Interdiffusion of hydrogen and alkali ions in glass surfaces
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Smit, W., & Stein, H. N. (1979). Interdiffusion of hydrogen and alkali ions in glass surfaces. Journal of
Non-Crystalline Solids, 34(3), 357-370. https://doi.org/10.1016/0022-3093(79)90022-X
DOI:
10.1016/0022-3093(79)90022-X
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Published: 01/01/1979
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Journal of Non-Crystalline Solids 34 (1979) 357-370 © North-Holland Publishing Company
I N T E R D I F F U S I O N O F HYDROGEN AND A L K A L I IONS IN GLASS SURFACES
W. SMIT and H.N. STEIN
Department o f General Chemistry, Eindhoven University o f Technology, Eindhoven, The Netherlands
Received 24 November 1978
Revised manuscript received 31 May 1979
The Boksay-Doremus theory on the interdiffusion of hydrogen and alkali ions in glasses was extended by taking into account the association of hydrogen ions and non-bridging oxygen ions. In the derivation of the interdiffusion coefficient the bulk composition of the glass is taken into account. Theoretical concentration profiles were compared with experimental data, reported in literature, of alkali-alkaline earth-silicate glasses and of a sodium-aluminium-sili- cate glass. An essential feature of this extended B-D theory is, that the relative order of chemi- cal durability of alkaline earth containing glasses is found by this theory without assuming excessive changes in individual diffusion coefficients.
1. Introduction
Doremus [1,2] e x t e n d e d the t h e o r y o f Boksay et al. [3,4] on concentration pro- files in the surface layer o f alkali containing glasses which have been in contact with water, b y introducing a concentration dependent interdiffusion coefficient o f the form
1~ = DADB/[CD A + (1 -- c) DB] , (1)
where D A and D B are individual diffusion coefficients o f ions A and B, and c is the a t o m fraction o f the alkali ion A originally present in the glass. Dore- mus compared calculated concentration profiles with experimental ones, reported in literature, and he showed that eq. (1) is superior to the assumption o f a constant diffusion coefficient. Nevertheless, the use o f (1) must be considered as a trial, because it strictly holds for free interdiffusing cations A and B, whereas the non- bridging oxygen ions in the silicate anion g r o u p s - SiO- (further d e n o t e d as O - ) trap and thus localize most o f the interdiffusing H + ions. In this paper we e x t e n d the B o k s a y - D o r e m u s t h e o r y b y taking this association into account. The relevant interdiffusion coefficient can be derived foUowing Helfferich [5]. Alkaline earth- ions, which do not participate in the ion-exchange process [3], increase the number o f non-bridging oxygen ions, whereas alumina decreases this number. Thus, the glass
358 w. Smit, H.N. Stein / Interdiffusion o f hydrogen and alkali ions in glass surfaces
composition must be taken into account in this derivation. It will be seen that by the present theory differences in durability between glasses can be explained with- out assuming excessive changes o f the individual diffusion coefficients.
