c of ,"a.ia~:af R3search

D 9 p a i t m e i i i o f the N a v y

C o n t r a c t N o n r - 2 2 0 ( 2 8 )

ION EXCHANGE KINETICS A NONLINEAR D I F F U S I O N PROBLEM

F. t - ie l f fer ich and M. S. Plesset

Eng inee r i ng D i v i s i o n

C4LIFCjRN1A I b ~ 5 S l I V b J ~ E Of JECHPNOLOG'at'

Pasadena, C a l i f o r n i a

Repcirt NO. 85-7 Sspern b e r 1957

A p p r o v e d b y M S, PIesset

Office of Naval Resea rch Department of the Navy Contract ~ o n r - 2 2 0 ( 2 8 )

ION EXCHANGE KINETICS

A Nonlinear Diffusion P rob lem

F. Helfferich and M. S, P l e s s e t

Reproduction in whole o r i n pa r t is permitted for any purpose of the United S ta tes Government.

Engineering Division California Institute of Technology

Pasadena, California

Repor t No. 85-7 September , 1957

Approved by M, S. P l e s s e t

ION EXCHANGE KINETICS

A Nonlinear Diffusion Problem

by

F, Helfferich and M. S. P les se t

Summary

Ideal limiting laws a r e calculated for the kinetics of particle dif-

fusion controlled ion exchange processes involving ions s f different

mobilities between spherical ion exchanger beads of uniform size and

a well- s t i r r ed solution, The calculations a r e based on the nonlinear

Nernst-Planck equations of ionic motion, which take into account the

effect of the e lec t r ic forces (diffusion potential) within the system.

Numerical resu l t s for counter ions of equal valence and s ix different

mobility rat ios a r e presented. They were obtained by use of a digital

computer. This approach contains the well-known solution to the cor -

responding linear problem a s a limiting case. An explicit empir ical

formula approximating the numerical resu l t s is given.

1, Introduction

An ion e'xsrchange r e s in consists essentially of a three-dimensional,

cross-linked network of hydrocarbon chains carrying fixed ionic groups,

the electr ic charge of which is compensated for by mobile ions of opposite

charge ("csuater ions"'), These counter ions axe f ree to diffuse within

the res in network, Tn contact with an electrolyte solution the res in takes

up solvent and some additional mobile ions (additional counter ions, and

l'co-ions" have the same charge sign a s the fixed ionic groups). In a n

ion exchange process the counter ion species present initially i s replaced

by another species,

We consider the ion exchange between spherical ion exchange resin

beads of uniform size containing the counter ion species A and a well-

s t i r r ed solution containing the counter ion species B, During the process

ions A diffuse out of the bead and a r e replaced by an equivalent amount of

ions B.

Id has been established that such processes a r e diffusion controlled:

the ra te determining mechanism i s the interdiffusion of the two species

A and B, either within the res in particles o r in a Nernst diffusion layer

("film"] adherent to the particle surface, which is not affected by stirring, 1

Limiting law8 for fi lm controlled exchange in ideal systems have been given

previously. in this paper ideal limiting laws for particle controlled

processes a r e calculated. Part ic le control is favored by concentrated solu-

tions, large diameter and high degree of cross-linking of the beads, and

efficient s t i r r ing. 3

The driving 'tforce" for the flux of a n ionic species is , in systems

without convection, the gradient of i t s general chemical potential, the

principal elements of which a r e the concentration gradient of the species

and the gradient of the electr ic potential. Even if no external e lectr ic

field is applied, a gradient of the electr ic potential will, a s a rule, be

built up by the diffusion process (diffusion potential). The corresponding

differential equations a r e nonlinear. The interdiffusion process con-

s idered here can be described in t e r m s of one interdiffusion coefficient

which, however, is not constant but changes with the concentration ratio

of the two counter ion species present. Hence, the mathematical t rea t -

ment is that of a diffusion problem with varying diffusion coefficient,

Hitherto only the corresponding linear problem, i. e. , with constant inter-

4 diffusion coefficient, has been solved and applied to ion exchange

kine t ics. In our approach the solution to the linear problem repre-

sents a limiting case: the exchange of counter ions of equal mobility.

