Quantitative Study of Enamel Dissolution Under Conditions of Controlled Hydrodynamics WESLEY WAHITE and GEORGE H. NANCOLLAS
Department of Chenmistry, State University of New York at Buffalo, Buffalo, New York 14214,.
UTSA A rotating disk method has been used to study the dissolution of bovine enamel. A diffusion model adequately describes the reaction for a wide range of conditions, and the rate is also hydrogen ion dependent. The solubility product of bovine enamel, 5.9 ± 1.8 X 10-59 moles9 1-9 at 25 C is derived by a method that minimizes the possibility of phase changes.
S;nce dental caries is a disease that involves the dissolution of dental enamel by bacterially produced acid, many studies have been made of the rates and mechanisms of the reaction.1-8 Because knowledge of the processes occurring in the mouth is the ultimate goal, a number of these studies have attempted to reproduce conditions found in the mouth.68 However, the complex nature of the fluid dynamics and coDditions of the oral environment makes the interpretation of suchi results difficult. In this study, conditions have been optimized for the measurement of dissolution rates accessible to treatment by fundamental theories of solution chemistry. Such rate laws may be used to interpret experiments designed to more exactly reproduce conditions found in nature or to quantitate the effects of natural or synthetic inhibitors. The exceptionally stable and reproducible mass transport conditions found at the surface of a rotating disk allow calculations of the hydrodynamic boundary layer dimensions to be made using the Levich theory9 for fluid flow at a rotating disk. In the present work, the dependence of the rate of dissolution of bovine enamel on the concentrations of hydrogen, calcium, and phosphate ions has been studied under a variety of hydrodynamic conditions. Materials and Methods Analytical grade chemicals, water prepaled by deionization followed by double distillation tinder nitrogen, and Grade A glassware
Ct.
Received for publication January 12, 1976. Accepted for publication July 9, 1976. * Dowex 50 ion exchanger, Dowex, Phiuipsburg, NJ. t Perkin Elmer Model 503, Perkin Elmer, Norwalk, +
524
';Orthocryl," Stratford-Cookson Co., Yeadon, Pa.
were used throughout. Solution calcium concentrations were determined by passing aliquots through a cation exchange resin* in the hydrogen form and by titrating the eluate with standard base. For solutions containing more than one cation, calcium analysis was made using an atomic absorption spectrometer.t Phosphate was determined spectrophotometrically as the phasphomolybdate.'0 The bovine teeth used in this study included both fully matured permanent incisors from animals 7 to 10 years old and erupted deciduous incisors. The selected bovine incisors were centrally set in an inert, self-setting, acrylic resinj and machined to the shape shown in Figure 1. This. shape has been shownIJ to meet the requirements of the hydrodvnamic boundary layer calculations of Levich.9 The exposed enamel surface (about 1 cM2) was carefully polished with 6-micron and finally with 1-micron diamond dust. The enamel surfaces were stored in Hanks' balanced salt solution'2 before and after polish-ing to avoid changes in the organic matrix. Before use in a rate experiment, they were washed twice in an ultrasonic cleaner with distilled water and thoroughly rinsed with distilled water between changes of bulk solution. The surfacer areas were determined as the geometric area by photographic reproduction and enlargement. Dissolution reactions were carried out in the presence of 0.02 M potassium nitrate using a thermostatically controlled (25.00 + 0.05 C) Pyrex glass cell of about 150 ml capacity which was purged with nitrogen presaturated by pass-ing through a 0.02 M potassium nitrate solu-
exposed
bovacrylic disk
expseti
exposed
enamel
etin of
~~~~~~~bovinetooth,
FIG 1. Rotating disk.
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VFol. 56 No. 5
ENAMEL DISSOLUTION
525
of acid
ml
_
A I
0
I_
I
2 4 6 8 10 12 14 time in min FIG 2.-Plots of acid uptake as function of tiimie. Ionic concenitration products = 1.77 X 10- 9. Circle, 903 rpnm: triangle. 401 rpm.