2. Theory
According to Boksay et al. [3,4] the diffusion in the glass must be considered with respect to a moving outer surface. If x is the distance o f a plane in the glass from its initial surface and a is the rate o f dissolution o f the surface, then y , the dis- tance from the actual surface at time t, m a y be expressed as
y = x - at (2)
where it is assumed that a is constant with time. The diffusion equation in y coor- dinates with a concentration-dependent interdiffusion coefficient b is [1 ] :
( O c l / a t ) y = (O/By) ~ ( O c l / O y ) t + a(Ocl/Oy)t , (3) Where cl is the concentration o f the diffusing alkali ion. In the steady-state (Ocl/ Ot)y = 0, and integration o f e q . (3) with d c l / d y -> 0 and cl -+ c o w h e n y -+ oo yields
D d c l / d y = a(c ° - c , ) . (4)
2.1. I n t e r d i f f u s i o n c o e f f i c i e n t o f silicate glasses containing alkaline earth ions
In a glass with bulk composition: p mol % (MI+)20, q mol % M2+O and r mol % SiO2, the mean number o f non-bridging oxygen atoms per silicon tetrahedron is given b y X = 2 [(p + q + 2r)/r] - 4. The total concentration ~o-(mol/1) o f the non- bridging oxygen ions is given by X C s, where Cs(mol/1 ) is the bulk concentration o f the silicate tetrahedra, which follows from the density and composition o f the glass. For convenience we divide ~ o - in parts ~ o ~ and ~ o ~ equivalent to the bulk con- centrations C~1 and C~2 o f the M ÷ and M 2÷ cations, respectively
= + ( 5 )
with
~ o f = cO and Cg~ = 2 C ° . (5a)
We assume that H ÷ ions can associate with any o f the non-bridging oxygen ions. Association o f H ÷ ions with non-bridging oxygen ions, which have M 2÷ ions in their vicinity, yielss groups, which as a whole carry a positive charge. This charge, how- ever, is compensated elsewhere b y a lack o f Na ÷, i.e. b y the negative charge o f a non-bridging oxygen ion, as required by the electroneutrality. In total, it remains true that Co~ non-bridging ions are required for the neutralization o f the non- moving divalent ions at any point y ; We write formally:
W. Smit, H.N. Stein / Interdiffusion o f hydrogen and alkali ions in glass surfaces 359
where
COH
iS the concentration o f associated H ÷ ions. The association o f H ÷ ions with the non-bridging oxygen ions is taken into account by use o f the degree o f dis- sociationa = CH+/CH (7)
where CH is used for the total concentration o f hydrogen.
In the following, H ÷ ions, which are not associated with non-bridging oxygen ions, are indicated b y "free". These ions, however, must be thought o f as bound to free electron pairs on bridging oxygen atoms and also to water molecules when present as in gel layers. This binding is much less tight than that o f H ÷ in silanol groups. Of course, alkali ions are also bound to negative oxygen ions, but this binding is less tight than that o f H ÷ in silanol-groups as well. Therefore the inter- diffusion can be described in terms o f "free" H ÷ and alkali ions. The equilibrium - SiOH ~ - SiO- + H ÷ is assumed to be instantaneous compared with the interdiffu- sion process.
The Nernst-Planck flux equations for M ÷ ions and f o r total hydrogen are [5] :
Jl = - D 1 [grad C1 + (CIF/RT) grad q~] (8)
JH = - D H [grad(aCH) + (~CHF/R 7) grad qS] (9)
where J is the flux, D is the individual (tracer) diffusion coefficient, F i s the Fara- day constant, T is the absolute temperature, ~ is the electrical potential and the subscripts 1 and H refer to M ÷ and hydrogen ions.
It has been assumed that coupling effects other than by electrical fields are negli- gible and activity gradients have been neglected. Moreover, the appearance o f the tracer diffusion coefficients in (8) and (9) amounts to assuming that the mobilities in a tracer diffusion experiment, in which there are no gradients in ionic concentra- tion, are equal to those in an interdiffusion experiment, involving concentration and electrical potential gradients. Doremus [6] found that the interdiffusion o f sodium and potassium ions in a sodium-lime glass at a temperature not to close to the transition temperature can be described b y eq. (1) with constant DA and DB. The problem whether or not a mixed alkali effect would cause D l and DH to vary with ionic concentration has not yet been solved. Just like Doremus [1 ] we assume that for interdiffusion o f hydrogen ions and alkali ions near room temperature D1 and D H are independent o f the changing ionic ratio in the glass.
Electroneutrality and absence o f electric current requires that
aCH + C1 + 2 C ° = Co- + C~)- 2
or with (5a)
~CH + C, = Co- (10)
and
360 W. Smit, H.N. Stein / lnterdiffusion o f hydrogen and alkali ions in glass surfaces
Combination o f eqs. (8) to (11) to eliminate a C H and grad ¢ gives
Jl = - {DHD1/[OH(C0- -- C1) + D I C 1 ] ) (Co- grad C1- C1 grad Co-). (12) Since it is assumed that H ÷ ions can associate with any o f the non-bridging oxygen ions, the mass action law with constant activity coefficients may be written as
(Co- + Co~) CH+/CoH = K o n = c o n s t a n t . (13) Strichtly speaking, the dissociation constant o f SiOH groups will be different for groups with a mobile alkali ion or with a much less mobile alkaline earth ion in their vicinity. However, for simplicity this difference is neglected. Thus, one disso- ciation constant KOH is considered to describe the dissociation o f --- SiOH groups.
