J . Phys. Chem. 1992, 96, 5478-5483
5478
Kinetics and Mechanisms of Octacaicium Phosphate Dissolution at 37 C Jingwu Zhang and George H. Nancollas* Department of Chemistry, State University of New York at Buffalo, Buffalo, New York 14214 (Received: November 19, 1991)
Octacalcium phosphate (OCP) is regarded as an important biomineralization precursor during the formation of hydroxyapatite. In the present work, the kinetics of dissolution of OCP has been studied at 37 "C over a range of undersaturations using the constant composition method. The kinetics data are analyzed in terms of recent crystal growth theories using a nonlinear least-squares procedure. A rate equation is derived for a spiral dissolution following a detachment-dwrption-volume diffusion mechanism at very low kink densities. Volume diffusion appears to provide little resistance to OCP dissolution compared with processes occurring at the crystal surface. It is suggested that the ions detached from the crystal steps undergo surface diffusion before escaping into the bulk solution.
I. Introduction Although described over a century ago,] the importance of octacalcium phosphate (Ca8H2(P04)6-5HZ0, OCP) in biological mineralization was not fully recognized until its structure was determined by Brown et al. in 1962.2 OCP crystals consist of alternating hydration and apatitic layers, the latter closely resembling hydroxyapatite (Ca10(OH)2(P04)6,HAP).Z It has been suggested that OCP acts as a precursor for the formation of the thermodynamically more stable HAPS3 Although OCP is frequently present as one of the crystalline components in human dental ~ a l c u l i no , ~ conclusive evidence has been found for its involvement in bone f o r m a t i ~ n .One ~ of the difficulties of demonstrating OCP as a precursor in vivo is that, having a higher solubility, it is exposed to an undersaturated environment in the presence of apatite, resulting in partial or complete dissolution. It has been proposed that the hydrolysis of OCP to HAP proceeds through both solid-state transformation and dissolutionfreprecipitation proces~es.~!~ Since the exchange of ions and water molecules between solid and solution phases is involved, the latter mechanism is more probable. Strong evidence in support of this suggestion is the marked dependence of transition rate upon pH,' and there is therefore considerable interest in elucidating the mechanism of OCP dissolution. A kinetics study of OCP dissolution at 25 OC has recently been made by Verbeeck and D e v e n y d using a pH-stat technique at relative undersaturations above 0.5. In solutions of stoichiometric calciumfphosphate molar ratio, a change in the effective dissolution order from 1 to 3 at u = 0.72 was observed as the driving force decreased. This was attributed to a change in mechanism from one of volume diffusion to surface nucleation or the possible formation of HAP a t lower OCP undersaturationsSs However, as noted by these authors, the transition appeared to occur over too narrow an undersaturation range to support the first suggestion. Moreover, no evidence was presented for HAP precipitation.8 The present study at physiological temperature, 37 OC, using the constant composition method was aimed at extending the undenaturation range considerably (0.03 < u < 0.9). In addition, in order to obtain more accurate thermodynamic driving forces for OCP dissolution, the solubility product of OCP was determined under identical experimental conditions. Crystal growth models were applied to analyze the kinetics data, and a spiral dissolution mechanism involving kink formation was derived for the combined detachment, desorption, and volume diffusion processes. 11. Solubility Determination
For OCP, the reported solubility p r o d u ~ t , ~pK, * l ~ (K, = (Ca),B(H),*(P04),6), has an uncertainty of about f0.4, corresponding to a variation coefficient in the K,value of up to 100%. The large differences between the results of different workers, as much as an order of magnitude,I0 may be partially attributed to imperfections of current electrolyte theories. The present solubility determination has been made in the same Ca(N-
TABLE I: Determination of OCP Solubility at 37.0 O
C
[NO,-]/ Tc,/mM
Tp/mM
[K+]/mM
mM
pH
PIM
pK,
4.389 2.416 1.347 0.8378
3.292 1.812 1.010 0.6284
87.22 93.06 96.01 97.50
92.43 95.70 97.34 98.18
5.638 6.027 6.427 6.815
0.1003 0.1005 0.1002 0.1003
98.00 97.97 98.00 97.89
a
Ionic strength.
