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Some new aspects of surface nucleation applied to the growth and dissolution of fluorapatite and hydroxyapatite
,. . . . . . . .
C R Y S T A L G R O W T H
ELSEVIER
Journal of Crystal Growth 163 (1996) 304-310
Some new aspects of surface nucleation applied to the growth and dissolution of fluorapatite and hydroxyapatite JCrgen Christoffersen *, Margaret R. Christoffersen, Thue Johansen Department of Medical Biochemistry and Genetics, Biochemistry Laborato o, A, Panum hzstitute, University o["Copenhagen, Blegdamsvej 3. DK-2200 Copenhagen N, Denmark
Received 25 July 1995; accepted 2 November 1995
Abstract The polynuclear mechanism of crystal growth and dissolution is revised using calcium concentrations instead of mean ion activities in the expression for the lateral growth of surface nuclei. The rate-limiting process for the lateral growth of surface nuclei for the growth and dissolution of fluorapatite, FAP, and for dissolution of hydroxyapatite, HAP, is suggested to be calcium ions performing a diffusion jump into a kink and simultaneously being partly dehydrated. The experimental values of vm, 105 s-1, for these processes hereby agree with the theoretical value for this process, except for growth of FAP at pH 5.0. A new mechanism, mononuclear in the direction of the minor axis and polynuclear in the direction of the major axis, is introduced to explain the rate of growth of FAP at pH 5.0. The rates of growth of HAP lead to low values of vm, about 103 s - 1. It is suggested that the building of hydroxyl ions into the crystal lattice in this case limits the rate. It is further illustrated that empirical rate expressions in general cannot be used to distinguish between rate-controlling processes.
I. Introduction Description of crystal growth and dissolution processes in terms of molecular events on the crystal surface opens for the possibility of well-planned interaction with in vitro and possibly also in vivo growth and dissolution processes. In the preceding paper [1], rates of growth and dissolution of fluorapatite, Caj0(PO4)oF2, FAP, were reported. These processes were shown to follow most likely a polynuclear mechanism. Both the rate of nucleation and the lateral growth rate of nuclei were expressed in terms of mean ion activities. This led to values of the frequency of an ion to enter a kink, ~%, for both
" Correspondingauthor.
growth and dissolution of FAP of the order l07-108 s - 1 If the bottle-neck for ion integration into a kink is a combination of calcium ions performing a diffusion jump and simultaneously partially dehydrating, the theoretical frequency for this event is Uin.Ca = 1.6 × 10 5 S- ], Ref. [2]. If the anions just follow calcium ions, ui. should be ( 1 8 / 1 0 ) • uio.ca = 2.9 × 10 5 s - z. Other reactions may cause ui. to be lower, but u m can hardly be expected to be higher than this value, as the dehydration step has to take place. The aim of this paper is to revise the polynuclear model in order to achieve better agreement between the value of ui. obtained experimentally from the study of rates of growth and dissolution of FAP [1] and that predicted from the model. Earlier results for growth [3] and dissolution [4] of hydroxyapatite, Cal0(PO4)6(OH) 2 are reanalysed and values of the
0022-0248/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0022-0248(95)00963-9
J. Christoffersen et al. / Journal o/'Crystal Growth 163 (1996) 304-310
surface tension, o- and rate constants are compared with those for FAP [1]. A mechanism is suggested to explain rates of growth of FAP at high supersaturation [5]. The overall aim is to reach a better understanding of the molecular events which take place at the crystal surface during growth and dissolution.
305
In[J/mo gp(C)] -13 m/m0 1.2 4 o A
0
0
-14
2. The polynuclear mechanism For definitions of symbols we refer to the list of symbols in Section 7 of the preceding paper [l]. We shall again express the rate of nucleation, I, in terms of mean ion activities. It may, however, be more correct to express the lateral growth of a nucleus in terms of a concentration and not an activity. This concentration could e.g. be the concentration of calcium ions, phosphate, or fluoride ions, all of which are proportional to the concentration of FAP, C a I 0 ( P Q ) 6 F 2, in solutions of stoichiometric composition. Instead of g(S) and gp(S) in [1], we shall use for growth
g( C) = ( d/xo)C~x
[(c/c,)
-'fl 1/6 a s1/3 S 1/3 C,2/3
1/3
-15 o
-16
pH 5.5 opH6.0 opH6.5 A
D
-17
i
-1.2
-1.0
I
I
-0.8
-0.6
-0.4
-Up
Fig. 1. ln[J/mogp(C)] plotted against - l/fl for growth experiments when m/m o = 1.2. The units of J i m o are mol(FAP)/s-g and go(C) has the units mol/m 3.
- 112% -°/3"
~gp(C)e -a/3~
(1)
and for dissolution g ( C ) = ( d / x o ) ~ - 1 / 3 / 3 ,/6als/3C?/3
=
x [1 -
=- gp( C)e -"/3~,
(2)
in the overall rate expression J = Idn(FAP)/dtl= (p/m)
= kjmoF(m/mo) g(C).
effect of the solution composition on the rate. To test the model with integration of calcium ions being the rate-determining step, we use calcium concentrations in the expression for g(C), Eqs. (1) and (2). In Eq. (3) kj = HPin 1.8Hr'in,C a, where H is a constant for the crystal preparation and Pi,.ca is the frequency for a calcium ion to enter a growth site. This is because the model assumes that integration of calcium ions into kinks is slower than integration of other ions. The mean distance between kinks into which calcium ions can enter is 1.8x 0. In Eqs. (1) and (2) x 0 should be replaced by 1.8x o. The two factors of 1.8 will cancel in the expression for J and we use k I = Hui,,,o,. The value of /'Jin,Ca is related to the frequency for a calcium ion to leave a kink by
aldr/dtl (3)
The morphology term F(m/mo) shall again be assumed equal to 1. The only term depending on the definition of C in Eqs. (1) and (2) is C,, as C/C, = a/a,. The value of C, is pH-dependent, and, to a lesser extent, dependent on ionic strength, whereas a,, used in go(S), is a constant, K~/18. The value of kj, and thereby the value of Pi,, the frequency for ion integration into a kink, will be very dependent on whether go(S) or go(C) is used, and, in the latter case, which concentration is used to describe the
2d3NACsPin,Ca = Pout,Ca"
(4)
In the following we shall reanalyse the data reported in Ref. [1] for growth and dissolution of FAP. In Fig. 1 ln[J/mogp(C)] is plotted against - 1 / / 3 for growth of FAP at m / m o = 1.2. A similar plot is given in Fig. 2 for dissolution of FAP at m / m o = 0.8.
