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Chapter-8__________________________________________________________
Shyamkumar G. Khambholja / Ph.D. Thesis/ Physics/ S.P. University/ April-2012
248
Chapter 8
Electrical Transport Properties of
Liquids
Proceedings of 5th NCTP (AIP CP) 1249 (2010) 194
Proceedings of 5th NCTP (AIP CP) 1249 (2010) 170
Proceedings of 55th DAE Solid State Physics Symposium (AIP CP) 1349 (2011) 945
8.1 Introduction 249
8.2 Theory 249
8.3 Results 259
8.4 Conclusions 272
8.5 References 275
Chapter-8__________________________________________________________
Shyamkumar G. Khambholja / Ph.D. Thesis/ Physics/ S.P. University/ April-2012
249
8.1 Introduction
The electrical transport properties of liquid metals and their alloys are of great
interest theoretically as well as experimentally. During last three decades numbers of
efforts are made to study electrical transport properties of liquid metals and their
alloys [1-5]. The main reason behind such study is that the study of electrical
transport properties viz; electrical resistivity and thermal conductivity determines
usefulness of materials at high temperatures. Especially the materials (which are
purely metallic or are made up of metallic elements) are found to have high electrical
conductivity due to presence of free electrons. In liquid phase, (near and above
melting point) the study of such properties is important theoretically also. The study
of electrical transport properties using model potential formalism judges the validity
of model potential used. Also, it works as a good test for the validity of theoretical
formulation used. Such study also provides an insight into transport properties in
liquids. Thus, the study of transport properties is important in many ways. In the
present work, we report the results of our theoretical study of electrical transport
properties of liquid aluminum and its three alloys Al-Cu, Al-Li and Al-Ni. The
temperature dependent electrical resistivity of aluminum is studied. On the other hand
for alloys, concentration dependent electrical transport properties at fixed temperature
are studied.
8.2 Theory
There are mainly two theoretical approaches to calculate the electrical
transport properties of liquid metals and their alloys. The first approach is the T-
matrix approach [6] and the second is the Faber-Ziman approach [7].
8.2.1 T-matrix approach
This approximation involves average T-matrix approximation or the coherent
potential approximation. In the T-matrix approximation, the electrical resistivity of
liquid metal is given by [6],
231
02 22
12T
k
qd
k
q
k FFF
oL
Ω= ∫
πρ (8.1)
Chapter-8__________________________________________________________
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250
Where, oΩ is the atomic volume. Fk is the Fermi wave vector. q is the reciprocal
lattice vector. 2
T is the average T-matrix and is given by [6],
22),()( qEtqaT F= (8.2)
Here, )(qa is the structure factor or the interference function of the liquid metal.
Experimental data of structure factor of most of the liquid metals are available [8]. On
the other hand, theoretically, the structure factor of liquid metals can be calculated
either using PY theory [9] or CHS theory [10] or OCP theory [11] or variational
modified hyperneated chain (VMHNC) theory [12]. The quantity ),( qEt F is given
by,
∑ +Ω
−=l
lll
Fo
F PilE
qEt )(cos)exp()sin()12()2(
2),(
2/1θηηπ
(8.3)
It represents the scattering of an electron from initial state 'Fk to a final state Fk by a
single muffin tin potential in liquid.
Here, FF kkq −= /
The quantity lη represents the phase shift at Fermi surface.
For binary alloys the average T-matrix is given by [6],
[ ] [ ] [ ]1)(..()(1)(1 *)*222−+++−++−= qattttCCqaCCtCqaCCtCT ijjijijijjjjjjiiiiiialloy
(8.4)
Where, Ci and Cj are concentration of ith and jth element respectively. ti and tj are the
average T-matrices for ith and jth elements respectively. The quantity aij(q) represent
the structure factor of the binary system. The structure factor of liquid binary alloys is
also available experimentally [8,13]. Theoretically, it can be calculated using
Ashcroft-Langreth approach [14]. Since, the T-matrix approach does not involve
Chapter-8__________________________________________________________
Shyamkumar G. Khambholja / Ph.D. Thesis/ Physics/ S.P. University/ April-2012
251
model potential in its calculation; we have not used this approach in present work.
