I ndian Journal of Pure & Applied Physics Vol. 3R. Septemher 2000. pp. 657-663

Empirical formulation of permittivity build-up of bound rutile samples �II

Ashutosh Prasada & Shubhtosh Bagehi"

aDepartment or Physics. TN B Col lege, Ti l kamanjhi Bhagalpur University. Bhagalpur X 1 2 007 hDepartment of Physics. MlII'arka Col lege. Su ltanganj, TM Bhagalpur Uni versity. Bhagalpur

Received 10 January 2000: revised 2 1 July 2000: accepted

.Using the experimental data for permittivity bui ld-up of resin. naphthalene and paraffin wax bound ruti le samples

ohtallled In the X-hand 01 microwave frequency ( at 9.6 GHz). two types of empirical relation \Tovernin!! the variation or permittivity with fractional weight of the hinders have been obtained. Parameters involved in �e fonm� lation have hecn evaluated using non-l inear regression method and correlation of parameters with the nature of the binders have heen sought. The present paper deals with the second type of formulation only.

C

1 Introduction

The present project is the second attempt at theoretical extension of the experimental and theoretical works of Prasad and Sharmal on the pressure-dependent as wel l as bi nder-dependent permitti vity bui ld-up of rut i le samples in the X-band of microwave frequency (at 9.6 GHz) . For finding the effective permittivity of the distributed inclusion in the ternary mixture of ruti le, binder and air, they have used Bottcher' s formula in the normalized form" for the air-binder binary mixture using air as the host and the same formula was used treating air bindeT composite as the host and the rut i le particles as the inclusion . The fractional weight of the binder ( x ) and packing fraction (f) for the two cases fol low from the composition of the sample. In the second attempt, Rother-Litchtenecker' s formula or the Logarithmic Law of Mixing was tried in both the cases and simi lar curves were obtained. In both types of curves for the permittivity of the distributed inclusion material under the influence of the binders is seen to pass through a peak val ue at a critical percentage of mixing of the binder, which i s different for different binders. Also, three types of curves obtained using three different binders show different sharpness .

In the present investigation a set of I I formu lae have heen tried . After evaluati ng the relat ive permittivity ( E, ) of the air-bi nder binary mixture taki ng air as the host (£11 = I ) and using it as £H for

the second computation, choosing any of the I I formul ae £1 or kp were computed . Here £H, £1 and £,

stand for the relati ve permittivity of the host, of the distributed inclusion 'material and of the mixture respectively, in each case.

Thus, taking data for all the three binders, 3 x ( I I x I I ) = 363 values of £1 or kp were obtained . Contrary to the expectations of gett ing coherent or identical values for each set, they gave quite different values. The value of relat ive permitti vity of ruti le in the powder form i s 2.366 and that in compact form is "" I 00. Taking these values into consideration, so many sets gave out of order and even -ive values of permittivity. These out of order and -ive values have been excl uded from further process of evaluation of parameters. Two types of formulation of empirical relation have heen tried and evaluation of parameters have been achieved using statistical non-l inear regression method . The present paper deals with the second type of formulation . The val ues of the parameters involved in the formulation show a defin ite trend, which tel l , a t least to a first degree of approximat ion, about the nature of the binders used.

The practical uti l ity of the present investigation is in the fabrication of a bound ruti le sample having a particular value of permittivity suited to a specific use in the range of X-band of microwave frequency .

6SX I N D I A N J PURE & APPL PHYS. VOL 38. SEPTEMBER 2000

2 Formulae for Permittivity of the Mixtures used

in the Present Investigation

Following are the formulae used :

(0) Taylor' s Formula for random angular distribution of discs'

( I ) Taylor' s Formula for random angular distribution of needles'

(2 ) Bottcher's Formula in normalized form2•

0 ) Lewin' s Formula4

(4) Sillar' s Formula�

where D = depolarizing factor = 0.2 in the present (, case

(5) Formula for Effective Medium Theory (EMT)7

(6) Rother-Litchtenecker' s Formula Logarithmic Law of MixingX

(7 ) Landau-Lifshitz ' s formulax

( 8 ) Beer's Formulax

(9) Lorentz-Lorentz Forlllulax

( 10) Weiner' s Forlllula')

or the

where II =Ct.orm-zahl or form no. = 5 in the present 1 0- 1 1 case

In all subsequent computations ./h and ./; were used in place of f in binary and ternary mixtures, respectively, where the h and f suffixes correspond to binder and titanium dioxide (rutile) respectively. Si milar to the l SI formulation, the different sets have been named as:

o.() . . . . . . . .. . . ... 0. I ( ) I . ( !. . . . . . . . . . . . . I . I ( )

1 0 .0 . . . . . . . . . 1 0. 10

respecti vely.

