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Kinetics of dissolution of calcium hydroxyapatite
Journal of Crystal Growth 47 (1979) 671 679 © North-Holland Publishing Company
KINETICS OF DISSOLUTION OF CALCIUM HYDROXYAPATITE II. Dissolution in non-stoichiometric solutions at constant pH Jq~rgenCHRISTOFFERSEN and Margaret R. CHRISTOFFERSEN Medicinsk-Kemisk Institut, Kqlbenhavns Universitet, Raadmandsgade 71, DK-2200 Copenhagen N, Denmark Received 24 April 1979
The influence of the total concentrations of calcium and phosphate ions on the rate of dissolution of calcium hydroxyapatite microcrystals in sub-saturated solutions at constant pH has been studied. The rate of dissolution is found to be a function of [CaIa[P0 4], where a is 3 for very dilute solutions. For more concentrated solutions a is close to 1.67. The results obtained explain the empirical results found in experiments in which a disc of calcium hydroxyapatite is dissolved in a rotating disc apparatus. The concentrations close to the dissolving surface have been calculated from the bulk concentrations, the diffusion coefficients, and the angular velocity of the disc. The “two-site dissolution” isshown to be an artifact due to mathematical curve fitting, which reproduces the experimental data quite well, but which has no relation to the surface processes governing the dissolution. 2/s and D 10_17 cm2/s for found D’ions 10-10 cm along pores in the enamel and fluoride diffusing in the crystallites respectively. Flim and Arends [5a1 45Ca in have measured diffusion for cm2/s or bovine enamel. The result coefficients was D 10_12 less. Arends [5b] has recently found a similar value for the diffusion coefficient of 32P0 4 in bovine enamel. For solutions of stoichiometric composition, i.e. Ca/P = 1.67, we have shown [3] that the rate of dissolution of HAP microcrystals at constant pH can be described as a function of m0, the mass of crystals used in an experiment, rn/in0 the fractinn of the mass remaining and C/Cs the ratio of the concentration of dissolved HAP to the solubility of HAP at the appropriate pH. The rate was shown to be controlled by surface processes, which means that the composition of the solution close to a dissolving surface is the same as the bulk composition. For non-stoichiometric composition of the solution, the rate was shown to depend more on the concentration of calcium than on the total concentration of phosphate. We found 3’3[P0 that the rate is a function of [Ca] 4]. We have, for a larger concentration range, investigated the influence of non-stoichiometric composition of the solution on the rate of dissolution of HAP microcrystals. We use the result of this study to show
1. Introduction The rate of dissolution of calcium hydroxyapatite Ca 10(P04)6(OH)2, calledofHAP, plays an important role in thehereafter understanding decalcification of bone and tooth material. In vivo carious lesions are characterized by an initial white spot formation in which sub-surface demineralization has occurred. To understand the formation of these white spots, at least the following problems should be elucidated: (a) how fast HAP crystals dissolve, and (b) how fast can dissolved material diffuse along pores or channels from the sub-surface lesion and into the surrounding medium, Jongebloed, Berg and Arends [1] have studied the change in morphology of large HAP crystals which were treated with a SM citric acid solution. Large holes formed along the c-axis of the crystals. Jongebloed, Molenaar and Arends [2] also showed that acid treatment of enamel caused hole formation in the HAP crystals. We too have observed hole formation in microcrystals of HAP partially dissolved in aqueous solutions at constant pH [3]. The importance of crystal defects in HAP has been reviewed by Jongebloed et al. [2]. Flim, Kolar and Arends [4] have measured the diffusion coefficients for fluoride ions penetrating dental enamel. They 671
672
J. Christoffersen, M.R. Christoffersen
/ Kinetics of dissolution of Ca1 0(P04) 6(OH)2
that Wu et al. [6], Fawzi eta!. [7], Fox et al. [8] and Griffith et al. [9] have not taken all of the factors influencing the rate into account in their analysis of their experiments with rotating discs of HAP. They have therefore not proved the reality of their so-called “new two-site model” for HAP dissolution in these experiments. We have reanalysed their experiments and show in the following that their model has no physical meaning and that their results can be explained by a combination of transport and surface processes. The dissolution process may of course be governed by different mechanisms, depending on the composition of the solution. We intend to discuss this in a forthcoming paper [10].
to be kept constant, by addition of potassium hydroxide or nitric acid. All chemicals used were of analytical grade. In experiments in which HAP dissolved in dilute potassium phosphate buffers, traces of inorganic pyrophosphate were enzymatically removed from the buffers. The rate of dissolution of HAP in normal analytical grade potassium phosphate solutions was found to be considerably lower than the rate in a pyrophosphate-free potassium phosphate solutions of the same concentration. In experiments in which crystals dissolved in solutions containing an excess of calcium ions, calcium nitrate was added to the solution before the addition of crystals. 3. Analysis of experiments with microcrystals
2. Experimental Crystals were prepared and analysed and the experiments were performed as previously described [3], except that a Radiometer pH-stat and Radiometer recorders were used. A combined glass and reference electrode was used, which enabled the thermodynamic value of pK2(H3PO4) 7.20 + 0.02 to be calculated from pH measurements with stirring in dilute phosphate buffers in the ionic strength and pH range to be used, Prior to the addition of crystals to the solution, the pH of the solution was adjusted to the pH value —
p[P0~]~”
In figs. Ia ic the results of dissolving HAP crystals in non-stoichiometric solutions are given. The composition of the solution is given by p[P04] log[P04] and p[Ca] log[Ca] where [PU4] and [Ca] are the total concentrations in mol/l of phosphate and calcium in solution. The shaded area in these plots corresponds to supersaturated solutions. The line with the slope 1 defines stoichiometric composition of the solution, i.e. [Ca]/[P04] = 1.67 or p[PO4] = p[Ca] + 0.22. For the non-stoichiometric solutions, we have determined f/rn0 for chosen values of pH and rn/mo. From an appropriate point in
P[PO41~~”
p[CQ]
P[PO~]~~”
p[Ca]
p[Co]
Fig. 1. Lines connecting pairs of points defining the composition of solutions for which the dissolution rates J/m0 are the same. pH — 6.77 and rn/rn0 is equal to 0.8 (a), 0.7 (b) and 0.6 (c). The lines drawn with slope 1 correspond to stoichiometric concen trations, i.e. cp = 1.67. The values of Jim0 and a, for each of the other lines are given in table 1. The slope of these lines is a.