One more drawback of T-matrix approach is that this approach can be used when the
resulting phase shifts are small enough. e.g. in case of alkali metals, where the ion-
electron interaction can be calculated using relatively soft pseudopotentials as the
wave functions are nearly plane wave. But in case of polyvalent metals and transition
metals, the interaction is quite complicated and hence it is less justifiable to use the T-
matrix approach in case of polyvalent metals, transition metals and their alloys.
8.2.2 The Faber-Ziman Theory
We have used the expression due to Ziman [15] to compute the electrical
transport properties of liquid aluminum and Faber and Ziman approach [7] to compute
the electrical resistivity and related transport properties of the binary alloys. In these
both formalisms, the valence electrons are treated as nearly free and the only potential
experienced by them is due to periodic arrangement of atoms. Due to such weak
effective potential the wave functions of the valence electrons are like plane waves.
This observation leads us to use pseudopotential to describe the electron-ion potential.
For the investigation of the transport properties of liquid metal alloys, the theory
proposed by Ziman [15] and Bardely [16] and by Faber and Ziman [7] with some
correction are being used. The main task of the theory is to solve the Boltzman
equation and to find out the relaxation time for the given system. The valence
electrons which are scattered by regularly placed ions are nearly free and hence we
can calculate the scattering probability per unit time from the Born approximation.
The scattering probability per unit time is given by [17],
2)(
2' krUqkP
kk+=
h
π (8.5)
We can write the relaxation time in terms of the scattering probability per unit time as
∫ −Ω= '
3)cos1(
)2(
1' dkP
kkk
θπτ
(8.6)
Chapter-8__________________________________________________________
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252
Here, Ω is atomic volume, θ the scattering angle and 'kkP is the probability of
scattering per unit time.
Here, it can be observed that when the scattering angle tends to zero, the relaxation
time tends to infinite. As a result, the collision frequency tends to zero. We can write
down the relaxation time in terms of matrix element as,
∫ +Ω=Fk
Fk
dqqkrUqkk
m2
0
32
33)(
4
.1
hπτ (8.7)
The electrical conductivity in terms of relaxation time is given by [18],
m
ne τσ2
= (8.8)
Therefore, the electrical resistivity is given by,
τσρ
2
1
ne
m== (8.9)
Where, n is the number density of the liquid metal, m is the atomic mass and e is
charge of electron. We can write down the expression for electrical resistivity in terms
of average structure factor and form factor as [19],
∫ +Ω
=Fk
F
oL dqqkrUqk
ke
m 2
0
32
623
2
)(4
3
h
πρ
dqqqSkquqkke
m Fk
F
oL
32
0
22
623
2
)()(4
3∫ +
Ω=∴
h
πρ (8.10)
Here, )(qS is the interference function or the structure factor of the liquid metal. For,
a liquid metal, it is given by [9],
[ ] 1)(1)( −−= qncqS (8.11)
Chapter-8__________________________________________________________
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253
Where, )(qnc represent the direct correlation function. It is given by [9],
∫ ++−=1
0
323 )()(
)sin(4)( dYYY
q
qYqnc γβα
σσπσσ
Where, Fk
qY
2=
2)21( ηα += , 4
2
)1(
)5.01(6
ηηηβ
−+−= ,
4
2
)1(
)21(
2 ηηηγ
−+=
Where, η is the packing density parameter. It is ratio of the volume occupied by hard
spheres to the total volume of the unit cell. It is given by [8],
3
6
1 σπη n= (8.12)
Here, n is the number density and σ is the hard sphere diameter. For the binary
alloys above formula of structure factor is modified by the introduction of partial
structure factor. When a solute is added to the solvent metal, the local density
fluctuations takes place in the solvent and as a result we have to take into account the
effect of solute and solvent atoms on each other in order to calculate the structural and
transport properties of such binary alloys.
For binary alloys the expressions for the partial structure factors 11S , 22S and 12S are
given by [14],
[ ] 1
2222122111111 )(1/)()(1
−−−−= yCnyCnnyCnS (8.13)
[ ] 1
1112122122222 )(1/)()(1
−−−−= yCnyCnnyCnS (8.14)
[ ] [ ] 12122122211112
2/12112 )()(1)(1)()()(
−−−×−= yCnnyCnyCnyCnnyS (8.15)
Chapter-8__________________________________________________________
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254
where, the quantities ijC represent the direct correlation functions of binary mixtures.
