)
(c) CdSe (52 nm)
1111213141516171819
0
235
853
936
123
5
155
9
185
3
229
4
252
9
Pressure (GPa)
K(P)K
o
(d) Ni-filled MWCNT
K(P)K
o
1
12
14
16
18
2
22
24
0 2
45 5 7 8 10
145
187
5
Pressure (GPa)MurnaghanUTE
SuzukiShanker
(e) Fe-filled MWCNT
1
102
104
106
108
11
112
114
116
02
5 57
5 1012
5 1517
5 2022
5 2527
5 3032
5 3537
5
Pressure (GPa)MurnaghanUTE
SuzukiShanker
K(P)K
o
(f) WC (25 nm)
Figure 2 Variation of relative Bulk modulus (119870(119875)1198700) with pressure (119875) for different materials
used to calculate the relative volume compression (1198811198810)
isothermal bulk modulus (119870(119875)1198700) and compression coef-
ficient (120572(119875)1205720) for the above nanomaterials are reported in
Table 1 with their corresponding referencesThe application of these equations of state is fully tested by
calculating the mechanical properties of nanomaterials citedpreviously The high pressure compression in the nanoma-terials is calculated by using isothermal equations (2) (6)(19) and (25) relative bulk modulus and relative isothermal
compression coefficient at different pressure is calculated byusing (3) (7) (20) (26) (4) (9) (21) and (27) respectively
The calculated results for all the nanomaterials havebeen shown in graphical form to show the validity of usedequations of state with available experimental data Figures1(a) to 1(f) show the relative compression with pressure fordifferent materials Figures 2(a) to 2(f) show the result ofrelative bulk modulus and Figures 3(a) to 3(f) representsthe result of relative compression coefficient The error bar
6 Physics Research International
075
08
085
09
095
1
105
0
251
755
105
141
Pressure (GPa)
120572(P)120572
o
(a) AlN (10 nm)
0
02
04
06
08
1
12
00
051
12 16
257
371 5
581
676
752 9 10
107
611
91
127
613
33
136
114
57
Pressure (GPa)
120572(P)120572
o
(b) Ge (13 nm)
0
02
04
06
08
1
12
0
085
265
365
475
575
645
Pressure (GPa)
120572(P)120572
o
(c) CdSe (52 nm)
0
02
04
06
08
1
12
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47
Pressure (GPa)
120572(P)120572
o
(d) Ni-filled MWCNT
0
02
04
06
08
1
12
0 2
45 5 7 8 10
145
187
5
MurnaghanUTE
SuzukiShanker
Pressure (GPa)
120572(P)120572
o
(e) Fe-filled MWCNT
082084086088
09092094096098
1102
02
5 57
5 1012
5 1517
5 2022
5 2527
5 3032
5 3537
5
MurnaghanUTE
SuzukiShanker
Pressure (GPa)
120572(P)120572
o
(f) WC (25 nm)
Figure 3 Variation of relative compression coefficient (120572(119875)1205720) with pressure (119875) for different materials
diagrams are shown in Figures 4(a) to 4(f) for the sake ofcomparison
4 Discussion
From graphical representation (Figures 1(a) to 1(f)) we seethat the 119881119881
0decreases with the increase in pressure 119875 We
have observed that all four formulations are in close agree-ment with their experimental values for all nanomaterialsat lower pressure range Error bar diagrams from Figures4(a) to 4(f) also explain this where E1 E2 E3 and E4show the deviation of theoretical values from experimentaldata which are calculated by Murnaghan UTE Suzuki andShanker EOS respectively This shows the validity of allthese equations of state in nanomaterials at lower pressure
Physics Research International 7
0
0001
0002
0003
0004
0005
00060
251
755
105
141
Erro
r
Pressure (GPa)
(a) AlN (10 nm)
0
0004
0008
0012
0016
00
051
12 16
257
371 5
581
676
752 9 10
107
611
91
127
613
33
136
114
57minus0004
Erro
r
Pressure (GPa)
(b) Ge (13 nm)
0
0005
001
0015
0
085
265
365
475
575
645
minus001
minus0005
Erro
r
Pressure (GPa)
(c) CdSe (52 nm)
0
0002
0004
0006
0008
001
0012
0014
0016
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47minus0002
Erro
r
Pressure (GPa)
(d) Ni-filled MWCNT
0
0002
0004
0006
0008
0 2
45 5 7 8 10
145
187
5
E1E2
E3E4
minus0012
minus001
minus0008
minus0006
minus0004
minus0002
Erro
r
Pressure (GPa)
(e) Fe-filled MWCNT
0
0001
0002
0003
0004
0005
0 5 10 15 20 25 30 35
minus0004
minus0003
minus0002
minus0001
Erro
r
Pressure (GPa)
E1E2
E3E4
(f) WC (25 nm)
Figure 4 Error (119884-axis) between experimental and calculated relative volume compression (1198811198810) at various pressures (119883-axis) for different
materials
8 Physics Research International
Table 1 Input parameters for different nanomaterials at referencetemperature and zero pressure Bulk modulus 119870
0and first order
pressure derivative of bulkmodulus11987010158400are taken from the references
which are given in square brackets
Material 1198700(GPa) 119870
1015840
0
AlN (10 nm) 321 [5] 4 [5]Ge (13 nm) 112 [3] 4 [3]CdSe (54 nm) 37 [4] 11 [4]Ni-filled MWCNT 1798 [8] 53 [8]Fe-filled MWCNT 167 [8] 85 [8]WC (25 nm) 416 [7] 4 [7]
range and shows that different materials are compressibleup to different extent At higher pressure range Murnaghanequation of state shows the best result however the usual TaitEOS is also near to experimental values but data obtained byShanker and Suzuki formulations show more deviation
It is clear that variations of relative isothermal bulkmodulus increase with the increase of pressure and variationsof relative compression coefficient decrease with the increaseof pressure It may be thus concluded that all these equationsexplain the compression behaviour of nanomaterials satis-factorily for lower pressure range From Figures 2(a) to 2(f)and Figures 3(a) to 3(f) we observe that at high pressureresults of Murnaghan and UTE coincide well and Shankergives somewhat closer results to Murnaghan and UTE butSuzuki deviates
The variation of relative isothermal bulk modulus andrelative isothermal compression coefficient with pressurecould not be comparedwith the experimental values since theexperimental data on these physical properties of nanomate-rials are under study and not are available so far
5 Conclusion
Thus it is clear that in all formulations there is an incrementin the ratio of isothermal bulk modulus with pressure whilerelative isothermal compression decreases with increasingpressure and different material is compressible up to differentlevels With the increase in pressures solid becomes morerigid This is well understood as at high pressure moleculescome closer and their electron clouds ripple each other Thepresent study shows that all these equations of state are validfor nanomaterials at lower pressure range It needs somemodification at high pressure range for nanomaterials
References
[1] J Z Jiang ldquoPhase transformations in nanocrystalsrdquo Journal ofMaterials Science vol 39 no 16-17 pp 5103ndash5110 2004
[2] S Ramasamy D J Smith PThangadurai et al ldquoRecent study ofnanomaterials prepared by inert gas condensation using ultrahigh vacuum chamberrdquo Pramana vol 65 no 5 pp 881ndash8912005