J. Christoffersen et al. / Journal of Crystal Growth 163 (1996) 304-310
306
ln[J/mogp(C)]
Similar results were obtained for m / m o = 1.4 and m / m o = 0.7. From the slopes of the lines, o- and x * can be calculated. From the rate constants, vi,,c a and Your,Ca can be determined. The results for growth and dissolution of FAP are summarized in Table 1, where it can be seen that, apart from vi,.c a for growth at pH 5.0, values of urn.Ca are in agreement with those expected from the model. Values of (7 and x* are almost the same as those obtained from the polynuclear model with mean ion activities [1]. In the following two sections we shall develop an alternative mechanism in an attempt to explain the high rate of growth of FAP at pH 5.0.
-9
m/mo = 0.8
-10
-11
-12 a pH ,~ pH o pH ~, pH
5.0 5.5 6.0 6.5
3. M o n o n u c l e a r m e c h a n i s m
I
-13 -3
-1
-2
0
Fig. 2. ln[J/mogv(C)] plotted against - 1 / / 3 for dissolution experiments when m / m o = 0 . 8 . The units of J / m o are m o l ( F A P ) / s - g and g p ( C ) has the units m o l / m 3.
If a nucleus formed at time zero on a surface with area A 0 at a later time 0 covers the total area A o without having inter-grown with other nuclei, we have, at least approximately,
AolO< 1,
(5)
Table 1 Results of plots of the function In[ J / m o g p ( C ) ] = ( o~/3)( - 1 / / 3 ) + ln(k l ) pH
v,,,,.c a ( s - I )
x*
7 13 0.4 0.7 1.0 1.4 0.5 2.0
13 25 0.4 0.6 0.4 0.5 0.1 0.4
14-26 16-28 5-12 6-15 6-15 6-17 5-12 7-14
1.6 1.6 1.8 1.4 1.0 0.8 0.6 0.4
2.7 2.7 1.3 1.1 0.4 0.3 0.1 0.1
m/m o
or/3
ln(k l)
r2
o" ( m J / m 2)
Vin.ca (105 s
1.2 1.4 1.2 1.4 1.2 1.4 1.2 1.4
7.86 8.42 4.82 5.54 6.68 7.19 6.45 8.99
- 8.09 -7.45 - 10.91 - 10.37 - 10.02 -9.71 - 10.73 - 9.32
0.98 0.99 0.86 0.93 0.96 0.97 0.90 0.90
124 129 97 104 115 119 113 133
0.8 0.7 0.8 0.7 0.8 0.7 0.8 0.7
0.80 0.84 1.02 0.93 0.88 0.77 0.98 0.91
-9.31 - 9.30 - 9.21 - 9.42 - 9.75 - 9.97 - 10.22 - 10.64
0.91 0.90 0.97 0.97 0.93 0.94 0.70 0.41
40 41 45 43 42 39 44 42
I)
Growth 5.0 5.0 5.5 5.5 6.0 6.0 6.5 6.5
Dissolution 5.0 5.0 5.5 5.5 6.0 6.0 6.5 (6.5
0.3-17 0.5-23 0.9-20 1.4-31 0.6-9 0.8-27 0.6-7 1.1-10)
J. Christqffersen et al. /Journal o{ Cp3'stal Growth 163 (1996) 304-310
307
The linear rate of growth is thus
where
0 ~ v'A~/dj.
(6)
Eqs. (5) and (6) lead to
a 3/2 < d j / l .
(V)
Using values from the analysis of the rate of growth of FAP according to the mononuclear mechanism [1] leads to
(a-~ < d(a~( S - l)e"/ts//3'/'-a) '/3 = 10 - 20d.
(s) This shows clearly that the mononuclear mechanism is unrealistic for description of the growth of FAP, with the crystal dimensions of the order 1800d by 330d.
d r / d t = d / v = d( t,~l, t) '/~-.
(11)
Combining equations for 1, t,_,_ and j [1] with Eq. (l 1) and using calcium concentrations in the expression for v~ gives, with x 0 = d, tt~r = d -3
dr/dt=
(2vi,,M/vp)d(~ll/d)'/2g(C).
(12)
Approximating the area on which the main growth takes place by a = m 0Asp 0 F ( m / m o),
(13)
the rate can be expressed in the form of Eq. (3) with g(C)
= (4;'/3'/2CsasS[(C/Cs)
-
-~ gmp( C ) e - " / 2 ~
(14)
and 4. Combination of mono- and polynuclear mechanisms Nucleation on a crystal surface with dimension lj × / 2 with 12>>l~ may cause nuclei to reach a diameter comparable to 1~ prior to intergrowth with other nuclei. In this case we have, at least approximately, 12 = ]~2 L~(~-- t) ll,l 2 dt,
(9)
in which z is the time required for deposition of a layer of thickness d. From Eq. (9) we obtain "r2 t~'~/l I = 1, or,
~-= ( , d , l ) -'/2
(10)
Table 2 Results of plots of the function In[ J/mll gmp(C)] = (o~/2)(
k~= (2/u)d(~/~l,/d)'/2a~poV,n =-Hv~°.