They are the Fourier transforms of the general solutions of the PY equations for the
mixtures of hard spheres.
The direct correlation functions in q-space are given by [14],
[ ]
++−−−+
−−−+−−=−
24)cos()2412()sin()244()(
)2)cos()2()sin(2())cos()(sin(
)(
24)(
121
4111
313
1
1
12111
1
11111
31
1111
yyyyyyy
yyyyy
yyya
yyCn
γ
βη
(8.16)
[ ]
++−−−+
−−−+−−=− −
24)cos()2412()sin()244()(
)2)cos()2()sin(2())cos()(sin(
)(
24)(
24331
3
221
32
222
yyyyyyy
yyyyy
yyya
yyCn
γα
βη
(8.17)
The remaining correlation function is given by,
[ ] [ ]
[ ]
[ ] [ ]
[ ]
−+−+
+
−−+−
+
++−−−+
−−−+−−−+
+−+−
++−+−+−+
×−+
−+−
−+−−=−
1
121
111
1
121
111
21
1
121
4111
312
1
1
11311
21
1
121
211112
41
121
4111
312
1
1
11311
21
1
121
211112
41
3
32/12/1
313
2/12/13
122/1
22/1
1
)sin(
2
11)sin()cos()sin(
)cos(1
2
1)cos()sin()cos(
24)cos()2412()sin()44(
)cos()6()sin()63(2)cos()2()sin(2)cos(
)sin()2412()cos()244(
6)sin()6()cos()63()sin()2()cos(2)sin(
)1(
)1(24
)cos()sin(
)1(
)1()1(3)(
y
y
y
yyyy
y
y
y
yyyy
y
a
yyyyyyy
yyyyyy
yyyy
y
y
yyyyyyy
yyyyyy
yyyy
y
y
xx
xx
y
yyya
xx
xxyCnn
αα
αα
γ
γβ
γ
γβ
ααη
αηα
λ
λ
λ
λ
λ
λλλ
(8.18)
Chapter-8__________________________________________________________
Shyamkumar G. Khambholja / Ph.D. Thesis/ Physics/ S.P. University/ April-2012
255
Here, 2
1
σσα = ( )10 ≤≤ α
2σqy = , 11 σqy =
21 ηηη +=
Here, 1η and 2η are the packing fractions of solvent and solute metallic element
respectively.
)( '
11 βρ
η∂∂=a , )( '
2
32 βρ
ηα
∂∂= −a
where,
[ ] 321
221
22
31
' )1()1(1)1(3)1)(( −−+++×−−+++= ηηαηαηηηηηαηβρ
1β And 2β can be found out in terms of functions11g , 22g and 12g as
1β =
++−= 212
22
211111 )1(
4
16 ggb ααηησ and
2β =
++−= − 212
231
222222 )1(
4
16 ggb ααηησ
( ) 122221112
2 )1(3 gggb ηηαασ ++−= − and
( )223
11311 aad ηαησγ +==
where, the parameters ijg are given by,
( ) ( )
( ) ( )
( ) 22112
21122
2211
1)()1(
)1(
2
3
2
11
112
3
2
11
,112
3
2
11
−
−−
−
−
−+−+
+=
−
−+
+=
−
−+
+=
ηηηααη
ηαηη
ηαηη
g
g
g
Chapter-8__________________________________________________________
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256
The total structure factor ( )qS of a binary mixture in terms of partial structure
factors ( )qS11 , ( )qS22 and ( )qS12 is given by [14],
( ) )()1()()1(2)( 11122/1
22 ySxySxxyxSqS −+−+= (8.19)
This total structure factor reduces to the expression for the structure factor of a liquid
metal when the concentration of the solute becomes zero. In terms of partial
structure factors, the expression for the resistivity of a binary alloy is given by [20],
( )∫ −= qkdqqqnke
mF
FL 2)(
4
3 3632
2
θλπρh
(8.20)
Here, the function )(qλ is given by,
)()()()1(2)()1()( 2222212211
2111 qVxSqVqVSSxxqVSxq +−+−=λ (8.21)
Here, the terms )(qVi represent the screened form factors for A and B elements. The
term )2( qkF −θ is the step function that cuts off the integration at Fkq 2= for
perfectly spherical Fermi surface.
