J . Phys. Chem. 1994,98, 1689-1694
1689
Unexpected pH Dependence of Dissolution Kinetics of Dicalcium Phosphate Dihydrate Jingwu Zhangt and George H. NancoUas' Department of Chemistry and Biomaterials, State University of New York at Buffalo, Buffalo, New York 14214 Received: August 24, 1993" The dissolution kinetics of dicalcium phosphate dihydrate ( C a H P 0 ~ 2 H 2 0 ) an , important biological mineral, has been studied using the constant composition method at 37 "Cover a range of pH and relative undersaturations (a). At u = 0.5, the dissolution rate, controlled mainly by volume diffusion, has an unexpected minimum at pH 5.0. This minimum occurs at higher pH values at lower driving forces where the rate is determined to a greater extent by surface processes. The pH dependence observed at u = 0.5 can be quantitatively interpreted by assuming independent desorption and diffusion of various ions in the presence of high concentrations of supporting electrolytes. Our theoretical model allows for the assessment of the relative importance of volume diffusion and adsorption solely from the kinetics data, which has been difficult in the past because both processes yield first-order kinetics. The rate minima at lower undersaturations may be attributed to the combination of proton catalysis at lower pH and solution stoichiometry under more basic conditions.
Introduction
Experiment
As compared with other calcium phosphate phases such as octacalcium phosphate [OCP, Ca~H2(P04)6.5H20) and hydroxyapatite (HAP, Calo(OH)2(P04)6],dicalcium phosphate dihydrate (DCPD, Brushite, CaHP04.2H20),with lower surface energy and higher degree of hydration, is the first crystalline phase to be precipitated in neutral and slightly acidic solutions.'q2 DCPD has consequently been suggested as an intermediate phase in the formation of bone and teeth., It has also been invoked in kidney stone, tartar, and pathological calcification.' On theother hand, because of the relatively high solubility of DCPD under physiological conditions, the solution is undersaturated with respect to this phase in the presence of other calcium phosphate minerals. There is therefore considerable interest in elucidating the mechanisms of DCPD dissolution. pH, an important factor in controlling the precipitation and dissolution of calcium phosphates, varies over a wide range in oral environments and in the initial stage of bone implantation. However, no systematic investigation has been made of the influence of pH on the kinetics of DCPD dissolution. In the experiments of Nancollas and M a r ~ h a l lthe , ~ continuous change in pH during dissolution made it difficult to reveal any pH effect. Although a more recent study by Christoffersen and Christoffersen was made using a constant-composition m e t h ~ d the , ~ influence of pH on the dissolution rate was not investigated at a given undersaturation. The lack of pH dependence observed in the latter work was contrary to the marked pH effect found for the dissolution rates of HAP and OCP.697 As will be discussed in the next section, our experimental results show that the dissolution rate depends strikingly on pH. At the same degree of undersaturation (cr = 0.5), a minimum rate is found at pH 5.0. This is quite surprising because the rate could be expected to increase with decreasing pH due to proton catalysis, as was the case for HAP dissolution.6 Although a minimum rate near neutral pH has been found for the dissolution of a number of clay minerals,E-lO the increase in rate at higher pH is attributed to an increased adsorption of OH- ions which promote the detachment of surface cations.8 However, in the present case, OH- adsorption is unlikely to be significant at pH values of 5.06.4, and a different theoretical approach is required to interpret the results.