With eqs. (6), (7) and (10), eq. (13) leads to a quadratic equation for C o - with the solution
Co - = ½(C1 - Cg3~ - KOH + A ) (14)
where
A = [(C1 - C~)~ - KOH) 2 + 4(KoHC~- + C ~ C , - C ~ K o H ) ] ,/2 . (15) Differentiation o f eq. (13) yields
grad Co- = ½ [1 + ( e l + c ~ - 2 - KOH)/A] grad C1 • (16)
Substituting C o - and grad C o - in eq. (12) b y eq. (14) and (16) one obtains, after rearrangement, the flux equation:
J1 = - / 3 grad C1 (17)
where
/3 = D1 [A (A - KOH - C~- 2) - C, (Cl + C ~ - K O H ) ] / [ A (A - KOH - C8~)
- C I A ( 1 - 2 D 1 / D H ) ] • ( 1 8 )
Eq. (18) reduces to Helfferich's [5] eq. 0 9 ) i f C ~ is equalized to zero.
Substitution o f b in eq. (4) yields, after some rearrangement, the differential equation
dC1/dz = (C o - C 1 ) [ A ( A - KOH - C~£) - C1Ab ] / [ A ( A - g o H - C ~ )
- C1 (C1 + C ~ - KOH)] ( 1 9 )
where b = 1 - 2 0 1 / 0 H and z = a y / D l are dimensionless parameters and A is given
by eq. (15).
2.2. l n t e r d i f f u s i o n coefficient o f a s o d i u m - a l u m i n i u m silicate glass
In the present paper we compared calculated steady-state concentration profiles with the experimental concentration profiles o f a leached s o d i u m - a l u m i u m silicate
W. Smit, H.N. Stein / Interdiffusion o f hydrogen and alkali ions in glass surfaces 361
glass reported by Boksay et al. [7]. The glass investigated by these authors has the composition 22 mol % Na20; 5.5 mol % A12Oa and 72.5 mol % SiO2. Assuming a density 2.5 × 103 kg m -3, the concentrations of these oxides are 8.77; 2.19 and 28.88 mol 1-1, respectively. The aluminium atoms were supposed by Boksay to form tetrahedra.
From the composition data it follows that the average number of non-bridging oxygen atoms per tetrahedron is 0.395. Thus, the concentration of non-bridging oxygen atoms is 13.14 tool 1-1. Boksay et al. [7] assumed that all the aluminium atoms in the glass investigated form tetrahedra and that none of the oxygen atoms attached to an aluminium ion are non-bridging oxygen ions. If it is assumed that these assemblies of atoms do not tend to take up protons like the non-bridging oxygen ions in the silicate anion groups, then the presence of aluminium would account for the retention o f 25% of the sodium ions in the glass phase during the treatment with water. However, since the experimental value is about 15%, Boksay et al. [7] assumed an interaction between tetrahedra and protons.
It should be noted that in the glass, investigated by Boksay, the occurrence of A13÷ in six-fold coordination with three bridging oxygens, one non-bridging and two passive oxygens, which are bridging oxygens belonging to adjacent tetrahedra, is not excluded [8,9]. This octahedral structure may conceivably have properties different from the A104-group. If it is assumed that both sodium ions, required to preserve electrostatic neutrality of such an octahedral group, are exchangeable by protons, and, moreover, that the non-bridging oxygen atoms are distributed at random over the Si and A1 ions, forming the octahedral structure with the latter ions, then a retention of 15% of the sodium ions results. The agreement between the calculated and experimental values of the retention can of course be fortuitous. We shall now leave aside the actual cause of the retention of the sodium ions.