Also, the subscripts R, N and P correspond to resin, naphthalene and paraffin-wax, similar to those used in the I " formulation.

3 Formulation of an Empirical Relation

Viewing the similarly in the nature of the ontained theoretical curves and the curve for binding energy of deuteron inside and outside the potential well having the equations of the form VI = A I sin kr and V2 = A l e-kr respectively, making the whole curve as continuous (where r = distance between the interacting particles), an equation of the form :

A ( . )IJ -II.. C Er = S i ll ru x e . + was chosen, taking into consideration the different sharpness of the curves and incorporating the permittivity of the rutile powder to have a value of C (z 2.366).

4 Designation of the Parameters

In consonance with the I sl formulation u and � may be designated as the binding parameter and di lution parameter, respectively. In the present investigation the term (sin ]txt', shows the binding capability of the binder i.e., the capability of the binder to increase the EI or kp depending on the values of u. Since 0 < x < I, (sin 1tx)IJ is a fraction and (fractionrivc , i.e. negative power of a fraction is a positive quantity and hence large negative value of u gives large value of ( sin ru)IJ, whereas a large positive value of u gives a small positive quantity and larger is the value of u, smaller is the value of ( sin ru)IJ. Similarly, a small positive value of u gives large positive value of (sin ru)I' . Further, e-'II corresponds to dilution effect of the binder. i.e. the capability of the binder to decrease the value of £1 or klh which depends on the value of � : larger the positive value of � , stronger is the dilution effect (and lower the value of EI or kp) and smal ler the positive value of � , weaker is the dilution effect, whereas the negative value of � corresponds to negative dilution effect, i.e. adding to the adhesion property of the binder. Also, larger the negative value of �, stronger is the negative dilution (adhesion) effect. A large value of ( sin ru)" and a

-Ill-very small value of e . corresponds to the sharpness of the x versus EI or kp curve, whereas a large value of (sin ru)IJ and also of e-'Ix corresponds to the flatness of the curve.

S Curve -Fitting and Evaluation of Parameters l 2

Taking y in place of E I or kp in subsequent discussions, the empirical formula obtained is :

y = A ( sin ru)" x e-ill + 2.366

= y -2.366 = A ( sin ru)" x e-(Ix

= In (y - 2.366) = In A + U In (s in rrx)-�x Taking In (y - 2.366) = Y -� = b and u = c

PRASAD & B AGEH I : EMPI R ICAL FOR M U LATION OF PER M ITTIV ITY 659

the above equation yields:

Y = a + hx + C I n (s in nx) •

A , Takmg D = I:( Y,- Y jt = I:( Yj- a-hxi - C I n sin 1tXi

The value of D is the least when :

dD dD dD - = 0' - = 0' - = 0 do ' dh . , dc

. . . ( I )

. . . (2)

The above three conditions gIven by Eq. (2) yield three normal equations of the form:

an + b Lxj + cI: In (sin 1tXj) = I:Yj

aI: Xj + hI: x/ + ('"xL xjl n (sinn.x,;) = Lxj Yj and L I n (s in 1tXj) + h L Xj I n s in 1tXj + Cx LI I n (s in 1tXj)f = L )Ji l n (s in 1tXj)

. . . (3a)

. . . (3b)

. . . (3c)

Compari ng the above three equations with three normal equations of the form:

tI lX + b ly + e l l = (/1

with the condition that

(/ 1 hi CI

,1 = m h2 c2 (n hJ c1

,1, = d 2 /n c2

d � /)". (".1

(l I d l C I

,1� = ([ 2 d '2 C 2

m d � C1

. . . (4)

