ON THE ELECTRIC CONDUCTIVITY OF A SUSPENSION OF ...gymarkiv.sdu.dk/MFM/kdvs/mfm 10-19/mfm-17-1.pdf · The increase of conductivity is constant, however, and has no influenc e on the

Det Kgl. Danske Videnskabernes Selskab.

Mathematisk-fysiske Meddelelser . XVII, 1 .

ON THE ELECTRI CCONDUCTIVITY OF A SUSPENSIONOF HOMOGENEOUS ELLIPSOIDS O F

REVOLUTION

WITH SPECIAL REFERENCE TO A N

ORIENTATION EFFEC T

B Y

JØRGEN E. THYGESEN

KØBENHAVN

EJNAR MUNKSGAARD

1939

Printed, ih- Denm a rk .Binncö Liüios I3öätrÿkkérï A/S .

INTROllUC,TIO N

The electrical conductivity of a suspension is dependen t

partly upon the direct conduction through the

suspending medium and the suspended particles, partly

upon the cataphoretic movement of the particles . The latter

phenomenon is caused by the setting-up of an electri c

double-layer between the suspended particles and th e

suspending medium . On establishment of an electric fiel d

the particles move and transport charges of electricity ; the

liquid layer, taking the opposite charge, also moves . The

corresponding contribution to the conductivity of th e

system depends upon the effective charges, the number an d

frictional resistance of the solid particles, as well as upo n

the charges of the moving liquid layer .

The present paper does not propose to deal with parti-

cular aspects of this cataphoretic phenomenon but ex-

clusively with the question as to how the aforementione d

direct conduction can be influenced by the orientation o f

particles of non-spherical form suspended in the medium .

The suspended particles are assumed to be solid con-

ductive ellipsoids, and the problem with which we ar e

here concerned is how the conductivity of the suspensio n

depends upon the orientation of the ellipsoid particles .

Let the liquid medium have a dielectric constant e o and

i

4

Nr . I . JØRGEN E . THYGESEN :

an electric conductivity ao, and let the suspended inelastic

("starr-elastische") rotation-ellipsoidal particles have a

dielectric constant e and a conductivity a .

Provided that the conductivity of the ellipsoids is differen t

from that of the suspending medium, the conductivity o f

the suspension will depend upon the orientation of th e

particles, assuming one extreme value (maximum) if al l

the ellipsoids are adjusted with their longitudinal axe s

parallel to the direction of the current, and the other extrem e

value (minimum) if they are orientated altogether irre-

gularly .

When e e or ao $ a or both, ponderomotive force s

will act on the ellipsoids, which will thereby become subject

to a moment of rotation and tend to attain a certain orient-

ation .

The problem will then be to follow more closely thi s

process of orientation during the application of an electri c

field, and the resulting changes in the conductivity .

We have therefore to consider the electric rotation

moment, the opposing moment of friction arising from th e

rotation of ellipsoids, and the influence of the Brownia n

movements upon an incipient orientation, and then cal-

culate the conductivity of the suspension for a given orient-

ation of ellipsoids .

I . The Electric Rotation Moment .

Suppose that the electrodes of the conductivity vesse l

are two vertical plates, only slightly apart in proportion t o

their extent .

If there are no ellipsoids between the electrodes, th e

field is homogeneous with horizontal lines of force .

On the Electric Conductivity of a Suspension .

If an ellipsoid is placed between the electrodes, the

field will be deformed locally, and the ellipsoid will be

affected by a rotation moment D.

We place a rectangular right-coordinate system in th e

ellipsoid, with the initial point in its centre and the x-axi s

coincident with the axis of symmetry (the short half-axi s

for planetary (oblate) ellipsoids of revolution, the lon g

half-axis for prolate ellipsoids) .

Since we are considering an ellipsoid of revolution, we ma y

choose the z-axis at right angles to the plane formed b y

the x-axis and the outer given field-vector, strength of field

E, so that E, = O . This gives the y-axis .

The acute angle between the symmetrical axis (x-axis )

and the field vector is called P . .

The moment of revolution exerted by the field upon th e

ellipsoid

was

calculated

by

terms we get :

R . FÜRTH 1 2 .

Retaining his

b9

aD

? E o E

1

2 s

(

in= (a2 -1)2)

, where

a° + ,~4C~° +ru

(a2 -x2)~

«x2 a

dx , =

and Xo and Yo are constants .

The moment of revolution is positive and tends t o

increase when a < b, and negative, tending to decrease

t, when a> b .

For oblate ellipsoids of revolution we have :

' R . FtiRTH : Zeit. f. Physik . 22 : 98, 1924.