The linear solution holds rigorously for isotopic exchange processes ,

whereas for other cases the deviations a r e considerable,

2. Model and Simplifying Assumptions

We use the simplest model conceivable and the simplest equations

proven to be adequate for the description of ionic diffusion processes, the

Nernst-Planck equations. 11-15

Without respect to i ts actual porous s t ructure the whole res in is

t reated a s a quasi-homogeneous phase. Individual diffusion constants for

the mobile ions present a r e defined correspondingly; they can be measured

by t r ace r techniques in equilibrium systems. 1' 15-18 The assumption is

niade with exper imenta l support 15' 16' l8 that these individual diffusion

constants a r e , essent ia l ly constant fo r a given res in , i, e . , independent

of ionic composition. We make use of the ideal Einste in re la t ion between

the individual diffusion constants and the e lectrochemical mobili t ies, We

neglect any coupling of ionic fluxes o ther than by e lec t r ic fa rces . We

a s s u m e a constant concentration of fixed ionic groups throughout the res in .

Any concentration change and flux of the co-ions i s d isregarded, since

they a r e s m a l l a s compared with those of the counter ions if the concen-

t ra t ion of the solution i s not too high and that of the fixed ionic groups not

too low. l3 W e neglect changes in ionic activity coefficients and swelling

condition of the r e s i n a s well a s the effect of gradients aP swelling p re s -

s u r e . Under the s imple boundary condition used here , the magnitude of

the selectivity coefficient i s i r re levant . l9 We r e s t r i c t ou r se lves to

r a d i a l diffusion, i. e . , spher ical symmetry. We a s s u m e that interdif-

fusion within the r e s i n beads i s the r a t e determining mechanism through-

out the process . Th i s l as t assumption i s not s t r i c t ly valid; i t leads to

exchange fluxes that a r e infinite initially, whe reas the possible diffusion

r a t e through the f i lm is finite. F o r a very sho r t init ial period f i lm

diffusion m u s t be the r a t e controlling step.

The assumptions and simplification l is ted above a r e those usually

made in the formulation of a model fo r ion exchange kinetics. 5- 10

We do not neglect the effect of the e l ec t r i c potential gradient on the

ionic f luxes; in th is r e spec t our approach dif fers f r o m previous theories.

3 , Formulation of the Problem

Under the assumptions mentioned, the fluxes of the two counter ion

species A and B a r e given by the Nernst-Planck equations 11-14

O B = - D B r ad CB -I zBCB (F/./RT) grad rg] , I lb )

where @ is the flux, 2) i s the individual diffusion constant, G i s the

molar concentration, z i s the electrochemical valence, is the

Faraday constant, R is the gas constant, T i s the absolute temperature,

i s the e lec t r ic potential, and the subscripts A and B refer to the

counter ion species ,

Electroneutrali ty requires that the total equivalent concentration of

counter ions is constant throughout the bead, since the concentrations of

both the fixed ionic groups and the co-ions a r e constant; thus

The absence of an electr ic cur rent in combination with (2) gives the con-

dition

By use of (2) and (31, Eqs. ( l a ) and ( lb) may be combined to give

2 D ~ D ~ ( ~ ~ ~ ~ + .BeB)

2 grad CA . D A A A z Z c + DBz BCB

%

'She quantity in brackets may be termed the interdiffusion coefficient

D~~ of the process , Its value depends not only on the individual dif-

fusion constants DA and Dg; it i s also a function of the concentration

rat io cA/cB and therefore a function of the time and space coordinates.

The dependence of the interdiffusion coefficient on ionic composition i s

shown in Fig. 1 for various values of the rat io D ~ / D * . F o r CA 6( CB9

DAB assumes the value DA, and for CB ( CA the value DB. The ion

present in smal le r concentration always has the stronger influence on

the interdiffusion coefficient. 33 l9 This simple rule i s a consequence of

of the fact that in (1) the concentration of the species en ters in the elec-

t r i c term. It is also physically evident since the e lec t r ic field ac ts on

every ion present and thus causes a large transference of the species

present in high concentration, whereas the flux of the species present in

low concentration is hardly affected.