(15)
For the present crystals, l~ is of the order 300d. This leads to H = 8.1 X 10 9 m 3 / g for this mechanism. In Fig. 3 ln[J/mogmp(C)] is plotted against - 1 / / 3 for growth of FAP at m / m o = 1.2. The results for or, ui,,,c~, Uo,t.c~ and x* are given in Table 2. It can be seen that the values of Um.c~ for pH 5.0 are now in better agreement with the model. We suggest therefore, that the mechanism of FAP growth can be described for 5.5 _< pH < 6.5 by the polynuclear mechanism w i t h g p ( C ) given by Eq. (1), with C being the concentration of calcium ions, and for pH 5.0 by the combined mono-polynuclear mechanism w i t h g m p ( C ) given by Eq. (14) and C being the concentration of calcium ions. This description leads to values of or, ui~.c~ and x* of the
1 / / 3 ) + ln(k I) for growth of FAP
pH
m/mll
a/2
ln(kj )
r2
o- ( m J / m 2)
J',,.c, (105 s
5.0 5.0 5.5 5.5 6.0 6.0 6.5 6.5
1.2 1.4 1.2 1.4 1.2 1.4 1.2 1.4
7.90 8.46 4.76 5.53 5.75 6.24 6.38 8.97
-6.74 - 6.09 - 9.78 -9.20 - 9.38 -9.08 - 9.86 - 8.42
0.98 0.99 0.86 0.93 0.94 0.96 0.90 0.90
102 105 79 85 87 91 92 109
1.5 2.8 0.07 0.12 0.10 0.14 0.06 0.27
I)
v,,ut.c~ ( s - I )
x
2.7 5 0.06 0.10 0.04 0.05 0.01 0.05
9-17 ll-19 3-8 4-10 3 8 4-10 3-8 5-10
308
J. Christq[fersen et a I. / Jott rna I o f Co,stal Growth 163 (1996) 3 0 4 - 3 1 0 In[J/mo gmp(C)]
dissolution o f F A P , Table 1, apart from the surface tension, which is clearly l o w e r for dissolution than for growth.
-12 mira0 1.2
/x o o
{3
5. Comparison of growth and dissolution kinetics of FAP and HAP
-13 D A
Z~
A
C
-14
-15
n pH 5.0
o
•"- pH 5.5 o pH 6.0 o pH 6.5 -16 -1.2
~
f
t
-1.0
-0.8
-0.6
-0.4
Fig. 3. l n [ J / m o g m p ( C ) ] plotted against - 1//3 for m / m o = 1.2. The units of J / m o are mol(FAP)/s-g and gmp(S) has the units mol/m 3.
same order of m a g n i t u d e for all pH values studied, a p p r o x i m a t e l y 110 m J / m z, 10 5 s -~ and 5 - 2 0 , respectively (Tables 1 and 2). Uout,Ca varies from about 5 s -~ at pH 5.0 to about 0.1 s-~ at pH 6.5. These results are in good a g r e e m e n t with results for the
A reanalysis of the data for growth o f H A P [3] and dissolution of H A P [4] according to the polynuclear m e c h a n i s m using concentrations of calcium in expressions (1) and (2) results in the values of ui°.c ~, rout,Ca and o- given in Table 3. S o m e interesting trends, which can help to elucidate m o l e c u l a r events taking place at the crystal surfaces, can be seen in this table. In Ref. [6] we suggested that the l o w e r value o f the surface tension d e t e r m i n e d e x p e r i m e n t a l l y for growth of H A P than the theoretical value, 240 m J / m 2, can be e x p l a i n e d by the imperfection of the surface nuclei. The surface tension found for both H A P and F A P dissolution is considerably l o w e r than that for growth and the surface tension found for growth of H A P is s o m e w h a t l o w e r than that for growth of F A P . This can be due to the reaction of hydrogen ions with phosphate groups and, for H A P , also with hydroxyl groups in the surface creating vacant calcium and, for H A P , also vacant hydroxyl sites. These reactions will w e a k e n bonds towards calcium and thus reduce the value o f o- and also the size of a critical nucleus. This will facilitate nucle-
Table 3 Vin.c~, ~,,ut.Ca and tr values calculated according to the polynuclear mechanism (except ": mono- polynuclear mechanism) from the following results: FAP growth, m / m o = 1.2, [1]; FAP dissolution, m / m o = 0.7 (except #: m / m o = 0.8) [1]; HAP growth, m / m o = 1.2 [2]; HAP dissolution, m / m l l = 0.7 [4] Growth FAP
Dissolution
pH
ui~,c~ (10 s s - t )
u.,~,.c. (s-t)
o" (ml/m 2)
pH
ui.,c . (10 5 s I)
u,,~t.c~ (s I)
o- (mJ/m 2)
5.0 * 5.5 6.0 6.5
1.5 0.4 1.0 0.5
2.7 0.4 0.4 0.1
102 97 115 113
5.0 5.5 6.0 6.5 #
1.6 1.4 0.8 0.6
2.7 1.1 0.3 0.1
41 43 39 44
6.0 6.4 6.8 7.2
0.01 0.01 0.02 0.03
0.02 0.003 0.004 0.004
86 78 73 91
5.0 5.5 6.3 6.8 7.2
0.4 0.8 0.8 0.6 0.5
6.0 4.0 0.7 0.2 0.1
50 50 45 43 44
HAP
309
J. Christq~ersen et at./Jourm.ll of Ct3,stal Growth 163 (1996) 304-310
ation for both growth and dissolution. For dissolution, formation of HPO 4- in the surface seems to be the most important reaction, as crdis.FAP = O'dis.HA PFor growth, the difference between O'rAP and crrtAP is larger than can be explained by the different solubilities. The difference seems thus due to formation of vacant hydroxyl sites in the HAP surface. Uo,,t.c. for both FAP and HAP is found to decrease with increasing pH, also in agreement with the importance of the reaction between hydrogen ions and phosphate groups. For dissolution Uo,t,ca of HAP is a factor of 2 larger than for FAP. This difference may be due to the reaction between hydrogen ions and surface hydroxyl groups. The values of ui,,.c~ for growth of FAP and dissolution of FAP and HAP are comparable to the theoretical value expected if the bottle neck for ion integration is the combined events of calcium ions making a diffusion jump and simultaneously partly dehydrating. The values of uin,c ~ and Uo~t.c, for HAP growth are significantly lower than for HAP dissolution and FAP growth and dissolution. We suggest that this could be explained by the building of hydroxyl ions into the HAP lattice strongly influencing the rate.