J. Christoffersen, M.R. Christoffersen / Kinetics of dissolution of Ca
1 0(P04) 6(OH)2
Table 1 Values ofJim0 and
a
for the lines in figs. la
Fig. la: rn/rn0
Expt.
=
C; pH =
6.77 Fig. ib: rn/rn0
0.8
J/m0 log a b c d e f g h
mol s
6.33 6.47 6.54 6.67 6.74 6.49 6.49 6.66 6.67 6.84
=
Fig. lc: rn/rn0
0.7
Jim0 g
a
log
2.9 2.3 2.0 2.2 2.0 3.6 3.1 2.1 2.4 1.9
6.53 6.76 6.91 7.16 7.26 6.58 6.62 6.80 6.90 7.12
figs. la lc defining the composition of a solution, a line is drawn to a point on the line defining stoichiometric composition. This latter point is determined as the point for a stoichiometric solution in which we have the same value of rn/mo, pH and f/mo as in the non-stoichiometric solution. Values of f/mo and the slopes of the lines in fig. 1 are given in table 1. From the slopes of the lines in fig. 1, the effect of [P04] and [Ca] on the rate of dissolution can be determined. For simplicity in the following discussion, we define the slope of these lines to be —a. The rate can easily0[P0 be seen to be constant for constant values of [Ca] 4]. From fig. 1 we see that a increases with increasing values of p[P04] and p[Ca]. Relatively close to the solubiity line, the lines for constant values of f/rn0 become parallel to the solubiity line. As shown [10], the rate of dissolution in this region becomes a function of the change in molar Gibbs energyfrom /.~Gd~ for the dissolution process. away the line, kinetic effects Further appear Iss, solubility to be important for the rate of dissolution in subsaturated solutions of non-stoichiometric composition. From the above we conclude that far from the solubiity line, the rate of dissolution depends on [Ca]a[P0 Thisofmeans rate of dissolution4]is with not aa~3. function L~Gforthat the the dissolution
mol s
1
[Ca] 1.67 [P0] r
1 1.67 ior~ 1
LCaJS
~‘‘-‘4]s
=
0.6
J/m0 g
1
a
log
2.5 2.0 1.6 1.8 1.7 2.7 2.7 1.7 1.7 1.6
6.81 7.22 7.40 7.85
2.5 1.4 1.4 1.7
6.72 6.80 7.03 7.24 7.59
2.1 2.1 1.5 1.6 1.7
mol s
1
g
1
a
if ion-pair formation can be neglected. We shall in the following relate this finding to observations obtained from rotating disc experiments.
4. General comments on the rotating disc experiments We give concentrations in the bulk index b, and interfacial concentrations, just outside the crystal or disc surface, index i. Index s refers to saturated solutions. The symbol w is the angular velocity of the disc. According to Levich [11], the rate of transport from the interface to the bulk, assuming steady state dissolution, is given by
i=
(D/6)(C~ Cb) (1) in which D is the diffusion coefficient and 6 is the thickness of a diffusion layer, given by 1D” \1/3( / ‘51/2 2 l”~” 6 1 61 ~ / ~‘ where v is the kinematic viscosity of the bulk. Gregory and Riddiford [12] have given the following expression for 6 ‘
— —
6
=
113(v/w)”2[l +O.35(D/v)°’36] 1.61(D/v) k 1 1.657 x 10-2 2/(rad ~l)l/2 cm = = ~“ .
process, which, proportional to for constant pH and ionic strength, is ln
673
(3)
In the following we shall use (3) whenever calculating a 6 value. The difference between (2) and (3) is about
J. Christoffersen, M.R. Christoffersen / Kinetics of dissolution of Ca
674
10(P04) 6(OH)2
3% in the calculated value of 6 for the experiments
discussed here. In a steady state the rate by which substance leaves the disc must be equal to the rate by which the dissolved substance is transported from the interface and to the bulk. We shall make use of the following approximations: (a) The diffusion 2/s.coefficient in solution of all relevant ionsEquilibria is l0~cminvolving ionic species in solution are (b) rapidly established, (c) The buffer capacity is so high that pH has the same value in the interface as in the bulk. (d) The ionic strength is so high that we can neglect any electrical potential differences in the solution even close to the interface. Combining (1) and (3) with the above assumptions enables us to express the rate of transport of calcium ions and of phosphate ion species in the following way DWV2 /Ca
=
([Ca]
[Ca]b)
i
(4)
pressed tablets of l’IAP having either two different sites of dissolution, or a very complicated dissolution mechanism which agrees with the two-site dissolution model presented [6 9]. Diffusion inside the disc is taken into account in the model presented. In the model the transport from the interface to the bulk is described by an equation identical to (I). Unfortunately analysis doesofnot information about thethecomposition thecontain interface. The transport equation (1) is combined with an equation describing the rate of dissolution R, in the inside of the disc. “To simplify the computational work, and for the lack of any compelling reasons for choosing another functional form”, R was assumed to be first order with respect to undersaturation, thus R
k(C
C~ Cb
k
Assuming
1
(7)
conditions gives the following expression for the flux, F, of HAP into the bulk,
Dw”2 =
C)
where C~ is the apparent solubility and C the concentration of HAP inside the disc. Combining (1) and (7) with appropriate boundary
([P04]1 congruent
04]b). [P dissolution,
(5)
(6/D) + (1/(kD’)112)’ (8) in which D’ is the diffusion coefficient of dissolved
F
/Ca/1p0
4 = 1.67, which is to be expected and is also found [6,7], enables us to calculate for the data [6,7] the interfacial concentrations [Ca]1 and [PU4]1and therefore also the calcium to phosphate ratio just outside the disc, cp1 = [Ca]1/[P04]1.
HAP inside the disc. In (8) there are two fitting parameters, C~and kD’. Taking the diffusion in the solid disc into account, two sets of fitting constants are obtained. C~is not given, but the corresponding solubiity products are given. Each set of fitting parameters are associated with a certain type of dissolution, called site one and site two dissolution. The values of the fitting parameters obtained are
5. Analysis of “the two-site model” Table 2
[6], Fawzi et al. [7], Fox et al. [8], and Griffith et al. [9] report data for dissolution of HAP obtained from experiments with a rotating disc. In these experiments the bulk solutions contain 0.1M aceticthe acid with pH 4.5 and and inertvarious electrolytes to raise ionic strength to 0.5M values of [Ca] and [P0 4]. The reported results appear to be accurate. Their main finding is that, for certain values2 of thea “kink” bulk concentrations, plots off against w~” have where the slope of plots of / against
Fitting parameters found [8j for eq. (8) and the “apparent solubiity” C~at the sites relative to the thermodynamic solubility C~ __________________________________________________ 5D (kD’)~2 pK 10 2/s) 5 C~/C~ (s/cm) (cm
w112 is not monotonically decreasing. In refs. [6 9] the data obtained are explained to be a result of the
Site 2 1.0 111.3 127.72 0.20 1.5 64.9 130.26 0.14 __________________________________________________
Wu et al.