03)z-KHzP04-KOH-KN03-Hz0 medium as that used in the dissolution experiments. Reagent grade chemicals and deionized distilled water were used to prepare stock solutions. Calcium nitrate solution was standardized by EDTA titration, and the concentrations of potassium monohydrogen phosphate and potassium nitrate solutions were determined by gravimetry. Potassium hydroxide solution was made in a carbon dioxidefree atmosphere using washed KOH pellets. All solutions were fdtered twice through 0.22-bm Millipore filters before use. Solutions, saturated in OCP according to the literature solubility product: were equilibrated with OCP crystals at 37.0 f 0.1 OC for 2 days. pH measurements were made in situ with a combination pH electrode (Orion 915500) calibrated against two standard buffer solutions:" 0.05 mol kg-l potassium hydrogen phthalate (pH = 4.022) and 0.025 mol kg-I potassium dihydrogen phosphate and 0.025 mol kg-I sodium hydrogen phosphate (pH = 6.841). The total calcium concentrations, Tca, were determined by titrating the filtrates against a standardized EDTA solution containing Zn-EDTA complex.12 Solid samples were examined by X-ray powder diffraction which confirmed the absence of phase transformation. The equilibrium conditions are summarized in Table I. The activities of the lattice ions were calculated using Davies' extended Debye-Huckel e q ~ a t i o n I ~from 3 ~ ~mass balance expressions for total calcium and total phosphate with appropriate equilibrium constants by successive approximation for the ionic strength. Values adopted for the dissociation constants of M,Is Kz = 6.58 X phosphoric acid were KI = 6.22 X M,I6 and K3 = 6.6 X 10-13,'7and the water ionic product was 2.40 X M2.18 The formation constants for the ion pairs CaH2PO4+,CaHP04, Capo4-, and CaOH+ were taken as 2.7,19 591, 1.35 X lo6, and 25 M-l,zorespectively. Table I lists the resultant pK, values. The mean value, 97.96 f 0.05, lies between 97.4 f 0.4 and 98.6 f 0.4 from refs 10 and 9, respectively. The pK, has been found to be considerably dependent on the selection of the equilibrium constants for the phosphoric acid dissociation and ion pair formation.I0 Using the equilibrium constants of Tung et a1.,I0 the present pK, value becomes 97.84 f 0.06, overlapping the uncertainty range reported by those authors. When the equilibrium constants of S h y et ai.' are adopted, the present data yield a value, 98.08 f 0.06, which does not significantly differ from the value obtained in ref 9. In spite of the apparent satisfactory agreement, use of the K, value
0022-3654/92/2096-5478%03.00/0 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 13, 1992 5479
Octacalcium Phosphate Dissolution
50
4 Corrected Rate
0-0
lh' 0
.
.
.
1
.
,
.
.
120
60
,
180
.
.
, L0 240
Time/min
Figure 2. Dissolution percentage and rates as a function of time. The corrected rates are calculated by assuming that the crystals retain their initial geometry during dissolution.
Figure 1. Scanning electron micrograph of OCP seed crystals.
of either of these previous studies would introduce substantial errors into the calculated relative undersaturations of the present work. The solubility product can be alternatively written K,' = (Ca2+),8(HP042-),Z(P043-)2 = K5/K32= 2.52 X MI4 (1) In accordance with Verbeeck and Devenyns,8 the relative undersaturation was calculated from the equation c = 1 - [(Ca2+)8(HP04)2(P043-)4/K,']1/14 (2) where parentheses denote the activities of the enclosed ion species.