;t o

and

ll t hl £i , ll2 bU/ 2 m bl dJ

where from applying Cramer' s ru le y ields :

x y z = = =

�2

Using the data of Lxj, n , I.x?, I: I n (sin 1tXj), I: Xj I n ( s in 1tXj). L Xj Yj, L Yj In ( sin 1tXj) and L 1 In (s in 1tXj)]2 from each set of acceptable; computed values out of 0.0, 0. 1 . . . . 0.1 0; 1 .0, I . I . . . . I . 10; . . . . 10:0. 1 0. 1 . . . . . . 1 0. 1 0 tables (Table I ) for each of the binders i .e . , out of 363 sets. The evaluated £1 or /.:." and,. also the parameters have been tabulated as below. �. '.

6 Results and Discussion

On the li nes of Prasad and Sharma' and using the First Empirical Formulation' i t i s expected

�,

barring a few exceptions, that naphthalene should be treated as the better of the two binders . Secondly .

'. paraffin wax should be treated as a better binder as compared to res in ; from the point of view that altho'ugh the maximum permitti vity ach ieved by a paraffin wax-bound sample is comparatively lower. x versus y curve is sharper at and about the critical ity of fractional weight of the bi nder (x) as compared to the curve of resin-bound samples.

On the same l i nes of the first formu lation, the second formu lation too, yielded incoherent values of the coeffic ients and parameters . Out of the I I formulae viz, D . . . . . I O only the formu lae n. 1 , 2, 6 and, to some extent, 9 and J O gave acceptable values of y and hence of the parameters. For the sake of brevity. let us cal l the ( s i n 7tt)" factor as the Binding Factor (B .F.) and exp(-�x) as the Di lut ion Factor (D.F.) henceforth and observe the tables giving acceptable value of y. A . u and � for seeking any correlation of the parameters with the nature of the bi nders, at least to a first degree of approximation, as obtained in the first formulation.

660 I N DIAN J PURE & APPL PHYS. VOL 38. SEPTEMBER 2000

Tablc I (T,,-Til,) -- Paramctcrs obtained using Model " only

( R= res in : N= naphthalcne: P= paraftln wax: A.u,/3 are the parameters: U= b i nding parameter: /3 = d i lution parameter)

Tahle No Tn

(J.O

n. I

0.2

0.6

n.7

O.X

(J .9

O. I ( )

40.5657

1 7 . 1 845

3R .7999

8 1 .6952

1 67.524

3 1 1 .79

NA

N A

Tahle N o T,

I .0

1 . 1

1 .2 1 .6

1 .7

I . X

1 .9

I . I ()

5 .39R45

8.8 1 8 1

1 5 .6 1 24

25.837

40. 1 324

58.9055

NA

NA

Tahle No T,

2.0

2 . 1

2 . 2 2 .6

2 .7

2.9

2 . I ( )

84. 1 772

1 1 8 .7()4

1 64.X57

224.83 I

30 1 .767

NA

NA

Tahle No T,

3.0 :\ . 1

3.2

3. 1 0

5 I 8.79X

676.273

X76.5()9

I 1 24.5')

1 427.64

1 7')O.7X

NA

N A

Valucs of A Values of v Values of /3

vp

-6R7367

1 .88807

4.08796

8. 1 7043

1 7 .4756

35.3749

25 . 1 794

1 .3640

(-)0.4672 1 9 (-)27 .5 1 5 t (-)3 .60392 7 .30 1 82c+l17 ( _)2.47452 ,'+117 5 ,875 1 8 ,."" •.