—
HAP Site 1
1.0 1.5 1.0 1.5
Unknown Unknown 1252.3 844.3
116.4
1,0
116.4
1.0
119.90 122.11
0.60 0.44
J. Christoffersen, M.R. Christoffersen / Kinetics of dissolution of Ca
1 0(P04) 6(OH)2
given in table 2. The thermodynamic solubiity product determined by Avnimelech, Moreno and Brown [13] and the “solubilities” of the different
sites relative to the thermodynamic solubility are also given in table 2. In ref. [6] the concentration profile
6~< l0~6
0.1 1um.
675
(10)
For enamel we obtain 6~<0.1 pm; for the disc 6~ was 5 10 pm. The reason for this difference is that
we have assumed diffusion in enamel to be a much
in the solid is given. The concentration becomes constant 5 10 pm away from the dissolving surface.
slower process than diffusion in the disc appears to be. We can conclude if the “two-site model” had been
6. Consequences of the two-site model
will from of the datafor[6,7], we find be theseen model alsoourtoanalysis be a poor model the dis-
One of the consequences of the applied model is that the apparent solubiities C~ at constant pH depend on the composition of the bulk and on the angular velocity. At a distance of 5 10 pm away from the dissolving surface of the disc, the concentration of HAP dissolved in the solid disc becomes constant and equal to either C~1or C~2,depending
solution of the disc, made of pressed HAP crystals.
on which site governs the dissolution. Much further away from the dissolving surface of the disc, say 200 pm away, the concentration of HAP must be very
constant. The rate of dissolution in this region must be very low dueis to a very low to ratetheof concentration the transporl process, which proportional gradient. Despite this, the concentration of HAP dissolved in this region, can for certain composition of the bulk, be changed from C~1to C~2= 0.3 C~1just by changing the angular velocity. We cannot imagine how any mechanism could have this effect. If the model, despite the above criticism, should be able to predict any important effect for demineralization and remineralization of enamel, the diffusion coefficient in the solid should be comparable to those found for the fluoride ions and calcium ions in enamel [4,5]. For the experiments reported, we can, for enamel, estimate the maximum value of the distance, over which the diffusion plays an important role, by demanding the rate of transport in the solid to be comparable to the rate of transport over the diffusion layer 6. We obtain D’(C~ Ci)>D(Ci
Cb),
(9)
a good model for dissolution of the disc, it would have been a poor model for dissolution of enamel. As
7. Analysis data from Wu Ct a!. [6] and Fawzi et a!. [7] We shall in the following reanalyse the experiments [6,7] and show that the kinetics of dissolution of HAP can be explained without introducing the “two-site model’.
In figs. 2a 2c data from Wu et al. [6] are plotted. In fig. 2a the flux-density, j, interfacial of HAP dissolved is 112. In fig. 2b concentraplotted against w tions, C 1, are plotted against ~~1/2 C~is [Ca]1if C~j~ 1.67 and C1 is [P04]~ if cpj> 1.67. In fig. 2c the interfacial value of the calcium to phosphate ratio is plotted against ~ In figs. 3a 3c similar plots are given for data from Fawzi et al. [7]. From these plots are seen: (1) A change in ~ causes changes not only in 1~but also in [Ca]~,[P04]1 and cp1. (2) The above changes depend on the composition of the bulk. The rate by which the surface processes take place are completely determined by the interfacial concentrations, which we can describe by C1 and cp1. The interfacial concentrations of course strongly depend on the bulk concentrations. The bulk concentration corresponding to C1 will in the following be called Cb. In the steady state the rate of dissolution can also be described by (4) or (5). Using C~as defined above, wehave 112
Dw
6
D’(Cs
C
1) D’c5 D~C1 Cb) 6
/
(C1 which df 112=~D d~ from
—
C’b
>
(11)
Cb)
[(~~ —
~b) + w~2
dC~ ~].
(12)
J. Christoffersen, M.R. Christoffersen / Kinetics of dissolution of Ca
676
1 0(P04) 6(OH)2
2s1
103C
motcm
[Ca]1
1
1;
[P04)
c.j’’/(rad SI’
-
~
:~ ~ u’/(rad s
I’
04]b as folFig. 2. Data from Wu et a!. [6]. Bulk composition: 0.1M acetate buffer, pH 4.5, ionic strength 0.5M [Ca]b and [P lows: Crosses: [Ca]b = [~O4]b 0; Squares: [Ca]b 2.2 X io—~M; [~O4]b 0; Triangles: [Ca]b = 0; l~O4]b— 5.5 X i0~M; Circles: [Ca]b = 5.0 x 10 ~ M; [~°4lb 0; Diamonds: [Ca]b = 0; L~°4lb = 110 2 M. (a) The flux-density, j, of HAP through the interface plotted against the square root of the angular velocity, w112. (b) The inter facial concentrations, C 112. If cpj > 1.67, C 112. ~ 1, plotted against 1 [P04]1, otherwise C~= [Cali. (c) Calcium to phosphate ratio, cp~,at the interface plotted against w
The term w 2dC 2 is never positive for the 1/do~’’ experiments discussed. If the interplay between the surface processes and the transport processes is such
‘
onically decreasing function of w1”2. In fig. 4 we give data extracted from fig. 2 and fig. 3, for which p[P0 4]1 could be plotted against p[Ca]1 for constant values off. From fig. 4 is seen: (1) For high values of p[P04]~ and low values of p[Ca] 1, i.e. with for cp1> rate on [CaJr[PU4]1 a 2,1.67, wherethe a is the depends slope of the
w high 2dC 2may be very important for the term w~ 1/dw” low2.values of w,high causing! veryterm fast with For very values to of decrease W, the same may w~ be important due to the high value of W. Hereby we have shown that d//dW”2 is not necessarily a monot-
tangents to the curves in fig. 4. This result is practically identical to the result shown in fig. 1. (2) For low values of p[PO 4]~ and high values of p[Ca]1 the rate depends on [Ca]r[P04]1 a of 1. This change in a can easily be due to with the use phosphate buffers containing a small amount of inorganic pyrophosphate.