[3] H Wang J F Liu Y He et al ldquoHigh-pressure structuralbehaviour of nanocrystalline Gerdquo Journal of Physics CondensedMatter vol 19 no 15 Article ID 156217 10 pages 2007
[4] S H Tolbert and A P Alivisatos ldquoThe wurtzite to rock saltstructural transformation in CdSe nanocrystals under highpressurerdquo Journal of Chemical Physics vol 102 no 11 article4642 15 pages 1994
[5] Z Wang K Tait Y Zhao et al ldquoSize-induced reduction oftransition pressure and enhancement of bulk modulus of AlNnanocrystalsrdquo Journal of Physical Chemistry B vol 108 no 31pp 11506ndash11508 2004
[6] G M Amulele M H Manghnani S Marriappan et alldquoCompression behavior of WC and WC-6Co up to 50GPadetermined by synchrotron x-ray diffraction and ultrasonictechniquesrdquo Journal of Applied Physics vol 103 no 11 ArticleID 113522 6 pages 2008
[7] Z Lin L Wang J Zhang H Mao and Y Zhao ldquoNanocrys-talline tungsten carbide as incompressible as diamondrdquoAppliedPhysics Letters vol 95 no 21 Article ID 211906 3 pages 2009
[8] H K Poswal S Karmakar P K Tyagi et al ldquoHigh-pressurebehavior of Ni-filled and Fe-filled multiwalled carbon nan-otubesrdquo Physica Status Solidi (b) vol 244 no 10 pp 3612ndash36192007
[9] O L Anderson ldquoThe use of ultrasonic measurements undermodest pressure to estimate compression at high pressurerdquoJournal of Physics and Chemistry of Solids vol 27 no 3 pp 547ndash565 1966
[10] S H Tolbert and A P Alivisatos ldquoSize dependence of a firstorder solid-solid phase transition the wurtzite to rock salttransformation in CdSe nanocrystalsrdquo Science vol 265 no5170 pp 373ndash376 1994
[11] S H Tolbert and A P Alivisatos ldquoHigh-pressure struc-tural transformations in semiconductor nanocrystalsrdquo AnnualReview of Physical Chemistry vol 46 pp 595ndash626 1995
[12] J Z Jiang J S Olsen L Gerward and S Morup ldquoEnhancedbulk modulus and reduced transition pressure in 120574-Fe
2O3
nanocrystalsrdquo Europhysics Letters vol 44 no 5 article 6201998
[13] J Z Jiang L GerwardD Frost R Secco J Peyronneau and J SOlsen ldquoGrain-size effect on pressure-induced semiconductor-to-metal transition in ZnSrdquo Journal of Applied Physics vol 86no 11 pp 6608ndash6610 1999
[14] J Z Jiang and L Gerward ldquoPhase transformation and conduc-tivity in nanocrystal PbS under pressurerdquo Journal of AppliedPhysics vol 87 no 5 article 2658 3 pages 2000
[15] B Chen D Penwell L R Benedetti R Jeanloz and M BKruger ldquoParticle-size effect on the compressibility of nanocrys-talline aluminardquo Physical Review B vol 66 no 14 Article ID144101 4 pages 2002
[16] Y He J F Liu W Chen et al ldquoHigh-pressure behavior ofSnO2nanocrystalsrdquo Physical Review B vol 72 no 21 Article
ID 212102 4 pages 2005[17] Z Wang L L Daemen Y Zhao et al ldquoMorphology-tuned
wurtzite-type ZnS nanobeltsrdquoNature Materials vol 4 pp 922ndash927 2005
[18] A San-Miguel ldquoNanomaterials under high-pressurerdquoChemicalSociety Reviews vol 35 no 10 pp 876ndash889 2006
[19] I Suzuki ldquoThermal expansion of periclase and olivine and theiranharmonic propertiesrdquo Journal of Physics of the Earth vol 23no 2 pp 145ndash159 1975
[20] I Suzuki K Seya H Tokai andY Sumino ldquoThermal expansionof single-crystal manganositerdquo Journal of Physics of the Earthvol 23 no 1 pp 63ndash69 1979
Physics Research International 9
[21] F D Murnaghan ldquoThe compressibility of media under extremepressuresrdquoProceedings of theNational Academy of Sciences of theUnited States of America vol 30 no 9 pp 244ndash247 1944
[22] J L Tallon ldquoThe thermodynamics of elastic deformation-IEquation of state for solidsrdquo Journal of Physics and Chemistryof Solids vol 41 no 8 pp 837ndash850 1980
[23] F Birch ldquoEquation of state and thermodynamic parameters ofNaCl to 300 kbar in the high-temperature domainrdquo Journal ofGeophysical Research vol 91 no 5 pp 4949ndash4954 1986
[24] J Shanker and S S Kushwah ldquoAnalysis of thermodynamicproperties for high-temperature superconducting oxidesrdquoPhys-ica Status Solidi (b) vol 180 no 1 pp 183ndash188 1993
[25] H Schlosser and J Ferrante ldquoLiquid alkali metals equation ofstate and reduced-pressure bulk-modulus sound-velocity andspecific-heat functionsrdquo Physical Review B vol 40 no 9 pp6405ndash6408 1989
[26] M Kumar ldquoHigh pressure equation of state for solidsrdquo PhysicaB vol 212 no 4 pp 391ndash394 1995
[27] M Kumar ldquoThermal expansivity and equation of state upto transition pressure and melting temperature NaCl as anexamplerdquo Solid State Communications vol 92 no 5 pp 463ndash466 1994
[28] J R Macdonald ldquoSome simple isothermal equations of staterdquoReviews of Modern Physics vol 38 no 4 pp 669ndash679 1966
[29] O L Anderson andD Issak ldquoThedependence of theAnderson-Gruneisen parameter 120575
119879upon compression at extreme condi-
tionsrdquo Journal of Physics and Chemistry of Solids vol 54 no 2pp 221ndash227 1993
[30] M P Toshi ldquoCohesion of ionic solids in the born modelrdquo SolidState Physics vol 1 pp 1ndash120 1964
[31] J Shanker B Singh and S S Kushwah ldquoOn the high-pressureequation of state for solidsrdquo Physica B vol 229 no 3-4 pp 419ndash420 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Physics Research International 5
09
095
1
105
11
115
12
0
251
755
105
141
Pressure (GPa)
K(P)K
o
(a) AlN (10 nm)
1
11
12
13
14
15
16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Pressure (GPa)
K(P)K
o
(b) Ge (13 nm)
K(P)K
o
1
15
2
25
3
35
1 2 3 4 5 6 7
Pressure (GPa)
(c) CdSe (52 nm)
1111213141516171819
0
235
853
936
123
5
155
9
185
3
229
4
252
9
Pressure (GPa)
K(P)K
o
(d) Ni-filled MWCNT
K(P)K
o
1
12
14
16
18
2
22
24
0 2
45 5 7 8 10
145
187
5
Pressure (GPa)MurnaghanUTE
SuzukiShanker
(e) Fe-filled MWCNT
1
102
104
106
108
11
112
114
116
02
5 57
5 1012
5 1517
5 2022
5 2527
5 3032
5 3537
5
Pressure (GPa)MurnaghanUTE
SuzukiShanker
K(P)K
o
(f) WC (25 nm)
Figure 2 Variation of relative Bulk modulus (119870(119875)1198700) with pressure (119875) for different materials
used to calculate the relative volume compression (1198811198810)
isothermal bulk modulus (119870(119875)1198700) and compression coef-
ficient (120572(119875)1205720) for the above nanomaterials are reported in
Table 1 with their corresponding referencesThe application of these equations of state is fully tested by
calculating the mechanical properties of nanomaterials citedpreviously The high pressure compression in the nanoma-terials is calculated by using isothermal equations (2) (6)(19) and (25) relative bulk modulus and relative isothermal