The retention of the sodium ions is taken into account by introducing the total exchangeable sodium concentration in the bulk C~l a
e~a = C~a - CN~r o (20)
0
where C ° a is the total concentration in the bulk and CNa,r iS the experimental con- centration of the retained sodium ions. In this special case C~a,r = 2.65 mol 1 -x and C ° a = 17.54 mol 1-1.
In fact, at least three association constants for H ÷ ions must be distinguished. However, just like in the previous sub-section, this difference is neglected and we introduce only one type of O- group with which H ÷ ions associate. The relation which replaces eq. (6) is
C8- = Co- +COH (21)
where Coil is again the concentration of associated H + ions and C ~ - is the total concentration o f O- in the bulk and equals C~a in magnitude (= 14.89 mol 1-1). On introducing CNa, the interdiffusion coefficient, which can be derived in the same way as in the previous subsection, is equal to the original Helfferich interdiffusion
362 w. Smit, H.N. Stein / Interdiffusion of hydrogen and alkali ions in glass surfaces
coefficient [ 5 ] . / 9 and A can be obtained from eqs. [18] and [15] by substitution Of CNa for C1 and by omitting the terms with CoE.
Numerical solutions o f eq. (19) and its sodium-aluminium-silicate glass analo- gue were obtained with the boundary condition C1 and CNa = 0, respectively, when y = 0, using the fourth-order R u n g e - K u t t a method on a Hewlett Packard HP-97
calculator.
3. Comparison with experimental data and discussion
3.1. Silicate glasses containing alkali earth ions
In table 1 experimental data, eeported in the literature, are compiled.
Curves o f concentration as a function o f the dimensionless parameter z = ay/Dx, calculated with the C~)- and C ~ values o f the Li glass, are given in fig. 1 for vari- ous values o f (D1/DH)K~31H . If KOH < 0.1 and 0.05 < (D1/DH)K()~H < 500 differ- ences in KOH do not noticeably influence the course o f the curves.
We cannot fit the experimental curve (ref. [10], curve g, fig. 5) exactly to one o f the theoretical curves. In fig. la a curve is shown which fits rather neatly to the lower part o f the experimental curve, but the upper part is too steep. From Zl/2, i.e. the z value for which C--~C , o f this curve and Baucke's [10] a andYl/2 val- _ 1 o ues, we calculate DLi = 2.4 X 10 -21 m 2 s -l. In order to obtain the same DLi value as calculated from the bulk resistivity, OLi = 1.6 X 10 -19 m 2 s -1 [9,1 ], z 1/2 must be equalized to 3.2 × 10 -3, corresponding to the theoretical curve with (DI/DH)KR~ 500. Like Doremus [1], we atrribute the discrepancy between experimental and theoretical curves to the small depth o f the surface layer (30 nm) and the limited resolution o f the technique ( ~ 4 nm), so that the lithium concentration profile appears to be more spread out than it actually is.