that for some compositions of the bulk, 2I IdC./d~’2
(13)
~‘
dC’dc~.~ 1! w
low
1
J. Christoffersen, M.R. Christoffersen / Kinetics of dissolution of Ca
1 0(P04) 6(OH)2
I
_________________________
mo!~~
I
677
I
,~:~11’1~
a
b 112/( rad s..I)I/2
w”’ /1 rad s —, I’”
w
‘~~=~-1-~
1.2-
10-
-
0.2-
o ~
rad s —,
Fig. 3. Data from Fawzi et a!. [71.Bulk composition: 0.1M acetate buffer, pH = 4.5, ionic strength 0.5M, [Ca]b and l~°4)bas follows: Solid circles: [Caib = [~°4]b = 8.52 X 10~ M; Triangles: [CaIb— [~O4lb = 1.30 X iO’3 M; Hexagons: LCalb = [P04]i, = 1.704 X iO~M; Crosses: [CaIb = [P0 4 lb = 3.524 X i0~ M; Squares: [CAIb = 8.536 X l0~ M; FPO4]b = 8.356 X 10~ M; 1”2. (b) Interfacial Open circles: [Ca]b = 1.48 x i0~ M; [~°4lb = 1.48 x 10—2 M. concentrations, C If cpj> 1.67,plotted C (a) The flux-density, j, of HAP through2.the interface, against the square root of the angular velocity w 1, plotted against 2. w” 1 = [P04l1,otherwise C4 = [Cal1. (c) Calcium to phosphate ratio, cp1, at the interface, plotted against w’-’
J. Christoffersen, M.R. Christoffersen / Kinetics of dissolution of Ca
678
1 0(P04} 6(OH)2
I
I
I
in analytical grade phosphate salts. For the non-
I
30
-
28
-
00
26
-
p[ P0 4]i
o
2/.
-
22
-
nO. (•,0
.05,10
20
I
cm s
mol 2
-
-
-
22
21. [Ca]
rate of dissolution can only be described as a function of ~Gdi5s if a = 1.67. In all other cases specific
independently the rate of dissolution is effectively a function of two concentration variables, meaning that the cannot be expressed a function of only one rate concentration parameter.asWe have here used [Ca]
-
I
described by one concentration parameter, which of course can be chosen as ~Gdjs5. For cp1 ~ 1.67, the
kinetic effects of the ions appear to be[PU4] important. In experiments where [Ca] and are varied
10
cm
I
stoichiometric experirrients [6,7] the rate of dissolution is not a function of ~GdisS. For a constant value of pl-I and Cp1 — 1.67, the rate of dissolution can be
I
I
26
2 8
p 11g. 4. Plot of the flux-density as function of p[P04]1 and p[Ca]~. The line drawn with the slope — I Lorresponds to stoichiornetric composition, i.e. cp1 1.67. Smooth curves 9 niol cm 2 value 1 Squares: / are drawn through points with the same of the flux density. 1.0 x 10 Circles: ~ niol cm / — 20.5s~. X i0 Solid symbols: data from Wu et al. [6]. Open symbols: data from l~awzieta!. [7].
1 and [PU4]1 or one of them together with cpj to describe the rate whereas Wu, FawLi, Fox et a!. have used a mean concentration C, in the bulk solution and ~‘, an apparent solubility in the disc. Their description is not satisfactory because C does not correspond to a real solubility. The two parameters C and c’ can, as shown [6,7], be used as mathematical parameters in a curve fitting procedure.
Acknowledgements We thank Professor Arne E. Nielsen for stimulating
8. Conclusion
discussions and for helpful comments on the manuscript. We are indebted to Niels Kj~rgaardfor technical assistance. The Danish Natural Science Research Council are thanked for the grant J. No. 511-8181
From the above analysis of our data and the reanalysis of data from Wu et a]. [6] and Fawzi et a]. [7] we conclude that all the observed dissolution rates both of a finely dispersed suspension and of a rotating disc can be described in the following man-
ner. The rate of dissolution is a function of [Ca]~’[PU 4], where —a is the slope of a line in fig. 1 or the slope of a tangent to the curves in fig. 4. For pH 7 a is 3 for very dilute solutions; for more saturated solutions a is close to 1.67. For a = 1.67 the rate of dissolution is simply a function of L’xGdISS. For the data [6,7] at pH 4.5, a is about 2 for solutions contaming an excess of calcium ions. For solutions contaming an excess of phosphates a is about 1. The change in a from 2 to 1 we think is due to the small
amount of inorganic pyrophosphate normally present
(and previous grants).
References [1] W.L. Jongebloed, P.J. van den Berg and J. Arends, Cakif. Tiss. Res. 15 (1974) 1. [2] W.L. Jongebloed, I. Molenaar and J. Arends, Calcif. Tiss. Res. 19 (1975) 109. [3] J. Christoffersen,M.R. Christoffersen and N. Kj~rgaard, J. Crystal 43 and (1978) 501. J. Bioeng. 2 (1978) [4] G.J. Film,Growth Z. Kolar J. Arends, [5] (a) G.J. Flim and J. Arends, Calcif. Tiss. Res. 24 (1977) 59 (b) J. Arends, [6] M. Wu, W.I. private Higuchi,communication. J.L. Fox and M. I riedman, J. Dental Res. 55 (1976) 496. [7] M.B. Fawzi, J.L. Fox, M.G. Dedhiya, W.l. Higuchi and J.J. Hefferren, J. Coiloid Interface Sci. 67 (1978) 304.
J. Christoffersen, M.R. Christoffersen I Kinetics of dissolution of Ca
1 0(P04)6(OH)2
[8] J.L. Fox, WI. Higuchi, M.B. Fawzi and M. Wu, J. Colbid Interface Sci. 67 (1978) 312. [91 F.N. Griffith, A. Katdare, J.L. Fox and W.I. Higuchi, J. Colloid Interface Sd. 67 (1978) 331. [10] J. Christoffersen, to be published.
[11] V.G.
679
Levich, Physicochemical Hydrodynamics (Prentice
Hail, Englewood Cliffs, NJ, 1962) ch. II. [12] D.P. Gregory and A.C. Riddiford, J. Chem. Soc. (1956) 375b.
[131Y. Avnimelech, E.C.(1973) Moreno NatI. Bur. Std. 77A 149.and
W.E. Brown, J.