Other method for calculating electrical resistivity using model potential formalism is
the “2kF” scattering model [21]. This approach is based on the multiple scattering of
electrons due to their collisions. This method includes the Debye temperature of
liquid as one of the input. The calculation of Debye temperature of liquid is obviously
a demanding job. Some authors have used this approach, but their work includes large
number of assumptions and fitting to the experimental observations [22, 23]. Thus, we
have not used this model in the present work.
8.2.3 Thermoelectric power
The absolute thermoelectric power of a binary alloy can be obtained by
differentiating the formula of electrical resistivity with respect to position of the Fermi
surface. The formula of the thermoelectric power at any temperature T is given by
[1,4,8],
Chapter-8__________________________________________________________
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257
IeE
TkQ
F
BL 3
22π= (8.22)
where, the term I is given by [1,4,8],
FEELF EEI =
∂∂−= ρln (8.23)
By simplifying the above equation we can write the expression for the thermoelectric
power, which is given by,
−=
LF
BL
q
eE
TkQ
ρλπ )(
233
22
(8.24)
8.2.4 Thermal conductivity
The expression for the thermal conductivity of a binary alloy in terms of the
electrical resistivity at particular temperature T is given by [24],
L
B
e
Tk
ρπσ 2
22
3= (8.25)
Here, the terms e and kB are charge of electron and Boltzman’s constant respectively.
Using the above equations we have calculated the electrical resistivity and thermal
conductivity of a liquid aluminum and its binary alloys Al-Cu, Al-Ni and Al-Li in the
whole range of concentration.
The ion-electron interaction in the present work is taken into account in the
present work using Ashcroft’s empty core model potential [25]. Five different forms
of local field correction functions namely Hartree (H) [26], Taylor (T) [27], Ichimaru
and Utsumi (IU) [28], Farid et al (F) [29] and Sarkar et al (S) [30] are used to
compute screened form factors. Transport properties are calculated using Ashcroft-
Langreth partial structure factors [14] in conjunction with the Faber-Ziman approach
[7].
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258
The input parameters for pure Al, Cu, Li and Ni are shown in table 8.1 below.
From the knowledge of inputs of pure elements, we have calculated the inputs for
their binary alloys as per their composition [31]. The core radius of all elements is
calculated using the formulation suggested by Jani et al [32].
Table 8.1 List of input parameters used in the present calculations [8].
Inputs Al Cu Li Ni
Z 3 3 3 1 1 1
effcr (au) 1.166 1.169 1.183 0.8279 1.841 1.341
eff0Ω (au)3 127.526 128.611 133.143 87.277 170.409 78.339
T (K) 943 1023 1323 1100 1023 1000
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259
8.3 Results
(1) Aluminum
In the present work, we have used the PY hard sphere structure factors [9] of
aluminum near and above its melting point in conjunction with the Ziman theory [15]
to compute transport properties. Figures 8.1-8.3 show the presently computed PY hard
sphere structure factors along with the corresponding experimental results [8] at three
different temperatures. A good agreement is observed between the presently
computed structure factors with the experimental results. In particular, the structure
factor at and around the first peak show good agreement. The position and height of
first and second peak in the structure factor show very good agreement with the
experimental results. This observation indicates that the structural information is
properly incorporated in the present work. At 1323 K temperature, the height of the
principal peak is overestimated as compared to experimental results. Further,
electrical resistivity is calculated using Ziman theory. The computed values of
temperature dependent electrical resistivities are shown in Figure 8.4 along with the
available experimental results [33]. The presently calculated value of electrical
resistivity increases with temperature, which indicates that aluminium possess
metallic behavior in liquid phase also. This observation is in agreement with the
experimental findings. The values computed using H function is slightly
underestimated in comparison to experimental findings. On the other hand, the values
computed using T, IU and F functions are overestimated. The experimental values lie
between those calculated using H and S functions. The computed values of
thermoelectric power are shown in Figure 8.5 at three different temperatures. The
thermoelectric power first decrease as we go from 943 K to 1023 K and decreases at
1323 K. The computed thermal conductivity is shown in Figure 8.6 at three different
temperatures. Thermal conductivity increases with temperature. This feature also
demonstrates the metallic behavior of liquid aluminum above its normal melting
point. No experimental data are found for comparison in case of thermoelectric power
and thermal conductivity of liquid aluminum at various temperatures.