2 R . FÜRTH : Zeit . f. Physik . 44 : 256, 1927 .

(

) U,c x

2i 1- ~2 - a 2

Nr . 1 . JØRGEN E . THYGESEN :

a = x >1 ; ~ = x2 2

13

2

2 ~/x 2 -1 arc tg Ux2 -1 + 3 x2

( _1)2

2 j/x 2 - l Vx2 -1

v'x2

Yo- 2x 2 _ 12+

n

arc tg

1 1x -1

x

2 vx 2-1 j/x 2 - 1

)./x2 -

For prolate ellipsoids of revolution :

b

-2a3 1 2x2

x 2

1 -v1-x2

= <1 ;,t =

22

}

{ loga

(1-x) 3

3

2v1 -x2

+v1-x 24 x2 1X° log -f- -x2

1-x2-1+

2~/1 -x2

1 -l/1-x2 J2x

2

1 log }

-x2_

1-x2 Lx2

-x2

'1-J/1-x2 -

We see that the expression for D is not dependent upo nthe ellipsoidal dielectric constant s, owing to the fact tha tthe electric field inside the ellipsoid is homogeneous .

The moment of revolution is proportional to the squar eof the field strength, so that D is independent of the directio nof the field .

As stated by FIRTH, the expression for D applies toquasistationary conditions only (cf . his discussion. withBuscn) .

As the moment of revolution results essentially fro maccumulation of electricity on the surface of the ellipsoid ,owing to the difference in the conductivity of the ellipsoidand the suspension medium, we have D = 0 for ao = cr .

On account of D's proportionality to sin 2 '0, the momen t

of revolution will have a maximum for 2= 2

or = 45° .

B . Ftwra : Phys . Zeit . 25 : 679, 1924 .

Xo = x2x 1 - ~1 - -- ~------ -1-

1

arc tg 1 -~

Yo =

On the Electric Conductivity of -a Suspension .

With 0 < P < 2 , D has a definite sign, positive for oblate ;

negative for prolate ellipsoids .

If the symmetrical axis is parallel to the direction o f

the lines of force, or is at right angles to these lines, we

have D = O .

Yo, a, b and 2. depend only on the geometrical form

of the ellipsoid ; with a = b, we have D = 0 .

So, independently of the magnitude of its electric con-

stants (r 6 0 , a co) the ellipsoid will tend to adjust

itself with its longitudinal axis parallel to the field .

In the present calculations no consideration has been

given to the phenomenon of cataphoresis .

The conductivity of a suspension is increased on accoun t

of the cataphoresis of the particles (SMoLucnowsxz l), as

the slip in the Helmholtz double layer causes a transfer

of electric charges .

For a suspension of spherical particles the cataphoretic

conductivity is

2 = 4 stv?l r (r -I- d) u 2

N 8

where A is the specific conductivity, v the number of part-

icles per c . c ., 77 the coefficient of inner friction, ô the thick-

ness of the double layer, r the radius of the particles, u

the cataphoretic mobility (cm/sec ./volt/cm), N the Avogadro-

Loschmidt number .

Whether the cataphoretic conductivity should be allowe d

for in measuring the conductivity of suspensions, has bee n

1 M. SiroLucuowsin : Anzeig . d . Akad. d . Wiss . Krakau . 1903, p . 185 .

Phys . Zeit . 6 : 529, 1905 .

8

Nr . 1 . JøncrN E . THYC.ESEN :

discussed by FREUNDLICH' and DE HEVESY 2 . The increase

of conductivity is constant, however, and has no influenc e

on the orientation effect, as the symmetrical distributio n

of the charges in the double layer cannot produce any

rotation of the ellipsoid .

There is no orientation of the particles in the directio n

of least hydrodynamic resistance, brought about by cata-

phoresis, electro-endosmotic flow or translational Brownian

movements, as in this sense there is no minimal principl e

of liquid resistance (R . GANS 3 and C . F. MARSHALL 4) .

In narrow vessels, however, the different velocities of

the liquid layers will be able to cause an orientation (cf .

orientation of particles by "streaming double refraction") .

The velocity of cataphoretic movement is low ; according

to HELMHOLTZ-SMOLUCHOWSKI ' S theory ' and FREUNDLICH and

ABRAMSON ' S experiments' it is independent of the form

of the particles, that is, independent of the orientation o f

the ellipsoids .

H. The Hydrodynamic Problem .

The moment of friction for an ellipsoid rotating about

a main axis in a viscous medium was first calculated b y

EDWARDES 7 .

i H . FREUNDLICH : Kapillarchemie . Akadem. Verlagsg . vol . I, p . 351 ,

1930 . vol . II,' p . 63, 1932 .2 G . DR HEVESY : Roll . Zeitschr . 21 : 136, 1917 .s R . GANS : Sitz . br . d . math .-phys . Klasse d . K . B . Akadem. d . Wiss .

zu lunchen, 1911, p . 191 .

' G. E . MARSHALL : Transact . of the Faraday Soc . 2G : 173, 1930 .

a L . GRAETZ : Handbuch der Elektrizität und d . Magnetismus . Ambr .

Barth, Leipzig . vol . II, p . 381, 1912-2.1 .a

H . FREUNDLICH and H . A . ABRAMSON : Zeit . f. phys. Chemie. 133 :

51, 1928 .

EDw . anFS : Quart . Journ . Math . 26 : 70, 1892 .

On the Electrie Conductivity of a Suspension .

