STUDY OF CERTAIN TECHNOLOGICALLY IMPORTANT...

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__________________________________________________________


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)

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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__________________________________________________________


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)

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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)

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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)

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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__________________________________________________________


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

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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],

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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].

Chapter-8__________________________________________________________


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

Chapter-8__________________________________________________________


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.

Chapter-8__________________________________________________________


260

Figure 8.1. Structure factor of aluminum at 943 K temperature.


Chapter-8__________________________________________________________


261


Figure 8.4. Electrical resistivity of aluminum as a function of temperature.

Chapter-8__________________________________________________________


262

Figure 8.5. Thermoelectric power of aluminum at three different temperatures.

Figure 8.6. Thermal conductivity of aluminum at various temperatures.

Chapter-8__________________________________________________________


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.

Chapter-8__________________________________________________________


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

Chapter-8__________________________________________________________


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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267

Figure 8.12. Electrical resistivity of Al-Li alloys.

Figure 8.13. Thermoelectric power of Al-Li alloys.

Chapter-8__________________________________________________________


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

Chapter-8__________________________________________________________


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

Chapter-8__________________________________________________________


270

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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271

Figure 8.16. Electrical resistivity of Al-Ni alloys.

Figure 8.17. Thermoelectric power of Al-Ni alloys.

Chapter-8__________________________________________________________


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

Chapter-8__________________________________________________________


273

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.

Chapter-8__________________________________________________________


274

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.

Chapter-8__________________________________________________________


275

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