tion. The acrylic disk xwith the polished tooth placed on the end of a Teflon-covx red spindle that xwas rotated by a synchlronious alternating current motor§ coupled to a pully assermnly that proxideod fixed rotationi sp)eds in the ranige of 400 to 2,500 i(pmii. A lMetrohm plI1 stat¶f with a sensitivity of 0.01 plH ullnit was used to (ontrol the pH Iy the adclitioni of 2.0 X 1t)- M HNlOu in 0.02 Al KNO5 froml a di,ital burette. The COLt of the reaction cotulc! be conveniently followed Ib observation of the r-ate of acic uiptake. T'lhe glass andI referenc electrodes xwere standardized in NBS phthalate anci phosphate bUffers" and hecked after each experimilenit; drifts in lfbUffer adjustmiienit Settinlgs of the instrtmnieint we eI alxw ay less thani 0.01 pH uinit. In txpical experiment blank acrv li( disk wvas firsit placed int the cell solution to sial)ilize the system. At this tim(', any nec essarx large pLI adjtistmlenits made with 0.10 ol 0.001 N KOH HNO. to m1inimi/(%t' v'olilule' changes; fiiial pH adjulstlltetit x-as mi,ade Il the pH-stat. The enamel SOuI-Ce WxaS tlenII place( in the c('II solutioni and the rate of eactioti was ohserved usinii a variety of totational s.peeds. The solution wxas thein remioxved for aIalNsis; a slightly imiore concenititated (calciumi phosphate solution in 0.02 Al potassiLlIli nitrate xas substituited,ainid the procedure repeated tintil the was
se
e
re
s
a
a
were
or
§ Type NSY-12 motor, Bodine Electric Co., Chicago, I 1.
; Combititrator 3D, Brinkmann InstruLments, Wesbtury. NY.
final soltution on(centration xwas close to calcittm phosphate satttration that the rtates beame too smalla for,accttrate m(easuement. Thsls mtethod of soltitioni additiont tl loxxed the rai'es of dissoluitioni to obt.tined over vide ranoIO of stthsattirtatiotis xw ithottt material lx chh:tn,itt c
so
he
the tootl
Lexich
xcii
fac e
theotxy.
h! droxvapatite
a
a
ouditio1
r0euictr((l
hv
the
(Calcultatiots of the amoutnt (f (I I.\Lr)remoxved Indicated tilht
\x0otlcl r(c(d((I bx nio 1mot e than -I X a complete Stele (i.Soucixh e\p(eriliet'its. Since a sill(otil Strfaxe is rTitica. disks tcre repol shed betxeenutse i'f there xW as the surfa-e
0-' ('11m
in
evidence of sutrfac e rtot-henitng. Soltitus of calcium phosphate xwith stoichtioltett i calciuIIt-phosphatte molar r atios of 5: 3 as req(luir de for HIAP as xwelIl as ratios of 1: 1 and :.5 xxe( uts(d in number of exp)erimuents andl the dleplendence of the rate of clissolutioiu On the hlt-
anyx
a
droeni iGol concentrationi xw
experiments at 5.30, anid 5.60.
as
detetrmined
pH valuCes of 4.4-0,
Results
anid
1.70,
bx
5.00.
DiscuissioIn
Typical plots of acid tiptake x erssus timlie in Figure 2. The slopcs of these lines yield the rate of dissolutioni R-) expressed - at eaCl as Imioles c:tlx itlm and phosphate concentration, aiid tihe liitial CtUrvatures reflect the timne lag for the pH stat to teachI a steadx state. Figutirc 3 shoxxws the variation of dlissolare shoxxin
sec
tion
in
rate
xith talc
iuLm1
phosplhate
conicenti-ation
solution.