N A

NA

6 1 .24 1 4

1 1 6.235

208.729

347. 1 64

541).M7

836.5 JI ) NA

NA

1 2 1 7 .25

1 71)4.54

2578.81)

3587.68

4XW.7

NA

NA

8437.98

1 0836.9

1 354 9

1 65 1 3 . ')

1 9758.7

NA

N A

NA

1 . 05098

3.953 1 5

1 1 .976R

NA

27. 1 668

44.2225

3 .05'

N A

80.3276

1 4 1 .27X

236.346

379.85

57 1 .634

59.82 1 6

NA

866.796

1 282.53

NA

2609.37

N A

NA

NA

N A

NA

N A

NA

NA

( -)0.00 1 437R2 0.006893RR 1 .9056

0.0.266233 ( - )0.480493 N A

0.522647 0.543222 384787

O. 1 9 1 642 0,548 1 68 (J.994365

0.432685 0.72708 1 .0648 1

N A N A N A

N A N A ( - )0.66560 1

(-) 1 .200 1 7

( - ) 1 .28544

(-) 1 .3492

(-) 1 .37263

(-) 1 .37829

( - ) 1 .37782

N A

N A

( - ) 1 .36383

(-) 1 .34982

( -) 1 .339 1 5

( - ) 1 .33268

(-) 1 .3290 1

N A

N A

( - ) 1 .32276

(-) 1 .32499

(-) 1 .33482

(-) 1 .3506 I

(-) 1 .3742 1

( - ) 1 .40 1 47

N A

N A

( - ) I . 1 863 0. 1 9 1 775

(-)2.23753 3.507

( -)3. 1 0783 NA

( - )3 .288 1 8 0.325083

(- )3 .32694 0.739 1 8

(-)3.2 1 3 1 8

N A 1 .26 1 55

N A 1 . 1 7 1 83

( - )3 .66 1 92 1 .0 1 659

( -)4.03203 ( -)0.5348

(-)4.4 1 988 (NA)

(-)4.47623 1 .05979

( -)4.32 1 8 1 1 . 1 5737

N A N A

NA 1 .4072

( - )3,93 1 98

(-)3 .76377

(-)3 544 1 5

( -)3.2(X ) J I

(-)2.75036

(-)2. 1 9 I

NA

NA

N A

N A

N A

N A

N A

N A

N A

N A

9R2.703

1 364.34

2027.22

3284.76

501 6.69

NA

N A

26.2536

20.3 1 (J4

25.3968

32.3202

40.8264

50.64 I

N A

NA

6 1 .972

76.23')

94.246

1 1 5.797

14 I .399

N A

N A

204.808

245 .96 I

295.665

353.6

420.284

4'J5.5O'J

N A

N A

1 5 .3275

24.7797

33 .7384

59.3549

1 0 1 .79 I

NA

N A

I 45'()53

2 I 8.507

323.3') I

453.3(1)

6 1 X .X43

X27.065

N A

N A

I 03!).5

1 3 10 .4 1

1 644.1)2

2029.74

2474. 1 4

NA

N A

3495.23

40()X.4X

4527.07

5030.7X

5530.65

6W7.74

NA

N A

1 . 1 50')

N A

3 .68627

1 0 . I 8()(l

27.84

N A

6 1 .24 I 3

76.22 1 4

1 .5243

NA

1 ()4.W3

1 48.529

207.8 1 4

289.238

368.8 1 5

3 .924

( NA )

469.26X

59X. ( )')

N A

952.407

N A

N A

NA

NA

N A

NA

NA

NA

COl1ld . .

PRAS A D & BAGEHI: EMPIRICAL FORMULATION OF PERM ITTIVITY 66 1

Tahle I (Tn-Tin) -- Parameters ohtained using Model " only . . . (Contd)

( R= rcsin: N= naphthalene: P= paraffin wax; A.u,� are the parameters; U= hinding parameter: � = di lution parametcr)

A�

Tahle No T.

4.0 222H.47

4. 1 275!U6

4.2 3379.04

4.6 4093 . 1 7

4.7 4907.2

4.X 5H24.9

4.1) N A

4. 1 0 N A

Tahle N o T,

5.0 6H75.7f> I

5 . 1 X077Xl

5 . 2

5.6

5 . 7

5.1)

5 . H)

94 1 1 .36

I 087X

1 24X7.5

NA

NA

Tahl'c No T"

6.0

6. 1

6.2

6.6

6.7

6.9

6. 1 0

1 62 1 0. X

1 R433.6

20880.4

2356 1 .5

26496.7

NA

N A

Tahlc No T,

7.0 332H9.5

7. 1 37436.1)

7.2 42 1 0 1 .1)

7.6 4730X.5

7.7 53 1 06.4

7.9 NA

7. 1 0 NA

V a lucs or A Values orv Value� of �

27240.1)