lTI. Dissolution Studies OCP seed crystals, kindly donated by Dr. W. E. Brown, were prepared by hydrolysis of dicalcium phosphate dihydrate (DCPD) in an acetate buffer.2' Chemical analysis22showed a molar calcium to phosphate ratio of 1.35 f 0.02 with characteristic OCP X-ray diffraction patterns and morphology (Figure 1). The specific surface area was 15.5 f 0.5 m2 g-' as determined by BET nitrogen adsorption (30:70 N2/He, Quantasorb 11, Q ~ a n t a c h r o m e ) . ~ ~ The dissolution experiments, initiated by the introduction of known amounts of OCP crystals, were conducted in magnetically stirred (450 rpm) double-jacketed vessels thermostated at 37.0 f 0.1 OC. Undersaturated solutions were prepared by slowly mixing Ca(N03)2, KH2P04, KN03, and KOH solutions. Nitrogen, saturated with water vapor at 37 OC, was purged through the reaction vessel to exclude carbon dioxide. The extent of OCP dissolution at a sustained undersaturation was followed using the constant composition t e c h n i q ~ e . ~A~glass . ~ ~ electrode (Corning 476022), coupled with a Bronsted type of Ag/AgCl reference electrode, was used as a probe to trigger titrant addition by means of a potentiostat (Metrohm Impulsomat E473, pH meter E5 12, Multi-Dosimat E41 5, Dosigraph E425, Brinkmann). Titrant solutions contained H N 0 3 and K N 0 3 with respective concentrations calculated from the equations [HN03], = 2[Ca(N03)2] - [KH2P0,] - [KOH] (3) [KNOJ, = [KH2PO4] + [KNOJ] + [KOH]
(4)
In all the experiments, the [Ca(N03)2]/[KH2P04]ratio was maintained at the OCP stoichiometric ratio, ionic strength 0.100 M, and pH = 5.66 f 0.01,
IV. Results The extent of dissolution at any instant can be determined from the recorded volume of titrant addition. A typical plot of the dissolution percentage (amount of OCP dissolved as a percentage of seed) as a function of time is shown in Figure 2. The overall dissolution rate, J, is defined as the number of moles of OCP
1E-114 . 0.03
.
.
. . . ,
0.10
0
1 .oo
Figure 3. Logarithmic plots of rates against u at different extents of dissolution.
dissolved per second divided by the total surface area of the crystals, AT J = -Tca/8 -
AT
dV dt
where dV/dt is the gradient of the titration curve. The initial value of AT is calculated from the specific surface area of the crystals, SA,and the subsequent values during dissolution are estimated from eq 6 assuming three-dimensional uniform dissolution,26
AT = m$A(m,/mo)2/3
(6)
where mo and m,are the masses of the crystals at time 0 and t, respectively. In Figure 2, the rates are plotted with and without correction for the surface area change. It can be seen that the corrected rate still decreases with time, suggesting a decrease in the crystal surface reactivity during dissolution. This may be interpreted by a decrease in the number of dislocations in the crystals, which markedly influences the rate dependence upon under~aturation.'~*~~ In order to avoid this complication, it is desirable to use the initial rates in analyzing the dissolution mechanisms as the dislocation densities are the same initially for experiments at all undersaturations. However, considerable errors are usually involved in the initial rate calculation. Thus, J values at 10% dissolution are used for the mechanistic determination. The rates at different extents of dissolution, summarized in Table 11, are normalized with respect to the initial seed surface area. The dependence of the dissolution rate on the relative undersaturation is represented as a logarithmic plot in Figure 3. Two straight lines are obtained from a least-squares regression of eq 7 using data at 0% and 20% dissolution. The slope of these regression lines is commonly referred to as the effective order, n, in the empirical rate equation where kd is a rate constant. n values at different percentages of dissolution are given in Table 111. However, it can also be seen
5480 The Journal of Physical Chemistry, Vol. 96, No. 13, 1992
Zhang and Nancollas
-
TABLE II: Summarv of Rates at Different Percentage of Dissolution J / 10-9 mol Tr./mM 0.325 0.675 1.33 2.36 3.05 3.50 3.80 3.98 4.10 -PI
a
0.922 0.839 0.683 0.442 0.28 1 0.176 0.106 0.065 0.038
0% 58.6 66.8 34.8 18.5 9.73 3.69 1.41 0.737 0.197
Aff
U
1.2 x 2.6 x 5.1 x 8.8 X 1.1 x 1.3 X 1.4 X 1.5 X 1.5 X
10-3 10-3 10-3 10-2
lo-*
5%
50.7 48.5 33.0 16.8 7.93 2.79 0.973 0.4 15 0.124
10% 42.3 37.4 30.6 15.2 6.90 2.23 0.657 0.217 0.064
s-I
15% 36.3 33.0 28.0 14.5 6.52 1.94 0.472 0.128
m-2 20% 34.4 30.7 25.3 12.8 6.03 1.78 3.10 0.100
30%
40%
20.6 10.8 5.35 1.53
16.9 8.36 4.27 1.23
0.075
0.038
Standard deviation in the relative undersaturation.