compression coefficient at different pressure is calculated byusing (3) (7) (20) (26) (4) (9) (21) and (27) respectively
The calculated results for all the nanomaterials havebeen shown in graphical form to show the validity of usedequations of state with available experimental data Figures1(a) to 1(f) show the relative compression with pressure fordifferent materials Figures 2(a) to 2(f) show the result ofrelative bulk modulus and Figures 3(a) to 3(f) representsthe result of relative compression coefficient The error bar
6 Physics Research International
075
08
085
09
095
1
105
0
251
755
105
141
Pressure (GPa)
120572(P)120572
o
(a) AlN (10 nm)
0
02
04
06
08
1
12
00
051
12 16
257
371 5
581
676
752 9 10
107
611
91
127
613
33
136
114
57
Pressure (GPa)
120572(P)120572
o
(b) Ge (13 nm)
0
02
04
06
08
1
12
0
085
265
365
475
575
645
Pressure (GPa)
120572(P)120572
o
(c) CdSe (52 nm)
0
02
04
06
08
1
12
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47
Pressure (GPa)
120572(P)120572
o
(d) Ni-filled MWCNT
0
02
04
06
08
1
12
0 2
45 5 7 8 10
145
187
5
MurnaghanUTE
SuzukiShanker
Pressure (GPa)
120572(P)120572
o
(e) Fe-filled MWCNT
082084086088
09092094096098
1102
02
5 57
5 1012
5 1517
5 2022
5 2527
5 3032
5 3537
5
MurnaghanUTE
SuzukiShanker
Pressure (GPa)
120572(P)120572
o
(f) WC (25 nm)
Figure 3 Variation of relative compression coefficient (120572(119875)1205720) with pressure (119875) for different materials
diagrams are shown in Figures 4(a) to 4(f) for the sake ofcomparison
4 Discussion
From graphical representation (Figures 1(a) to 1(f)) we seethat the 119881119881
0decreases with the increase in pressure 119875 We
have observed that all four formulations are in close agree-ment with their experimental values for all nanomaterialsat lower pressure range Error bar diagrams from Figures4(a) to 4(f) also explain this where E1 E2 E3 and E4show the deviation of theoretical values from experimentaldata which are calculated by Murnaghan UTE Suzuki andShanker EOS respectively This shows the validity of allthese equations of state in nanomaterials at lower pressure
Physics Research International 7
0
0001
0002
0003
0004
0005
00060
251
755
105
141
Erro
r
Pressure (GPa)
(a) AlN (10 nm)
0
0004
0008
0012
0016
00
051
12 16
257
371 5
581
676
752 9 10
107
611
91
127
613
33
136
114
57minus0004
Erro
r
Pressure (GPa)
(b) Ge (13 nm)
0
0005
001
0015
0
085
265
365
475
575
645
minus001
minus0005
Erro
r
Pressure (GPa)
(c) CdSe (52 nm)
0
0002
0004
0006
0008
001
0012
0014
0016
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47minus0002
Erro
r
Pressure (GPa)
(d) Ni-filled MWCNT
0
0002
0004
0006
0008
0 2
45 5 7 8 10
145
187
5
E1E2
E3E4
minus0012
minus001
minus0008
minus0006
minus0004
minus0002
Erro
r
Pressure (GPa)
(e) Fe-filled MWCNT
0
0001
0002
0003
0004
0005
0 5 10 15 20 25 30 35
minus0004
minus0003
minus0002
minus0001
Erro
r
Pressure (GPa)
E1E2
E3E4
(f) WC (25 nm)
Figure 4 Error (119884-axis) between experimental and calculated relative volume compression (1198811198810) at various pressures (119883-axis) for different
materials
8 Physics Research International
Table 1 Input parameters for different nanomaterials at referencetemperature and zero pressure Bulk modulus 119870
0and first order
pressure derivative of bulkmodulus11987010158400are taken from the references
which are given in square brackets
Material 1198700(GPa) 119870
1015840
0
AlN (10 nm) 321 [5] 4 [5]Ge (13 nm) 112 [3] 4 [3]CdSe (54 nm) 37 [4] 11 [4]Ni-filled MWCNT 1798 [8] 53 [8]Fe-filled MWCNT 167 [8] 85 [8]WC (25 nm) 416 [7] 4 [7]
range and shows that different materials are compressibleup to different extent At higher pressure range Murnaghanequation of state shows the best result however the usual TaitEOS is also near to experimental values but data obtained byShanker and Suzuki formulations show more deviation
It is clear that variations of relative isothermal bulkmodulus increase with the increase of pressure and variationsof relative compression coefficient decrease with the increaseof pressure It may be thus concluded that all these equationsexplain the compression behaviour of nanomaterials satis-factorily for lower pressure range From Figures 2(a) to 2(f)and Figures 3(a) to 3(f) we observe that at high pressureresults of Murnaghan and UTE coincide well and Shankergives somewhat closer results to Murnaghan and UTE butSuzuki deviates
The variation of relative isothermal bulk modulus andrelative isothermal compression coefficient with pressurecould not be comparedwith the experimental values since theexperimental data on these physical properties of nanomate-rials are under study and not are available so far
5 Conclusion
Thus it is clear that in all formulations there is an incrementin the ratio of isothermal bulk modulus with pressure whilerelative isothermal compression decreases with increasingpressure and different material is compressible up to differentlevels With the increase in pressures solid becomes morerigid This is well understood as at high pressure moleculescome closer and their electron clouds ripple each other Thepresent study shows that all these equations of state are validfor nanomaterials at lower pressure range It needs somemodification at high pressure range for nanomaterials
References
[1] J Z Jiang ldquoPhase transformations in nanocrystalsrdquo Journal ofMaterials Science vol 39 no 16-17 pp 5103ndash5110 2004
[2] S Ramasamy D J Smith PThangadurai et al ldquoRecent study ofnanomaterials prepared by inert gas condensation using ultrahigh vacuum chamberrdquo Pramana vol 65 no 5 pp 881ndash8912005
[3] H Wang J F Liu Y He et al ldquoHigh-pressure structuralbehaviour of nanocrystalline Gerdquo Journal of Physics CondensedMatter vol 19 no 15 Article ID 156217 10 pages 2007
[4] S H Tolbert and A P Alivisatos ldquoThe wurtzite to rock saltstructural transformation in CdSe nanocrystals under highpressurerdquo Journal of Chemical Physics vol 102 no 11 article4642 15 pages 1994
[5] Z Wang K Tait Y Zhao et al ldquoSize-induced reduction oftransition pressure and enhancement of bulk modulus of AlNnanocrystalsrdquo Journal of Physical Chemistry B vol 108 no 31pp 11506ndash11508 2004
[6] G M Amulele M H Manghnani S Marriappan et alldquoCompression behavior of WC and WC-6Co up to 50GPadetermined by synchrotron x-ray diffraction and ultrasonictechniquesrdquo Journal of Applied Physics vol 103 no 11 ArticleID 113522 6 pages 2008
[7] Z Lin L Wang J Zhang H Mao and Y Zhao ldquoNanocrys-talline tungsten carbide as incompressible as diamondrdquoAppliedPhysics Letters vol 95 no 21 Article ID 211906 3 pages 2009