From fig. 1 it is apparent that the inflexion in the curves disappears when(DK/ DH)K~IH < 0.2. Figure 2 shows the theoretical curves for Boksay's [3] potassium glass 4. The experimental curve can be fitted to the curve with (DK/DH)K~D1H = 0.05. We have also plotted in fig. 2 t h e OlD K values for this curve. From the 21/2,
Yl12 and a values we calculate D K = 3.6 X 10 -13 m 2 h -1. The DID K values are nearly constant. This explains why Boksay [3] could fit the experimental data with a theoretical curve based on a constant diffusion coefficient. With/9/DK ~ 1.6, we find the same diffusion coefficient/~ = 5.8 X 10 -13 m 2 h -1 as calculated by Boksay. So far our treatment is equivalent to the use o f the interdiffusion coefficient eq. (1). However, with eq. (1), the different concentration profiles (Yu2 values) o f the glasses which differ only in the percentages o f the composing oxides, can only be explained by assuming different Dt/D H values a n d / o r a values. From the experi- mental curves (ref. [1 I ] , figs. 1 - 4 , we deduce approximately the same a values for the glasse $1-$4. Figure 3 shows the theoretical steady-state curves for Wikby's [11] glasses and fig. 4 shows those for Boksay's [3] glasses 2 and 3, all calculated
Table 1 Compilation of experimental data and composition of glasses. C O _ b) Ref. Glass Glass composition C~3- a)O~ Yl/2 Temp. Hydration a (tool %) (tool 1-1 ) (#m) °C Time (h) (nm h -1 ) [10] Li 25 Li20 q K2+O r SiO 2 27 4.4 0.018 50 15 0.10 [11,12] S 1 22 Na20 10 CaO 68 SiO 2 26.6 8.3 0.3 25 190 $2 22 Na20 6 CaO 72 SiO2 23.3 5.0 0.8 25 190 S 3 21.4 Na20 6.4 CaO 72.2 SiO2 23.1 5.3 0.5 25 190 0.02 c) S 4 22 Na20 4 CaO 74 SiO2 21.6 3.3 1.3 25 190 [3] B 2 28 Na20 4 SrO 68 SiO2 25.7 3.2 50 40 144 B 3 20 Na20 12 SrO 68 SiO 2 24.4 9.1 8 40 288 B 4 20 K20 12 SrO 68 SiO 2 22.2 8.3 4 40 >240 88.7 [13] Dole 21.4 Na20 6.4 CaO 72.2 SiO2 23.1 5.3 0.8 20 312 2.4 40 312 r.q ~a a) Calculated with d = 2.3 g cm -3 and q = 5 for the Li glass and d = 2.5 g cm-3 for the other glasses. 1 b) Distance at which C 1/C~1 = 5' deduced from the experimental concentration profiles or equalized to the reported gel layer thickness [12]. c) V1 ofref. [14]. 2"
364 W. Smit, H.N. Stein / lnterdiffusion of hydrogen and alkali ions in glass surfaces / / --C l / --C o / ° 0.6 0.125 0.4 O2. I J I I I 0 0.2 I 0.4 0.6 0.8 a y / D 1 1.0
"° I
F28
32
c,,c:[ {:'1
{
/
O. 8
/I II
0.6 I 0.4 //
0.2 / , b / O~ 0.0 , ' 1 0.04Fig. 1. a and b. Concentration profiles for the lithium silicate glass of table 1, calculated with different values of k = (D 1/DH)Ko1H (on curves). The experimental curve (curve g, fig. 5 of ref. [ 10] is fitted with the point Cz/C ° = 0.5 to calculated curves with k = 2 and 512 respectively.
with the same value o f 50 for (DNa/DH)K() ~. The sodium bulk concentrations o f Wikby's glasses are all nearly the same, y e t , the calculated curves have different zz/2 values. The relative position o f the calculated curves is independent o f the value o f (DN ~/DH)K~)IH.
The order o f the z u 2 values, $4 > $2 > $3 > $1 and B2 > B3, is the same as the order o f the experimental gel layer thickness and YU2 values. If it is assumed that DNa is not very different for the four S glasses then it follows from the theoretical curves that in order to attain and to maintain the steady state, the outflow o f alkali is such that $1 < $3 < $2 < $4 o r the chemical durability order is $1 > $3 > $2
I¢. Smit, H.N. Stein / Interdiffusion of hydrogen and alkali ions in glass surfaces 365 1.0 / c C K Cp O~ O~ O 4 0 2 0 __ 0.5 I 1.0 I 1.5 t 2.0 J 2.5 1 3 0 1 3.5 I 4.0 I 4.5 I 5.0 I to a y / D K
Fig. 2. Concentrating profiles for the potassium strontium silicate glass B4 of table 1, calculated
with different values o f k = (DK/DH)K~31H ( o n curves). T h e e x p e r i m e n t a l curve [3] is fitted with t h e p o i n t CK/C ~ = 21- to t h e calculated curve with k = 0.05 for w h i c h t h e DID K c u r v e is also shown.