6. R o u g h g r o w t h
(17)
dr/dt = d.j/2.
The factor 2 is introduced because the w)lume from which an ion can jump onto a flat part of the crystal is d ~ and not 2 d -~, as used in the expression for j. Combining Eqs. (3) and (17) gives J=ad(S-
1),
1)a.,.Uin/u=kpa(s-
(18)
with kp
10
mol/m- - s
for FAP with
ui. = 3 X 105/s.
(19)
Iog(J/mo)
-10
Amjad et al. [5] found for growth of FAP at high values of S, 10-30, a slope of 1.3 for a l o g ( J / A o) - log(S - 1) plot (Fig. 5 in Ref. [1]). High supersaturation often causes crystal growth to be diffusion controlled, particularly if the crystals are larger than about 10 /zm. For crystals so small that they do not move relative to the solution, despite stirring, the diffusion equation is of the form dr/dt = DM( C-
the nucleation barrier due to an increase in S does not change other energy barriers, such as dehydration of cations entering the surface. For growth controlled by the polynuclear mechanism the slope of plots of l o g ( J / m o) against l o g ( S - 1) should approach 1 as S becomes so large that x* approaches 1 and the nucleation barrier is removed. All positions on the surface act as kinks and we have
C')/pr,
-20
/ #
,/
-30
"1 -40
I
(16)
where C' is the concentration at the crystal surface. For purely diffusion-control, C ' = C S. If the rate by which ions are incorporated in the crystal is much slower than the diffusion rate, C' will approach C and the rate becomes surface-controlled. If the supersaturation is so high that x* is 1 or less, there is no nucleation barrier to be overcome. For the transfer of x building units from solution to the crystalline phase, A G ( x ) < 0 for all values of x. Reduction of
I
c~/3=6
I
-50
-
-
- -
-60 -1
-
c~/3=8
--o:/3=11
r
I
0
1
2
Zog(Sq)
Fig. 4. Logarithmic plot of values of J/ml) (in mol(FAP)/s.g) against S-1. J / m o is calculated from the formula J/rail = ktgv(S) e ,,/3~ for ln(kj)= - 6 and 3 typical values of o~/3.
310
J. Christoffersen et al. / Journal ol" Crystal Growth 163 (1996) 3 0 4 - 3 1 0
For diffusion,
J=ADas(S- l)/~r=akD(S-
1),
(20)
with
k D = Das/ur= 10 -7 m o l / m 2. s,
(21)
for FAP with r - 0.1 ~ m and D --- 10 -9 m2/s. If the rates are expressed in concentration terms, i.e. a S replaced by C S, the values of kp and kt) will change, but the ratio of kp to kD, 10 -2, does not change. It is thus possible that growth of small crystals controlled by a polynuclear mechanism at very high supersaturation does not change over to purely bulk diffusion-controlled; dehydration of cations is important. The results of Amjad et al. [5] for the growth of FAP at relatively high supersaturation can possibly be explained by the rate expression Eq. (18). From the plot in Fig. 5 from Ref. [1], we obtain for Amjad et al.'s results kp in Eq. (18) to be 2 × 10 - j ° m o l / m 2 - s . This value is close to the theoretical value for kp given in Eq. (19). Distinction between the spiral mechanism and the polynuclear mechanism is often made on the background of the slopes, p, of plots of log(J/m o) against l o g ( S - 1) for rates corrected for changes in the surface area. For spirals p < 2. For the polynuclear mechanism p is often assumed or expected to be larger than 3. In Fig. 4 log(J/m o) is plotted against l o g ( S - 1), with J / m o calculated from Eqs. (1) and (3) for three typical values of c~/3. This plot shows that p is high for low values of S and that p approaches 1 for high values of S for the polynuclear mechanism, where rough growth takes over. Similarly, x* may be of the order 1 for the
polynuclear dissolution mechanism at very high degrees of undersaturation. As long as the activation energy for a building unit to leave the edge of a dissolution nucleus is high enough, the rate will be controlled by the surface reaction, and not by bulk diffusion.
Acknowledgements
We acknowledge valuable scientific discussions over nearly three decades with the late Professor Arne E. Nielsen. We are most grateful for the technical assistance of Ms. Mette Kja~r Schou and Ms. Julita Kuzimska. We thank Dr. H.E. Lundager Madsen for providing the ion speciation program Ionics. Financial support from the Danish Health Science Research Council, from Novo Nordic's Foundation and from the Vera and Carl Johan Michaelsen Foundation is gratefully acknowledged.
References
[1] J. Christoffersen, M.R. Christoffersen and T. Johansen, J. Crystal Growth 163 (1996) 295. [2] J. Christoffersen and M.R. Christoffersen, J. Crystal Growth 87 (1988) 41. [3] M.R. Christoffersen and J. Christoffersen, J. Crystal Growth 121 (1992) 617. [4] J. Christoffersen and MR. Christoffersen, J. Crystal Growth 57 (1982) 21. [5] Z. Amjad, P.G. Koutsoukos and G.H Nancollas, J. Colloid Interface Sci. 82 (1981) 394. [6] J. Christoffersen and MR. Christoffersen, J. Crystal Growth 121 (1992) 608.