Res.
KINETICS OF DISSOLUTION OF CALCIUM HYDROXYAPATITE II. Dissolution in non-stoichiometric solutions at constant pH Jq~rgenCHRISTOFFERSEN and Margaret R. CHRISTOFFERSEN Medicinsk-Kemisk Institut, Kqlbenhavns Universitet, Raadmandsgade 71, DK-2200 Copenhagen N, Denmark Received 24 April 1979
The influence of the total concentrations of calcium and phosphate ions on the rate of dissolution of calcium hydroxyapatite microcrystals in sub-saturated solutions at constant pH has been studied. The rate of dissolution is found to be a function of [CaIa[P0 4], where a is 3 for very dilute solutions. For more concentrated solutions a is close to 1.67. The results obtained explain the empirical results found in experiments in which a disc of calcium hydroxyapatite is dissolved in a rotating disc apparatus. The concentrations close to the dissolving surface have been calculated from the bulk concentrations, the diffusion coefficients, and the angular velocity of the disc. The “two-site dissolution” isshown to be an artifact due to mathematical curve fitting, which reproduces the experimental data quite well, but which has no relation to the surface processes governing the dissolution. 2/s and D 10_17 cm2/s for found D’ions 10-10 cm along pores in the enamel and fluoride diffusing in the crystallites respectively. Flim and Arends [5a1 45Ca in have measured diffusion for cm2/s or bovine enamel. The result coefficients was D 10_12 less. Arends [5b] has recently found a similar value for the diffusion coefficient of 32P0 4 in bovine enamel. For solutions of stoichiometric composition, i.e. Ca/P = 1.67, we have shown [3] that the rate of dissolution of HAP microcrystals at constant pH can be described as a function of m0, the mass of crystals used in an experiment, rn/in0 the fractinn of the mass remaining and C/Cs the ratio of the concentration of dissolved HAP to the solubility of HAP at the appropriate pH. The rate was shown to be controlled by surface processes, which means that the composition of the solution close to a dissolving surface is the same as the bulk composition. For non-stoichiometric composition of the solution, the rate was shown to depend more on the concentration of calcium than on the total concentration of phosphate. We found 3’3[P0 that the rate is a function of [Ca] 4]. We have, for a larger concentration range, investigated the influence of non-stoichiometric composition of the solution on the rate of dissolution of HAP microcrystals. We use the result of this study to show
1. Introduction The rate of dissolution of calcium hydroxyapatite Ca 10(P04)6(OH)2, calledofHAP, plays an important role in thehereafter understanding decalcification of bone and tooth material. In vivo carious lesions are characterized by an initial white spot formation in which sub-surface demineralization has occurred. To understand the formation of these white spots, at least the following problems should be elucidated: (a) how fast HAP crystals dissolve, and (b) how fast can dissolved material diffuse along pores or channels from the sub-surface lesion and into the surrounding medium, Jongebloed, Berg and Arends [1] have studied the change in morphology of large HAP crystals which were treated with a SM citric acid solution. Large holes formed along the c-axis of the crystals. Jongebloed, Molenaar and Arends [2] also showed that acid treatment of enamel caused hole formation in the HAP crystals. We too have observed hole formation in microcrystals of HAP partially dissolved in aqueous solutions at constant pH [3]. The importance of crystal defects in HAP has been reviewed by Jongebloed et al. [2]. Flim, Kolar and Arends [4] have measured the diffusion coefficients for fluoride ions penetrating dental enamel. They 671
672
J. Christoffersen, M.R. Christoffersen
/ Kinetics of dissolution of Ca1 0(P04) 6(OH)2
that Wu et al. [6], Fawzi eta!. [7], Fox et al. [8] and Griffith et al. [9] have not taken all of the factors influencing the rate into account in their analysis of their experiments with rotating discs of HAP. They have therefore not proved the reality of their so-called “new two-site model” for HAP dissolution in these experiments. We have reanalysed their experiments and show in the following that their model has no physical meaning and that their results can be explained by a combination of transport and surface processes. The dissolution process may of course be governed by different mechanisms, depending on the composition of the solution. We intend to discuss this in a forthcoming paper [10].
to be kept constant, by addition of potassium hydroxide or nitric acid. All chemicals used were of analytical grade. In experiments in which HAP dissolved in dilute potassium phosphate buffers, traces of inorganic pyrophosphate were enzymatically removed from the buffers. The rate of dissolution of HAP in normal analytical grade potassium phosphate solutions was found to be considerably lower than the rate in a pyrophosphate-free potassium phosphate solutions of the same concentration. In experiments in which crystals dissolved in solutions containing an excess of calcium ions, calcium nitrate was added to the solution before the addition of crystals. 3. Analysis of experiments with microcrystals
2. Experimental Crystals were prepared and analysed and the experiments were performed as previously described [3], except that a Radiometer pH-stat and Radiometer recorders were used. A combined glass and reference electrode was used, which enabled the thermodynamic value of pK2(H3PO4) 7.20 + 0.02 to be calculated from pH measurements with stirring in dilute phosphate buffers in the ionic strength and pH range to be used, Prior to the addition of crystals to the solution, the pH of the solution was adjusted to the pH value —
p[P0~]~”
In figs. Ia ic the results of dissolving HAP crystals in non-stoichiometric solutions are given. The composition of the solution is given by p[P04] log[P04] and p[Ca] log[Ca] where [PU4] and [Ca] are the total concentrations in mol/l of phosphate and calcium in solution. The shaded area in these plots corresponds to supersaturated solutions. The line with the slope 1 defines stoichiometric composition of the solution, i.e. [Ca]/[P04] = 1.67 or p[PO4] = p[Ca] + 0.22. For the non-stoichiometric solutions, we have determined f/rn0 for chosen values of pH and rn/mo. From an appropriate point in
P[PO41~~”
p[CQ]
P[PO~]~~”
p[Ca]
p[Co]
Fig. 1. Lines connecting pairs of points defining the composition of solutions for which the dissolution rates J/m0 are the same. pH — 6.77 and rn/rn0 is equal to 0.8 (a), 0.7 (b) and 0.6 (c). The lines drawn with slope 1 correspond to stoichiometric concen trations, i.e. cp = 1.67. The values of Jim0 and a, for each of the other lines are given in table 1. The slope of these lines is a.