$4, as was also found experimentally [11 ]. The agreement is only qualitative. For a quantitative agreement some variations in
(DNa/DH)K~IH
must be assumed. In fig. 4 it is shown that, if it is assumed that the a and DNa values o f the B2 and B3 glasses are the same, an increase o f(DNJDH)KJH
o f the B3 glass with a factor 4 is suffici- ent to make the ratio o f theZll 2
values o f the B2 and B3 glasses more in agreement with the ratio o f the experimentalYl/2
values. However, for the B2 and B3 glasses the a values may also be different and in general variations in DNa may occur too.We shall now estimate D H in the case o f the Dole glass ($3), assuming that the diffusion coefficient DNa in the transition layer is equal to that in the bulk. Taking for the resistivity p o f the quenched samples a value about 10 times lower than for the annealed samples and using the Einstein equation DNa =
RT/pF2CNa,
we find DNa = 6 X 10 -20 m 2 s -1 at 20°C. WithY1/2
= 0.8/am and a about 0.02 nm h -1 the value 0.075 results forzl/2 . This
value corresponds to a curve with(DNa/DH)K?9~H
8. The value o f D n depends strongly on the value chosen for KOH. According to Dugger et al. [ 15 ] the dissociation constant o f surface silanol groups is 1 0 - 6 - 1 0 -8 . With KOH = 10 -7 the value D n = 7.4 × 10 -14 m 2 s -1 follows.
However, D H can be several orders higher or lower. From our radio-tracer experi- ments [16], in which concentration profiles in silica gel films were measured after different immersion times in alkaline 24NaS2Br solutions, we found with KOH =
10 - 7 D H = 10 - 1 4 m 2 s -1 ( a n d D N a = 1 0 - 1 8 - - 1 0 - i s m 2 s - l ) .
This D H value was also evaluated taking into account the association o f H ÷ ions with non-bridging oxygen ions as in the present paper. We note that the DH values, calculated in the present paper, are several orders in magnitude greater than the value O H estimated b y Doremus [1] from Baucke's [10] lithium silicate glass data.
Doremus, DH, however, must be considered to be an apparent diffusion con- stant, referring to total hydrogen.
366 W. Smit, H.N. Stein / lnterdiffusion o f hydrogen and alkali ions in glass surfaces 1.0 CNa leNa 0.8 0.6 0.4 0.~ $1 $3 $2 ~ 4 ~ , , . . - - 1 5 ° °
_
/
//
I
//
/
i
I
I/
/
'~ .\ s3,s2 I I I I I-,\\\
/
/ /
/
-~2oo
•
...',\
I
//
/
I
s,, ..~
~
/
/
/ /
//
~
/
_Jl
/ 1oo
\ ~ "~'~ ~ ~ ~"-"--~"~ . . . 0 0.01 0.02 ay/DNa 0.03Fig. 3. Concentration profiles ( ) and resistivity curve ( . . . ) calculated for Wikby's
[ 11,12 ] glasses S 1 - $ 4 o f table 1 w i t h k = (DNa/DH)K~31H = 50.
If in eq. (9), in which Jn is the flux for total hydrogen and D H stands for DH,,free,,, a is assumed to be constant so that grad(aCH) = a gradCH, then Jn can be written in terms ofDapp. = otO,,free,,.
The constancy o f a holds in the case o f self-diffusion in a traced chelating resin [17], but o f course not in the present case. The above is merely to give an indica- tion why o u r D H values are much greater than D n values o f Doremus. Moreover, the values o f Dalkali are found to be smaller than the D H values at variance with Doremus. As D n refers to the free H ÷ ions, this can be related to the smaller dimen-
1.0 / o CNa CNa 0.8 0.6 o.41 0 . 2 0 I B
i i ... /
B 3' B 3/ -
B 2 O.O1 0.02 ay/DNa 0.03 2 0 0Cpo
100Fig. 4. Concentration prof'des ( ) and resistivity curve ( . . . ) calculated for B o k s a y ' s
[31 sodium glasses B 2 and B 3 o f table 1 with k = (DNa/DH)K~)IH = 50. Curve B~ ( . . . ) has
IV. Smit, H.N. Stein / Interdiffusion of hydrogen and alkali ions in glass surfaces 367
sions o f the protons compared with those o f the alkali ions. Consequently, the free protons encounter less resistance in moving through the lattice. Of course, although Dalkali * ( ( D H , "]alkali + JH = 0 on account o f the coupling with the electrical field set up to maintain electrical neutrality.