TABLE 111: Effective Order at Different Extents of Dissolution dissolution, % 0 5 10 15 20 30 40
n
1.77
1.89
2.06
2.13
2.21
2.42
2.61
from Figure 3 that, with the exception of initial rates, data at other reaction extents do not follow eq 7 with a single n value. At 10% dissolution, for instance, there appears to be a change in the slope at a = 0.5 from unity (n = 1.1 f 0.1) to 2.2 f 0.1 as the undenaturation decreases. This order change and the relatively larger n values at greater extents of dissolution are probably manifestations of greater deceleration at lower driving f ~ r c e s . ' ~ . ~ ' Thus, it does not necessarily indicate a change in ratedetermining processes. The n values at low dissolution percentages, with little influence from the surface reactivity change, suggest that OCP follows a spiral dissolution m e c h a n i ~ m . ~ ~ - ~ O V. Discussion A. Spiral Dissolution Models. As in the case of growth, crystal dissolution may proceed by surface nucleation and spiral mecha n i s m ~ . ~However, ~ - ~ ~ unlike growth, dissolution may also be initiated at crystal edges.33 This, along with the promotion effect of etch pit formation by elastic stress around dislocations, is primarily responsible for the commonly observed asymmetry between growth and dissolution rates.34 Recent kinetics studies of the growth and dissolution of dicalcium phosphate dihydrate (DCPD) suggested that dissolution from screw dislocations dominated the overall rate.'9*3s Thus, in the following discussion, the factors leading to the asymmetry are neglected and the rate equations developed for spiral growth are adapted to describe the dissolution process. During a spiral dissolution process, lattice ions detached from steps originating from a screw dislocation source may follow two simultaneous paths to reach the bulk solution: (1) direct detachment into the solution phase or (2) diffusion along the surface away from the steps before entering the ~ o l u t i o n . ~In ~ -the ~~ following, the dissolution rates from each path will be discussed separately. The dissolution rate of a crystal surface via the former path (direct detachment) is given by36338
where D,is the coefficient of volume diffusion, C,the solubility, yo the step distance in a spiral, a the size of a growth unit, and 6 the diffusion layer thickness. The parameter A,, defined by eq 9, characterizes the impedance for exchanging ions between the A, = a exp[(E[ - E,)/kT] (9) solution and steps,36 where E[ and E, are the activation free energies for direct integration to the step and for volume diffusion, respectively. The distance between two adjacent steps in a spiral, yo, can be calculated by solving the equation36 6 = Y O C Y O / ~ -~ Pl)[A,/a ~ + (I/..) In Cvo/a)l (10) where p c is the radius of a critical surface nucleus2* pc = -ya/kT In S (1 1)
S is the saturation ratio (S = 1 - a), y the edge free energy per ionic site, k the Boltzmann constant, and T the absolute temperature. For ions dissolving by the latter path (indirect detachment), resistances will be encountered from detachment, surface diffusion, desorption, and volume diffusion. These may be represented respectively by the terms in the denominator of eq 1239*40 J=
DvCsa Ai.Yo/2Kada + A((-Y~/~x,) coth bo/2X,) - 1)
+A +6 (12)
In eqs 12 and 13, Kadis the adsorption coefficient?0 x, is the mean surface diffusion distance, A and Ai are characteristic impedances for adsorption and integration, respectively A = a exp[(Ea - E,)/kT] Ai = a exp[(Ei - E,)/kT]
(15)
where E, and Ei are the activation free energies for adsorption and indirect integration. Equation 12 is normally valid for growth or dissolution in aqueous solution as the coupling between surface and volume diffusion is usually weak. Moreover, it can be shown that the first term in the denominator of eq 12, Ago/2Kada, is identical to the corresponding term in eq 3.7 of ref 39 using the parameter definitions given in Table 1 of ref 40. In the above rate equations, steps are regarded as continuous line sinks, which is normally valid for steps with very high kink densities. This assumption also holds when x, is much larger than the distance between kinks, xo, for a surface diffusion controlled mechanism.29 However, if detachment is the rate-determining step, xo becomes a function of the driving force at very low kink den~ities.~'We will thus derive an equation for a limiting case of indirect detachment in which the surface diffusion is not rate determining and the steps have very small kink densities. In this case, the saturation ratio, S', in the adsorption layer is approximately constant across the crystal terrace. Under such conditions, the average distance between kinks along a step may be written as41 xo = (1/2)a(2