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526
WHITE & NANCOLLAS
Quantitative represent ation of ioin transport processes involvititu spe ies stuch as calc imlli, hydrogen, and phosphate ioiis is coImlplicatedl 1)oth bx the e';SteneC of h1om0o1en0onLs eri1illbrialt iln the diffusion laver and bx- the differencltS in ionic charges anld dijfti,ion oefficiel-tS which (an lead t) harge as well as co( enttrationg aralients in the dliffn sion lax er. 1hle lassical methods of Nernst based oni Fick's first lawN are inapl)lical te if c harge giradients atre pieseni I omvever, the addition of a relatix (eI la-rge ount of ani inert' salt minimi/es the dcvelopmnunt of c harge gradients and unrlde- ( ertaimi conditioisl.t. permits a tIllthemati(cal representation of the conceuntration gradients hx the Lex\iIh thcorx f(r fluitii floxs ait a rttating tlisk. Unider the ptesensl expeitmental conditions, the uise of 0.02 .A KNO as a stlp)p)ortilg ele( trolx te has )eenl shoxswn to sumpprses charge gradients, and experinents durimug this study lretonhfime(
these findlings. Ac( ordiln, to the Lexit t theorv tlle dtepentlence of R-j, tIhe steadx state dissoltition r-ate, on tlec stirrin!g rate is of the form' R(= tt, r1 wxhere co is the ang-ular velocitv anld 0 is a comllplex integral, dlepeudent on the solution anld
J Dent Res Mlay 1977surfac e cou entration of ea li ion in the Nernst bounldar lav er, bttt niot ani explicit fuinctioni of W. IF r laminar flows, a x=w-hereas for tur< a < 1.1 Plots of the dissoltibtllet floHtw tionI ratt, R, explressed as imilliliters of acid added 1)t-r t inn1te-, against I 1 at several uincletnst-t.t'aios for a single (lisk, shoxsn in Figt1 . onftirsI that Iuder the coniditionis of thefl e experiments both- the conditions of diffiisioit contitol and latmtinar flows are met. It has been replsrtetl that larger caltltil
at
higlher sin speeds and
ttlncentratioins Imtpor taut a-
Itn tilits of sut ftwe Contrtol arc .llcotutecr1e. If thet riac tion waas ompletelx sturfat ontrolledI, the tate wxould not be an e\l)litit ftiutction tf thie thic kiess of the Nernst boundaiv laver andt Fi2cItt xi - twtld shotl a series of hom,ono Iital i, S. If tuitbtl entt floss xw as tate deterltttittin,' as thle anl(Ular walocits xas incr teased, Figtjtr+e 4 woldd shosw a sr i's tf ur'vs t conc(\ax (: Upwatrd. .itim l is rept (sent ative of the tresultis of all the txl)erimnttts made at the v artolis ioict elilt 1ratiotns of t alt iut anid phlosphatt aindi at thlec vartious totatioI trat(e itn the Ipresent xvttk. Sill e (litftl.sioll SeeIlls to be tiate controllinlg ritdeAr contlitiots itt \vbith hlar(e g-radiemnts hase beetn stlppl-essed. it is apl)Olpopriate to U-se-
0.3
Rd ml/min acid
% HAP saturation
FIG 3. Variation of dissolution rate with perceittage HAP satuiration in soluition. Ttiangle, 1.603 rpm: circle, 903 rpm: plts sign, 401 rpm. Downloaded from jdr.sagepub.com at Bobst Library, New York University on May 9, 2015 For personal use only. No other uses without permission.
ENAMEL DISSOLUTION
Vol. 56 No. 5
527
Rd ml/min acid
0.20
10 '2 W
in (radians/sec)
'/2
FIG 4. Plots of dissolution rate against w')2 at variouse calcium phosphate concentrations expressed as percentage HAP saturation, pH- 5.00.
Fick's lawv to represent the nieasured rate of dissolution. Such an equtation could be derived in terms of the separate diffusion of each ionic species if the ionic diffusion is not charge coupled; however, in the present iiistance these diffusion processes are coupled both by stoichic-inetry and by solubility limllitatioins. For diffusion control, the concentrations at the enamiiel surface are coupled by the solubility product constant, K,P, since the linit of the Kspi mulst !)e approached for diffu.sioni control. In adclition, for the stead) state, the ratios at wshich specific ions leave the diffusion layer mllust be in the correct stoichiometric ratios for the substance dissolving. In terms of these coonsiderations, an enmpirical equation F2] can be written relating the rate of diffnIsion of hydroxyapatite in "moles" sec-' cm-, R1, to solution concentrations: 16,17 Rd =V
d (HAP)
.-SDHAP
103 di [2] (HAP ) {(HAPo) In equation F2] V is the volume of the cell solution in liters; S, the area of enaimiel exposed in centimters'; DII AP the "overall" diffusion coefficienit of HAP in centinmeters2 plr seconid; S., the thickness in centimeters of the Nernst diffusion layer; (HAP), .he nmolar activity of aquated HAP in the bulk solution given by -
).
equation 3; and (HAP0), the value of the activitv a" the enamllel surface corresponding to satturationi.