The theoretical curves differ in two important aspects from the experimental concentration profiles o f the sodium glasses. The CI/C ° values in the outer section o f the experimental concentration profiles are higher and the slope lower compared with the theoretical curves (which were calculated with the condition C~ = 0 when z = 0). The second difference refers to the resistivity curves. In fig. 3 and 4 we have plotted the resistivity curves, which were calculated using the Einstein equation and with the relation p-1 = ~ C i X i / l O 0 0 ' where ~k i is the equivalent conductivity o f
ion i. The use o f these relations is compatible with the assumptions leading to eq. (18) which ignore the possibility o f a mixed alkali effect. Assuming that D~ in the surface layer is equal to D O in the bulk glass, the relation
P/Po = [C,/C ° + (CH+/C°)/KoHk ] -' (26)
can be derived, where P0 is the bulk resistivity and k = (Da/DH)KJH. The values o f CH ÷ follows from eqs. (7), (10) and (14). Calculations show that CH + K o ~ is nearly independent o f the value o f KOH used in the calculation o f the concentra- tion profiles for KOH < 0.01.
According to the theoretical resistivity curves the resistivity increases sharply from the region where the alkali concentration rises rapidly to its bulk value, towards the solid/electrolyte interface. This increase o f the resistivity increases with increasing durability o f the glass. This is in agreement with the experimental facts [11]. However, Wikby [11] located the surface resistance in a small region only. Between the electrolyte/solid interface and the region o f the surface resistance, the resistivity o f the surface layer is according to Wikby et least a factor 5 lower than that o f the bulk glass *
Both deviating aspects o f the theoretical curves can be explained by Boksay's [3] assumption that a structural transformation starts when the alkali ion concen- tration in the layer has been diminished to a certain low value. This transformation is the breaking o f silicon-oxygen bonds, which leads to a gel layer with a loosened structure. This gel layer formation is the cause o f a fall-back o f the resistivity curve after the initial rise and o f the low concentration gradient o f the outer region. According to Boksay and kengyel [17] a certain proportion o f alkali ions cannot diffuse out o f the layer because o f the existence o f impasses in glasses, which remain intact to some extent when the network disintegrates.
• We can obtain P/Po curves with a maximum at the foot of the rising part of the C1/C ° curves, when D 1 and D H of eq. (8) and eq. (9) are special functions of the changing ionic ratio in the glass, which simulate the mixed-alkali effect. The relative positions of the concentration pro- •es of the S and B glasses do not change and the maximum of the P/Po curve increases with the durability. However, when the mobilities of the M ÷ ions at the surface are chosen not higher than in the bulk glass, the O[Po curves increase again with diminishing C1/C~1.
368 W.
Smit, H.N. Stein / lnterdiffusion of hydrogen and alkali ions in glass surfaces
0.5 1.0 d (IJrn) 1,5 I k:2.5 2 1.51 1.25 1 0!75 1 . 0 - - / o CNa CNa 0 . 8 - - 0,~ 0.4O'~'~'J
e
I
~
I
I 0 0.5 1,0 - N T : a y / D - a " 1~,Fig. 5. Concentration profiles for Boksay's [7] sodium-aluminium silicate glass calculated with different values of k =
(DNa/DH)K~IH
(on curves). The experimental results [7] of the speci- men indicated by the full circles has been fitted to the curve with k = 1.25.3.2. Sodium aluminium silicate glass
Figure 5 shows calculated sodium profiles with various k =
(DNJDH)K~31H
valuesf o r the s o d i u m - a l u m i n i u m glass discussed in sub-section 2.2. These curves were cal- culated with KOH = 10 -7. However, curves with the same k b u t different K o H are practically the same for KOH < 0.01. This also holds for the
CH÷/KoH
curves. The curves with k = 1.25 is a good fit to the experimental values o f the specimen indi- cated b y full circles. Since the value o f a is n o t k n o w n DNa cannot be calculated. I f a is taken 0.1 nm h -1, D N a = 3 × 10 -2o m 2 s -1 would result. The value o f D H d e p e n d s on the value o f KOH and will be several orders o f magnitude greater t h a nDNa.