9

The coefficient of resistance for rotation of an ellipsoi d

of revolution about an axis at right angles to the symmetrica l

axis is

16nn

a 2 -I-- b 23

a 2 A-I-b 2 B

where a og b are the lengths of the semi-axes, 17 the coeffi-

cient of the internal friction of the medium, and A and B

are constants expressed b y

dsA

X„=

(a 2 -F s) (b 2 + s) 1/a2 -{-- s

2 ab 2

~

ds Yo 2

B --_

0 (b 2 -}- s)2 ya 2 + s 2 ab 2

and Yo are the integrals given in Section I .

EDWARD ES ' concluding formula is incorrect, as th e

numerical factor 32/ 5 must be replaced by 16/3 when the coeffi-

cient of the last term of the preceding equation is correcte d

to 4 .

III . Calculation of a Distributive Function for Orientatio nof the Ellipsoids in Relation to Time .

If a polar coordinate system (0, cp) is introduced int o

the sphere of unit with the direction of the field for pola r

axis, and if from the initial point we draw for each ellipsoi d

a radius vector parallel to the momentary direction of th e

symmetrical axis, the density of intersections on the surfac e

of the sphere will, for symmetrical reasons, statistically,

depend only on the polar angle B, the angle between the

field vector and the symmetrical axis, and on the time, bu t

not on q) .

ZTo

10 Nr. 1 . JOBGEN E . TIIYGESEN :

Giving theterm fdQ to the relative number of ellipsoid swith the symmetrical axis within a solid angle dS2 aroundthe direction (0, y), the distributive function f (t, '0) will

be altered, partly due to the electric field which causes th eorientation of the ellipsoids, partly due to the Brownianrotation movement .

This alteration of f can be calculated by a metho danalogous to that employed by DEBYE ' for calculation o fthe orientation of polar molecules in electric fields .

Instead of the simple MAXWELL-BOLTZMANN distributio nfunction, which applies to statistical equilibrium onl yP. DEBYE-on the basis of EINSTEIN ' S theory concerningthe Brownian movement-derived a partial differentialequation determining the distribution function for orient-ation of polar molecules .

Applied to the present case, this method will only b eroughly outlined . The number of ellipsoids, the directio nof whose symmetrical axes in the time interval 61 run smore into than out of cif?, is given by the equatio n

Stof (In = 41 +4 2

where 4 1 signifies the part arising from the orientatio neffect of the field, 4 2 the part due to the Brownian rotatio nmovements .

If O is the angle between the axes of dS2 and an adjacentsolid angle dQ', the following expression applies :

02 [cos 7%' f + a2 f ~4 2 = dsa 4 sin~ 80 a02 .

From this expression it is evident that the Brownian rotatio n

a P . DEBYE : Polare Molekeln . Hirzel, Leipzig . 1929, p . 88 .


1 1

movements have no influence if f is independent of '0 ,

i . e ., in an altogether irregular distribution . As soon as the

electric field induces an orientation (f dependent on 0) ,

the Brownian rotation movements influence the distribution .

The rotation moment D, acting on the individual ellipsoid

(mentioned in Section I) and causing it to rotate round a n

axis at right angles to the direction of the symmetrical axi s

and the field, tends to alter O .

Under the influence of a constant rotation moment, the

friction would make an ellipsoid rotate with a constan t

angular velocity, determined by the equatio n

D_e "

dt

where is the coefficient of resistance mentioned in Sec-

tion II .

If the rotation moment is positive, the angular velocity

is also positive, and vice versa .

Applying this equation to our case with a variabl e

rotation moment, we shall again assume that the acceleratio n

effect is negligible .

For 4i we then have :

4 t

(2zf~åisinz9) d

Introducing 4, and A2 and (JD = 2 n sin td'O in the

original equation we obtain

af = 1 a

r o2 af »at

i

1 4åt

~, .

For determination of the characteristic constantÔt

we

12 Nr . 1 . .TØRGEN E . THYGESliN :

make use of the fact that the MAXWELL-BOLTZMANN ex-

pression

f = c1 . e- rt 1'

where u is the potential energy, k the BOLTZMANN constan t

(gas constant per single molecule), and T the absolut e

temperature, in the special case of f = 0 (constant field )

must be a solution of the differential equation .

As in the stationary cas e

it can be shown that the MAXWELL-BOLTZMANN expressio n

is a solution when :

O J

k T

4ôt

~

Introduction of this expression gives the final differentia l

equation for determination of the distribution function

f (t, 6), which can then be found for an arbitrary perio d

of the regulating moment of revolution .

f

_8at

sine

sin 19, (kT

a

)

(1 )

Solution of the Differential Equation and Determinatio nof the Relaxation Time of the Particles .

We divide equation (1) by kT and transfer the term

with å to the left side . If then we express the complicate d

formula for the rotation moment (mentioned in Section I) as

D = Do sin 2i9


1 3

2and put zo = kT and a = kD°

where a may be a function of the time, the differentia lequation for our problem is as follows :

(1) z° atf

1

a[sin

0 sin a~1

sin 19 19 [sin 0 sin 2a fl .

We first consider the homogeneous equation resultin gfrom (1) by putting a = 0 :

af 1 a

at.(2)z° at

sin [stn 9a~9]

= 0 .