[8] H K Poswal S Karmakar P K Tyagi et al ldquoHigh-pressurebehavior of Ni-filled and Fe-filled multiwalled carbon nan-otubesrdquo Physica Status Solidi (b) vol 244 no 10 pp 3612ndash36192007
[9] O L Anderson ldquoThe use of ultrasonic measurements undermodest pressure to estimate compression at high pressurerdquoJournal of Physics and Chemistry of Solids vol 27 no 3 pp 547ndash565 1966
[10] S H Tolbert and A P Alivisatos ldquoSize dependence of a firstorder solid-solid phase transition the wurtzite to rock salttransformation in CdSe nanocrystalsrdquo Science vol 265 no5170 pp 373ndash376 1994
[11] S H Tolbert and A P Alivisatos ldquoHigh-pressure struc-tural transformations in semiconductor nanocrystalsrdquo AnnualReview of Physical Chemistry vol 46 pp 595ndash626 1995
[12] J Z Jiang J S Olsen L Gerward and S Morup ldquoEnhancedbulk modulus and reduced transition pressure in 120574-Fe
2O3
nanocrystalsrdquo Europhysics Letters vol 44 no 5 article 6201998
[13] J Z Jiang L GerwardD Frost R Secco J Peyronneau and J SOlsen ldquoGrain-size effect on pressure-induced semiconductor-to-metal transition in ZnSrdquo Journal of Applied Physics vol 86no 11 pp 6608ndash6610 1999
[14] J Z Jiang and L Gerward ldquoPhase transformation and conduc-tivity in nanocrystal PbS under pressurerdquo Journal of AppliedPhysics vol 87 no 5 article 2658 3 pages 2000
[15] B Chen D Penwell L R Benedetti R Jeanloz and M BKruger ldquoParticle-size effect on the compressibility of nanocrys-talline aluminardquo Physical Review B vol 66 no 14 Article ID144101 4 pages 2002
[16] Y He J F Liu W Chen et al ldquoHigh-pressure behavior ofSnO2nanocrystalsrdquo Physical Review B vol 72 no 21 Article
ID 212102 4 pages 2005[17] Z Wang L L Daemen Y Zhao et al ldquoMorphology-tuned
wurtzite-type ZnS nanobeltsrdquoNature Materials vol 4 pp 922ndash927 2005
[18] A San-Miguel ldquoNanomaterials under high-pressurerdquoChemicalSociety Reviews vol 35 no 10 pp 876ndash889 2006
[19] I Suzuki ldquoThermal expansion of periclase and olivine and theiranharmonic propertiesrdquo Journal of Physics of the Earth vol 23no 2 pp 145ndash159 1975
[20] I Suzuki K Seya H Tokai andY Sumino ldquoThermal expansionof single-crystal manganositerdquo Journal of Physics of the Earthvol 23 no 1 pp 63ndash69 1979
Physics Research International 9
[21] F D Murnaghan ldquoThe compressibility of media under extremepressuresrdquoProceedings of theNational Academy of Sciences of theUnited States of America vol 30 no 9 pp 244ndash247 1944
[22] J L Tallon ldquoThe thermodynamics of elastic deformation-IEquation of state for solidsrdquo Journal of Physics and Chemistryof Solids vol 41 no 8 pp 837ndash850 1980
[23] F Birch ldquoEquation of state and thermodynamic parameters ofNaCl to 300 kbar in the high-temperature domainrdquo Journal ofGeophysical Research vol 91 no 5 pp 4949ndash4954 1986
[24] J Shanker and S S Kushwah ldquoAnalysis of thermodynamicproperties for high-temperature superconducting oxidesrdquoPhys-ica Status Solidi (b) vol 180 no 1 pp 183ndash188 1993
[25] H Schlosser and J Ferrante ldquoLiquid alkali metals equation ofstate and reduced-pressure bulk-modulus sound-velocity andspecific-heat functionsrdquo Physical Review B vol 40 no 9 pp6405ndash6408 1989
[26] M Kumar ldquoHigh pressure equation of state for solidsrdquo PhysicaB vol 212 no 4 pp 391ndash394 1995
[27] M Kumar ldquoThermal expansivity and equation of state upto transition pressure and melting temperature NaCl as anexamplerdquo Solid State Communications vol 92 no 5 pp 463ndash466 1994
[28] J R Macdonald ldquoSome simple isothermal equations of staterdquoReviews of Modern Physics vol 38 no 4 pp 669ndash679 1966
[29] O L Anderson andD Issak ldquoThedependence of theAnderson-Gruneisen parameter 120575
119879upon compression at extreme condi-
tionsrdquo Journal of Physics and Chemistry of Solids vol 54 no 2pp 221ndash227 1993
[30] M P Toshi ldquoCohesion of ionic solids in the born modelrdquo SolidState Physics vol 1 pp 1ndash120 1964
[31] J Shanker B Singh and S S Kushwah ldquoOn the high-pressureequation of state for solidsrdquo Physica B vol 229 no 3-4 pp 419ndash420 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
6 Physics Research International
075
08
085
09
095
1
105
0
251
755
105
141
Pressure (GPa)
120572(P)120572
o
(a) AlN (10 nm)
0
02
04
06
08
1
12
00
051
12 16
257
371 5
581
676
752 9 10
107
611
91
127
613
33
136
114
57
Pressure (GPa)
120572(P)120572
o
(b) Ge (13 nm)
0
02
04
06
08
1
12
0
085
265
365
475
575
645
Pressure (GPa)
120572(P)120572
o
(c) CdSe (52 nm)
0
02
04
06
08
1
12
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47
Pressure (GPa)
120572(P)120572
o
(d) Ni-filled MWCNT
0
02
04
06
08
1
12
0 2
45 5 7 8 10
145
187
5
MurnaghanUTE
SuzukiShanker
Pressure (GPa)
120572(P)120572
o
(e) Fe-filled MWCNT
082084086088
09092094096098
1102
02
5 57
5 1012
5 1517
5 2022
5 2527
5 3032
5 3537
5
MurnaghanUTE
SuzukiShanker
Pressure (GPa)
120572(P)120572
o
(f) WC (25 nm)
Figure 3 Variation of relative compression coefficient (120572(119875)1205720) with pressure (119875) for different materials
diagrams are shown in Figures 4(a) to 4(f) for the sake ofcomparison
4 Discussion
From graphical representation (Figures 1(a) to 1(f)) we seethat the 119881119881
0decreases with the increase in pressure 119875 We
have observed that all four formulations are in close agree-ment with their experimental values for all nanomaterialsat lower pressure range Error bar diagrams from Figures4(a) to 4(f) also explain this where E1 E2 E3 and E4show the deviation of theoretical values from experimentaldata which are calculated by Murnaghan UTE Suzuki andShanker EOS respectively This shows the validity of allthese equations of state in nanomaterials at lower pressure
Physics Research International 7
0
0001
0002
0003
0004
0005
00060
251
755
105
141
Erro
r
Pressure (GPa)
(a) AlN (10 nm)
0
0004
0008
0012
0016
00
051
12 16
257
371 5
581
676
752 9 10
107
611
91
127
613
33
136
114
57minus0004
Erro
r
Pressure (GPa)
(b) Ge (13 nm)
0
0005
001
0015
0
085
265
365
475
575
645
minus001
minus0005
Erro
r
Pressure (GPa)
(c) CdSe (52 nm)
0
0002
0004
0006
0008
001
0012
0014
0016
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47minus0002
Erro
r
Pressure (GPa)
(d) Ni-filled MWCNT
0
0002
0004
0006
0008
0 2
45 5 7 8 10
145
187
5
E1E2
E3E4
minus0012
minus001
minus0008
minus0006
minus0004
minus0002
Erro
r
Pressure (GPa)
(e) Fe-filled MWCNT
0
0001
0002
0003
0004
0005
0 5 10 15 20 25 30 35
minus0004
minus0003
minus0002
minus0001
Erro
r
Pressure (GPa)
E1E2
E3E4