30724.3

33454.5

35669. 1

37.7285

31)X55.7

NA

NA

42339.7

43X20.0

442X2.1)

44370.3

4450 1 .4

NA

NA

4571)X.4

46 1 4 1 .3

45X47.2

453X2.3

( -) 1 1).322X

N A

NA

( -)70.54 1 I) ( - ) 1 30.5(1)

( - )220.977

44630.X

(-)503 .467

NA

NA

35 1 H.51)

N A

N A

4648.55

51)54.9 1

7427.95

N A

9099.54

I OH96.9

N A

N A

1 2 1 27.7

1 2753

NA

1 35 1 X,7

1 407X

1 4.7399

N A

1 3936.5

1 3344.2

NA

1 2429.2

N A

N A

NA

NA

N A

N A

NA

(-) 1 .43593

(-) 1 .47874

( - ) 1 .529H6

(-) 1 .5835

(-) 1 .64506

(-) I . 70945

N A

N A

( - ) 1 .78074

(-) 1 .85565

(-) 1 .93 1 37

(-)2 JX)24 7

(-)2 .07492

NA

N A

( - )2 .2 1 936

(- )2.292

( - )2.36 1 X3

(- )2.42509

(-)2.48834

N A

NA

( -)2 .6 t 1 76

(-)2.6702

(-)2.7232 1

(-)2. 76785

( -)2.80956

N A

N A

(-) 1 .73755

( - ) 1 .2X88 1

0.9 1 4 1 1 H

f-)0.627044

(-)0.36575 1

(-)0. 1 0 1 585

NA

N A

0. 1 1 4379

0.2 1 5 1 33

V,.

1 . 1 53 1 3

N A

N A

0.906567

0.7583 5 1

0.78 1 6 1 5

N A

0.73389 1

0.5035 1 6

N A

0.207434 N A

0. 1 2 1 249 0.097 1 X97

0.()4 1 8489 (-)0.28 1 297

NA N A

N A (-)0.5 1 1 76X

( -)(>.033487 1 (-)0.663776

(- )0. 1 2 1 964 3 .57 1 4

( -)0.26 1 62 NA

( -)0.43672 (-)0.957643

0.892975 (-)1 .24853

N A NA

NA

( -)O.785H 1 6

1-)3.82724

()5.7H I 35

H.I , I 6 1 2 1

(�)(}.2 1 053

NA

N A '

(-) 1 .374 1 4

N A

N A

N A

N A

NA

N A

NA

13,{

578.975

672.606

776.244

8X8.67 I

J 009.X6

1 1 39.22

NA

NA

1 276.05

1 420.95

1 573.4 1

1 732.53

I 89x'1) I

N A

N A

2252.53

2443.26

2645.53

2X59J)4

3084.7X

NA

NA

357 1 .X

3X45.38

4 1 47.55

447 1 . 1

4X35.41)

N A

NA

64X 1 36

6688.71)

67 1 2.36

6636.45

6543.7

6473.57

NA

NA

M I I .XX

6 1 66.X

5 H 1 9.1)7

54(1).1) 1 1

5 1 65.611

N A

NA

4729.9X

4475 .1)5

4 1 92.65

31)25.26

(-)40.30X7

N A

N A

3350.27

( -) 1 2 I . H7 3

(-) 1 64.6 1 (,

2825.82

(- )270. 963

N A

N A

1 1 32.21)

NA

NA

1 3 1 0.27

1 492.76

1 675.9

NA

I X66.75

20 1 7.21)

NA

NA

20 1 6. 1 6

1 1)23.6 1

NA

1 7 1 9. 1 5

1 640.

O.6X/)5

NA

1 483.4 1

1 306.X2

NA

I 04X.67

N A

NA

NA

NA

NA

NA

NA

Contd . . .