Putting in the usual way,

l = ( t)"Y (~)

where 99 (t) is a function of t alone and y 0) a functio nof 19 alone, we get :

1 a[sin

a~ l(3) °

- ~ _

siu ~

'19'a14 J

2 is a constant .(3) divides into two equations, of which we conside r

1 1

a

~ _ - ,P sin ~9 a .-d,[sin z9

o~~9 ~ .

This is LEGENDRE ' S equation, which allows only regula rsolutions for the proper value s

î = n (n + 1)

(with n = 0, 1, 2, )

with the appertaining proper functions ( ''Eigenfunktionen") ;

the normalized spherical harmonics of the first kind :

14

Nr . 1 . JØRGEN E, THYGESEN :

=/2r1-- 1

y~ n - ~ 2 P (x)

x = cos 7i .

After this, the second half of (3) gives :

2

r' (n+ 1) _

1 'c o_ - -

- as -g7

to --

To

n

n

I1 (l'-F 1) ~

The solution of this equation i s

/

t

Tn = C n e Tn .

Then the general solution of (2) is :

(n+i> t

yi 1L (cos 0) e T„

y„ (cos 0) e

To

By a suitable selection of the coefficients Cn it may b eadjusted to any function for the time t = 0, and we se ethat, no matter what solution we start with, in the courseof time we arrive at the equal distribution, all the termsexcept the first one being eliminated, and Po (x) = 1

f(t, 0) is given for the time t = 0, for exampl ef (0,0) - F(~) where F is a given function :

F ('O)=G

(co s (cos '0) .n= o

In order to solve (1) with this initial condition we mak ea calculation of perturbation, considering cc as a smal lvalle, and put :

(4) f = f o + fi

where f o is that solution of the homogeneous equation (2 )which for t= 0 is. equal to the assumed f for t . = O . This


15

means that f' = 0 at t = 0 for all 19' ; this makes f° a givenknown function .

As both a and fl are small, the products are disregarded,so that substitution of (4) in (1) gives :

(z) r0 a1 sin c~~ [sin z9' e t l = - sin z9 sin sin 2 O f° .J

~

]

(5) is a nön-homogeneous differential equation in whic hthe right side is a given function of f'° and V .

For any time we may expand fi(t, 9) in terms of theproper functions of the homogeneous equation :

(6)

f,t (t, ?9)

> Q7n (t) 1~Jn (cos ti') .

n °

Substitution of (6) in (5) gives for the left side :

t)

~

_

dt( vn (cos ~)-sin z~a~(sin~aa~

9'n( t)vn (co s ~) _

1

1

a r

61p n(cos 4%) 1

~n (t) sin ~ ôz9 Lsin D' a ~ I

-n (n -I-1) 1P„ (cos ,9.)

and makes the whole equation :

[ro d~dt(t)

rz (n + 1)99 2 (t)]

yin (cos 19.) _

2 sin ?9,

[ sinz9sin2Dr] .

Introduction of x = cos z9', as a new variable instead of # ,finally gives :

dT ril ()

da°dt

+ n (n +1) 9'n (t)] y'n (x) = -}- adx

[x (1- x2) f0 ]

= o

N"'-Id 99 n'- (t)dt - Vu (cos #) -

n = oi°

16 Nr . 1 . .JØRGEN E . THYGESEN :

By multiplication of this equation with tPm (x) andintegration over x from - 1 to +1, employing the ortho-

gonal relation of spherical harmonics, we ge t

d Ønß (t)(7)

t ° dt-+ In (m + 1) pD, (t) = gm (t)

+where g ,n (t) - + a

'p ,,, (x) dx [x (1- x2) f° (t, x)l dxu--- i

and 1n = 0, 1, 2, 3, . . . o0

In order to solve (7), using again index n instead o fnt, we put :

_ i

n0:--1) t

~n (t) - e T„ gn (t) = e

T0 gn (t) .

By substitution we ge t

z° dg„ ( t )dt -

g„ (t) eTTin

(9) can be integrated directly, since yn, like tpn and fl ,will vanish when t = O . We then have

t

t'

gn (t) =

g n (t') e2 " dt'~ ' °

where t ' is a variable of integration, or according to (8 )

t

. t

t'

( 11 )

p! (t) = l eTn~ g

n (t')e'n dt' .z °

c 0

Substituting this expression in (6) we have f l as a functionof t and s or of t and &

The formulae derived so far, which are quite general ,

will now be applied to a particular simple initial distri-bution .

(8)

(9)

(10)

On the Electric .Conductivity of a .Suspension .

1 7

If f for the time t = 0 is independent of then f° = 1

for t = 0 ; and as f° = constant is a solution of the homo-

geneous equation, f° = 1 is valid for all t .For gn (t) we then have :

~- i

tin (t) = + a (t) tpn (x) (1 - 3 x') dx .. -- i

But we know that

1P2 (x) = (3 x z -1) and. Tp,

(x) .

Therefore

(12) g,, ( t) = - 2 j~-2

a (t) yin (x)'VZ(x) dx .

5

e -1

So, for this simple initial condition (equal distributio n

of direction) all g n (t) = 0, except Mt), which is equal t o

2-2 j -a (t> .