(f) WC (25 nm)
Figure 4 Error (119884-axis) between experimental and calculated relative volume compression (1198811198810) at various pressures (119883-axis) for different
materials
8 Physics Research International
Table 1 Input parameters for different nanomaterials at referencetemperature and zero pressure Bulk modulus 119870
0and first order
pressure derivative of bulkmodulus11987010158400are taken from the references
which are given in square brackets
Material 1198700(GPa) 119870
1015840
0
AlN (10 nm) 321 [5] 4 [5]Ge (13 nm) 112 [3] 4 [3]CdSe (54 nm) 37 [4] 11 [4]Ni-filled MWCNT 1798 [8] 53 [8]Fe-filled MWCNT 167 [8] 85 [8]WC (25 nm) 416 [7] 4 [7]
range and shows that different materials are compressibleup to different extent At higher pressure range Murnaghanequation of state shows the best result however the usual TaitEOS is also near to experimental values but data obtained byShanker and Suzuki formulations show more deviation
It is clear that variations of relative isothermal bulkmodulus increase with the increase of pressure and variationsof relative compression coefficient decrease with the increaseof pressure It may be thus concluded that all these equationsexplain the compression behaviour of nanomaterials satis-factorily for lower pressure range From Figures 2(a) to 2(f)and Figures 3(a) to 3(f) we observe that at high pressureresults of Murnaghan and UTE coincide well and Shankergives somewhat closer results to Murnaghan and UTE butSuzuki deviates
The variation of relative isothermal bulk modulus andrelative isothermal compression coefficient with pressurecould not be comparedwith the experimental values since theexperimental data on these physical properties of nanomate-rials are under study and not are available so far
5 Conclusion
Thus it is clear that in all formulations there is an incrementin the ratio of isothermal bulk modulus with pressure whilerelative isothermal compression decreases with increasingpressure and different material is compressible up to differentlevels With the increase in pressures solid becomes morerigid This is well understood as at high pressure moleculescome closer and their electron clouds ripple each other Thepresent study shows that all these equations of state are validfor nanomaterials at lower pressure range It needs somemodification at high pressure range for nanomaterials
References
[1] J Z Jiang ldquoPhase transformations in nanocrystalsrdquo Journal ofMaterials Science vol 39 no 16-17 pp 5103ndash5110 2004
[2] S Ramasamy D J Smith PThangadurai et al ldquoRecent study ofnanomaterials prepared by inert gas condensation using ultrahigh vacuum chamberrdquo Pramana vol 65 no 5 pp 881ndash8912005
[3] H Wang J F Liu Y He et al ldquoHigh-pressure structuralbehaviour of nanocrystalline Gerdquo Journal of Physics CondensedMatter vol 19 no 15 Article ID 156217 10 pages 2007
[4] S H Tolbert and A P Alivisatos ldquoThe wurtzite to rock saltstructural transformation in CdSe nanocrystals under highpressurerdquo Journal of Chemical Physics vol 102 no 11 article4642 15 pages 1994
[5] Z Wang K Tait Y Zhao et al ldquoSize-induced reduction oftransition pressure and enhancement of bulk modulus of AlNnanocrystalsrdquo Journal of Physical Chemistry B vol 108 no 31pp 11506ndash11508 2004
[6] G M Amulele M H Manghnani S Marriappan et alldquoCompression behavior of WC and WC-6Co up to 50GPadetermined by synchrotron x-ray diffraction and ultrasonictechniquesrdquo Journal of Applied Physics vol 103 no 11 ArticleID 113522 6 pages 2008
[7] Z Lin L Wang J Zhang H Mao and Y Zhao ldquoNanocrys-talline tungsten carbide as incompressible as diamondrdquoAppliedPhysics Letters vol 95 no 21 Article ID 211906 3 pages 2009
[8] H K Poswal S Karmakar P K Tyagi et al ldquoHigh-pressurebehavior of Ni-filled and Fe-filled multiwalled carbon nan-otubesrdquo Physica Status Solidi (b) vol 244 no 10 pp 3612ndash36192007
[9] O L Anderson ldquoThe use of ultrasonic measurements undermodest pressure to estimate compression at high pressurerdquoJournal of Physics and Chemistry of Solids vol 27 no 3 pp 547ndash565 1966
[10] S H Tolbert and A P Alivisatos ldquoSize dependence of a firstorder solid-solid phase transition the wurtzite to rock salttransformation in CdSe nanocrystalsrdquo Science vol 265 no5170 pp 373ndash376 1994
[11] S H Tolbert and A P Alivisatos ldquoHigh-pressure struc-tural transformations in semiconductor nanocrystalsrdquo AnnualReview of Physical Chemistry vol 46 pp 595ndash626 1995
[12] J Z Jiang J S Olsen L Gerward and S Morup ldquoEnhancedbulk modulus and reduced transition pressure in 120574-Fe
2O3
nanocrystalsrdquo Europhysics Letters vol 44 no 5 article 6201998
[13] J Z Jiang L GerwardD Frost R Secco J Peyronneau and J SOlsen ldquoGrain-size effect on pressure-induced semiconductor-to-metal transition in ZnSrdquo Journal of Applied Physics vol 86no 11 pp 6608ndash6610 1999
[14] J Z Jiang and L Gerward ldquoPhase transformation and conduc-tivity in nanocrystal PbS under pressurerdquo Journal of AppliedPhysics vol 87 no 5 article 2658 3 pages 2000
[15] B Chen D Penwell L R Benedetti R Jeanloz and M BKruger ldquoParticle-size effect on the compressibility of nanocrys-talline aluminardquo Physical Review B vol 66 no 14 Article ID144101 4 pages 2002
[16] Y He J F Liu W Chen et al ldquoHigh-pressure behavior ofSnO2nanocrystalsrdquo Physical Review B vol 72 no 21 Article
ID 212102 4 pages 2005[17] Z Wang L L Daemen Y Zhao et al ldquoMorphology-tuned
wurtzite-type ZnS nanobeltsrdquoNature Materials vol 4 pp 922ndash927 2005
[18] A San-Miguel ldquoNanomaterials under high-pressurerdquoChemicalSociety Reviews vol 35 no 10 pp 876ndash889 2006
[19] I Suzuki ldquoThermal expansion of periclase and olivine and theiranharmonic propertiesrdquo Journal of Physics of the Earth vol 23no 2 pp 145ndash159 1975
[20] I Suzuki K Seya H Tokai andY Sumino ldquoThermal expansionof single-crystal manganositerdquo Journal of Physics of the Earthvol 23 no 1 pp 63ndash69 1979
Physics Research International 9
[21] F D Murnaghan ldquoThe compressibility of media under extremepressuresrdquoProceedings of theNational Academy of Sciences of theUnited States of America vol 30 no 9 pp 244ndash247 1944
[22] J L Tallon ldquoThe thermodynamics of elastic deformation-IEquation of state for solidsrdquo Journal of Physics and Chemistryof Solids vol 41 no 8 pp 837ndash850 1980
[23] F Birch ldquoEquation of state and thermodynamic parameters ofNaCl to 300 kbar in the high-temperature domainrdquo Journal ofGeophysical Research vol 91 no 5 pp 4949ndash4954 1986