F o r the t reatment of time dependent processes the equation of con-

tinuity must be introduced

In previous work on time dependent processes , only the limiting

l inear case has been considered, to which (4) and (5) reduce when a

constant interdiffusion coefficient DAB is assumed. Analytical solutions

to the linear problem have been derived for a number of systems including

those with spherical symmetry with various boundary conditions. 4, 20

However, Eq. (4) and Fig. 1 show that the assumption of a constant inter-

diffusion coefficient holds only i f the individual diffusion constants DA and

D a r e equal, a s is the case for example in isotopic exchange processes. B

This paper d e a l s with the general, nonlinear problem and will t reat

s p e ~ i f i c a l l y sys tems with spherical symmetry. Upon substitution of (4)

into (5) an expression i s obtained which can be written conveniently in

dimensionless form with the introduction of the following dimensionless

var iables and parameters:

a s z D / z D - 1 ; A A B B b = aA/zB - 1

where t is time, r is the radial coordinate, and ro is the radius of the

beads. The resulting equation

was derived in a preliminary note. The fraction qA of the species A

s t i l l present in the sphere a t the dimensionless t ime T is expressed in

t e r m s of the solution y ( p , .c ) of Eq. (7) a s follows:

The ion exchange ra te , deiined as the decrease with time of the amount

QA of the species A (in moles) present in the spheres, may be written a s

where V is the total volume of ion exchanger used.

If the problem is limited to the exchange of counter ions of equal

valence (zA = zB)$ SO that b = 0. the general equation (7) may be

simplified by the iniroduction of the new variable

to be come

which is a convenient form for numerical integration. The resu l t s

reported a t the present time will be for this case. Determination of the

solutions of Eq, (7) for non-zero values of b a r e planned for a la ter time.

We use the simple s t initial and boundary conditions possible, assuming

that initially the beads contain counter ions A only and the solution counter

ions B only, and that no concentration changes occur in the solution,

The boundary condition (12b) corresponds to infinite solution volume o r

to continuous renewal of the solution, The selection of initial and bound-

a r y conditions other than (12) would require the introduction of further

parameters which, for the numerical evaluation we car r ied out, would

have to be chosen a rb i t r a r i ly and have led to resu l t s applicable in special

cases only. In contrast , by using the conditions (121, more general l imit-

ing laws a r e obtained, which can be approached fair ly closely by experi-

mental techniques.

Even under the simple conditions (12) a n analytical solution to the

Eqs. (7) o r (11) was not obtained, We have evaluated (11) numerically for

six values of the ratio D ~ / D ~ of the individual diffusion constants:

The result; a r e given in section (5).

4. Calculation Procedure

F o r the numerical evaluation, (11) was approximated by the finite

difference form

- R- ( P ) [u(P, 7 - u(P-AP,.I)]

where

g(p, = exp - 4 ~ 3 T If A 7 / ( A P ) ~ ;

By use of (13), U(P, z + h 7 ) can be calculated f rom u ( p + ~ p , T ),

u(pI 1~ ), and u ( p - 4 , T ). F o r the calculation the initial condition and the

boundary conditions for p = 0 and p = ]I a r e required. The initial

condition is obtained f rom (12a) and (10):

\

The boundary conditions a t the bead surface ( P = 1) i s given by (12b) and

It can be shown that the gradients of y and u vanish a t the center of the

sphere:

( a u / a P I P ' 0 = o .

Hence the condition a t p = 0 becomes

By choice of Ap an upper limit is given for A T by the stability

condition f o r the numerical integration. 22 In case of (131, this condition

The function u(p, 7 ) was calculated numerically f rom (13) and (14)

in accord with the requirement (15).

F o r the numerical integration corresponding to (a), which leads to

the fraction qA st i l l present in the sphere, S impsonfs rule was used:

where

The dimensionless ra te , -dqA/d r, was approximated by

The ca l cuh t ion s t a r t s out with a step' function, Therefore, a very

fine subdivision of space and t ime is requi red initially, The init ial

spacing of p used was Ap = l/640, The spacing of AT was

selected i n accordance with the stability condition (15). As the cal-

culation proceeded, Ap was increased stepwise to 1/320, 1/160,

1/80. 1/40 ( a t about qA = 0 . 9 ) , 1/20 ( a t about qA = 0.6), and 1/10 (at

qA 4 0.1). AT was increased correspondingly. The estimated max-

4- irnurn e r r o r in qA is t 0.01; the e r r o r i n dqA/d.r i s about - 2%.