This means T,1-, (t) = 0 for all n } 2, an d

1

. t

i ,

(13) 4'2( t') = -~ e Tz~

a(t~)e'' dt ' ., °

If now we further consider the case that a constant field

is established at the time t = 0, a becomes constant in th e

time, and then we have : cp 2

6(t) = -2

!/

2a

\1 - e r'

-

t

a S ~

Ltt' = ti t (e '~-- 1 ) and

=6

ZO

, °

As 1'1 = 94 (0 9)2 (cos V.) and y, (cos

-1 )

Vidensk . selsk.Ajat h .-fys. Medd. XVn, L

2

18 Nr . 1 . JOBGEN E . THYGESEN :

we get :

(14)

f = f°+fl = 1- 6 (1-e 2 ') (3 cos' 19' 1) .

With t = oc formula (14) becomes the simple MAXWEL L

distribution, as in this case we get :

f = 1- ~jcos 2 ~ -,+-3I .

This means that, apart from values of higher orders o fmagnitude in a we have :

-4 (cos 2,9 . + 3)

-~ cos2~.f o..) e

CV const e

N const e

where

a = 2 D0 and D = D° sin 2 = ou the last expres-kT

äk ,sion by integration giving

u = 0 cos2 ~ = 4kT cos2O.

From (14) the relaxation time for the particles is see nto be :

=T0 er2

=

6

6kT '

If a is a function of the time a (t) (for example, whenthe field results from a sine tension), with the usual symbolsfor angular velocity and phase-displacement we get :

E = ED sin (co t + 6) and D = Do sin 215 sin e (cot + 6)[e i

+ (1') _ e-i(wt+d') 2D° sin' (w t+ S) = D° _2 i

i(2wt+2~f)0 2D Real part of {e

that is


1 9

a(t)

2Dosin 2 (wt-F- 6 ) _ Do-DoIi, e i(2wt+2d)

kT

1cT kT

Introducing this a(t) into the expression (13) found beforefor 991(t), and remembering that :

f~ =

(t) y 2 (cos 19)

we get by fairly simple calculations :

D

(

cos [2wt-}-2ô]+ 2wr2 sin[2wt+2S]- +f''=1 -

6-k9-7'(3 cos' '0- 1) 1- 1 } (2 wr,) 2

cos [2 8]+ 2cor2 sin [2 S] }1+(2 wr2)2

1- e

f = 1 - 0~7,(3 cos' ~- 1 cos[2wt--2b- -+ b ]+ (2 on-2) 2

f

_-e zx -

+ 1

(2 wr2)2cos [2 8 1 MI}v1

f - l -6~~,(3cos2~-1)G(t)

where b = are tg (- 2 co-r2) signifies a phase-displacement,and G(t) an abbreviation for the {} parenthesis .

We find that the distribution of direction varies withtwice the frequency of the field strength . This is easy t ounderstand, as the rotation moment (cf . Section. I) dependson the square of the field strength .

For co = 0 å = 2 we get G(t) = 2 1-e n ) and f

equal to expression (14) with

2 Doa const = k T

For (wr2 ) >> 1 we get2 k

20 Nr . 1 . JøßGCN E . TxYGEsr_-N

:/

t

f = 1-6DkT(3eosL#-1)(1 -e T~)l_(3cO s 2 _1) (1 _e T=)

that is, when the oscillation periods are small in proportionto the time of relaxation, the expression for the deviationfrom the equal distribution is exactly one half of the cor -responding expression for direct current .

IV . Calculation of the Conductivity of a Suspensio nfor a Given Time.

H. FRicI E 1 has calculated the conductivity of a suspen-sion of ellipsoids, assuming a random orientation of th eellipsoids .

As we are here interested in the orientating effect of th eelectric field on the conductivity of the suspension, thi smethod must be generalized .

We consider again (see Section I) a coordinate systems, y, z fixed in the ellipsoid, where the x-axis coincide swith the symmetrical axis and the z-axis is perpendicula rto the direction of the applied external electric field . Wefurther introduce the coordinate system x', y', z', havingthe same point of origin and z-axis as the x, y, z system an dthe x '-axis situated in the direction of the field .

The components of the applied field in the direction sof the x-, y- and z-axes are :

Ex = Ecosm

Ey = EsinØ E - = 0

as previously, signifying the acute angle between th esymmetrical axis (x-axis) and the direction of the field(x'-axis) .

For the componenLs of the field strengtet inside an' H . Fruear : Phys . Rev . 24 : 575, 1924 .

N1 = - -

1-I- N 4 Y

°

1-{ P 4

Yo

E-

0 - - - - - = 01---4 1Zo

1+kt4 Zo

E


2 1

ellipsoid calculated according to directions of the mai n

axes (x, ry, z) we have (cf. FüRTH 1 p . 102) :

E a

= -~ - Yz

ax

1

â y

whereEx

Ecos v

1+ /z4

Xo

2

dt

dt

X0 = 2 ab

Ylo(a2-I-t)T(t)

o = Zab°(b2-f-t)7,(f)

T (t) = Va2 + t (b 2 -1- t)

the x' component of the intensity of the field inside th e

ellipsoid Ex, can be calculated by means of the formula

A S = As cos (s, x) + Ay cos (s, y) + A z cos (s, z)

which gives

(1) Ex, = -a l cos 2i -fil sinP =Ecos 2 t~ + Esin 2 0

1 -} P-

P4

1Xo 1 -{- 41 Y°

The intensity of the field outside the ellipsoids will b e

called E° . In the immediate proximity of the ellipsoid E 0

differs from E, but at great distances from the ellipsoid s

the two values are identical .