C R Y S T A L G R O W T H
ELSEVIER
Journal of Crystal Growth 163 (1996) 304-310
Some new aspects of surface nucleation applied to the growth and dissolution of fluorapatite and hydroxyapatite JCrgen Christoffersen *, Margaret R. Christoffersen, Thue Johansen Department of Medical Biochemistry and Genetics, Biochemistry Laborato o, A, Panum hzstitute, University o["Copenhagen, Blegdamsvej 3. DK-2200 Copenhagen N, Denmark
Received 25 July 1995; accepted 2 November 1995
Abstract The polynuclear mechanism of crystal growth and dissolution is revised using calcium concentrations instead of mean ion activities in the expression for the lateral growth of surface nuclei. The rate-limiting process for the lateral growth of surface nuclei for the growth and dissolution of fluorapatite, FAP, and for dissolution of hydroxyapatite, HAP, is suggested to be calcium ions performing a diffusion jump into a kink and simultaneously being partly dehydrated. The experimental values of vm, 105 s-1, for these processes hereby agree with the theoretical value for this process, except for growth of FAP at pH 5.0. A new mechanism, mononuclear in the direction of the minor axis and polynuclear in the direction of the major axis, is introduced to explain the rate of growth of FAP at pH 5.0. The rates of growth of HAP lead to low values of vm, about 103 s - 1. It is suggested that the building of hydroxyl ions into the crystal lattice in this case limits the rate. It is further illustrated that empirical rate expressions in general cannot be used to distinguish between rate-controlling processes.
I. Introduction Description of crystal growth and dissolution processes in terms of molecular events on the crystal surface opens for the possibility of well-planned interaction with in vitro and possibly also in vivo growth and dissolution processes. In the preceding paper [1], rates of growth and dissolution of fluorapatite, Caj0(PO4)oF2, FAP, were reported. These processes were shown to follow most likely a polynuclear mechanism. Both the rate of nucleation and the lateral growth rate of nuclei were expressed in terms of mean ion activities. This led to values of the frequency of an ion to enter a kink, ~%, for both
" Correspondingauthor.
growth and dissolution of FAP of the order l07-108 s - 1 If the bottle-neck for ion integration into a kink is a combination of calcium ions performing a diffusion jump and simultaneously partially dehydrating, the theoretical frequency for this event is Uin.Ca = 1.6 × 10 5 S- ], Ref. [2]. If the anions just follow calcium ions, ui. should be ( 1 8 / 1 0 ) • uio.ca = 2.9 × 10 5 s - z. Other reactions may cause ui. to be lower, but u m can hardly be expected to be higher than this value, as the dehydration step has to take place. The aim of this paper is to revise the polynuclear model in order to achieve better agreement between the value of ui. obtained experimentally from the study of rates of growth and dissolution of FAP [1] and that predicted from the model. Earlier results for growth [3] and dissolution [4] of hydroxyapatite, Cal0(PO4)6(OH) 2 are reanalysed and values of the
0022-0248/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0022-0248(95)00963-9
J. Christoffersen et al. / Journal o/'Crystal Growth 163 (1996) 304-310
surface tension, o- and rate constants are compared with those for FAP [1]. A mechanism is suggested to explain rates of growth of FAP at high supersaturation [5]. The overall aim is to reach a better understanding of the molecular events which take place at the crystal surface during growth and dissolution.
305
In[J/mo gp(C)] -13 m/m0 1.2 4 o A
0
0
-14
2. The polynuclear mechanism For definitions of symbols we refer to the list of symbols in Section 7 of the preceding paper [l]. We shall again express the rate of nucleation, I, in terms of mean ion activities. It may, however, be more correct to express the lateral growth of a nucleus in terms of a concentration and not an activity. This concentration could e.g. be the concentration of calcium ions, phosphate, or fluoride ions, all of which are proportional to the concentration of FAP, C a I 0 ( P Q ) 6 F 2, in solutions of stoichiometric composition. Instead of g(S) and gp(S) in [1], we shall use for growth
g( C) = ( d/xo)C~x
[(c/c,)
-'fl 1/6 a s1/3 S 1/3 C,2/3
1/3
-15 o
-16
pH 5.5 opH6.0 opH6.5 A
D
-17
i
-1.2
-1.0
I
I
-0.8
-0.6
-0.4
-Up
Fig. 1. ln[J/mogp(C)] plotted against - l/fl for growth experiments when m/m o = 1.2. The units of J i m o are mol(FAP)/s-g and go(C) has the units mol/m 3.
- 112% -°/3"
~gp(C)e -a/3~
(1)
and for dissolution g ( C ) = ( d / x o ) ~ - 1 / 3 / 3 ,/6als/3C?/3
=
x [1 -
=- gp( C)e -"/3~,
(2)
in the overall rate expression J = Idn(FAP)/dtl= (p/m)
= kjmoF(m/mo) g(C).
effect of the solution composition on the rate. To test the model with integration of calcium ions being the rate-determining step, we use calcium concentrations in the expression for g(C), Eqs. (1) and (2). In Eq. (3) kj = HPin 1.8Hr'in,C a, where H is a constant for the crystal preparation and Pi,.ca is the frequency for a calcium ion to enter a growth site. This is because the model assumes that integration of calcium ions into kinks is slower than integration of other ions. The mean distance between kinks into which calcium ions can enter is 1.8x 0. In Eqs. (1) and (2) x 0 should be replaced by 1.8x o. The two factors of 1.8 will cancel in the expression for J and we use k I = Hui,,,o,. The value of /'Jin,Ca is related to the frequency for a calcium ion to leave a kink by
aldr/dtl (3)
The morphology term F(m/mo) shall again be assumed equal to 1. The only term depending on the definition of C in Eqs. (1) and (2) is C,, as C/C, = a/a,. The value of C, is pH-dependent, and, to a lesser extent, dependent on ionic strength, whereas a,, used in go(S), is a constant, K~/18. The value of kj, and thereby the value of Pi,, the frequency for ion integration into a kink, will be very dependent on whether go(S) or go(C) is used, and, in the latter case, which concentration is used to describe the
2d3NACsPin,Ca = Pout,Ca"
(4)
In the following we shall reanalyse the data reported in Ref. [1] for growth and dissolution of FAP. In Fig. 1 ln[J/mogp(C)] is plotted against - 1 / / 3 for growth of FAP at m / m o = 1.2. A similar plot is given in Fig. 2 for dissolution of FAP at m / m o = 0.8.
