Electromigration in Metals, Salts, and Aqueous Solutions - Advances

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12

Electromigration in Metals, Salts, and Aqueous Solutions

A. KLEMM and K. HEINZINGER

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Max Planck-Institut für Chemie (Otto Hahn-Institut), Mainz, Germany

The progress achieved in the field of isotope electromigration in metals, salts, and aqueous solutions since the meeting on isotope separation in Paris in 1963 is reported. It is shown that the temperature dependence of the isotope effect in liquid metals leads to the conclusion that it is a result of classical atom-atom interactions. Isotope effects in molten salts are smaller than in classical ionic gases. A three stage model is proposed for an explanation of the temperature dependences of the isotope effects in molten salts. The available data of the relative difference in mobilities of isotopes in aqueous solutions are summarized.

T he purpose of this article is to report on the progress achieved i n the field of isotope electromigration since the 12th annual reunion of the Societe de Chimie Physique i n Paris i n 1963 (21). W e shall restrict ourselves to the isotope effects themselves and shall not discuss experimental techniques, multiplication of the elementary effects, and technical applications. Earlier work w i l l only be cited occasionally. (See References 21, 24, and 49.)

/

1

A

Metals Isotope electromigration in pure liquid metals ("Haeffner-effect") has been observed up to now i n L i (50), K (44), R b (60), Z n (43), C d (43), G a (47), In (44), Sn (43), and H g (5, 15). Invariably the migration of the lighter isotopes relative to the heavier ones is towards the anode, and the effect increases with rising temperature. A measure of the effect is the mass effect of electromigration N / m — m x N _ / \(m + m )' 1 2

/

x

_

t

1

+

x

2

2

248

Spindel; Isotope Effects in Chemical Processes Advances in Chemistry; American Chemical Society: Washington, DC, 1969.

12.

K L E M M AND HEINZINGER

249

Electromigration

In this expression m and m are the masses of two isotopes 1 and 2, x is the number of isotopes of kind 1 divided by the number of all isotopes present and N is the number of isotopes 1 transported through a unit cross section attached to isotope 2, if N. = — N _ electrons (—) are transported through a unit cross section attached to the atomic nuclei ( + )• /x has the advantage of being dimensionless and independent of the two isotopes 1 and 2 chosen from a multicomponent isotope mixture—e.g., tin. Typical values of /* are given i n Table I. 2

1

x

12

+

+

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Table I. m.p. °C.

Temp. °C.

Ref.

M • 10

s

K

63.7

81 227

1.1 3.7

44

Rb

39.0

76 390

2.3 9.6

60

Ga

29.9

29.9 300

1.68 5.94

47

In

156.4

190 580

1.35 8.4

44

For R b and G a at low temperatures the apparent activation energy E/x = d(liifx)/d(l/RT) agrees with that of self-diffusion ( E ) . E is approximately 2 kcal./mole for Rb. A t higher temperatures E» > E {47,60). D

B

D

Phenomenologically the mass effect of electromigration can be written, i n a linear approximation i n relative isotopic differences, as (25) _ DF c ~ ~RT\ 2

/A

/&z _ A r _ \ m_ \ V ~r^7)~Km~ +

Z

where D = self diffusion coefficient, K = specific electric conductance, F = Faraday's constant, c = molar concentration, z = number of electrons, contributed by an atom to the conduction electrons, r _ = friction coefficient, describing the friction between the conduction electrons and the atoms, m = atomic mass; As, A r . and A m are isotopic differences. +

+

From measurements of /x, D , c, T, and K it follows that z Ai* \ tn °f d e r of 10" to 10" . Compared with this order of magnitude the term Az • m/Am is negligible, because z for a given atom is a function of its ionization energy which in turn is a function of the reduced mass m = m.m/(m. -f- m ) , where m = electronic mass, and because therefore Az • m/Am is of the order zm./m. i s

t

n

e o r

2

1

l e d

Spindel; Isotope Effects in Chemical Processes Advances in Chemistry; American Chemical Society: Washington, DC, 1969.