Materials and Methods. The preparation and standardization of stocksolutions, Ca(N03)2,KH2P04, KNO,, KOH, and HN03, using Reagent grade chemicals and deionized CO2-free water, have been described elsewhere." DCPD crystals, prepared by the method of Marshall and Nancollas,12 were aged as a slurry at pH 5 for 2 months before filtering and drying at room temperature. The crystals were platelike with an equivalent spherical diameter of 14 pm and a specific surface area of 1.2 m2 g-l. The kinetics of DCPD dissolution was studied using the constant-composition method.13.14 At any given pH and undersaturation, the required solution composition, containing equal molar total calcium and phosphate, was computed taking into account phosphate protonation and ion pair formation equilibria. Activity coefficientswere calculated using the Davies equation.15 The undersaturated solutions were prepared by mixing stock solutions in a water-jacketed Pyrex vessel thermostated at 37.0 f 0.1 "C. pH was measured using glass and calomel reference electrodes (Corning) calibrated with at least two standard buffer solutions: potassium hydrogen tartrate (saturated at 25 "C, pH = 3.548 at 37 "C), 0.05 m potassium hydrogen phthalate (pH = 4.022), and a mixture of 0.025 m KH2P04 and 0.025 m Na2H P 0 4 (pH = 6.841).16 During solution preparation and dissolution experiments, atmospheric carbon dioxide was excluded from the vessel by a continuous flow of nitrogen presaturated with water vapor. Dissolution was initiated by the introduction of known amounts of DCPD powder. A uniform suspensionwas ensured by magnetic stirring at a rate of 450 rpm unless otherwise stated. In a closed system, DCPD dissolution would lead to an increase in pH as well as calcium and phosphate concentrations. Thus, a titrant containing HNO, and K N 0 3 was used in order to maintain the solution composition constant.l1 A glass electrode coupled with a Bronsted type Ag/AgCl referenceelectrode was used to monitor the pH during dissolution experiments, and a threshold of change of about 0.003pH was sufficient to trigger titrant addition through a pH state (Metrohm, pH meter E5 12, Impulsomat E473, MultiDosimat E41 5, Brinkmann). The dissolution rate was calculated from the recorded volume of titrant. Results. The relative undersaturation, u, is defined by eq 1:
t Present address: Department of Inorganic, Analytical and Applied Chemistry, UniversityofGeneva,Sciences II,30Quai Ernest-Ansermet, 121 1 Geneva 4, Switzerland. Abstract published in Advance ACS Abstracts. January 15, 1994.
0022-365419412098-1689$04.50/0
u=
1 - [(Ca2+)(HPO~-)/K,,]'/2
(1) where (Ca2+) and (HP04*-) are the activities of Ca2+ and HP0d2- ions. Speciation calculations were based on consecu0 1994 American Chemical Society
1690 The Journal of Physical Chemistry, Vol. 98, No. 6, 1994 TABLE 1: Summary of Equilibrium Constants equilibrium constant 6.22 X lo-' mol L-I 6.59 X 10-8 mol L-I 6.6 X mol L-I 2.7 L mol-I 591 L mol-' 1.35 X lo6 L mol-I 25 L mol-' 2.40 X mol2L-2 2.20 x 10-7 mol2 L-2
ref 17 18 19 11 21 21 21 20 11
tive dissociation constants of phosphoric acids, the activity product of water, and the ion pair formation constants of CaOH+, CaH2PO4+,CaHP04, and CaPOd-. The solubility activity product of DCPD, Ksp,determined at an ionic strength of 0.1 M (KN03) was 2.20 X lo-' (mol L-I)2.11 The equilibrium constants are summarized in Table 1. For experiments at pH 3.74, where the ionic strength was probably too high to use the Davies equation to calculate the activity coefficients, the solubility was estimated by using the following "kinetic" interpolation. While DCPD crystals dissolved at total calcium concentration, Tca = 6.85 X M, extensive growth occurred at Tca= 7.15 X M; X-ray diffraction confirmed that the growing phase was DCPD. Thus, the solubility could be estimated by averaging these two values, with an error less than 2%. The driving force at this pH was calculated from eq 2:
where Teas was the equilibrium total calcium concentration. It should be noted that eqs 1 and 2 were not identical due to ion association. However, the difference in u was only about 1% over the entire undersaturation range studied. The experimental conditions and dissolution rates are summarized in Table 2. The rates were calculated from titration curves as described previously.22 The standard deviations in u, given in parentheses in Table 2, were estimated based on a pH variance of 0.005 in the undersaturated solution. The relative error in the rate measurement was estimated to be 5%. Previously published data at pH = 5.66 were also included in the present analysis. I 1 Figure 1 shows the dissolution rates as function of time for typical experiments at two different pH values. The broken curves represent data corrected for surface area changes assuming isotropic dissolution. As was previously observed at pH 5.66, the corrected rate decreased with time," suggesting a decrease in the number of dislocations.22 In order to avoid the increasing influence of this deceleration at larger extents of dissolution as well as greater uncertainties in determining the initial rate, the rate at 10% dissolution was used for computations. Logarithmic plots of rate against u at three pH values are given in Figure 2. Data at other pH values are excluded for clarity. The linearity of these plots suggests that the rate can be expressed by an empirical equation of the form: J = kdun
(3)
where kd is the rate constant and n is commonly referred to as the effective order. These values, summarized in Table 3, are calculated by using a rigorous nonlinear least-squares method considering the uncertainties in both undersaturation and rate d a t a . l I ~At ~ ~several undersaturations, the rate is calculated from eq 3 and plotted as a function of pH in Figure 3. The error bars represent twice the standard deviations estimated from those in both k d and n values. They are considerably greater than the 5% estimated from rate measurement alone" because the uncertainties in u are also included. At pH 4.4 and pH 5.17, an error of 15% was assumed since the number of data points was too few for meaningful statistical estimations. It is extremely interesting to note that the dissolution rate has a minimum value at pH of
Zhang and Nancollas TABLE 2 Summary of DCPD Dissolution Experiments at 37 O C in Solutions with Tc~. = Tp Using KNOJ as the Supporting Electrolyte' exPtb T d m M PH ZIM U Jlmol m-2 s-I 35 36 37 38. 39 40 41 42 43 44 45 46 47* 48 49 50* 51 52 53 54' 55 56 57 58 59 60 61 62 63 64 65 66
27.0 24.0 36.8 36.8 49.0 58.0 65.0 68.3 12.8 22.0 22.0 22.0 22.0 8.95 8.10 8.10 10.5 13.4 15.3 15.3 9.96 5.52 8.19 1.42 1.82 2.08 2.22 2.34 2.37 2.41 2.43 2.44
3.74 3.74 3.74 3.74 3.74 3.74 3.74 3.74 4.10 4.10 4.10 4.10 4.10 4.40 4.68 4.68 4.68 4.68 4.68 4.68 5.07 5.17 5.17 6.46 6.46 6.46 6.46 6.46 6.46 6.46 6.46 6.46
0.258 0.108 0.263 0.263 0.271 0.277 0.282 0.285 0.103 0.107 0.107 0.107 0.107 0.101 0.101 0.101 0.102 0.103 0.104 0.104 0.102 0.101 0.101 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100
0.614 (0.008) 0.508 (0.003) 0.474 (0,010) 0.474 (0,010) 0.300 (0.014) 0.171 (0.016) 0.071 (0.019) 0.024 (0.020) 0.593 (0.002) 0.312 (0.004) 0.312 (0.004) 0.312 (0.004) 0.3 12 (0.004) 0.595 (0.002) 0.494 (0.003) 0.494 (0.003) 0.348 (0.004) 0.173 (0.005) 0.060 (0.005) 0.060 (0.005) 0.041 (0.005) 0.399 (0.003) 0.1 14 (0.005) 0.427 (0.002) 0.272 (0.003) 0.173 (0.003) 0.120 (0.003) 0.074 (0.003) 0.063 (0.004) 0.048 (0.004) 0.040 (0.004) 0.037 (0.004)
1.74 X 1.19 X 1.04 X 7.00 X 8.06 X 5.22 X 2.33 X 9.40 X 5.93 x 2.67 X 2.68 X 2.58 X 1.54 X 3.59 x 1.99 X 1.48 X 1.53 X 5.40 X 2.26 X 2.18 X 1.21 x 1.40 X 3.14 X 2.71 X 1.28 X 8.59 X 5.22 X 2.21 x 2.28 X 1.17 X 7.72 X 9.92 X
10-4 10-4 10-4 10-5 1k5 10-5 10-6 10-5 lk5 10-5 le5 10-5
10-6
10-6 10-6 10-6 10-5
10-6 10-5
10-6 10-6
10-6 10-6 10-6 10-7 lo-'
a The stirring speed was maintained at 450 rpm except those runs marked withasterisks (* at 225 rpm). The rates weredetermined at 10% dissolution without corrections for surface area changes. * Experiments 1-34 at pH = 5.66 can be found in Table 3 of ref 11.