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Figure 8.1. Structure factor of aluminum at 943 K temperature.
Figure 8.2. Structure factor of aluminum at 1023 K temperature.
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261
Figure 8.3. Structure factor of aluminum at 1323 K temperature.
Figure 8.4. Electrical resistivity of aluminum as a function of temperature.
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262
Figure 8.5. Thermoelectric power of aluminum at three different temperatures.
Figure 8.6. Thermal conductivity of aluminum at various temperatures.
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263
(2) Al-Cu system
The aluminum based binary alloys are of great technological importance due
to their exceptional physical properties like low mass, high hardness etc. Due to these
properties, their various physical properties are studied much. Khajil et al [34] have
studied the electrical transport properties of Al-Cu alloys using MHS model [21] in
conjunction with model potential formalism. Experimentally, Romanov et al [35]
have studied the transport properties of Al-Cu alloys. Bretonnet et al [36] have studied
the transport properties of Al-Cu system using four probe method. No other
theoretical results are available for Al-Cu system. In the present work, we report the
results of our theoretical study of Al-Cu system using model potential formalism in
conjunction with the Faber-Ziman approach. The input parameters are shown in Table
8.1.
Figure 8.7. AL partial structure factor of Al50Cu50 system.
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264
Figure 8.8. Electrical resistivity of Al-Cu alloys.
Figure 8.9.. Thermoelectric power of Al-Cu alloy.
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265
Figure 8.10. Thermal conductivity of Al-Cu alloys.
The Ashcroft-Langreth (AL) partial structure factors of binary Al-Cu system
at equi- atomic concentration is shown in Figure 8.7. Figure 8.8 represent the results
of presently calculated electrical resistivity of Al-Cu system along with the
experimental and other theoretical results. The presently computed values of electrical
resistivity are found to be in very good agreement with the experimental results
reported by Bretonnet et al [36] in the whole range of concentration. The maximum in
the electrical resistivity occurs at 62.94 % Cu concentration using H, IU, F and S
functions. On the other hand, electrical resistivity calculated using T function shows
maximum at 86 % Cu concentration. This observation is found to be similar with
other copper based alloys [37]. The experimental results of Romanov et al [35] are
available up to limited range of concentration. On the other hand, other theoretical
results show comparatively much deviation from the experimental results compared to
our results. Looking to the overall picture, it is observed that our present approach is
suitable for computing electrical resistivity of binary alloys. Further, regarding
exchange-correlation effects, it is observed that H function gives lowest values of
electrical resistivity among all five local field correction functions. On the other hand,
T function gives largest values. Results computed using S function is very close to the
experimental results. S function satisfy compressibility sum rule in the long
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266
wavelength limit more precisely as compared to all other four local field correction
functions, used in the present work. This may be the possible reason that it generates
better results. Figure 8.9 shows the presently computed thermoelectric power of Al-
Cu alloys in the whole range of Cu concentrations. Computed thermoelectric power
shows minimum at Cu concentration, near which the computed electrical resistivity
shows its peak. Figure 8.10 shows the computed thermal conductivity of Al-Cu alloys
as a function of Cu concentration. No experimental or other theoretical results are
found for comparison for thermoelectric power or thermal conductivity. It is observed
that H function gives highest values of thermal conductivity, while T gives lowest
values. In the absence of any results experimental and theoretical for comparison, we
could not put any concrete remark on our results of thermal conductivity.
(3) Al-Li system
Binary aluminum-lithium system is studied much in solid phase using
different theoretical and experimental approaches [38-39]. But in liquid phase, studies
on Al-Li system are rare. Kiselev [40] has studied the electrical transport properties of
Al-Li system using variational parameter approach. No other experimental results are
available for Al-Li system. In the present work, we have studied the electrical
transport properties of binary Al-Li alloys in liquid phase. Input parameters are shown
in Table 8.1.