J. Christoffersen et al. / Journal of Crystal Growth 163 (1996) 304-310
306
ln[J/mogp(C)]
Similar results were obtained for m / m o = 1.4 and m / m o = 0.7. From the slopes of the lines, o- and x * can be calculated. From the rate constants, vi,,c a and Your,Ca can be determined. The results for growth and dissolution of FAP are summarized in Table 1, where it can be seen that, apart from vi,.c a for growth at pH 5.0, values of urn.Ca are in agreement with those expected from the model. Values of (7 and x* are almost the same as those obtained from the polynuclear model with mean ion activities [1]. In the following two sections we shall develop an alternative mechanism in an attempt to explain the high rate of growth of FAP at pH 5.0.
-9
m/mo = 0.8
-10
-11
-12 a pH ,~ pH o pH ~, pH
5.0 5.5 6.0 6.5
3. M o n o n u c l e a r m e c h a n i s m
I
-13 -3
-1
-2
0
Fig. 2. ln[J/mogv(C)] plotted against - 1 / / 3 for dissolution experiments when m / m o = 0 . 8 . The units of J / m o are m o l ( F A P ) / s - g and g p ( C ) has the units m o l / m 3.
If a nucleus formed at time zero on a surface with area A 0 at a later time 0 covers the total area A o without having inter-grown with other nuclei, we have, at least approximately,
AolO< 1,
(5)
Table 1 Results of plots of the function In[ J / m o g p ( C ) ] = ( o~/3)( - 1 / / 3 ) + ln(k l ) pH
v,,,,.c a ( s - I )
x*
7 13 0.4 0.7 1.0 1.4 0.5 2.0
13 25 0.4 0.6 0.4 0.5 0.1 0.4
14-26 16-28 5-12 6-15 6-15 6-17 5-12 7-14
1.6 1.6 1.8 1.4 1.0 0.8 0.6 0.4
2.7 2.7 1.3 1.1 0.4 0.3 0.1 0.1
m/m o
or/3
ln(k l)
r2
o" ( m J / m 2)
Vin.ca (105 s
1.2 1.4 1.2 1.4 1.2 1.4 1.2 1.4
7.86 8.42 4.82 5.54 6.68 7.19 6.45 8.99
- 8.09 -7.45 - 10.91 - 10.37 - 10.02 -9.71 - 10.73 - 9.32
0.98 0.99 0.86 0.93 0.96 0.97 0.90 0.90
124 129 97 104 115 119 113 133
0.8 0.7 0.8 0.7 0.8 0.7 0.8 0.7
0.80 0.84 1.02 0.93 0.88 0.77 0.98 0.91
-9.31 - 9.30 - 9.21 - 9.42 - 9.75 - 9.97 - 10.22 - 10.64
0.91 0.90 0.97 0.97 0.93 0.94 0.70 0.41
40 41 45 43 42 39 44 42
I)
Growth 5.0 5.0 5.5 5.5 6.0 6.0 6.5 6.5
Dissolution 5.0 5.0 5.5 5.5 6.0 6.0 6.5 (6.5
0.3-17 0.5-23 0.9-20 1.4-31 0.6-9 0.8-27 0.6-7 1.1-10)
J. Christqffersen et al. /Journal o{ Cp3'stal Growth 163 (1996) 304-310
307
The linear rate of growth is thus
where
0 ~ v'A~/dj.
(6)
Eqs. (5) and (6) lead to
a 3/2 < d j / l .
(V)
Using values from the analysis of the rate of growth of FAP according to the mononuclear mechanism [1] leads to
(a-~ < d(a~( S - l)e"/ts//3'/'-a) '/3 = 10 - 20d.
(s) This shows clearly that the mononuclear mechanism is unrealistic for description of the growth of FAP, with the crystal dimensions of the order 1800d by 330d.
d r / d t = d / v = d( t,~l, t) '/~-.
(11)
Combining equations for 1, t,_,_ and j [1] with Eq. (l 1) and using calcium concentrations in the expression for v~ gives, with x 0 = d, tt~r = d -3
dr/dt=
(2vi,,M/vp)d(~ll/d)'/2g(C).
(12)
Approximating the area on which the main growth takes place by a = m 0Asp 0 F ( m / m o),
(13)
the rate can be expressed in the form of Eq. (3) with g(C)
= (4;'/3'/2CsasS[(C/Cs)
-
-~ gmp( C ) e - " / 2 ~
(14)
and 4. Combination of mono- and polynuclear mechanisms Nucleation on a crystal surface with dimension lj × / 2 with 12>>l~ may cause nuclei to reach a diameter comparable to 1~ prior to intergrowth with other nuclei. In this case we have, at least approximately, 12 = ]~2 L~(~-- t) ll,l 2 dt,
(9)
in which z is the time required for deposition of a layer of thickness d. From Eq. (9) we obtain "r2 t~'~/l I = 1, or,
~-= ( , d , l ) -'/2
(10)
Table 2 Results of plots of the function In[ J/mll gmp(C)] = (o~/2)(
k~= (2/u)d(~/~l,/d)'/2a~poV,n =-Hv~°.