The calculations were c a r r i e d out o n the "Datatronl' digital com-

puter (manufactured by Electrodata, Inc. , Pasadena, California) in the

computing cen te r a t the California Institute of Technology. Floating

point numbers were used. The total t ime required was about 35 hours.

5. Resul ts and Discussion

The calculation yields the dimensionle s s functions u ( ~ , 7 ), qA( s ),

and -dqA( T ) / d l . The f i r s t i s readily converted to y (p. .r ), which

gives the rad ia l concentration profiles of the species A. In tabulating

and plotting the resu l t s , we have followed general practice and given

the f ract ional at tainment of equilibrium, ~ ( s ), r a the r than qA(7 ).

Under the init ial and boundary conditions (121, this quantity is given by

F ( 7 ) = 1 - q A ( 7 ) (18)

The representat ion of the data i s kept in dimensionless form, since the

actual values of the concentrations, the fractional attainment of equilib-

r iurn, and the r a t e s a t a given time t depend a lso on D A# ro, C, and V.

They can be obtained frqm the tables and graphs by substituting numcri-

ca l values for these quantities and using ( 6 ) and (9).

Table I gives the fractional attainment of equlibriurn, F ( T ), and

the dimensionless rate , -dq / d q l for the various cases . Figure 2 A

shows a plot of F ve r sus T and includes the solution to the linear ease

4 (DA = DB) which is well known to be

2 2 cup(-n a T) . (19) F ( T ) = f - - mi! n = ~ n

It should be emphasized that the time coordinate T ~ o n t a i n s the

quantity DA; hence, these resu l t s can be used fo r d i rec t comparison of

different processes only if the ion A initially present in the beads is the

same in a l l cases . F igure 2 shows that the curves for different rat ios

D ~ / D ~ can not be transformed into one another by a linear t ransforma-

tion of the time. If the ion B is s b w e r , the process i s slower, but not

by a constant factor, because the interdiffusion coefficient increases with

decreasing concentration of A, On the other hand, i f B is fas te r , the

process i s faster , but the interdiffusion coefficient decreases during the

process.

I t is of in te res t to compare the r a t e s for forward and backward

exchange of two counter ion species. In Fig, 3 we have plotted the f rac-

-I- tional attainment of equilibrium for the exchange processes H against

+ -I- 13' and L i against H , assuming a ratio D ~ / D ~ ~ = 10. F o r direct

comparison we have chosen D t / r2 a s the dimensionless time H o

coordinate in both cases , i t is seen that the process is faster if H', the

fas te r ion, *is initially present in the resin, The half t imes of exchange

differ by about the factpr 2, the time required for 90% exchange by about

Che factor 3. According to previous theoiies, which assume a constant

interdiffusion coefficient, the two curves should coincide.

The radial concentrakion profiles of the species A for 25%, 50%,

75%, and 90% exchange a r e shown in Fig, 4. T h e profiles for the linear

case (D* = ~ g ) given by 4

2 2 sin (nnp) exp (-n n T ) (20) T P , = 1 n

a r e included. The shape of the profiles depends strongly on the ratio

DA/Dg If the ian present in the sphere initially is much the faster one,

a comparatively sharp boundary moves in toward the center of the bead

(I?ig. 4a). If the ion present in the bead initially is much the slower one,

the boundary is diffuse and the process reaches the center rapidly

(Fig. 4b). Again, the explanation is straightforward. In the f i r s t case

the concentration of the faster ion is sma l l near the surface and large

near the center of the bead. Hence, the interdiffusion coefficient

decreases toward the center. Therefore the outer shells of the bead a r e

exhausted rapidly, whereas the exchange near the center remains slow,

In the second case, the opposite holds, The interdiffusion coefficient

increases toward the canter ; therefore the exhaustion is more uniform.

Previous theories lead to one se t of curves for a l l ca ses (Fig. 4g).