Figure 8.11. AL partial structure factor of Al50Li 50 system.
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Figure 8.12. Electrical resistivity of Al-Li alloys.
Figure 8.13. Thermoelectric power of Al-Li alloys.
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268
Figure 8.14. Thermal conductivity of Al-Li alloys.
Figure 8.11 shows the computed AL partial structure factors of binary Al-Li
alloys at equi- atomic composition. The principal peak in S11(q) occurs at 2.64 Å-1.
Figure 8.12 shows the presently computed electrical resistivity of Al-Li alloy along
with the theoretical results of Kiselev et al [40]. Kiselev et al [40] have performed the
theoretical study using variational scheme and generated the inputs using self
consistent calculations. On the other hand, our work involves the model potential
formalism along with well established local field correction functions. Presently
computed electrical resistivity is underestimated when compared to results of Kiselev
et al [40]. The possible reason lies in the different model potential used. However, the
trend in the electrical resistivity is accurately reproduced. The maximum in the
resistivity occurs at 80 % Li concentration. This trend is similar with the other
aluminum based transition metal alloys like Al-Cu. Lithium is having complicated
structure, which is difficult to explain even at ambient condition. At temperatures
beyond melting, there is obviously increase in the complication. This may be another
possible reason behind the difference in the present results and reported by Kiselev et
al [40] near equi-atomic concentration. However, in the absence of experimental
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269
results, present results serve as a useful set of data for further research. Figure 8.13
shows the computed thermoelectric power of Al-Li alloys. Computed thermoelectric
power, which is related to scattering probability per unit time near Fermi surface, is
negative in the whole range of Li concentration. Earlier, it is observed that
thermoelectric power of pure aluminum is positive. However, in case of Al-Li alloy, it
is negative. Thus, it is concluded that alloying of Al with Li drastically affects the
Fermi surface of the resultant alloy. Figure 8.14 shows the presently calculated
thermal conductivity of Al-Li alloys. The minimum in the thermal conductivity occurs
at 0.8 % Li concentration, near which the electrical resistivity shows its maximum.
No experimental or other theoretical results are available for comparison for
thermoelectric power of thermal conductivity.
(4) Al-Ni system
Binary Al-Ni system is of large technological importance due to its variety of
properties which other metals or alloys do not possess. This system is studied much
during last two decades. Structural, vibrational and atomic transport properties have
been reported [41-43]. However, electrical transport properties of binary Al-Ni alloys
are not reported till date. In view of all these facts, we report the theoretical study of
electrical transport properties of Al-Ni binary alloys. The input parameters are shown
in Table 8.1. Figure 8.15 shows the presently computed AL partial structure factors of
Al-Ni system at equi-atomic composition. Unlike Al-Li and Al-Cu system, where
there is not much change in the position of the first peak in S11(q) and S22(q), there is
notable difference between the position of the first peak in S11(q) and S22(q). This
difference may be attributed to the difference between the hard sphere diameters as
well as the ion-ion interaction in liquid phase. Figure 8.16 shows the presently
computed electrical resistivity of Al-Ni alloys in the whole range of Ni concentration.
It is observed that the maximum in the electrical resistivity occurs near equi-atomic
composition. This observation is also different from that observed in other two
aluminum based binary alloys (Al-Cu and Al-Li). Also, compared to the Al-Cu and
Al-Li alloys, electrical resistivity of Al-Ni alloy is small in magnitude. The smaller
value of electrical resistivity indicates high electrical conductivity and high thermal
conductivity. These properties makes Al-Ni alloy, a prominent candidate for
applications at extreme conditions. In the absence of any experimental or theoretical
results, we could not make any comparison of our results. However, present results
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would be certainly useful for further research in this field. Figure 8.17 shows the
presently computed thermoelectric power of Al-Ni alloys. In case of this alloy,
thermoelectric power shows minimum near 80% Ni concentration. However, the
computed resistivity shows maximum nearly at equi-atomic composition. Such
feature is interesting. Figure 8.18 shows the computed thermal conductivity of Al-Ni
alloys. The minimum in the thermal conductivity occurs near 0.65 % Ni
concentration. No experimental or theoretical results are available for comparison.