[24] J Shanker and S S Kushwah ldquoAnalysis of thermodynamicproperties for high-temperature superconducting oxidesrdquoPhys-ica Status Solidi (b) vol 180 no 1 pp 183ndash188 1993
[25] H Schlosser and J Ferrante ldquoLiquid alkali metals equation ofstate and reduced-pressure bulk-modulus sound-velocity andspecific-heat functionsrdquo Physical Review B vol 40 no 9 pp6405ndash6408 1989
[26] M Kumar ldquoHigh pressure equation of state for solidsrdquo PhysicaB vol 212 no 4 pp 391ndash394 1995
[27] M Kumar ldquoThermal expansivity and equation of state upto transition pressure and melting temperature NaCl as anexamplerdquo Solid State Communications vol 92 no 5 pp 463ndash466 1994
[28] J R Macdonald ldquoSome simple isothermal equations of staterdquoReviews of Modern Physics vol 38 no 4 pp 669ndash679 1966
[29] O L Anderson andD Issak ldquoThedependence of theAnderson-Gruneisen parameter 120575
119879upon compression at extreme condi-
tionsrdquo Journal of Physics and Chemistry of Solids vol 54 no 2pp 221ndash227 1993
[30] M P Toshi ldquoCohesion of ionic solids in the born modelrdquo SolidState Physics vol 1 pp 1ndash120 1964
[31] J Shanker B Singh and S S Kushwah ldquoOn the high-pressureequation of state for solidsrdquo Physica B vol 229 no 3-4 pp 419ndash420 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Physics Research International 7
0
0001
0002
0003
0004
0005
00060
251
755
105
141
Erro
r
Pressure (GPa)
(a) AlN (10 nm)
0
0004
0008
0012
0016
00
051
12 16
257
371 5
581
676
752 9 10
107
611
91
127
613
33
136
114
57minus0004
Erro
r
Pressure (GPa)
(b) Ge (13 nm)
0
0005
001
0015
0
085
265
365
475
575
645
minus001
minus0005
Erro
r
Pressure (GPa)
(c) CdSe (52 nm)
0
0002
0004
0006
0008
001
0012
0014
0016
00
592
356
478
539
129
3610
88
123
513
82
155
916
76
185
321
47
229
424
41
252
926
47minus0002
Erro
r
Pressure (GPa)
(d) Ni-filled MWCNT
0
0002
0004
0006
0008
0 2
45 5 7 8 10
145
187
5
E1E2
E3E4
minus0012
minus001
minus0008
minus0006
minus0004
minus0002
Erro
r
Pressure (GPa)
(e) Fe-filled MWCNT
0
0001
0002
0003
0004
0005
0 5 10 15 20 25 30 35
minus0004
minus0003
minus0002
minus0001
Erro
r
Pressure (GPa)
E1E2
E3E4
(f) WC (25 nm)
Figure 4 Error (119884-axis) between experimental and calculated relative volume compression (1198811198810) at various pressures (119883-axis) for different
materials
8 Physics Research International
Table 1 Input parameters for different nanomaterials at referencetemperature and zero pressure Bulk modulus 119870
0and first order
pressure derivative of bulkmodulus11987010158400are taken from the references
which are given in square brackets
Material 1198700(GPa) 119870
1015840
0
AlN (10 nm) 321 [5] 4 [5]Ge (13 nm) 112 [3] 4 [3]CdSe (54 nm) 37 [4] 11 [4]Ni-filled MWCNT 1798 [8] 53 [8]Fe-filled MWCNT 167 [8] 85 [8]WC (25 nm) 416 [7] 4 [7]
range and shows that different materials are compressibleup to different extent At higher pressure range Murnaghanequation of state shows the best result however the usual TaitEOS is also near to experimental values but data obtained byShanker and Suzuki formulations show more deviation
It is clear that variations of relative isothermal bulkmodulus increase with the increase of pressure and variationsof relative compression coefficient decrease with the increaseof pressure It may be thus concluded that all these equationsexplain the compression behaviour of nanomaterials satis-factorily for lower pressure range From Figures 2(a) to 2(f)and Figures 3(a) to 3(f) we observe that at high pressureresults of Murnaghan and UTE coincide well and Shankergives somewhat closer results to Murnaghan and UTE butSuzuki deviates
The variation of relative isothermal bulk modulus andrelative isothermal compression coefficient with pressurecould not be comparedwith the experimental values since theexperimental data on these physical properties of nanomate-rials are under study and not are available so far
5 Conclusion
Thus it is clear that in all formulations there is an incrementin the ratio of isothermal bulk modulus with pressure whilerelative isothermal compression decreases with increasingpressure and different material is compressible up to differentlevels With the increase in pressures solid becomes morerigid This is well understood as at high pressure moleculescome closer and their electron clouds ripple each other Thepresent study shows that all these equations of state are validfor nanomaterials at lower pressure range It needs somemodification at high pressure range for nanomaterials
References
[1] J Z Jiang ldquoPhase transformations in nanocrystalsrdquo Journal ofMaterials Science vol 39 no 16-17 pp 5103ndash5110 2004
[2] S Ramasamy D J Smith PThangadurai et al ldquoRecent study ofnanomaterials prepared by inert gas condensation using ultrahigh vacuum chamberrdquo Pramana vol 65 no 5 pp 881ndash8912005
[3] H Wang J F Liu Y He et al ldquoHigh-pressure structuralbehaviour of nanocrystalline Gerdquo Journal of Physics CondensedMatter vol 19 no 15 Article ID 156217 10 pages 2007
[4] S H Tolbert and A P Alivisatos ldquoThe wurtzite to rock saltstructural transformation in CdSe nanocrystals under highpressurerdquo Journal of Chemical Physics vol 102 no 11 article4642 15 pages 1994
[5] Z Wang K Tait Y Zhao et al ldquoSize-induced reduction oftransition pressure and enhancement of bulk modulus of AlNnanocrystalsrdquo Journal of Physical Chemistry B vol 108 no 31pp 11506ndash11508 2004
[6] G M Amulele M H Manghnani S Marriappan et alldquoCompression behavior of WC and WC-6Co up to 50GPadetermined by synchrotron x-ray diffraction and ultrasonictechniquesrdquo Journal of Applied Physics vol 103 no 11 ArticleID 113522 6 pages 2008
[7] Z Lin L Wang J Zhang H Mao and Y Zhao ldquoNanocrys-talline tungsten carbide as incompressible as diamondrdquoAppliedPhysics Letters vol 95 no 21 Article ID 211906 3 pages 2009
[8] H K Poswal S Karmakar P K Tyagi et al ldquoHigh-pressurebehavior of Ni-filled and Fe-filled multiwalled carbon nan-otubesrdquo Physica Status Solidi (b) vol 244 no 10 pp 3612ndash36192007
[9] O L Anderson ldquoThe use of ultrasonic measurements undermodest pressure to estimate compression at high pressurerdquoJournal of Physics and Chemistry of Solids vol 27 no 3 pp 547ndash565 1966
[10] S H Tolbert and A P Alivisatos ldquoSize dependence of a firstorder solid-solid phase transition the wurtzite to rock salttransformation in CdSe nanocrystalsrdquo Science vol 265 no5170 pp 373ndash376 1994
[11] S H Tolbert and A P Alivisatos ldquoHigh-pressure struc-tural transformations in semiconductor nanocrystalsrdquo AnnualReview of Physical Chemistry vol 46 pp 595ndash626 1995