F o r practical purposes, an explicit expression for F ( .c ) which

approximafes the numerical resu l t s will be useful. The relation

~ ( 2 ) = 2 - exp Cn ( f l ( a ) e t f 2 ( a ) z 2 + f 3 ( a ) 'z .~)J] 1/2

where a =: D ~ / D ~ and the coefficients f a), f2( a ) , and f 3 ( a ) a r e

given by

t f i t s a l l the numerical resu l t s within a n e r r o r of - 6%. I t should hold

equally well for intermediate values of the rat io D ~ / D ~ . Equation (21)

was developed a s an extension of a simpler but l e s s accurate approxi-

mation given by vermeulenZ3 for the linear case.

We wish to emphasize again that the resu l t s represent ideal limiting

laws only, f rom which the behavior of actual sys tems may be expected to

show more o r l e s s pronounced deviations. However, the comparison with

the solution fo r the linear case demonstrates conclusively that the effect

of the e lec t r ic potential included in ou r approach is an essent ial feature of

the process which no theory can omit.

4. Acknowledgments

We a r e indebted to Drs . J. Franklin, C. B. Ray, and R. Nathan

fox helpful suggestions concerning the numerical integration. We also

wish to express our gratitude to Mr . D. Brouillette and Miss Z. Lindberg

for their valuable assis tance in the computations.

FRACTIONAL ATTAINMENT OF EQUILIBRIUM, ~ ( t ) , AND DIMEI\TSIONLESS

RIZTE, - aqA(*)/a+ , FOR SIX D ~ W I T VALUES OF THE RATIO D ~ / D ~

TABLE I ( ~ o n t ' d , )

FRACTIONAL ATTAJXMENT OF EQUILIBRIUM, F ( * ) , AND DIMEXVSIONLESS

RATE, - dqA( r ) i d s , FOR SIX DIFFERWT VALUES OF 'PHE RRTIO ~ ~ 1 %

TABLE I (~ont'd.) %

FRACTIONAL ATTAUNMENT OF EQUILIBRIUM, F( * ) , AND DLMENSIOIEZSS

RATE, - dqA( ( )/ d z , FOR SIX D~~ VALUES OF THE RATIO ~ ~ 1 %

TABLE I (C!ontfd,) 1

FRACTIONAL ATTAINMENT OF EQUILIBRIUM, F( ) , AND DIMENSIONLESS

RATE, - dqA( l )/ d~ FOR SIX DIFFEBEXQ VALUES OF THE RATIO D ~ / D ~

a) ~ ~ 1 % - 1/2

TABLE I (cont9d.) x

FRACTIONAL ATTAINMENT OF EQUILIBRIUM, F(Z ) , AND DIMENSIONLESS

UTE, -aql((% ) / a~ , FOR SM ~rn- VALUES OF TIE WTO D ~ / D ~

FRACTIONAL ATTAIJSTMENT OF EQUILIBRIUM, F(*E' ), AND DLMENSIONLESS

RATE, - aqA(-z )/ , FOR SIX DIFFERENT VALUES OF THE RATIO D ~ / D ~

EQUIVALENT FRACTION OF B EQUIVALENT FRACTION OF B

Fig. 1 . Dependence of the interdiffusion coefficient DAB on the ionic composition of the exchanger, calculated from Eq. (4), for the interdiffusion of ions of equal valence (left) and of a uni- valent and a bivalent ion (right). The different curves cor- respond to different ratios D * / D ~ .

CH-pA

O 0.2 u .4 u

$"1& b""; V

.*-+ 2 2% 2 , ZJ+\ 4 +h X IT.; <

0) 2 ; f.9; U &I-* *v c d * k+?* e - c W CrXJ d.2

1

(a) DA /DB=IO

W -I a (el DA/DB= 2 > - 1.0 3 0 W

0.5

0 0 0.5 1.0 0 0 . 5 1.0

RADIAL COORDINATE p '

RADIAL COORDINATE p

RADIAL COORDINATE p

Fig. 4. Concentration profiles of the species A in the bead for 2570, 5070, 7570, and 90% exchange (F = 0 . 2 5 , 0 . 5 0 , 0 . 7 5 , and 0 .90) for seven different rat ios DA/DB. The profiles for the linear case (DA = DB) were calculated f rom Eq. ( 2 0 ) .

<

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