The thermal conductivity calculated using T, IU and F functions are nearly same. On
the other hand, thermal conductivity computed using H and S functions is nearly
same. However, thermal conductivity is not influenced much by different exchange-
correlation functions.
Figure 8.15. AL partial structure factor of Al50Ni50 system.
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Figure 8.16. Electrical resistivity of Al-Ni alloys.
Figure 8.17. Thermoelectric power of Al-Ni alloys.
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272
Figure 8.18. Thermal conductivity of Al-Ni alloys.
8.4 Conclusions
In the present 8th chapter, we have computed structural and electrical transport
properties of aluminum and its three alloys namely Al-Cu, Al-Li and Al-Ni using
model potential formalism. To compute the electrical transport properties of
aluminum, we have used the Ziman approach and to compute the transport properties
of binary alloys, we have make use of Faber-Ziman approach. Following conclusions
emerge out of the present work.
1) The structure factor of aluminum at three different temperatures are computed
using PY hard sphere theory and the results show good agreement with the
corresponding experimental results near and around the first and second peak
as well as at large q-values. These observations indicate that the structural
information is properly incorporated using PY theory. Further, the electrical
resistivity is computed at three different temperatures using Ziman theory.
Electrical resistivity computed using T, IU and F functions is higher as
compared to those computed using H and S functions as well as the
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experimental results. The S function, which satisfies compressibility sum rule
in the long wavelength limit precisely as compared to other three functions (T,
IU and F) generates results, which are near to experimental findings. Thus the
present results show the importance of exchange-correlation effects in the
electrical transport properties.
2) In the present work, we have the approach of Faber and Ziman to compute
electrical resistivity of binary alloys. Computed values of electrical resistivity
of Al-Cu alloys do not such much deviation from experimental findings. On
the other hand, the values computed using MHS model and T-matrix
formulation are also in line with our results. Thus, it is observed that presently
used approach is successful even without including quantum effect. Using the
same formulation, we have computed electrical resistivity of Al-Li and Al-Ni
alloys.
3) In case of Al-Cu system, the transport properties are computed using Faber-
Ziman approach. The computed electrical resistivity shows very good
agreement with the experimental results. The trend of variation as well as the
magnitude is in nice agreement with the experimental findings. Further, our
results are also in better agreement with the experimental results as compared
to the other theoretical results. In case of Al-Cu alloy also, S function
generates results, which are in very good agreement with the experimental
findings as compared to other four local field correction functions. Values of
thermoelectric power and thermal conductivity are also predicted.
4) The results of electrical resistivity of Al-Li alloys are underestimated as
compared to the other theoretical findings. However, the trend is exactly
obtained. The maximum occurs at 0.8 % Li concentration.
5) In case of Al-Ni alloys, the computed values of resistivity show maximum
near equi-atomic composition. This observation is different from the other two
Al based binary alloys namely Al-Cu and Al-Li, in which the maximum
occurs near 0.8 % (Cu/Li) concentration.
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6) Thermoelectric power is also computed for aluminum and its three binary
alloys. Computed thermoelectric power of aluminum is positive.
Thermoelectric power shows minimum at (Li and Cu) concentrations, where
the electrical resistivity shows peak. In case of Al-Ni system, minimum in
thermoelectric power occurs at slight different Ni concentration. No
experimental finding is available in literature for comparison.
7) Thermal conductivity is computed for aluminum as well as for its three alloys.
No experimental or theoretical data are available for comparison. However,
presently computed results will serve as a useful set of data for further
research in this field. In case of Al-Cu alloy, a kink is observed at 0.2 % Cu
concentration. This feature is not observed in other Al based binary alloys.
Electrical resistivity computed at such composition does not show any such
features. Thus, further study of thermal conductivity of Al-Cu alloy will be
interesting.
8) Present approach of computing electrical transport properties of liquid metals
and their alloys does not include any kind of fitting with the experimental
quantities. The parameter of the potential are obtained using theoretical
methods and are not fitted. Only the experimental values of density are
adopted. Thus, present approach is found capable to explain electrical
transport properties of liquids, without any fitting procedure.
9) The present approach may be extended to study transport properties of other
binary and ternary systems of transition metals with aluminum and other
metals.
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