[12] J Z Jiang J S Olsen L Gerward and S Morup ldquoEnhancedbulk modulus and reduced transition pressure in 120574-Fe
2O3
nanocrystalsrdquo Europhysics Letters vol 44 no 5 article 6201998
[13] J Z Jiang L GerwardD Frost R Secco J Peyronneau and J SOlsen ldquoGrain-size effect on pressure-induced semiconductor-to-metal transition in ZnSrdquo Journal of Applied Physics vol 86no 11 pp 6608ndash6610 1999
[14] J Z Jiang and L Gerward ldquoPhase transformation and conduc-tivity in nanocrystal PbS under pressurerdquo Journal of AppliedPhysics vol 87 no 5 article 2658 3 pages 2000
[15] B Chen D Penwell L R Benedetti R Jeanloz and M BKruger ldquoParticle-size effect on the compressibility of nanocrys-talline aluminardquo Physical Review B vol 66 no 14 Article ID144101 4 pages 2002
[16] Y He J F Liu W Chen et al ldquoHigh-pressure behavior ofSnO2nanocrystalsrdquo Physical Review B vol 72 no 21 Article
ID 212102 4 pages 2005[17] Z Wang L L Daemen Y Zhao et al ldquoMorphology-tuned
wurtzite-type ZnS nanobeltsrdquoNature Materials vol 4 pp 922ndash927 2005
[18] A San-Miguel ldquoNanomaterials under high-pressurerdquoChemicalSociety Reviews vol 35 no 10 pp 876ndash889 2006
[19] I Suzuki ldquoThermal expansion of periclase and olivine and theiranharmonic propertiesrdquo Journal of Physics of the Earth vol 23no 2 pp 145ndash159 1975
[20] I Suzuki K Seya H Tokai andY Sumino ldquoThermal expansionof single-crystal manganositerdquo Journal of Physics of the Earthvol 23 no 1 pp 63ndash69 1979
Physics Research International 9
[21] F D Murnaghan ldquoThe compressibility of media under extremepressuresrdquoProceedings of theNational Academy of Sciences of theUnited States of America vol 30 no 9 pp 244ndash247 1944
[22] J L Tallon ldquoThe thermodynamics of elastic deformation-IEquation of state for solidsrdquo Journal of Physics and Chemistryof Solids vol 41 no 8 pp 837ndash850 1980
[23] F Birch ldquoEquation of state and thermodynamic parameters ofNaCl to 300 kbar in the high-temperature domainrdquo Journal ofGeophysical Research vol 91 no 5 pp 4949ndash4954 1986
[24] J Shanker and S S Kushwah ldquoAnalysis of thermodynamicproperties for high-temperature superconducting oxidesrdquoPhys-ica Status Solidi (b) vol 180 no 1 pp 183ndash188 1993
[25] H Schlosser and J Ferrante ldquoLiquid alkali metals equation ofstate and reduced-pressure bulk-modulus sound-velocity andspecific-heat functionsrdquo Physical Review B vol 40 no 9 pp6405ndash6408 1989
[26] M Kumar ldquoHigh pressure equation of state for solidsrdquo PhysicaB vol 212 no 4 pp 391ndash394 1995
[27] M Kumar ldquoThermal expansivity and equation of state upto transition pressure and melting temperature NaCl as anexamplerdquo Solid State Communications vol 92 no 5 pp 463ndash466 1994
[28] J R Macdonald ldquoSome simple isothermal equations of staterdquoReviews of Modern Physics vol 38 no 4 pp 669ndash679 1966
[29] O L Anderson andD Issak ldquoThedependence of theAnderson-Gruneisen parameter 120575
119879upon compression at extreme condi-
tionsrdquo Journal of Physics and Chemistry of Solids vol 54 no 2pp 221ndash227 1993
[30] M P Toshi ldquoCohesion of ionic solids in the born modelrdquo SolidState Physics vol 1 pp 1ndash120 1964
[31] J Shanker B Singh and S S Kushwah ldquoOn the high-pressureequation of state for solidsrdquo Physica B vol 229 no 3-4 pp 419ndash420 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
8 Physics Research International
Table 1 Input parameters for different nanomaterials at referencetemperature and zero pressure Bulk modulus 119870
0and first order
pressure derivative of bulkmodulus11987010158400are taken from the references
which are given in square brackets
Material 1198700(GPa) 119870
1015840
0
AlN (10 nm) 321 [5] 4 [5]Ge (13 nm) 112 [3] 4 [3]CdSe (54 nm) 37 [4] 11 [4]Ni-filled MWCNT 1798 [8] 53 [8]Fe-filled MWCNT 167 [8] 85 [8]WC (25 nm) 416 [7] 4 [7]
range and shows that different materials are compressibleup to different extent At higher pressure range Murnaghanequation of state shows the best result however the usual TaitEOS is also near to experimental values but data obtained byShanker and Suzuki formulations show more deviation
It is clear that variations of relative isothermal bulkmodulus increase with the increase of pressure and variationsof relative compression coefficient decrease with the increaseof pressure It may be thus concluded that all these equationsexplain the compression behaviour of nanomaterials satis-factorily for lower pressure range From Figures 2(a) to 2(f)and Figures 3(a) to 3(f) we observe that at high pressureresults of Murnaghan and UTE coincide well and Shankergives somewhat closer results to Murnaghan and UTE butSuzuki deviates
The variation of relative isothermal bulk modulus andrelative isothermal compression coefficient with pressurecould not be comparedwith the experimental values since theexperimental data on these physical properties of nanomate-rials are under study and not are available so far
5 Conclusion
Thus it is clear that in all formulations there is an incrementin the ratio of isothermal bulk modulus with pressure whilerelative isothermal compression decreases with increasingpressure and different material is compressible up to differentlevels With the increase in pressures solid becomes morerigid This is well understood as at high pressure moleculescome closer and their electron clouds ripple each other Thepresent study shows that all these equations of state are validfor nanomaterials at lower pressure range It needs somemodification at high pressure range for nanomaterials
References
[1] J Z Jiang ldquoPhase transformations in nanocrystalsrdquo Journal ofMaterials Science vol 39 no 16-17 pp 5103ndash5110 2004
[2] S Ramasamy D J Smith PThangadurai et al ldquoRecent study ofnanomaterials prepared by inert gas condensation using ultrahigh vacuum chamberrdquo Pramana vol 65 no 5 pp 881ndash8912005
[3] H Wang J F Liu Y He et al ldquoHigh-pressure structuralbehaviour of nanocrystalline Gerdquo Journal of Physics CondensedMatter vol 19 no 15 Article ID 156217 10 pages 2007
[4] S H Tolbert and A P Alivisatos ldquoThe wurtzite to rock saltstructural transformation in CdSe nanocrystals under highpressurerdquo Journal of Chemical Physics vol 102 no 11 article4642 15 pages 1994
[5] Z Wang K Tait Y Zhao et al ldquoSize-induced reduction oftransition pressure and enhancement of bulk modulus of AlNnanocrystalsrdquo Journal of Physical Chemistry B vol 108 no 31pp 11506ndash11508 2004