250

ISOTOPE EFFETS IN CHEMICAL PROCESSES

Theory thus has to explain why — z

is of the order of 10 f+_ Am to 10" . In a classical theory of electron-atom collisions the reduced mass m again enters into the calculations and the theoretical isotope effect becomes orders of magnitude smaller than the experimental one. AT m But ~^jr~~^ could be a quantum isotope effect, having its origin in electron-atom interactions (11) or in atom-atom interactions. If this were the case, it should be proportional to T" in the high temperature range. In this context it is interesting to look at the temperature dependence of - £ = —z From data in the literature one finds that the " (\ TK I 1\ temperature coefficient of TK/C is small M d l n - ^ / c U n T ~ g V Also, 2

1

r e d

rri

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2

F+

A

M

as stated above, for R b and G a the temperature coefficient of fx/D is zero at low and positive at high temperatures. Thus, -y^-

could only be

a quantum isotope effect, if z were proportional to about T . Such a strong increase of z with temperature is quite unlikely, and one is led 2

to the conclusion that —— is not essentially a quantum isotope f+. Am effect. Since it is also no classical isotope effect in electron-atom interactions, it must be chiefly a classical isotope effect in atom-atom interactions. To specify this idea it has been suggested (25) that in liquid metals there exists an electromigration of the more disordered atoms relative to the less disordered atoms towards the anode (electro-self-transport), and that this transport is faster the smaller the mass of the more disordered atoms, because light atoms in an atomic surrounding are generally more mobile than heavy ones. Electro-self-transport is a well known property of solid metals. In a solid the "less disordered" atoms form a lattice that can be marked—e.g., by a scratch on the surface of the material. If direct current passes the metal, material is transported relative to the mark and can be measured. In most solid metals ( L i (48), N a (65), U (10), T i (69), Zr (69), N i (16), C u (64), A g (17), A u (46), Zn (46), C d (46), A l (46), In (46), Sn (46), and Pb (46)) self-transport is directed towards the anode. Only W (59), Pt (46), Fe (46), and C o (18) show self-transport towards the cathode. The nine liquid metals known for their isotope electro-transport chemically belong to the group of metals that show self-transport towards the anode in the solid state, and it is quite probable that in these metals a kind of self-transport towards the anode also happens in the liquid state.

Spindel; Isotope Effects in Chemical Processes Advances in Chemistry; American Chemical Society: Washington, DC, 1969.

12.

K L E M M AND HEINZINGER

251

Electromigration

The self-transport number of solid metals is defined as * i=N o where N i is the number of atomic nuclei ( + ) transported through a unit cross section attached to the lattice (1), if N. = — N _ electrons ( —) are transported through a unit cross section attached to the atomic nuclei. By definition t + t = 1, where f_ (=iV. /]V_ ) is the transport number of the electrons with respect to the lattice. +

+

+

+l

1

A

1

+

+

Typical values of t i in solid metals are given i n Table II. +

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Table II. m.p. (°C.)

Temp. (°C.)

- t i • io

Reference

26.6

46

9

+

Zn

419.5

350

In

156.4

113 135

2.76 8.90

46

Sn

231.9

181 206

3.4 5.6

46

The magnitude of the self-transport numbers t i of liquid metals necessary to explain the observed mass-effects can be estimated by writing +

N N/

/

1 2

*

+1

Xl

+l

m -m Um^m ) 1

2

2

and estimating the factor to the right of f to be —10" . F o r liquid mercury ^ 10" ) this yields t i « —10" . The self-mobility 1

+1

5

4

+

u = t /cF +l

+lK

of liquid mercury (K = 10 Q" cm." , c = 0.067 moles/cm. ) would then be u i ^ —1.5 10" c m . / V sec. 4

+

4

1

1

3

2

If the quasiAattice of current carrying liquid metals would remain stationary with respect to solid walls contacting the liquid, it would be possible to observe self-transport i n liquid metals by studying electroosmosis of the liquid i n capillaries (26, 31) or electrophoresis of small particles suspended i n the liquid (36). But the quasi-laXXice of liquid is itself electro-transported with respect to solid walls, because near walls the forces acting on the c/wasi-lattice, electron friction and electrostatic force, are unbalanced, the electron drift being attenuated owing to inelastic electron scattering at the wall. For example, for liquid mercury at 3 0 ° C , the external mobility was found to be 2.5 • 10' c m . / V sec. (26, 31) as compared with the self-mobility —1.5 • 10" cm. /V sec. as estimated from the observed mass effect /x « 10" of electromigration i n mercury. (The external mobility is the velocity of the liquid relative to 3

4

2

2

5

Spindel; Isotope Effects in Chemical Processes Advances in Chemistry; American Chemical Society: Washington, DC, 1969.