0
10
20
30
40
Timehin. Figure 1. Typical plots of dissolution rates as function of time: (0) experiment 57, pH = 5.17, u = 0,114; (A)experiment 65, pH = 6.46, u = 0.040. The corresponding rates, corrected for surface area change, are given as broken lines.
about 5 a t u = 0.5. Moreover, the minimum occurs at higher pH values as the driving force decreases. The dissolution rate was found to be sensitive to stirring speed at higher undersaturations (Table 2). The influence of solution hydrodynamics was extensively studied a t pH 5.66.1LA stronger dependency was observed as the undersaturation increased. Similar trends were apparent at other pH values. Thus at pH 4.68, a reduction in the stirring speed from 450 to 225 rpm resulted in a 26% rate decrease at u = 0.494 (experiments 49 and 50, Table 2), while no significant stirring effect was detected at a lower undersaturation, u = 0.057 (experiments 53 and 54). These results suggested that the dissolution of DCDP was controlled by
The Journal of Physical Chemistry, Vol. 98, No. 6,1994 1691
Dissolution Kinetics of Dicalcium Phosphate Dihydrate
calculation for electrolyte solutions containing more than three components.27J* However, this difficulty is remedied for systems involving a high concentration of indifferent electrolytes in which ionic diffusions can be considered to be independent. Such a diffusion model has been applied to dissolution of sparingly soluble electrolytes29JOand will now be modified to include the adsorption/ desorption process as well. For a dissolution rate controlled by volume diffusion and desorption, solubility equilibrium is attained in the adsorption layer due to rapid surface diffusion and ion detachment from kinks. The flux of species i into the bulk solution may be written as
1 o-.
-0 E
>
10-6
lo-' 0.01
1
0.1
J , = ki([ilO- [ i ] )
U
Figure 2. Logarithmic plots of rate against u at pH = 3.74 (a), pH = 5.66 and 5.68 (0)and , pH = 6.46 (A).
I
1 10.4:
(4)
where [ i ] is bulk concentration of species i and [ i ] is ~ its value at saturation. Since solubility equilibrium is maintained in the adsorption layer, we may practically regard [ i ]as~ the interfacial concentration although it is actually the concentration of a saturated solution that would be in thermodynamic equilibrium with the adsorption layer. The rate constant, ki, is given by eq 5.
N
k, = Di/(6 + A,)
-
E
'v)
Here, Di is the diffusion coefficient of species i, 6 is the diffusion layer thickness, and A , with the same dimension as 6, represents the resistance for ions to enter the adsorption layer.
105:
E
5
(5)
:
,
hi = Di/XiYiad 3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
PH Figure 3. Dependence of rate on pH at 17 = 0.1,0.3,and 0.500.
TABLE 3: Effective Dissolution Orders and Rate Constants PH kd/mol s-1 m-2 n 0.86 i 0.12 3.74 (2.23f 0.24)X 10-4 ~
(6)
where Viad and Xi are the adsorption frequency and jump distance. Assuming that the latter is the same as the jump distance during diffusion, a, A, is related to the activation free energies for adsorption, Ea&, and for diffusion, Evilby eq 7:
(7)
A, = a exp((Eadi - Ev,>/kT)
~~~~~~~
4.10 4.40 4.68 5.17 5.66 6.46
(1.14k0.43)X 6.69 X lW5 (4.37k 0.62) X 4.19 X (5.88 0.26)X (8.50f 0.90)X
*
10-4
lW5 lW5 lW5
1.26k 0.04 1.2(est) 1.10f 0.09 1.19 1.42i 0.03 1.37 & 0.05
volume diffusion and surface processes, the latter becoming more dominant as the driving forces decreased. Modeling. Our previous kinetics study" at pH 5.66 suggests that DCPD dissolution follows a spiral mechanism and that the lattice ions undergo the following processes: (1) detachment from a kink site, (2) surface diffusion away from crystal steps, (3) desorption from the surface, and (4) diffusion into the bulk solution. At higher driving forces, Q > 0.2, the dissolution rate is controlled mainly by volume diffusion and desorption. However, at lower undersaturation, the rate is controlled by surface diffusion and/or detachment. A comprehensive model for spiral growth, incorporating the four elementary processes, has been developed by Gilmer et al.24 and by van der Eerden.25 However, the model was based on a monocomponent crystal. Recently, we have found that the growth rate of ionic crystals, in general, depends on the relative concentrations and rate constants for individual lattice ions.26 The present system is more complex because of the influence of pH on the concentration of HP042-. In order to understand the pH dependence observed here, it seems necessary to take into account of characteristics of all ions involved during dissolution. While such a general model considering both volume diffusion and surface processes remains to be developed, a systematic method for calculating coupled ionic diffusion has formally been established although it is still difficult, in practice, to make the