[6] G M Amulele M H Manghnani S Marriappan et alldquoCompression behavior of WC and WC-6Co up to 50GPadetermined by synchrotron x-ray diffraction and ultrasonictechniquesrdquo Journal of Applied Physics vol 103 no 11 ArticleID 113522 6 pages 2008
[7] Z Lin L Wang J Zhang H Mao and Y Zhao ldquoNanocrys-talline tungsten carbide as incompressible as diamondrdquoAppliedPhysics Letters vol 95 no 21 Article ID 211906 3 pages 2009
[8] H K Poswal S Karmakar P K Tyagi et al ldquoHigh-pressurebehavior of Ni-filled and Fe-filled multiwalled carbon nan-otubesrdquo Physica Status Solidi (b) vol 244 no 10 pp 3612ndash36192007
[9] O L Anderson ldquoThe use of ultrasonic measurements undermodest pressure to estimate compression at high pressurerdquoJournal of Physics and Chemistry of Solids vol 27 no 3 pp 547ndash565 1966
[10] S H Tolbert and A P Alivisatos ldquoSize dependence of a firstorder solid-solid phase transition the wurtzite to rock salttransformation in CdSe nanocrystalsrdquo Science vol 265 no5170 pp 373ndash376 1994
[11] S H Tolbert and A P Alivisatos ldquoHigh-pressure struc-tural transformations in semiconductor nanocrystalsrdquo AnnualReview of Physical Chemistry vol 46 pp 595ndash626 1995
[12] J Z Jiang J S Olsen L Gerward and S Morup ldquoEnhancedbulk modulus and reduced transition pressure in 120574-Fe
2O3
nanocrystalsrdquo Europhysics Letters vol 44 no 5 article 6201998
[13] J Z Jiang L GerwardD Frost R Secco J Peyronneau and J SOlsen ldquoGrain-size effect on pressure-induced semiconductor-to-metal transition in ZnSrdquo Journal of Applied Physics vol 86no 11 pp 6608ndash6610 1999
[14] J Z Jiang and L Gerward ldquoPhase transformation and conduc-tivity in nanocrystal PbS under pressurerdquo Journal of AppliedPhysics vol 87 no 5 article 2658 3 pages 2000
[15] B Chen D Penwell L R Benedetti R Jeanloz and M BKruger ldquoParticle-size effect on the compressibility of nanocrys-talline aluminardquo Physical Review B vol 66 no 14 Article ID144101 4 pages 2002
[16] Y He J F Liu W Chen et al ldquoHigh-pressure behavior ofSnO2nanocrystalsrdquo Physical Review B vol 72 no 21 Article
ID 212102 4 pages 2005[17] Z Wang L L Daemen Y Zhao et al ldquoMorphology-tuned
wurtzite-type ZnS nanobeltsrdquoNature Materials vol 4 pp 922ndash927 2005
[18] A San-Miguel ldquoNanomaterials under high-pressurerdquoChemicalSociety Reviews vol 35 no 10 pp 876ndash889 2006
[19] I Suzuki ldquoThermal expansion of periclase and olivine and theiranharmonic propertiesrdquo Journal of Physics of the Earth vol 23no 2 pp 145ndash159 1975
[20] I Suzuki K Seya H Tokai andY Sumino ldquoThermal expansionof single-crystal manganositerdquo Journal of Physics of the Earthvol 23 no 1 pp 63ndash69 1979
Physics Research International 9
[21] F D Murnaghan ldquoThe compressibility of media under extremepressuresrdquoProceedings of theNational Academy of Sciences of theUnited States of America vol 30 no 9 pp 244ndash247 1944
[22] J L Tallon ldquoThe thermodynamics of elastic deformation-IEquation of state for solidsrdquo Journal of Physics and Chemistryof Solids vol 41 no 8 pp 837ndash850 1980
[23] F Birch ldquoEquation of state and thermodynamic parameters ofNaCl to 300 kbar in the high-temperature domainrdquo Journal ofGeophysical Research vol 91 no 5 pp 4949ndash4954 1986
[24] J Shanker and S S Kushwah ldquoAnalysis of thermodynamicproperties for high-temperature superconducting oxidesrdquoPhys-ica Status Solidi (b) vol 180 no 1 pp 183ndash188 1993
[25] H Schlosser and J Ferrante ldquoLiquid alkali metals equation ofstate and reduced-pressure bulk-modulus sound-velocity andspecific-heat functionsrdquo Physical Review B vol 40 no 9 pp6405ndash6408 1989
[26] M Kumar ldquoHigh pressure equation of state for solidsrdquo PhysicaB vol 212 no 4 pp 391ndash394 1995
[27] M Kumar ldquoThermal expansivity and equation of state upto transition pressure and melting temperature NaCl as anexamplerdquo Solid State Communications vol 92 no 5 pp 463ndash466 1994
[28] J R Macdonald ldquoSome simple isothermal equations of staterdquoReviews of Modern Physics vol 38 no 4 pp 669ndash679 1966
[29] O L Anderson andD Issak ldquoThedependence of theAnderson-Gruneisen parameter 120575
119879upon compression at extreme condi-
tionsrdquo Journal of Physics and Chemistry of Solids vol 54 no 2pp 221ndash227 1993
[30] M P Toshi ldquoCohesion of ionic solids in the born modelrdquo SolidState Physics vol 1 pp 1ndash120 1964
[31] J Shanker B Singh and S S Kushwah ldquoOn the high-pressureequation of state for solidsrdquo Physica B vol 229 no 3-4 pp 419ndash420 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Physics Research International 9
[21] F D Murnaghan ldquoThe compressibility of media under extremepressuresrdquoProceedings of theNational Academy of Sciences of theUnited States of America vol 30 no 9 pp 244ndash247 1944
[22] J L Tallon ldquoThe thermodynamics of elastic deformation-IEquation of state for solidsrdquo Journal of Physics and Chemistryof Solids vol 41 no 8 pp 837ndash850 1980
[23] F Birch ldquoEquation of state and thermodynamic parameters ofNaCl to 300 kbar in the high-temperature domainrdquo Journal ofGeophysical Research vol 91 no 5 pp 4949ndash4954 1986
[24] J Shanker and S S Kushwah ldquoAnalysis of thermodynamicproperties for high-temperature superconducting oxidesrdquoPhys-ica Status Solidi (b) vol 180 no 1 pp 183ndash188 1993
[25] H Schlosser and J Ferrante ldquoLiquid alkali metals equation ofstate and reduced-pressure bulk-modulus sound-velocity andspecific-heat functionsrdquo Physical Review B vol 40 no 9 pp6405ndash6408 1989
[26] M Kumar ldquoHigh pressure equation of state for solidsrdquo PhysicaB vol 212 no 4 pp 391ndash394 1995
[27] M Kumar ldquoThermal expansivity and equation of state upto transition pressure and melting temperature NaCl as anexamplerdquo Solid State Communications vol 92 no 5 pp 463ndash466 1994
[28] J R Macdonald ldquoSome simple isothermal equations of staterdquoReviews of Modern Physics vol 38 no 4 pp 669ndash679 1966
[29] O L Anderson andD Issak ldquoThedependence of theAnderson-Gruneisen parameter 120575
119879upon compression at extreme condi-
tionsrdquo Journal of Physics and Chemistry of Solids vol 54 no 2pp 221ndash227 1993
[30] M P Toshi ldquoCohesion of ionic solids in the born modelrdquo SolidState Physics vol 1 pp 1ndash120 1964
[31] J Shanker B Singh and S S Kushwah ldquoOn the high-pressureequation of state for solidsrdquo Physica B vol 229 no 3-4 pp 419ndash420 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
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