R . FORTH : Zeit. f. Physik : 22 : 98, 1924 .

Y1 = -

and

22

Nr . LJØAGEN E .Ti-zYGESEN :

The volume between the electrodes is called 2, and the

total volume of the ellipsoids Q 1 .

Let F express the strength of the electric field at a n

arbitrary point of the suspending medium . For a definite

assumed distribution and orientation of the ellipsoids w e

now form the mean value taken over the space outside th e

ellipsoids, and call i t

(2)

F = Fo outside ellipsoids .

We know that it is a vector, at right angles to the electrodes ,

that is

(3)

Fx,=FandFy,=F2,=O .

Inside the ellipsoids there is a certain intensity of elec-

tric field F which, strictly speaking, varies from point t o

point within the ellipsoids .

Considering all the ellipsoids whose symmetrical axis

(x-axis) forms an angle between and 0+ db' with the

direction of the field (x'-axis), and forming the mean valu e

of F. taken over the volume of these ellipsoids, we get a

vector E, O), that is :

(4)

FO) = F,9, ,9+a .

It is natural to assume that this mean strength of the field

F 0) is the same as the constant field strength in a single

ellipsoid placed in a suitably chosen external constant fiel d

E, and whose symmetrical axis forms the angle with th e

direction of this field .

Now the problem is how we are to choose this field E .

The condition must be that the mean value of E° (see

above) taken over a volume (Q - Q 1) outside the ellipsoid

is equal to the F found above .


23

It is found That if the volume Q is large in proportio nto the volume of a single ellipsoid, the mean value of x°within this region will be approximately equal to E, thatis, that we have to put

( 5 )

E=Farad E ' = F(7i)

(F1 (O)'s components ex, etc . )

In order to get the connection between F and V (th epotential between the electrodes), we note that the mea nvalue of the true field strength taken over the entire volum e

VQ must be equal to ! , where l is the distance between

the plates, i . e . ,

(6)

(F.) „, , (Q -Q1 ) + (F1)x,outside ellipsoids

Vinside ellipsoids `

which is equal to

(6 ')

F (Q -D1) + D1 \ ( F1 (9)) x., N 09) d 79. = Se'V

.,y .= o

where N(?9) d'O means the relative number of ellipsoid swithin 2, whose symmetrical axes forms an angle betwee n

and +do with the x '-axis, and normalized so tha t

(~TL

1V(9')(0 = 1

By employing (1) and (5) we get, instead of (6 ' ) :

(7) (2-Q1)F+Q1F`7t-

cos2 z9

+_ sin2z:

N(,r9) dD . = VQ0 1+

4 1X0 1+141Yo

1

This relation replaces FRICKE ' S 1 equation 7, p . 580 .L H . FSicHE : Phys . Rev. 24 : 575, 1924 .

=n

24 Nr . 1 . Jf7ßGBN E . THYGESEN :

When p means the volume concentration =

weget by dividing (7) by Q :

~ cos' 0 sin2 ~9

_ V(8) F(1-n) I F~ ~ A -{ B~N(~)d V0

wher e

(8 ' )

A= 1-f- '1 ~ Xo B 4

When we know the mean value of the field strength, w ealso know the mean value of the density of the current u ,as it applies to every point tha t

u = a E

so that the mean value of u through a region of space wherea is constant is equal to a times the mean value of th efield strength .

So, for the mean value of u outside the ellipsoids, we get

0

0

_ o ,= no outside ellipsoids = Cr Fo outside ellipsoids = c F .

u° is perpendicular to the electrodes, and equal to a°F .The mean value of the x ' component of the current -

density inside the ellipsoids, the direction of whose sym -metrical axes is between 19 and 't9' + dO, is likewise :

E '(u i (é9)) x, = a (F (0))x, =cos' z9 + sin' i9

= a E ( A

B ~

_ 6 rC

cos'z9 + sin' i9l

A

B 1

Hence the mean value of the x' component of the current -density taken through the entire region of space Q become sequal to :


25

(9) a°F(S2-S2 i )-{-S2 i ~oF ~co~ ~ + si ►~~) N (~9) dP = u S2

~T!

6°F(1 - P) {

F._ o

cos' 29 i slll 2 19. )A T

B

N (~) d

u.

, .(9' )(9')

The conductivity of the suspension a I ,equation :

Vu x , = 6,,, .