(HAP)
AVithin
{[Ca"+] f,/5)
9
t{P043 ] f3/3}'9
{[OH. f1 } /!. equatioin [3], squaire brackets
[-I
denote coincentrationis and braces, activities; fz is the actixity coefficient of an ion of charge z. The latter was calculated using the extended form of the Debye Jijickel equation proposed by Davies.18 In all the experiments, the ionic strenigth wvas effecti-.,ely constant, 0.02 AS, mraintained b)y the relatively large potassiuim nitrate concentration. The corresponding value of f, xwas 0.85 + 0.01.
Phosphate activities were calculated by stanidard alpha" fractions-'3 using activity coefficienits. No corrections were made for the small quanitities (less ltan 2% of analytical calcium) of CaH.PPO4+ and CaHPO4 present.'4 Values of the phosphoric acid protonation constants w-ere pK,- 2.16; pK, = 7.13; and pK, = 12.30.19 WVith a pH of 5.00, the stoichiometry of the dissolution is represented by equation 4 as follows: Ca5 (PO4) :OH + 6.976H+ - 5Ca-+ + H.O + 0.012 (3) HPO4 2- +0.987 (3) [4] HPO-4 + 0.001 (3)H3PO4. It caln be seen in Figure 5 that eqtiation
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WHITE & NANCOLLAS
528
I Dent
Res
May
1977
[2] satisfactorily represents the data within experimental accuracy with the exception of a small nonzero intercept. This probably indi6 cates an error in the value, 3.7 + 0.5 X 10-5& XX0 used for Ksp which was taken as the value for -Rd + HAP synthetic HAP reported by Moreno, Gregory, moles per cm-40 and Brown.20 There has been considerable debate over the solubility product of both < i+ ° 2 HAP20-25 and enamel,26 and a recent model proposes more than one Ksp value27-28 for enamel. A corrected solubility product for 0 2 4 6 8 10 bovine enamei, which would require the lines (HAPo) (HAP) mules per liter X107 in Figure 5 to pass through the origin was calFIG 6.-Plots of rate of dissolution against culated at 5.9 + 1.8 X 10-59. As can be seen in - (HAP) according to equation [2] using (HAP0) Figure 6, the use of this value gave satisfactory = 10-59 for bovine enamel. convergence of the data with little change of Ksp 5.9 X -the straicrht lines from those ofJX6%Fimre 5. It is In this equation, v is the kinematic viscosity in %1LW notew ,orthy that the same value of the solucentimeters2in per second and Dj is the diffusion bilitv product was obtained with disks prepared coefficient centimeters2 per second of the jth using different teeth. In contrast, Brown, Patel, Diffusion coefficients were calculated and C hoW.29 in a study of human tooth enamel species. .. from conductances30 s equivalent n p except for the powd( i phosphate ion value, which was estimated e varied found tha solubility 10-.6 cm2 sec-1) from the mobility of simimulative amount of dissolution. TThe aad- la(8Xrvln the cuimulative vanta£ge of the present method is that the solurvalen is. 1\alues used species 1~~ ~ ~ ~ were Do V iOnl D03 bility product can be calculated from~~e the 1- for re-5.13X = 7.94X cm2 other sec1 and sults oif experiments in which there is effectively a of th soiufc. Thr.sltl no etc ;bin hi 10-6 cm2 sec-1. Nernst31 shown that for surface phase i-odifica- electrolvtes of type CA, thehas proper average diffor surface chanc(e, ttherefore, for tion dluring the dissolution experiments. The fusion coefficient is given by 2DCDA/(Dc + error limits indicated in this report are those DA), where DC and DA are the diffusion cocorres' ponding to one SDj SNof equation [2] can efficients of the individual species C and A. can and neglecting Using a similar relationship onetheLDjofequahtieony be dettonding trom from Levich theory as112modiinteraction terms, the average diffusion cofied b)y Riddiford'0 and is given by equation 5 : trCintrs h a-rg ifso o n i efficient for HAP calculated be to a may first (Ddo13 -1g2 [ s =N 1 80.5 approximiation as 1.4X 10-5 cm2 sec-l from E0.8934 + 0.316 p co equation 6: (Dj)0.36 [51 2DOH (2 DCaDpO/[Dca + Dp0j]) V *J D =6] -
.1
I.
sporouct woltit amolunilityprofdisstion the
os
phasemi lfica-
fore,
~ermined
A / / 0/ o ^//0
6 -Rd moles HAP per cm-sec
X
+
101,
A
+
+0
2
6 2 4 8 6 i0 (HAP.) - (HAP) moles per iter Xil7 FI,G 5.-PlotS of rate of dissolution against 0
0
*ration
subsatu (HAP) - (HAP) according to equatic)n [2] using K5p = 3.7 X 10 58 for bovine enamell.