J. Christoffersen, M.R. Christoffersen / Kinetics of dissolution of Ca
1 0(P04) 6(OH)2
Table 1 Values ofJim0 and
a
for the lines in figs. la
Fig. la: rn/rn0
Expt.
=
C; pH =
6.77 Fig. ib: rn/rn0
0.8
J/m0 log a b c d e f g h
mol s
6.33 6.47 6.54 6.67 6.74 6.49 6.49 6.66 6.67 6.84
=
Fig. lc: rn/rn0
0.7
Jim0 g
a
log
2.9 2.3 2.0 2.2 2.0 3.6 3.1 2.1 2.4 1.9
6.53 6.76 6.91 7.16 7.26 6.58 6.62 6.80 6.90 7.12
figs. la lc defining the composition of a solution, a line is drawn to a point on the line defining stoichiometric composition. This latter point is determined as the point for a stoichiometric solution in which we have the same value of rn/mo, pH and f/mo as in the non-stoichiometric solution. Values of f/mo and the slopes of the lines in fig. 1 are given in table 1. From the slopes of the lines in fig. 1, the effect of [P04] and [Ca] on the rate of dissolution can be determined. For simplicity in the following discussion, we define the slope of these lines to be —a. The rate can easily0[P0 be seen to be constant for constant values of [Ca] 4]. From fig. 1 we see that a increases with increasing values of p[P04] and p[Ca]. Relatively close to the solubiity line, the lines for constant values of f/rn0 become parallel to the solubiity line. As shown [10], the rate of dissolution in this region becomes a function of the change in molar Gibbs energyfrom /.~Gd~ for the dissolution process. away the line, kinetic effects Further appear Iss, solubility to be important for the rate of dissolution in subsaturated solutions of non-stoichiometric composition. From the above we conclude that far from the solubiity line, the rate of dissolution depends on [Ca]a[P0 Thisofmeans rate of dissolution4]is with not aa~3. function L~Gforthat the the dissolution
mol s
1
[Ca] 1.67 [P0] r
1 1.67 ior~ 1
LCaJS
~‘‘-‘4]s
=
0.6
J/m0 g
1
a
log
2.5 2.0 1.6 1.8 1.7 2.7 2.7 1.7 1.7 1.6
6.81 7.22 7.40 7.85
2.5 1.4 1.4 1.7
6.72 6.80 7.03 7.24 7.59
2.1 2.1 1.5 1.6 1.7
mol s
1
g
1
a
if ion-pair formation can be neglected. We shall in the following relate this finding to observations obtained from rotating disc experiments.
4. General comments on the rotating disc experiments We give concentrations in the bulk index b, and interfacial concentrations, just outside the crystal or disc surface, index i. Index s refers to saturated solutions. The symbol w is the angular velocity of the disc. According to Levich [11], the rate of transport from the interface to the bulk, assuming steady state dissolution, is given by
i=
(D/6)(C~ Cb) (1) in which D is the diffusion coefficient and 6 is the thickness of a diffusion layer, given by 1D” \1/3( / ‘51/2 2 l”~” 6 1 61 ~ / ~‘ where v is the kinematic viscosity of the bulk. Gregory and Riddiford [12] have given the following expression for 6 ‘
— —
6
=
113(v/w)”2[l +O.35(D/v)°’36] 1.61(D/v) k 1 1.657 x 10-2 2/(rad ~l)l/2 cm = = ~“ .
process, which, proportional to for constant pH and ionic strength, is ln
673
(3)
In the following we shall use (3) whenever calculating a 6 value. The difference between (2) and (3) is about
J. Christoffersen, M.R. Christoffersen / Kinetics of dissolution of Ca
674
10(P04) 6(OH)2
3% in the calculated value of 6 for the experiments
discussed here. In a steady state the rate by which substance leaves the disc must be equal to the rate by which the dissolved substance is transported from the interface and to the bulk. We shall make use of the following approximations: (a) The diffusion 2/s.coefficient in solution of all relevant ionsEquilibria is l0~cminvolving ionic species in solution are (b) rapidly established, (c) The buffer capacity is so high that pH has the same value in the interface as in the bulk. (d) The ionic strength is so high that we can neglect any electrical potential differences in the solution even close to the interface. Combining (1) and (3) with the above assumptions enables us to express the rate of transport of calcium ions and of phosphate ion species in the following way DWV2 /Ca
=
([Ca]
[Ca]b)
i
(4)
pressed tablets of l’IAP having either two different sites of dissolution, or a very complicated dissolution mechanism which agrees with the two-site dissolution model presented [6 9]. Diffusion inside the disc is taken into account in the model presented. In the model the transport from the interface to the bulk is described by an equation identical to (I). Unfortunately analysis doesofnot information about thethecomposition thecontain interface. The transport equation (1) is combined with an equation describing the rate of dissolution R, in the inside of the disc. “To simplify the computational work, and for the lack of any compelling reasons for choosing another functional form”, R was assumed to be first order with respect to undersaturation, thus R
k(C
C~ Cb
k
Assuming
1
(7)
conditions gives the following expression for the flux, F, of HAP into the bulk,
Dw”2 =
C)
where C~ is the apparent solubility and C the concentration of HAP inside the disc. Combining (1) and (7) with appropriate boundary
([P04]1 congruent
04]b). [P dissolution,
(5)
(6/D) + (1/(kD’)112)’ (8) in which D’ is the diffusion coefficient of dissolved
F
/Ca/1p0
4 = 1.67, which is to be expected and is also found [6,7], enables us to calculate for the data [6,7] the interfacial concentrations [Ca]1 and [PU4]1and therefore also the calcium to phosphate ratio just outside the disc, cp1 = [Ca]1/[P04]1.
HAP inside the disc. In (8) there are two fitting parameters, C~and kD’. Taking the diffusion in the solid disc into account, two sets of fitting constants are obtained. C~is not given, but the corresponding solubiity products are given. Each set of fitting parameters are associated with a certain type of dissolution, called site one and site two dissolution. The values of the fitting parameters obtained are
5. Analysis of “the two-site model” Table 2
[6], Fawzi et al. [7], Fox et al. [8], and Griffith et al. [9] report data for dissolution of HAP obtained from experiments with a rotating disc. In these experiments the bulk solutions contain 0.1M aceticthe acid with pH 4.5 and and inertvarious electrolytes to raise ionic strength to 0.5M values of [Ca] and [P0 4]. The reported results appear to be accurate. Their main finding is that, for certain values2 of thea “kink” bulk concentrations, plots off against w~” have where the slope of plots of / against
Fitting parameters found [8j for eq. (8) and the “apparent solubiity” C~at the sites relative to the thermodynamic solubility C~ __________________________________________________ 5D (kD’)~2 pK 10 2/s) 5 C~/C~ (s/cm) (cm
w112 is not monotonically decreasing. In refs. [6 9] the data obtained are explained to be a result of the
Site 2 1.0 111.3 127.72 0.20 1.5 64.9 130.26 0.14 __________________________________________________
Wu et al.