- sq1l2exp(e/k7)
(16)
where e is the kink formation energy. The net flux of ions dissolving from each kink can be derived by analogy to the corresponding growth process30 j , = 2v0 exp(-Ei/kr)a3KadC,(1
- sq
(17)
where vo = kT/h and h is Planck's constant. Assuming that the lattice ions jump a distance corresponding to their size, a, in each diffusion move in the solution, the diffusion coefficient may be written30 D, = a2voexp(-E,/kT)
Combination of eqs 15, 17, and 18 leads to eq 19:
(18)
Octacalcium Phosphate Dissolution
-1
The Journal of Physical Chemistry, Vol. 96, No. 13, 1992 5481
‘I
!i
6
The step distance may be expressed as42343 yo = -4rya/kT In S’
(20) Since the kink density on the surface is l/xoyo, the dissolution rate, J, will be jk/xoyo:
The difference in the saturation ratios between the adsorption layer and bulk solution, S’ - S, serves as the driving force for ion desorption and volume diffusion. The corresponding flux density should be identical to J and is given by the equation J = D,C,(S’- S)/(6 + A) (22) Combination of eqs 19, 21, and 22 leads to eq 23:
which is similar to eq 12 but with the introduction of the xo/a term and the absence of the term for the surface diffusion resistance. At very high driving force when the term containing yo becomes negligible, a linear rate law will be obtained. On the other hand, at very low driving force, the domination of the detachment term will yield a parabolic rate law. The interfacial saturation ratio, S’, can be obtained from eq 24, a combination of eqs 16 and 20-22.
B. btimation of Parameters. Some of the physical parameters in the rate equations (8), (12), and (23) may be estimated more accurately from other sources and then used to calculate the remaining parameters from the dissolution kinetics data. In an electrolyte solution, it is reasonable to regard the individual ions as growth units, the elementary building blocks for the crystal. There are at least two selections of ions (growth units) to build up a formula unit of OCP: (1) 8Ca2+, 2H+, and 6PO:-; (2) 8Ca2+,2HP04, and 4P04+. In a slightly acidic solution as encountered in the present work, the second selection is more reasonable since HP02- is much more abundant than PO:-. Thus, from the molecular volume of OCP (6.96 X m3) containing 14 ions, the average size of the growth units is found to be 3.7 X m, the edge length of a cubic block. It has been implicitly assumed in eqs 12 and 17 that the thickness of the adsorption layer is of the same order of magnitude. The solubility, C,, in the rate equations has not been well-defined for systems in which the concentrations of lattice ions do not match the crystal stoichiometry as in the present case.” Since an OCP formula unit contains 8Ca2+, 2HPOd2-, and 4P043- ions, the solubility, Cs, may have the dimension of (K:)’/l4 = 5.53 X mol m-j. Although consistent with the definition of the relative undersaturation, eq 2, it is insensitive to the composition of an OCP formula unit. A more rigorous selection would be possible if the concentrations of the lattice ions were stoichiometric; the mean concentration in terms of the OCP “molecule” would have become (K,’/882244y2sv22v34)1/14 = 3.64 X mol m-3, where y2 and y 3 are the activity coefficients of the ions with 2 and 3 charges, respectively. Since the latter condition is not fulfilled and the resulting C, value is nevertheless similar, (K,’)l/14 will be used as C, for simplicity. From OCP growth kinetics which follows a polynucleation mechanism, Lundager Madsen” estimated the edge free energy to be 3.06 X lo-” J m-I, the height of the edge being 6.79 X m. The corresponding surface free energy is thus 45 X J m-2. With the size of the growth units as 3.7 X m, y / k T is found to be 1.9. In a homogeneous nucleation study, Boistelle and Lopez-Valer~~~ have estimated that the surface free energy of OCP is roughly twice as large as that of DCPD. The highest surface free energy for DCPD so far r e p ~ r t e d ~is ~73, ~X ~ , ~ J~m-2,
I>
2
1 0 0.0
0.2
0.4
0.6
0.8
0
Figure 4. Fitting experimental data by the direct detachment model with an estimated 6 value, 5 pm (-), or using 6 as a variable parameter (---, 6 = 60 i 10 pm). The curves, calculated using parameters based on y / k T = 1.9 and 5 , are indistinguishable.