662 I NDIAN J PURE & ApPL PHYS: VOL 38. SEPTEMBER 2000

Tahle I (TI)-T",) -- Parameters obtained using Model " only . . . (Contti ) ( R= resin : N= naphthalene: P= paraftln wax: A.u,� are the parameters: U= binding parameter; � = di lution parameter)

Values of A Values of v Values of �

-----------------------------------------.- ---------------- - -- -- --- ----------------------- ------------------------------------AR AN AI' VR

Tahlc No T"

9.0 44.7626 6.8624 9.3 1 06 0.6233

1). 1 29.96 5 .687X 1 .20X4 0.455

1).2 45.43 1 1 NA NA 0.7877

9.6 NA 40.54 1 X 1 . 1 200 NA 9.9 NA NA NA NA 9. 1 ( ) NA NA 2.2333 NA Table No Til ,

1 0.0 44.6 1 57 6.555X 8.4203 0.6232

1 0. 1 29.6877 4.949 2X.R4 0.4309

1 0.2 2X.7465 NA NA 0.7886

I D.6 NA 0.3022 42.6X 1 6 NA I D.9 NA NA NA NA I D. IO NA NA 43 .0045 NA

Thus, barring a few exceptions, it transpires from the tables that (B .F')N > (B .F.)p > (B .F')R and also that (D.F ')N > (D .F.) , , > (D.F. k

In some cases the negative val ues of � have been obtai ned. This negat ive value of � can be interpreted in the sense that instead of producing the d i lution effect , it adds to the binding action and it is only due to the nature of the Sinusoidal function (sin 1tX ) for values of .x between Y2 and I that the value of permitti vity decreases causing the curve to fal l fl at ly .

Thus. testing a l l the data of parameters on the criterion for binding or di l ution effects the strongest binding action of naphthalene is wel l established on the basis of both types of formu lations, but more sharpness of the permittivity versus weight fraction curve for naphthalene. expected on the physical grounds and corroborated ana lyticatl y by the first formulation, cou ldn ' t be achieved . The anomaly may be either due to the l imi tations of the appl icabi l ity of the formu lae used or due to the lack of coherence in both types of formu lations. At this juncture, therefore. the project needs either replacement of the formu lae used or some

VN VI' �K �N 13"

1.246 1 4.4522 1 .9763 ( -)0.5535 0.236 1

1 .327X 4.9697 1 .3 1 ( - )0.9657 (-)4.X455

NA NA 5.5383 NA NA 1 .7665 (-)5 .5 1 7 1 NA (-)0.3823 (-)4.293<)

NA NA NA NA NA NA (-)6.8 1 78 NA NA ( - )5 .5767

1 .2266 4.4723 1 .985 1 (-)0.606() O. I 1 55

1 .2675 4.8788 1 .3264 ( - ) 1 . 1 4 1 0 X9.65

NA NA 5. 1006 NA NA (-)5.3677 6.5326 NA (-)X 7402 1 . 1 70X

NA NA NA NA NA NA 8. 1 33 1 NA NA 1 . 1 633

modification in the empi rical relation, wh ich may form the basis of subsequent works .

7 Practical Utitity of the Investigation

Using the assigned values of A , v, � & C for a

bound ruti le sample with a particu lar binder, the value of the fractional weight of the binder (x) to be added to the host (ruti le) to get a particu lar value of permittivity is ascertained on the bas is of the empirical re lation governing the variatIon of permitt ivity with weight fraction of the binder. Now if a patticu lar binder is properly mixed with the host in the estimated proport ion, the bound sample with the required value of bulk permittivity is expected to have been ach ieved if the empirical relation is to hold good over the whole range of x. In other words, tai lori ng of permittivity is expected to be achieved using the empirical formu lation, which wou ld help in the fabrication of bound sample with requis ite value of permittivity for a particu lar use in the range of x-band of Microwave Frequency such as in microwave phase shifters, 'fi lters, MASERS, etc .

Acknowledgement

The authors are highly thankfu l to Prof K L

Mishra (Retd), Prof. Department of Statistics,

PRASAD & BAGEHI : EMPIRICAL FORM ULATION OF PERMITTIV ITY

Ti lkamanjh i Bhagalpur University, Bhagalpur, for his valuable guidance and help in computational works. Thanks are also due to Dr B K Das, .Reader, Department of Statistics, T M Bhagalpur University, and Prof S C Pathak, Reader, Department of Physics, TNB College, Bhagalpur, for their co-operation in the same computational works.

.

References

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