—
HAP Site 1
1.0 1.5 1.0 1.5
Unknown Unknown 1252.3 844.3
116.4
1,0
116.4
1.0
119.90 122.11
0.60 0.44
J. Christoffersen, M.R. Christoffersen / Kinetics of dissolution of Ca
1 0(P04) 6(OH)2
given in table 2. The thermodynamic solubiity product determined by Avnimelech, Moreno and Brown [13] and the “solubilities” of the different
sites relative to the thermodynamic solubility are also given in table 2. In ref. [6] the concentration profile
6~< l0~6
0.1 1um.
675
(10)
For enamel we obtain 6~<0.1 pm; for the disc 6~ was 5 10 pm. The reason for this difference is that
we have assumed diffusion in enamel to be a much
in the solid is given. The concentration becomes constant 5 10 pm away from the dissolving surface.
slower process than diffusion in the disc appears to be. We can conclude if the “two-site model” had been
6. Consequences of the two-site model
will from of the datafor[6,7], we find be theseen model alsoourtoanalysis be a poor model the dis-
One of the consequences of the applied model is that the apparent solubiities C~ at constant pH depend on the composition of the bulk and on the angular velocity. At a distance of 5 10 pm away from the dissolving surface of the disc, the concentration of HAP dissolved in the solid disc becomes constant and equal to either C~1or C~2,depending
solution of the disc, made of pressed HAP crystals.
on which site governs the dissolution. Much further away from the dissolving surface of the disc, say 200 pm away, the concentration of HAP must be very
constant. The rate of dissolution in this region must be very low dueis to a very low to ratetheof concentration the transporl process, which proportional gradient. Despite this, the concentration of HAP dissolved in this region, can for certain composition of the bulk, be changed from C~1to C~2= 0.3 C~1just by changing the angular velocity. We cannot imagine how any mechanism could have this effect. If the model, despite the above criticism, should be able to predict any important effect for demineralization and remineralization of enamel, the diffusion coefficient in the solid should be comparable to those found for the fluoride ions and calcium ions in enamel [4,5]. For the experiments reported, we can, for enamel, estimate the maximum value of the distance, over which the diffusion plays an important role, by demanding the rate of transport in the solid to be comparable to the rate of transport over the diffusion layer 6. We obtain D’(C~ Ci)>D(Ci
Cb),
(9)
a good model for dissolution of the disc, it would have been a poor model for dissolution of enamel. As
7. Analysis data from Wu Ct a!. [6] and Fawzi et a!. [7] We shall in the following reanalyse the experiments [6,7] and show that the kinetics of dissolution of HAP can be explained without introducing the “two-site model’.
In figs. 2a 2c data from Wu et al. [6] are plotted. In fig. 2a the flux-density, j, interfacial of HAP dissolved is 112. In fig. 2b concentraplotted against w tions, C 1, are plotted against ~~1/2 C~is [Ca]1if C~j~ 1.67 and C1 is [P04]~ if cpj> 1.67. In fig. 2c the interfacial value of the calcium to phosphate ratio is plotted against ~ In figs. 3a 3c similar plots are given for data from Fawzi et al. [7]. From these plots are seen: (1) A change in ~ causes changes not only in 1~but also in [Ca]~,[P04]1 and cp1. (2) The above changes depend on the composition of the bulk. The rate by which the surface processes take place are completely determined by the interfacial concentrations, which we can describe by C1 and cp1. The interfacial concentrations of course strongly depend on the bulk concentrations. The bulk concentration corresponding to C1 will in the following be called Cb. In the steady state the rate of dissolution can also be described by (4) or (5). Using C~as defined above, wehave 112
Dw
6
D’(Cs
C
1) D’c5 D~C1 Cb) 6
/
(C1 which df 112=~D d~ from
—
C’b
>
(11)
Cb)
[(~~ —
~b) + w~2
dC~ ~].
(12)
J. Christoffersen, M.R. Christoffersen / Kinetics of dissolution of Ca
676
1 0(P04) 6(OH)2
2s1
103C
motcm
[Ca]1
1
1;
[P04)
c.j’’/(rad SI’
-
~
:~ ~ u’/(rad s
I’
04]b as folFig. 2. Data from Wu et a!. [6]. Bulk composition: 0.1M acetate buffer, pH 4.5, ionic strength 0.5M [Ca]b and [P lows: Crosses: [Ca]b = [~O4]b 0; Squares: [Ca]b 2.2 X io—~M; [~O4]b 0; Triangles: [Ca]b = 0; l~O4]b— 5.5 X i0~M; Circles: [Ca]b = 5.0 x 10 ~ M; [~°4lb 0; Diamonds: [Ca]b = 0; L~°4lb = 110 2 M. (a) The flux-density, j, of HAP through the interface plotted against the square root of the angular velocity, w112. (b) The inter facial concentrations, C 112. If cpj > 1.67, C 112. ~ 1, plotted against 1 [P04]1, otherwise C~= [Cali. (c) Calcium to phosphate ratio, cp~,at the interface plotted against w
The term w 2dC 2 is never positive for the 1/do~’’ experiments discussed. If the interplay between the surface processes and the transport processes is such
‘
onically decreasing function of w1”2. In fig. 4 we give data extracted from fig. 2 and fig. 3, for which p[P0 4]1 could be plotted against p[Ca]1 for constant values off. From fig. 4 is seen: (1) For high values of p[P04]~ and low values of p[Ca] 1, i.e. with for cp1> rate on [CaJr[PU4]1 a 2,1.67, wherethe a is the depends slope of the
w high 2dC 2may be very important for the term w~ 1/dw” low2.values of w,high causing! veryterm fast with For very values to of decrease W, the same may w~ be important due to the high value of W. Hereby we have shown that d//dW”2 is not necessarily a monot-
tangents to the curves in fig. 4. This result is practically identical to the result shown in fig. 1. (2) For low values of p[PO 4]~ and high values of p[Ca]1 the rate depends on [Ca]r[P04]1 a of 1. This change in a can easily be due to with the use phosphate buffers containing a small amount of inorganic pyrophosphate.