Thus, Ai may be considered as the distance that an ion would travel in solution during the adsorption relaxation time. If 6 >> Ai, ki becomes the rate constant for a diffusion-controlled process, but when 6
In addition, the following assumptions are made to facilitate the calculation of dissolution rate: (1) Solubility equilibrium is
Zhang and Nancollas
1692 The Journal of Physical Chemistry, Vol. 98, No. 6,1994
attained in the adsorption layer. (Ca2+),(HP0;-),
2.0~10~
= Ksp
(12)
where (Ca2+)o and (HPO42-)o are activities in terms of bulk solution concentration. The validity of this assumption, in the case of spiral dissolution, requires that (i) the exchange of ions at kink sites is sufficiently rapid and (ii) the mean surface displacement of ions is much larger than the step distance. These requirements are largely satisfied for DCPD dissolution at high undersaturations.' (2) In the adsorption layer, equilibria are instantaneously established between various phosphate species and between ion pairs and related ions. Since the lattice ion, HP042-, is more basic over the pH range studied, the major reaction is one of protonation, which is diffusion controlled.31 Due to the exceedingly high proton mobility, it is reasonable to assume that the equilibration between phosphate ions is instantaneous. The ion pairs are introduced in our model in order to calculate the ionic activities, it is uncertain if they are in contact and behave as a kinetic unit. Thus, it is justified to assume an instantaneous attainment of ion pair formation equilibrium. (3) The adsorption resistances for H+ and OH- ions are negligible (AH = AOH= 0); those for other ions are the same (Ai = A). The former assumption is reasonable because of the unique transfer mechanisms of proton and hydroxyl ions, but the latter is not as easy tojustify. For adsorption to take place, the following steps seem to be necessary: (i) dehydration of the surface site, (ii) partial dehydration of the ion, and (iii) a diffusive jump toward the surface. Of course, intermediate states may also be involved such as one with water molecules between the surface site and the adsorbing ion, contact adsorption taking place when the sandwiched water molecules escape. But for simplicity, we will consider that the three steps occur simultaneously during adsorption. The activation energy may then be written as
where EdchS, Ed&, and &if are the activation free energies for dehydration of the surface site, partial dehydration of the ion, and the diffusive jump, respectively. As for the activation energy of volume diffusion, Edifisexpected to be of the same magnitude for all ions. Since calcium ions are adsorbed on phosphate sites which require much smaller activation energy for dehydration, the adsorption resistance for Ca2+ions arises mainly from their dehydration. For adsorption of phosphate ions, although their dehydration energy may be negligible, the dehydration of surface calcium sites may require considerable energy. In addition, the adsorption frequency data for calcium and phosphate ion are not available. Thus, we will assume that the adsorption resistances are the same for all ions with the exception of H+ and OH-. There are only two unknown parameters in our model, 6 and A, which will be determined from rate data at Q = 0.5 using a nonlinear least-squares method.' The computational procedure is outlined below: (i) Initial estimates are made of the parameters 6 and A. (ii) The rate constant, ki,is calculated for each species from eq 5 and its diffusion coefficient (to be discussed later). (iii) The interfacial concentrations [il0 are computed by first expressing them as a function of [H+]o, [CaZ+lo,and [HP042-lo with suitable equilibrium constants between various species (Table 1). Substitution ofeqs 4intoeqs lOand 11 yields twosimultaneous equations involving the three species. Combining them with eq 12, [H+]o,[Ca2+]o,and [HP042-],can bedeterminednumerically. The interfacial concentrations of other species can then be calculated from equilibrium constants. (iv) The flux of individual species is obtained from eq 4 and the theoretical dissolution rate, J*, from eq 8 or 9.
1.5~10~
?J
E
7
-0
1.0~10~
v)
E
5
5.0~10~
0.0 I
4
5
6
I 7
PH Figure 4. Model fitting of dissolution data at u = 0.5.