In fig. 6
D/DNa, CH÷/KoH
andp/po
have been p l o t t e d as a function o f the dis- tance p a r a m e t e r z for k = 1.25. The calculation o fpip °
was p e r f o r m e d with eq. (26) neglecting the c o n t r i b u t i o n s o f the retained sodium ions to p-1 and ( p o ) - l .The calculated O/O ° curve shows a m a x i m u m and decreases to about 0.015 at the surface. This m a x i m u m is related to the sharp increase o f
CH÷/KoH
towards the surface.The ~ z ) curve, calculable from eqs. (8), (9) and ( 1 1 ) w i t h use o f the calculated values o f CH + and CNa and with the condition ~ = 0 in the b u l k o f the solid, has the same shape as the
P/Oo
curve.The function
l(z)= fz== (O/Po- 1)dz
has a faint m a x i m u m near the surface, which is contrary to the experimental surplus resistance curve [7].F u r t h e r on, however, the
I(z)
curve is in agreement with the experimental curve. F r o m the theoretical curves three experimental facts, apart from the sodium con-W. Smit, H.N. Stein / Interdiffusion o f hydrogen and alkali ions in glass surfaces 369 / DNa 2 CH+/KoH 0.5 1.0 Z = ay/DNa - - 1 . O l CH* / KOH ,clO "2' - - 0 . 5 J 0
Fig. 6. Calculated resistivity (P/Po), surplus resistance (/), interdiffusion coefficient (/9/DNa) and free hydrogen ion concentration (CH*/KoH) curves, calculated with k = 1.25 for the glass of fig. 5.
centration profile, can be explained: (i). The surface conductance o f the rod shaped specimens which considerably exceeds that o f the bulk conductance, (ii) this sur- face conductance stems for H ÷ and not from sodium ions; (iii) there is a barrier layer in the surface layer. Boksay et al. [7] attributed the surface conductivity, which is constant up to 60% humidity in the measuring vessel, to a stratum behind the barrier layer, and the additional term at higher humidities to a layer exposed directly to the influence o f the atmosphere. We can only agree with the explanation o f the additional term.
Our treatment o f the interdiffusion o f hydrogen and alkali ions in glass surfaces is o f course still only an approximation to the true situation. Stress introduced into the glass by the interdiffusion, violating the assumption o f constant individual dif- fusion coefficients, and changes in activity coefficients have been neglected. It is to be hoped that our treatment has indeed revealed one o f the other unknown influ- ences o n / 9 as Doremus wished examined [1].
Acknowledgement
We acknowledge the stimulating discussions with Prof. J.M. Stevels.
References
[I] R.ft. Doremus, J. Non-Crystalline Solids 19 (1975) 137. [2] R.H. Doremus, J. Non-Crystalline Solids 25 (1977) 261.
370 W. Smit, H.N. Stein / Interdiffusion o f hydrogen and alkali ions in glass surfaces
[3] Z. Boksay, G. Bouquet and S. Dobos, Phys. Chem. Glasses 8 (1967) 140. [4] Z. Boksay, G. Bouquet and S. Dobos, Phys. Chem. Glasses 9 (1968) 69. [5] F. Helfferich, J. Phys. Chem. 69 (1965) 1178.
[6] R.H. Doremus, J. Am. Ceram. Soc. 57 (1974) 478.
[7] Z. Boksay, M. Varga and A. Wikby, J. Non-Crystalline Solids 17 (1975) 349. [8] E.D. Lacy, Phys. Chem. Glasses 4 (1963) 234.
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[15] D.L. Dugger, J.H. Stanton, B.M. lrby, B.L. McConneU, W.W. Cummings and R.W. Maat- man, J. Phys. Chem. 68 (1964) 757.
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