estimated from the y/kTvalue, 2.7, by assuming a DCPD growth unit size of 3.9 X m.19 Thus, the upper limit of the y / k T value for OCP is about 5. Since there is lack of independent methods to estimate the edge free energy, a range of y/kTvalues from 1.9 to 5 has been used in this study. In the present calculation, the diffusion coefficient 0,is taken as 1 X 10” m2 s-’ with the activation energy for volume diffusion of 17.5 kJ m01-l.l~ The adsorption coefficient Kat estimated from the formation constant of the CaHP04 ion pair, is about 120.’9,47 The use of the data for CaHPO, rather than Capo4- is justified since at the pH studied, most of the phosphate ions at the surface are protonated. The average size of OCP crystals, estimated from the electron micrographs (Figure l ) , is about 5 gm which may serve as an approximation to the thickness of the diffusion layer.30348 Values of the estimated parameters are summarized in Table IV. C. Model Fitting. A general least-squares method’9,*5z is used to find values of the unknown parameters which offer the closest agreement between the model prediction and experimental results. The validity of these values will serve as a measure of the applicability of the fitted model. To apply this procedure, it is necessary to assign appropriate weights to both the rate and undersaturation data. The standard deviation in J from the present experiments probably does not exceed 10%. The error in the calculated undersaturation arises primarily from an uncertainty in pH determination of about 0.01, and this has been assumed in calculating the standard deviation in the relative undersaturation, Au (Table 11). The weights for J and u are taken as the reciprocals of the squares of the corresponding standard deviations.sl In fitting the direct detachment model, eq 8, after substitution of the estimated parameters in Table IV, only the 4value remains to be determined. In the present application of the least-squares method, the finite-difference method is used to supply required derivative^.^^ During each iteration, the step spacing, yo, is calculated from eq 10 using the secant methods3and then substituted into eq 8 to calculate improved parameters. At the optimal A, values (Table V), the model fails to yield an adequate fit as demonstrated in Figure 4 (solid line). Moreover, a change in y / k T from 1.9 to 5 does not significantly alter the goodness of fit. If the diffusion layer thickness is also considered as a variable parameter, a better agreement between the kinetics data and the fitted curve is obtained as expected (Figure 4, dashed line). In spite of this agreement, however, the resulting diffusion layer thickness, 61 f 10 pm (Table V), is too large compared with the size of the OCP crystals. Thus, the direct detachment model cannot offer an adequate description of OCP dissolution kinetics. Bennemaj7 and van der Eerden36 also suggested that a direct incorporation is not normally favored for crystal growth. For the indirect detachment model, eq 12, after substitution of the estimated parameters, A, x,, and Ai remain to be determined. By varying these parameters, one can achieve excellent fit to the
5482 The Journal of Physical Chemistry, Vol. 96, No. 13, I992 TABLE I V Summary of Estimated Parameters a/m C,/mol m3 3.7 x 10-10 5.53 x 10-3
YlkT 1.9-5
Zhang and Nancollas
s/m
E,/kJ mol-'
Kad
5 x 10"
17.5
120
D,/m2 s-I 1
x 10-9