(15)
For the present crystals, l~ is of the order 300d. This leads to H = 8.1 X 10 9 m 3 / g for this mechanism. In Fig. 3 ln[J/mogmp(C)] is plotted against - 1 / / 3 for growth of FAP at m / m o = 1.2. The results for or, ui,,,c~, Uo,t.c~ and x* are given in Table 2. It can be seen that the values of Um.c~ for pH 5.0 are now in better agreement with the model. We suggest therefore, that the mechanism of FAP growth can be described for 5.5 _< pH < 6.5 by the polynuclear mechanism w i t h g p ( C ) given by Eq. (1), with C being the concentration of calcium ions, and for pH 5.0 by the combined mono-polynuclear mechanism w i t h g m p ( C ) given by Eq. (14) and C being the concentration of calcium ions. This description leads to values of or, ui~.c~ and x* of the
1 / / 3 ) + ln(k I) for growth of FAP
pH
m/mll
a/2
ln(kj )
r2
o- ( m J / m 2)
J',,.c, (105 s
5.0 5.0 5.5 5.5 6.0 6.0 6.5 6.5
1.2 1.4 1.2 1.4 1.2 1.4 1.2 1.4
7.90 8.46 4.76 5.53 5.75 6.24 6.38 8.97
-6.74 - 6.09 - 9.78 -9.20 - 9.38 -9.08 - 9.86 - 8.42
0.98 0.99 0.86 0.93 0.94 0.96 0.90 0.90
102 105 79 85 87 91 92 109
1.5 2.8 0.07 0.12 0.10 0.14 0.06 0.27
I)
v,,ut.c~ ( s - I )
x
2.7 5 0.06 0.10 0.04 0.05 0.01 0.05
9-17 ll-19 3-8 4-10 3 8 4-10 3-8 5-10
308
J. Christq[fersen et a I. / Jott rna I o f Co,stal Growth 163 (1996) 3 0 4 - 3 1 0 In[J/mo gmp(C)]
dissolution o f F A P , Table 1, apart from the surface tension, which is clearly l o w e r for dissolution than for growth.
-12 mira0 1.2
/x o o
{3
5. Comparison of growth and dissolution kinetics of FAP and HAP
-13 D A
Z~
A
C
-14
-15
n pH 5.0
o
•"- pH 5.5 o pH 6.0 o pH 6.5 -16 -1.2
~
f
t
-1.0
-0.8
-0.6
-0.4
Fig. 3. l n [ J / m o g m p ( C ) ] plotted against - 1//3 for m / m o = 1.2. The units of J / m o are mol(FAP)/s-g and gmp(S) has the units mol/m 3.
same order of m a g n i t u d e for all pH values studied, a p p r o x i m a t e l y 110 m J / m z, 10 5 s -~ and 5 - 2 0 , respectively (Tables 1 and 2). Uout,Ca varies from about 5 s -~ at pH 5.0 to about 0.1 s-~ at pH 6.5. These results are in good a g r e e m e n t with results for the
A reanalysis of the data for growth o f H A P [3] and dissolution of H A P [4] according to the polynuclear m e c h a n i s m using concentrations of calcium in expressions (1) and (2) results in the values of ui°.c ~, rout,Ca and o- given in Table 3. S o m e interesting trends, which can help to elucidate m o l e c u l a r events taking place at the crystal surfaces, can be seen in this table. In Ref. [6] we suggested that the l o w e r value o f the surface tension d e t e r m i n e d e x p e r i m e n t a l l y for growth of H A P than the theoretical value, 240 m J / m 2, can be e x p l a i n e d by the imperfection of the surface nuclei. The surface tension found for both H A P and F A P dissolution is considerably l o w e r than that for growth and the surface tension found for growth of H A P is s o m e w h a t l o w e r than that for growth of F A P . This can be due to the reaction of hydrogen ions with phosphate groups and, for H A P , also with hydroxyl groups in the surface creating vacant calcium and, for H A P , also vacant hydroxyl sites. These reactions will w e a k e n bonds towards calcium and thus reduce the value o f o- and also the size of a critical nucleus. This will facilitate nucle-
Table 3 Vin.c~, ~,,ut.Ca and tr values calculated according to the polynuclear mechanism (except ": mono- polynuclear mechanism) from the following results: FAP growth, m / m o = 1.2, [1]; FAP dissolution, m / m o = 0.7 (except #: m / m o = 0.8) [1]; HAP growth, m / m o = 1.2 [2]; HAP dissolution, m / m l l = 0.7 [4] Growth FAP
Dissolution
pH
ui~,c~ (10 s s - t )
u.,~,.c. (s-t)
o" (ml/m 2)
pH
ui.,c . (10 5 s I)
u,,~t.c~ (s I)
o- (mJ/m 2)
5.0 * 5.5 6.0 6.5
1.5 0.4 1.0 0.5
2.7 0.4 0.4 0.1
102 97 115 113
5.0 5.5 6.0 6.5 #
1.6 1.4 0.8 0.6
2.7 1.1 0.3 0.1
41 43 39 44
6.0 6.4 6.8 7.2
0.01 0.01 0.02 0.03
0.02 0.003 0.004 0.004
86 78 73 91
5.0 5.5 6.3 6.8 7.2
0.4 0.8 0.8 0.6 0.5
6.0 4.0 0.7 0.2 0.1
50 50 45 43 44
HAP
309
J. Christq~ersen et at./Jourm.ll of Ct3,stal Growth 163 (1996) 304-310
ation for both growth and dissolution. For dissolution, formation of HPO 4- in the surface seems to be the most important reaction, as crdis.FAP = O'dis.HA PFor growth, the difference between O'rAP and crrtAP is larger than can be explained by the different solubilities. The difference seems thus due to formation of vacant hydroxyl sites in the HAP surface. Uo,,t.c. for both FAP and HAP is found to decrease with increasing pH, also in agreement with the importance of the reaction between hydrogen ions and phosphate groups. For dissolution Uo,t,ca of HAP is a factor of 2 larger than for FAP. This difference may be due to the reaction between hydrogen ions and surface hydroxyl groups. The values of ui,,.c~ for growth of FAP and dissolution of FAP and HAP are comparable to the theoretical value expected if the bottle neck for ion integration is the combined events of calcium ions making a diffusion jump and simultaneously partly dehydrating. The values of uin,c ~ and Uo~t.c, for HAP growth are significantly lower than for HAP dissolution and FAP growth and dissolution. We suggest that this could be explained by the building of hydroxyl ions into the HAP lattice strongly influencing the rate.