252

ISOTOPE EFFECTS IN CHEMICAL PROCESSES

the wall in a distance from the wall where the velocity becomes uniform, divided by the electric field.) Direct observation of self-transport in liquid metals by studying external electro-transport is thus impossible. It would be interesting if in a solid metal both the electro-self-transport and the electro-isotope-transport could be measured. Lodding has tried to do this by subjecting In at 137 °C. to a direct current of 5000 amps./cm. for about eight months. H e preliminarily reported a selftransport and a transport of the light isotope towards the cathode (45); meanwhile in more elaborate measurements he found the reverse direction for the self-transport in solid indium (46). It remains a question i n which direction the isotope migration in solid indium goes, because the isotope effect reported in Reference 45 was at the limit of measurability.

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2

Salts The low price of separated lithium isotopes ($23 per gram of 99% L i ; $1 per gram of 99.99% L i ) permits the study of the transport properties of lithium and its compounds at varying isotopic compositions. The electrical conductances of pure liquid L i C l , L i C l , L i N 0 , and L i N 0 (42), and of pure solid and liquid L i S 0 and L i S 0 (37) have been measured. The change of the molar volumes of these salts with isotopic composition can be neglected, thus the conductances measured are proportional to the eigen-mobilities u% and u° of the isotopes (the attribute "eigen" and the superscript o are adjoined to indicate the presence of only one isotope in the substance.) 6

7

G

7

6

3

2

7

6

7

4

2

3

4

7

The eigen-mass-effects of lithium electromigration, defined by (u° ~ u° ) (u° - + " V )

o

6

7

e

(m + ro ) ' (me-m ) 6

7

7

(w°e.7 = eigen-mobility of L i with respect to the anions, m< = of L i ) can be seen in Table III. 6

7

i7

mass

6 7

Table Substance Molten L i C l Solid L i S 0 Molten L i S 0 Molten LiNO 2

Temp. t(°C.)

4

2

4

tH

620-780 575-860 860-930 280-440

III. Ref.

tf

Li

0,335 + 1.38 • 10" (* 0.28 0.28 0,167 + 1.63 • 10" (f 4

610)

4

254)

22, 42 37 37 22

It is interesting to compare these results with the mobilities of isotopes in classical ionic gases. These can be calculated as follows.

Spindel; Isotope Effects in Chemical Processes Advances in Chemistry; American Chemical Society: Washington, DC, 1969.

12.

K L E M M AND HEINZINGER

253

Electromigration

W e start with the equations of linear irreversible thermodynamics (27) N - grad fH-ei grad cf> = % y r v ; k

ik

ik

r = r ; ik

ki

N

These equations imply that in a system of N components i with molar fractions y and friction coefficients r a set of relative velocities v = v% ~ v w i l l establish itself i n the stationary state, if on a mole of component i the chemical force — grad m and the electrical force — e grad is exerted, where ^ and are the chemical and electrical potentials, respectively and e is the charge per mole. It is typical that the friction coefficients make no distinction between the chemical and electrical force. Therefore, for our model calculation of r , we shall choose a situation where grad cf> = 0, i n order to get r i d of accelerations caused by the external electrical force. t

ik

ik

k

x

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x

ik

Let us now look at a gas so dilute that only two-body collisions must be considered and the distribution of free particles i n space is absolutely random, not withstanding their different electrical charges (22). Then evidently the r are positive, proportional to the overall particle concentration, and independent of the molar fractions, and each r only depends on its corresponding type of collision ik. ik

ik

W e consider spherically symmetric particles of masses mi and m with an interaction potential