TABLE V Parameters from the Direct Detachment Model
Aclrm Glum 5 (input) 61
10
r l k T = 1.9
rlkT = 5
7.2 i 1.3 2.1 i 0.4
2.8 f 0.5
0.8 f 0.2
TABLE VI: Results from Indirect Detachment with A, = 0 (Submodel i)
Glm
Alum
5 (input)
76 f 9
E.lkJ mol-' 49.5 f 0.3
xs r l k T = 1.9 r 1 k T = 5
(17 f 4)a
(41
i1o)o
kinetics data. However, large standard deviations in the parameters also result, suggesting that an adequate fit is possible with one of these parameter values fned. It is interesting to investigate the possibility of further distinguishing between surface diffusion and detachment as the rate-determining process. The indirect detachment model is thus divided into two submodels by assuming (i) Ai = 0 and (ii) x, >> yo. In the former, detachment from the step is not rate determining, while in the latter, surface diffusion is not rate determining. The parameters obtained by fitting both submodels are summarized in Tables VI and VII, respectively. The agreement between the experimental data and fitted curves is satisfactory as shown in Figure 5 . If submodel i with Ai = 0 applies, the saturation ratio near the step, S', will be unity.29 The kink distance xois calculated from eq 16 to be 4a and 74a when the e/kT is approximated by y / k T = 1.9 and 5, respectively. Table VI reveals that the inequality x, >> xo is not satisfied in the upper range of the y / k T values. Thus, the treatment of the steps as continuous sinks may not be valid. However, it is not sufficient to reject the general indirect detachment model based on such a mathematical oversimplification which may be improved by the introduction of a kink retardation factor.28 Unlike xs,the validity of the A value may be assessed by evaluating, from eq 14, the corresponding activation free energy, E,, for adsorption. The result is also included in Table VI where the standard deviation is based on that of A without considering the uncertainties in the estimated E, value. The resulting E, value is close to the sum of activation energies for Ca2+ion dehydration, 26 kJ mol-', and volume diffusion, 18 kJ mol-', two necessary processes for ions to enter the adsorption layer.19 Here we have assumed that the adsorption rate is mainly governed by the more hydrated Ca2+rather than HP0,2- ions. The E, value is also in line with those obtained from the growth of inorganic crystals following a surface diffusion m e c h a n i ~ m . ~ ~ It is interesting to note that this activation energy is considerably higher than that of DCPD, 41 kJ mol-', obtained from dissolution kinetics data.lg This is to be expected since DCPD is more hydrated and less dehydration is required for ions to enter the adsorption layer. With the parameters A and Ai derived from submodel ii with x, >> yo (Table VIIE), the corresponding activation energies E, and E, can be obtained from eqs 14 and 15, respectively. The E, value, similar to that of submodel i, should be viewed as reasonable. The Eivalue arises from integration within the adsorption layer which probably requires additional dehydration followed by an immediate surface diffusion jump. Thus, the magnitude of E, also seems to be reasonable. Therefore, this submodel describes another probable mechanism for OCP dissolution. The results summarized in Tables VI and VI1 at y / k T = 5 differ only slightly from those reported previously using the definitions u = 1 - ((Ca2+)8(H)2TABLE VII: Results from Indirect Detachment with x.