TABLE 4: Diffusion Coefficients at 37 OC (in mz 9-l) H+ H3PO4 1 CaH2P04+ 1 11.2 OHH2P041.2 CaHP04 1 6.79 Ca2+ 1.06 HP0421.0 CaP041 PO4' 1 CaOH+ 1 (v) The sum of squared residuals of the rates is determined over all the pH values: k
where the weight, W,, is taken as the reciprocal of the squared standard deviation of the experimental rate, J k . The initially estimated parameters are readjusted until the sum of squared residuals reaches a minimum value using a nonlinear least-squares procedure. I In the presence of high concentrations of supporting electrolytes, the movements of the ions are no longer correlated. Thus, the diffusion coefficients used in our calculations may be estimated from the limiting conductivities based on the Nernst-Einstein relation. The equivalent conductivities of hydrogen and calcium ions at 37 O C are 405.2 and 76.5 f2-l cm2equiv-l, calculated from data at 25,35, and 45 OC by a linear interpolation.20 The diffusion coefficients are thus 1.12 X 10-8 and 1.06 X 10-9 m2 s-l, respectively. The conductance of the hydroxyl ion, extrapolated fromdataat 18and25 OC,is245 Wcm2equirl,andthediffusion coefficient is 6.79 X m2 S - ~ . ~ O From their corresponding equivalent conductivities, 33 and 57 Q cm2 equirl,32 the diffusion coefficients of H2PO4- and HP042- are calculated to be 8.8 X 10-10 and 7.6 X 10-I0m2 s-I at 25 O C . Using the mean activation energy for ionic diffusion, 16 kJ mol-I, the diffusion coefficients of HzP04- and HP042-at 37 OC are found to be 1.2 X and 1.0 X m2 s-1, respectively. The diffusion coefficients of H3P04 and ion pairs are unknown and taken as 1 X m2 s-l. All the diffusion coefficients are summarized in Table 4. The best agreement between theory and experiment is found at 6 = 22.9 f 0.5 pm and A = 0.34 f 0.22 pm. This result shows that the adsorption resistances for calcium and phosphate ions are negligible as compared with the diffusion layer thickness. Therefore, another calculation is performed by setting A to zero for all ions; the resulting value is 6 = 23.0 f 0.4 pm. The theoretical curve based on the latter calculation is presented in Figure 4, which is nearly indistinguishable from the computed data using two parameters. Figure 5 shows the ratio of interfacial to bulk concentrations at u = 0.5. As can be seen, there is a depletion of H+ and H3P04 at the interface with pH about 0.3 higher than the bulk value. This causes H+and H3P04to diffuse toward the surface while the other ions diffuse outward. The fluxes of several major species are plotted as a function of pH in Figure 6; the fluxes of all the species are given in Figure 7. With the exception of H3P04, all the phosphate species including ion pairs contribute to the transfer
The Journal of Physical Chemistry, Vol. 98, No. 6,1994 1693
Dissolution Kinetics of Dicalcium Phosphate Dihydrate E
-0-Ca -t-HZP
4-
0'
3-
+S % -!i!-*-¶.-!I
0 , 3
I
1
I
4
5
6
7
PH Figure 5. Concentration ratio of adsorption layer to the bulk solution at u = 0.5. 2.0x10-4 1.5x10'4
1
l.oxlo-4{
\\
-0 E
5
-2.0~10~ 3
5
4
6
7
PH Figure 6. Fluxes of various ions as a function of pH at u = 0.5. 10 3
cu
E
7
v)
107
3
4
5
6
7
PH Figure 7. Logarithmic plot of the ionic fluxes against pH.
of the dissolved phosphate ions to the bulk solution. At pH < 5 , H2PO4- has the major contribution, but the HP042- flux becomes greater at pH > 5 . The minimum rate occurs approximately at the pH value where JHP= J ~ 2 p . Discussion It is quite remarkable that the present model with only one or two parameters offers excellent agreement with experiment at higher undersaturations. Let us first examine the validity of the resulting parameters. The activation free energy for adsorption is given by eq 13. For the calcium ion, the activation free energy
for dehydration, Ed&,is 26 kJ m ~ l - l , ~and ' that for dehydration of a surface site (a phosphate ion), Edeb,must be negligibly small. It is often assumed that the activation energy for the diffusive jump toward the surface, after dehydration of both ion and adsorption site, is the same as for volume diffusion (&if = E, = 16 kJ mol-l).33 However, because a tunnel to the surface has already been created through dehydration, a smaller &if value should be expected. By assuming that the ion jumps a distance of about the diameter of a water molecule (3 X m), we find from eq 7 that the A value, 0.34 pm, corresponds to Edif = 10 kJ mol-', which is not unreasonable. The small A value also justifies assumption 3 regardless of the differences in the adsorption resistances among various ions. However, the diffusion layer thickness (6 = 23.0pm) ismuchgreater thanthe5Spmestimated using the sedimentation velocity of a spherical particle with equivalent volume.' 