that for some compositions of the bulk, 2I IdC./d~’2
(13)
~‘
dC’dc~.~ 1! w
low
1
J. Christoffersen, M.R. Christoffersen / Kinetics of dissolution of Ca
1 0(P04) 6(OH)2
I
_________________________
mo!~~
I
677
I
,~:~11’1~
a
b 112/( rad s..I)I/2
w”’ /1 rad s —, I’”
w
‘~~=~-1-~
1.2-
10-
-
0.2-
o ~
rad s —,
Fig. 3. Data from Fawzi et a!. [71.Bulk composition: 0.1M acetate buffer, pH = 4.5, ionic strength 0.5M, [Ca]b and l~°4)bas follows: Solid circles: [Caib = [~°4]b = 8.52 X 10~ M; Triangles: [CaIb— [~O4lb = 1.30 X iO’3 M; Hexagons: LCalb = [P04]i, = 1.704 X iO~M; Crosses: [CaIb = [P0 4 lb = 3.524 X i0~ M; Squares: [CAIb = 8.536 X l0~ M; FPO4]b = 8.356 X 10~ M; 1”2. (b) Interfacial Open circles: [Ca]b = 1.48 x i0~ M; [~°4lb = 1.48 x 10—2 M. concentrations, C If cpj> 1.67,plotted C (a) The flux-density, j, of HAP through2.the interface, against the square root of the angular velocity w 1, plotted against 2. w” 1 = [P04l1,otherwise C4 = [Cal1. (c) Calcium to phosphate ratio, cp1, at the interface, plotted against w’-’
J. Christoffersen, M.R. Christoffersen / Kinetics of dissolution of Ca
678
1 0(P04} 6(OH)2
I
I
I
in analytical grade phosphate salts. For the non-
I
30
-
28
-
00
26
-
p[ P0 4]i
o
2/.
-
22
-
nO. (•,0
.05,10
20
I
cm s
mol 2
-
-
-
22
21. [Ca]
rate of dissolution can only be described as a function of ~Gdi5s if a = 1.67. In all other cases specific
independently the rate of dissolution is effectively a function of two concentration variables, meaning that the cannot be expressed a function of only one rate concentration parameter.asWe have here used [Ca]
-
I
described by one concentration parameter, which of course can be chosen as ~Gdjs5. For cp1 ~ 1.67, the
kinetic effects of the ions appear to be[PU4] important. In experiments where [Ca] and are varied
10
cm
I
stoichiometric experirrients [6,7] the rate of dissolution is not a function of ~GdisS. For a constant value of pl-I and Cp1 — 1.67, the rate of dissolution can be
I
I
26
2 8
p 11g. 4. Plot of the flux-density as function of p[P04]1 and p[Ca]~. The line drawn with the slope — I Lorresponds to stoichiornetric composition, i.e. cp1 1.67. Smooth curves 9 niol cm 2 value 1 Squares: / are drawn through points with the same of the flux density. 1.0 x 10 Circles: ~ niol cm / — 20.5s~. X i0 Solid symbols: data from Wu et al. [6]. Open symbols: data from l~awzieta!. [7].
1 and [PU4]1 or one of them together with cpj to describe the rate whereas Wu, FawLi, Fox et a!. have used a mean concentration C, in the bulk solution and ~‘, an apparent solubility in the disc. Their description is not satisfactory because C does not correspond to a real solubility. The two parameters C and c’ can, as shown [6,7], be used as mathematical parameters in a curve fitting procedure.
Acknowledgements We thank Professor Arne E. Nielsen for stimulating
8. Conclusion
discussions and for helpful comments on the manuscript. We are indebted to Niels Kj~rgaardfor technical assistance. The Danish Natural Science Research Council are thanked for the grant J. No. 511-8181
From the above analysis of our data and the reanalysis of data from Wu et a]. [6] and Fawzi et a]. [7] we conclude that all the observed dissolution rates both of a finely dispersed suspension and of a rotating disc can be described in the following man-
ner. The rate of dissolution is a function of [Ca]~’[PU 4], where —a is the slope of a line in fig. 1 or the slope of a tangent to the curves in fig. 4. For pH 7 a is 3 for very dilute solutions; for more saturated solutions a is close to 1.67. For a = 1.67 the rate of dissolution is simply a function of L’xGdISS. For the data [6,7] at pH 4.5, a is about 2 for solutions contaming an excess of calcium ions. For solutions contaming an excess of phosphates a is about 1. The change in a from 2 to 1 we think is due to the small
amount of inorganic pyrophosphate normally present
(and previous grants).
References [1] W.L. Jongebloed, P.J. van den Berg and J. Arends, Cakif. Tiss. Res. 15 (1974) 1. [2] W.L. Jongebloed, I. Molenaar and J. Arends, Calcif. Tiss. Res. 19 (1975) 109. [3] J. Christoffersen,M.R. Christoffersen and N. Kj~rgaard, J. Crystal 43 and (1978) 501. J. Bioeng. 2 (1978) [4] G.J. Film,Growth Z. Kolar J. Arends, [5] (a) G.J. Flim and J. Arends, Calcif. Tiss. Res. 24 (1977) 59 (b) J. Arends, [6] M. Wu, W.I. private Higuchi,communication. J.L. Fox and M. I riedman, J. Dental Res. 55 (1976) 496. [7] M.B. Fawzi, J.L. Fox, M.G. Dedhiya, W.l. Higuchi and J.J. Hefferren, J. Coiloid Interface Sci. 67 (1978) 304.
J. Christoffersen, M.R. Christoffersen I Kinetics of dissolution of Ca
1 0(P04)6(OH)2
[8] J.L. Fox, WI. Higuchi, M.B. Fawzi and M. Wu, J. Colbid Interface Sci. 67 (1978) 312. [91 F.N. Griffith, A. Katdare, J.L. Fox and W.I. Higuchi, J. Colloid Interface Sd. 67 (1978) 331. [10] J. Christoffersen, to be published.
[11] V.G.
679
Levich, Physicochemical Hydrodynamics (Prentice
Hail, Englewood Cliffs, NJ, 1962) ch. II. [12] D.P. Gregory and A.C. Riddiford, J. Chem. Soc. (1956) 375b.
[131Y. Avnimelech, E.C.(1973) Moreno NatI. Bur. Std. 77A 149.and
W.E. Brown, J.
Res.