0
La .- . @... I
0.0
0.2
5 (input)
A l w 56 f 10
0.6
0.8
1 .o
(e-)
(P043-)6/Ks)1/16 and a = 3.5 X m, consistent with 16 growth units in an OCP formula.27 When detachment is rate determining (x, >> yo), at y / k T = 5 , the kink density is so small that kink formation may play an important role in governing the overall dissolution Thus, submodel iii, described by eq 23, may be applicable. The fitted curve, indistinguishable from those of submodels i and ii shown in Figure 5 , is again satisfactory. The parameters are summarized in Table VIII. The ratio of the Ai values of submodel ii over submodel iii, 68, can be accounted for by the factor x o / a in eq 23. However, the adsorption resistance, A, in submodel iii is markedly reduced probably due to the dependence of xo on S. Thus, significant errors may be introduced in the calculated parameter values when the involvement of kink formation is ignored. Although at such a low kink density submodel iii is conceptually superior to submodel ii, it is difficult to further distinguish these submodels based on the present kinetics data. Nevertheless, it is possible to deduce that the indirect detachment path is more favored for the dissolved ions. Examination of Tables VI-VI11 indicates that the volume diffusion resistance, 6, is much less than that of adsorption, A, suggesting that volume diffusion is not rate determining even at high driving forces. Moreover, from Figure 5 , the fitted curves for the three submodels of indirect detachment do not show an obvious change in the J-u dependence at u = 0.5. The theoretical curves between u = 0.5 and 0.95 yield an effective order of 1.7. The experimentally observed value of unity in this range could be due to uncertainties in the measured rates. Moreover, the apparent order change shown in Figure 3 may also be attributed to the fact that the dissolution deceleration becomes greater at lower driving force^.'^^^^
VI. Conclusion Although volume diffusion for crystal dissolution has often been emphasized, the dissolution of OCP in well-stirred solutions, in agreement with many other sparingly soluble electrolytes,25~5s~s6 is controlled mainly by processes occurring at crystal surface. As in the case of solution growth,54the direct detachment model is inadequate for interpreting the dissolution kinetics data. However, in addition to the good agreement between the experimental data and fitted curves,the indirect detachment models yield reasonable parameter values. Although it is difficult to further assess the relative importance of detachment and surface diffusion as the
>> Y , (Submadel ii)
E,/kJ mo1-I
48.2 f 0.5
a
Figure 5. Agreement between experimental data and the theoretical curves from submodels of indirect detachment: (-) submodel i, (---) submodel ii, submodel iii.
4lrm 6/lrm
0.4
y/kT = 1.9 505 f 101
EJkJ mol-' y/kT = 5 196 f 37
y/kT = 1.9 53.9 t 0.5
y/kT = 5 51.5 f 0.5
J. Phys. Chem. 1992, 96, 5483-5481 TABLE VIII: Results from Indirect Detachment with x, >> yo and xo Given by Eq 16 (Submodel iii) rlkT S/um Alum E./kJ mol-’ Ai/” &/kJ mol-’ 5 5 (input) 32 f 8 46.8 f 0.7 2.9 f 0.4 40.6 f 0.4
rate-determining step in the indirect detachment model, the involvement of surface diffusion is strongly suggested. Acknowledgment. The National Institute of Dental Research is acknowledged for a grant (DE 03323) in support of this project.
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Electric Birefringence of Poly(tetrafluoroethy1ene) “Whiskers” Kenneth R. Foster,**+Amanda J. OsbomJ and Michael S. Wolfe* Department of Bioengineering, University of Pennsylvania, 220 South 33rd Street, Philadelphia, Pennsylvania 191 04-6392, and DuPont Co., P.O. Box 80356, Wilmington, Delaware 19880-0356 (Received: November 21, 1991)
T h e alternating-current electric birefringence was measured from 68 Hz t o 6.8 MHz for suspensions of rod-like poly(tetrafluorcethylene) (PTFE) microcylinders of approximately 20-nm diameter and 0.4-pm length, in dilute KCl. The birefringence spectra shows a large, broad dispersion centered near 100 kHz whose amplitude is strongly dependent on the conductivity of the medium. A simple model can account for these properties, based on a calculation of the potential energy of homogeneous ellipsoids immersed in lossy dielectric media. The model suggests that the dispersion arises from a combination of two effects: an increase with frequency in the surface conductance of the particles and a Maxwell-Wagner effect. T h e estimated surface conductance of the PTFE cylinders is comparable t o values previously reported for other colloidal particles in suspension.
Introduction Poly(tetrafluoroethy1ene) (PTFE) can be formed into rodlike particles by polymerization under the proper conditions and suspended stably in electrolyte.’ The rods are strictly uniform in diameter (20-nm diameter), and have a mean length of about + University
of Pennsylvania. DuPont Co.
0022-3654/92/2096-S483%03.00/0
0.4 pm. At high concentrations, the suspensions are birefringent, suggesting liquid crystallinity; below a critical concentration they show little if any birefringence, which suggests t h e absence of strong ordering effects. This is the second of two papers that explore the dielectric and electrooptic properties of this material. The PTFE cylinders are an interesting model system to s t u d y counterion polarization effects, which are pronounced in t h e dielectric properties of such 0 1992 American Chemical Society