We put

(10)

is defined by th e

cos 2 P' sin z 0A ' B

NO) d 'd.7e1o

I =

and instead of (8) and (9 ' ) we get

(11)

F ( 1 -e)+C FI = 1

(12)

a°F(1 -C) -F ae FI = a 771 7V , which gives

a

a°+e(aI-a° )=

Now, obviously, the relative number of ellipsoids, th edirection of whose symmetrical axes falls between an dD. + ds9, is proportional t o

f (0) sin '19d 1

that is

N 09) d s9 = C f (?9-) sin 29d

where the constant C can be determined by the normalizin gequation :

N 09) d ~9 = C f (0) sin z9 d ~ = 1 .~ o

(13)1+e(I-1)

ç

26

Nr . 1 . JøRGliN E . THYGESP;N :

The function f(V) is normalized so that the integral, bein g

extended over all directions, is equal to 4 7r, since we started

with an equal distribution ; and this integral is constant

in the course of time, that is :

727L ÇT 7L

4nc= f(~9)d.Q=`

f(t9)dq~sin29dz9= 2 Tr 1('O) sin i9 el &t sphere of emit

o

0

T

his means that C

2or

N (~9) d=9

f (~9) sin d 79

and thus we get

I

Çi(co

A

z9 + si

B

) 2f (09) sin 79 d ~ .

Substitution of x = cos z9, as a new variable of integra-

tion gives :2 1

=-(-

s-l+ 1 Bx l

2f(x)dx .

i

For equal distribution of direction it holds tha t

f = f° = 1 ; the corresponding I is called I °

1

1 +i = dx

= 1 +°

2 I(A

B )

3 (A B)

This value introduced in (13) then gives an, for a random

distribution of direction, in agreement with H . FRICKE' .

For a independent of the time t (w -> 0) we foun d

above (p .18) :t 1

f = 1-- (3cos' 19-1)(1- e

By By again introducing x = cos 19 as a variable of integra-

tion, eve find for the corresponding I

1 H . FR]CKE : Physical Review . 24 : 575, 1924 .


27

CC ooust . = Ip

-et) y(X2 -x 2

3

~

4 (

-

1 ~( 2 _I

-+

2a

t 1 1 \= Ip -45 (1-é

72 tt-B) .

We see that I decreases from the value lo at the origina l

state of equal distribution to the value la constant .

The time of relaxation i . e . the time that passes befor e

the conductivity assumes its new constant value-i s

-n = z'°, as stated already on p . 18 .

In the case of ap> > al, we see from (13) that a

decrease of I means an increase of the conductivity i n

the course of time .

If on the other hand, a is so great in proportion to a o

that ap can be considered small in proportion to al, adecrease of I means a decrease in the conductivity of the

suspension .

For a dependent on the time, cc (t), we found (p. 19)

f = 1 - 4 (x2 3) G (t) .

So the corresponding I becomes

Iafl)=

Ip-8G(t)(~Å

+1

8x2) \x2-3

Jdx

= 1045{~-8~1 1 ~cos[2wt-I-26---b ]

V 1 -{- (2 wz2 )

- ~3[1 1 cos [2 S +el/1 -{- (2 cor,)

For (cow2 ) >> 1 we get :

la(I) =Ip-45

(Å-~)(1

dx

28 Nr . 1 . JØRGEN E . TRYGESI3N :

where the term containing a is half as great as the cor -responding term in the expression for Ill c°nst .

V. Numerical Examples of the Preceding Calculations .

The effect of orientation depends especially on the termcharacteristic of our theory

2 D °

field energy per ellipsoida_

kT

kT

)

that is, of s ° , E, ,u = ô, a, b and T, (X° and P° dependa

only on x = b also on a ; 4 = 3 (A +B

and (IA--I )

depend on fc and x), besides on the volume concentratione (formula (13), Section IV . )

The duration of the effect is determined by the time o frelaxation

-r 2

6 k T

which depends on )7, besides on a, b and T.

From the formula for the rotation moment D (se eSection I) it is evident that we have to distinguish betwee ntwo main cases :

I . ,u> > 1 and II . ,u < < 1, where we may obtain con-siderable effects in the first case, and only insignificant one sin the second, as the value of the denominator in the ex -pression for D chiefly depends on ,u .

For the sake of simplicity, examples are calculated fo rrod-shaped particles (elongated ellipsoids of revolution)only .

I . u> > 1

a = 1x10 4 rec.Ohm .cm_i

a = 4X10-4 cma° = 1 x 10-6 rec. Ohm . cmi b = 0,5X 10-4 cm

= a = 100

= a = $

T = 291 Kelvin (18°C . )

e


2 9

Sine current 1000 Hertz . Eo = Maximal tension/c m

2 volts/cm =300 electrostatic unit/cm .

u = -2,4284

la ° Oust . = 0,26358 + 0,03939 • (1 - eFor n = 0,05 Poise we have r2 = 159,4 seconds .

6m 105

1, 25 -

1,20 -

1,15 -

1,10

t O

1

2

3

5t2

By calculating .I, const . and substituting it in formul a

(13), Section IV, we get the conductivity for various length sof time .

Fig. 1 gives a graphic presentation of the orientatio n

effect when the volume concentration of the suspensio n

= 0,3, that is, when it amounts t o

= about 7x10 particles/c .c .

3ab 2

II . r<<1

a = 1X10 'rec.Ohm.cm a = 4X10-' cm

u = 1 x 1o-4 rec. Ohm. cm 1 b = 0,5x 10-4 c m

1

1

~` = 100

x = 8

T = 291 Kelvin (18°C .)