1,34 This discrepancy may be attributed to, at least, the following two factors: First, the platelike DCPD crystals will certainly sediment at a lower rate compared with spherical particles,3s leading to a higher 6 value. Second, our experimental rate is calculated based on the total surface area of crystals. As they have several different faces, the overall rate is expected to be dominated by the faster dissolving faces which are probably controlled by desorption and volume diffusion at high undersaturations.11 Therefore, the experimental rate, averaged over all faces, may be considerably lower, resulting in an apparently higher 6 value from our model fitting. However, a quantitative assessment of either factor is difficult at the present time in the absence of more diffusion layer thickness data of irregular particles and the dissolution rates of individual DCPD crystal faces. Nevertheless, the close agreement between theory and experiment suggests that the present model must have included the most important factors governing the observed p H dependence at high undersaturations. At lower undersaturations, minimum dissolution rates also exist but occur a t higher pH values (Figure 3). Because of the involvement of other rate-determining processes (such as surface diffusion and detachment), our model is not directly applicable. At present, we can only give a qualitative explanation for the occurrence of rate minima a t lower undersaturations. An increase in rate with decreasing pH has been observed for the surfacecontrolled dissolution of OCP and HAP,6>' which has been attributed to the increased detachment frequencies due to protonation of interfacial phosphate ions. This is in general agreement with our experimental data at lower pH values (Figure 3). But the effect of proton catalysis alone cannot explain the increase in rate at higher pH's. Zhang and Nancollas have shown that the rate of crystal growth/dissolution is a function not only of the thermodynamic driving force but also of the solution stoichiometry or, more precisely, the kinetic ionic ratio.26 The latter is defined as theconcentration ratioof lattice ions multiplied by their relative rate constants of the rate-determining process. For a crystal of symmetrical electrolytes, the rate has a maximum value when the kinetic ionic ratio is unity. If we assume that at very low undersaturation DCPD dissolution is largely controlled by detachment of Ca2+ and HP042-ions from kinks and their detachment frequencies are the same, a maximum rate would occur when their concentrations are approximately equal. In the present experiment with Tca= Tp,as pH increases, [HP0d2-]/ [CaZ+]will become closer to unity; the rate would be approaching the maximum if there were no proton catalysis. Therefore, the observed minimum is probably due to the interplay of proton catalysis a t lower pH and solution stoichiometric effects at higher pH values. As both volume diffusion and adsorption lead to a linear rate l a ~ , only ~ 3 the parameter combination, 6 + A, can be determined from kinetics data using classic crystal growth theory. Further assessment of their relative importance usually involves estimation of the diffusion layer thickness based on a hydrodynamic calculation. The present approach offers a method to calculate both diffusion layer thickness and the average adsorption
1694
The Journal of Physical Chemistry, Vol. 98, No. 6, 1994
resistance simultaneously from rates a t different pH's. In addition, it avoids the difficulty of selecting the solubility, C,, in a nonstoichiometric solution as encountered in using a monocomponent crystal growth theory." In our previous analysis of the kinetics of DCPD dissolution at pH 5.7,l' C, was taken as K,p1/2(=4.69 X 1V M). Using a detachmentaesorption-volume diffusion model, we obtained (A 6) = 7.0 pm, considerably smaller than the present value, 23 pm. If the latter value and the experimental rate (at pH = 5.66 and u = 0.5) are substituted into the rate equation of a monocomponent crystal
+
J = C,D,u/(J+ A)
(15)
we will obtain C, = 1.1 X 10-3 M, which is 2.3 times as large as Ksp1/*. Had the higher solubility value been used in our previous analysis, (A 6) would be 16 pm, much closer to the present value. The resistances for surface diffusion and detachment in our previous analysis would also be increased by a factor of 2. However, this would only result in an increase in the calculated activation energies of less than 2 kJ mol-'. Since our previous analysis is largely based on the calculated activation energies at lower undersaturations, the main conclusions about the dissolution mechanisms are not affected except that the adsorption resistance should be much smaller compared with that of volume diffusion in the light of the present results.
+
Acknowledgment. We thank the National Institute of Health for a grant (DE03223) in support of this work. References and Notes (1) Johnsson, M. S. A.; Nancollas, G. H. Crit. Rev. Oral Biol. Med. 1992, 3, 61.
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