6. R o u g h g r o w t h
(17)
dr/dt = d.j/2.
The factor 2 is introduced because the w)lume from which an ion can jump onto a flat part of the crystal is d ~ and not 2 d -~, as used in the expression for j. Combining Eqs. (3) and (17) gives J=ad(S-
1),
1)a.,.Uin/u=kpa(s-
(18)
with kp
10
mol/m- - s
for FAP with
ui. = 3 X 105/s.
(19)
Iog(J/mo)
-10
Amjad et al. [5] found for growth of FAP at high values of S, 10-30, a slope of 1.3 for a l o g ( J / A o) - log(S - 1) plot (Fig. 5 in Ref. [1]). High supersaturation often causes crystal growth to be diffusion controlled, particularly if the crystals are larger than about 10 /zm. For crystals so small that they do not move relative to the solution, despite stirring, the diffusion equation is of the form dr/dt = DM( C-
the nucleation barrier due to an increase in S does not change other energy barriers, such as dehydration of cations entering the surface. For growth controlled by the polynuclear mechanism the slope of plots of l o g ( J / m o) against l o g ( S - 1) should approach 1 as S becomes so large that x* approaches 1 and the nucleation barrier is removed. All positions on the surface act as kinks and we have
C')/pr,
-20
/ #
,/
-30
"1 -40
I
(16)
where C' is the concentration at the crystal surface. For purely diffusion-control, C ' = C S. If the rate by which ions are incorporated in the crystal is much slower than the diffusion rate, C' will approach C and the rate becomes surface-controlled. If the supersaturation is so high that x* is 1 or less, there is no nucleation barrier to be overcome. For the transfer of x building units from solution to the crystalline phase, A G ( x ) < 0 for all values of x. Reduction of
I
c~/3=6
I
-50
-
-
- -
-60 -1
-
c~/3=8
--o:/3=11
r
I
0
1
2
Zog(Sq)
Fig. 4. Logarithmic plot of values of J/ml) (in mol(FAP)/s.g) against S-1. J / m o is calculated from the formula J/rail = ktgv(S) e ,,/3~ for ln(kj)= - 6 and 3 typical values of o~/3.
310
J. Christoffersen et al. / Journal ol" Crystal Growth 163 (1996) 3 0 4 - 3 1 0
For diffusion,
J=ADas(S- l)/~r=akD(S-
1),
(20)
with
k D = Das/ur= 10 -7 m o l / m 2. s,
(21)
for FAP with r - 0.1 ~ m and D --- 10 -9 m2/s. If the rates are expressed in concentration terms, i.e. a S replaced by C S, the values of kp and kt) will change, but the ratio of kp to kD, 10 -2, does not change. It is thus possible that growth of small crystals controlled by a polynuclear mechanism at very high supersaturation does not change over to purely bulk diffusion-controlled; dehydration of cations is important. The results of Amjad et al. [5] for the growth of FAP at relatively high supersaturation can possibly be explained by the rate expression Eq. (18). From the plot in Fig. 5 from Ref. [1], we obtain for Amjad et al.'s results kp in Eq. (18) to be 2 × 10 - j ° m o l / m 2 - s . This value is close to the theoretical value for kp given in Eq. (19). Distinction between the spiral mechanism and the polynuclear mechanism is often made on the background of the slopes, p, of plots of log(J/m o) against l o g ( S - 1) for rates corrected for changes in the surface area. For spirals p < 2. For the polynuclear mechanism p is often assumed or expected to be larger than 3. In Fig. 4 log(J/m o) is plotted against l o g ( S - 1), with J / m o calculated from Eqs. (1) and (3) for three typical values of c~/3. This plot shows that p is high for low values of S and that p approaches 1 for high values of S for the polynuclear mechanism, where rough growth takes over. Similarly, x* may be of the order 1 for the
polynuclear dissolution mechanism at very high degrees of undersaturation. As long as the activation energy for a building unit to leave the edge of a dissolution nucleus is high enough, the rate will be controlled by the surface reaction, and not by bulk diffusion.
Acknowledgements
We acknowledge valuable scientific discussions over nearly three decades with the late Professor Arne E. Nielsen. We are most grateful for the technical assistance of Ms. Mette Kja~r Schou and Ms. Julita Kuzimska. We thank Dr. H.E. Lundager Madsen for providing the ion speciation program Ionics. Financial support from the Danish Health Science Research Council, from Novo Nordic's Foundation and from the Vera and Carl Johan Michaelsen Foundation is gratefully acknowledged.
References
[1] J. Christoffersen, M.R. Christoffersen and T. Johansen, J. Crystal Growth 163 (1996) 295. [2] J. Christoffersen and M.R. Christoffersen, J. Crystal Growth 87 (1988) 41. [3] M.R. Christoffersen and J. Christoffersen, J. Crystal Growth 121 (1992) 617. [4] J. Christoffersen and MR. Christoffersen, J. Crystal Growth 57 (1982) 21. [5] Z. Amjad, P.G. Koutsoukos and G.H Nancollas, J. Colloid Interface Sci. 82 (1981) 394. [6] J. Christoffersen and MR. Christoffersen, J. Crystal Growth 121 (1992) 608.