30 Nr . 1 . JØRGEN E . THYGESEN :

Sine current 1000 Hertz Eo = Maximal tension/cm = 300

electrostatic units/cm.

a = - 0,03226 la(Il

= 1,6502 - 0,0007 1 - e =2 . For

= 0,05 Poise we have r2 = 159,4 seconds .

Here the orientation effect becomes insignificant, a s

the conductivity increases only from 5,8987 X 10 5 to

5,8996X 10 5 rec. Ohm . cm-l that is, 0,0009X 1 0-5 rec. Ohm.

cm from the time 0 to the Lime Do .

The slight rotation moment in case II explains why

FREUNDLICH and ABRAMSON ] and ABRAMSON 2 in cata-

phoretic experiments found no orientation of the red bloo d

corpuscles of horse (which may approximately be con -

sidered oblate ellipsoids of revolution), either when situate d

singly or in rouleaux ("piles of coins") .

The conductivity of suspensions of blood corpuscles ,

with special reference to the phenomenon demonstrated b y

H . Fiucxr and S . MORSE3 and by MC .CLENDON 4-that the

conductivity of blood changes in the course of time whe n

the stirring up or agitation movement have ceased-wil l

be discussed in a subsequent paper .

As D depends on a 3 , the orientation effect will be very

slight for very small particles, even in case I . (it > > 1) .

In the experiment performed by KRUYT5 with a colloi d

solution of vanadium pentoxide (a = 0,5 micron) it must b e

assumed, therefore, that the particles are orientated throug h

cataphoretic liquid currents in the very shallow observatio n

chamber (as, eeteris paribus, a decrease of a from 4 X 10-4

to 0,5 X 10 4 will reduce D by 64/0,125 = 512 times) .

H . FREUNDLICH and H . A . ADRAMSON : Zeit . f. phys . Chem . 133 : 51,1928 .

2 H . A . ABRAMSON : Proceed . Soc . Exp . Biol . and Med . 26 : 147, 1928-29 .a H . FRICKE and S . Mouse : Journ . of Gen . Physiol . 9 : 153, 1926 .

4 MCCLENDON : Journ . Biol . Chem. 68 : 653, 1926 .

KBEY'r : Kolloidz . 19 : 161, 1916 .


3 1

The different views held by French physicists (COTTO N

and MOUTON, MESLIN and CHAUDIER) and by Germa n

colloid chemists (FREUNDLICH and collaborators, KRUYT) a s

to the cause of the orientation of the particles by "electrica l

double refraction" in suspensions have been discussed by

G. E. MARSHALL . '

Summary .

In a suspension of homogeneous ellipsoids it is assume d

that a change in the electrical conductivity takes place i n

the course of time, owing to an orientation of the ellipsoid s

produced by that moment of rotation with which the elec-

tric field acts upon the ellipsoids .

In the present paper this effect is calculated for ellipsoid s

of revolution.

For the electric rotation moment acting on an ellipsoi d

of revolution I have used R . Fi,RTH's formula, which i s

valid if quasi-stationary conditions be assumed .

For the antagonistic moment of friction I have use d

EDWARDES' formula .

By means of a method analogous to the one employe d

by P . DEBYE for polar molecules, a function of distribution

is calculated, which describes the orientation of the ellip-

soids under the influence of an arbitrary electric field an d

of the Brownian molecular movements .

Finally, in agreement with H . FRICKE, a formula is cal-

culated for the conductivity of the suspension when th e

ellipsoids of revolution have a given distribution of direc-

tion, so that: it is possible to estimate the conductivity at a

given time and follow its change in the course of time.

For a quite random distribution of the ellipsoids th e

C . E . MARSHALL : Transact . of the Faraday Soc . 26 : 173, 1930 .

32 Nr . 1 . JGJIIGEN E . T1-lYCE91$S

formula for the conductivity is identical with the one worke d

out by H . FRICKE .

For a constant field the conductivity changes with th e

time, and after some length of time (relaxation time) it

assumes a new constant value that differs somewhat fro m

the one that corresponds to equal distribution of direction .

For alternating voltage, the periods of which are smal l

in proportion to the time of relaxation, the change in con-

ductivity (orientation effect) is found to be in simple relation

to the change observed under constant voltage .

For suspensions of ellipsoids whose conductivity is man y

times greater than that of the medium, considerable effect s

may be obtained . If the conductivity of the suspendin g

medium is many times greater than that of the particles ,

only a slight effect is obtained .

When the particles are very small, the effect is insigni-

ficant in every instance .

I wish to acknowledge my indebtedness to Mr . CHR .

NIØLLER, Ph. D . for his kind interest and valuable assistanc e

during my work on this paper .

Finsen Laboratory ,

Finsen Institute and Radium Station ,

Copenhagen .

Indleveret til 'trykkeriet den 31 . Februar 1939 .Færdig Ira Trykkeriet den 3 . Juni 1939 .

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ON THE ELECTRIC CONDUCTIVITY OF A SUSPENSION OF ...gymarkiv.sdu.dk/MFM/kdvs/mfm 10-19/mfm-17-1.pdf · The increase of conductivity is constant, however, and has no influenc e on the