HAP DOH + 2D0aDpO4/[Dea + DPO4] Setting v l.OX 10-2 cm2 sec-1 in equation [6], the Nemst boundary layer may be solved for
different rotation rates and ihis in turn can be Lised in equation [2] to calculate a theoretical reaction rate. It is found that there is a dependence of Rd on the hydrogen ion activity that is not explained by the simple model of equation [2]. This is clearly seen in Figure 7 in which the ratio of experimental to calculated rates is plotted as a function of hydroaen ion activity. A straight line with slope, 2.112X 106 M-', and intercept of 1.044 fits the data over the whole experimental range with a correlation coefficient if 0.9996. There is clearly a theoretical justification for considering the dissolution as resulting from a coupled diffusion of hydrogen
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Vol. 56 No. 5
ENAMEL DISSOLUTION
529
100
Rdexp. Rdcalc.
(H+) X 10 FIG 7. Plots of ratio of experimental to calculated dissolution rates as function of hydrogen ion activity.
ions inward and dissolution products outward. The term SN must be considered for each ion and in the case of phosphate and hydroxide, the actual diffusion distance may be shortened by reactions resulting in the formation of water and the hydrogen phosphate ion, respectively. The effcctive distance from the surface in which these reactions are essentially complete will depend inversely on the rate of inward diffusion of hydrogen ions, a term first order in bulk hydrogen ion concentration. This model involving a decrease in the diffusion distance explains the observed dissolution rate, which is more rapid than diffusion control for a single species and less mobile than a hydrogen ion. This observation will be covered in greater detail in a subsequent publication. In the light of the results presented, the overall dissolution rate may be expressed by equation [7]: -SDHAP H S{ (HAPO)-(HAP) }[a (H+) Rd _=
+ b],
[7]
where the constants are a = 6.469X 106 and
b = 32.99. Equation [7] reduces to a form similar to equation [2] at constant hydrogen ion concentration.
Conclusions The results of studies of the dissolution rate for several samples of bovine dental enamel under reproducible, well-defined hydrodynamic conditions lead to a simple equation based on fundamental solution chemistry and diffusion theory that adequately describes the reaction over a wide range of conditions. At the present time, although the equation is still largely empirical, it contains a term related to a coupled diffusion model that forms the basis of a rigorous representation of the reaction. Conditions of pH, ionic concentration, and ionic strength have been used that are typical of those in regions of carious lesions, and a value of Ksp for bovine enamel has been determined by a method that minimizes the possibility of phase changes. This confirms that dissolution of bovine enamel can be inter-
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530
WHITE & NANCOLLAS
preted in terms of conventional thermodynamic methods in a manner similar to that for synthetic HAP.
References 1. LINGE, H.G., and NANCOLLAS, G.H.: A Rotating Disc Study of the Dissolution of Den-
tal Enamel, Calcif Tissue Res 12:193-208, 1973. 2. GRAY, J.A.: Kinetics of the Dissolution of Human Dental Enamel in Acid, J Dent Res 41:633-645, 1962. 3. HIGUCHI, W.I.; GRAY, J.A.; HEFFERREN, J.J.; and PATEL, P.R.: Mechanism of Enamel Dissolution in Acid Buffers; J Dent Res 44:330-341, 1965. 4. DEDHIYA, M.G.; YOUNG, F.; and HIGUCHI, W.I.: Mechanism of Hydroxapatite Dissolution: The Synergistic Effects of Solution Fluoride, Strontium, and Phosphate, J Phys Chem 78:1273-1279, 1974. 5. NILSON, K.G., and HIGUCHI, W.I.: Mechanism of Fluoride Uptake by Hydroxyapatite from Acidic Fluoride Solutions: I. Theoretical Considerations, I Dent Res 49:15411548, 1970. 6. PIGMAN, W.; ELLIOTT, H.C; and LAFFRE, R.O.: An Artificial Mouth for Caries Research, J Dent Res 31:627-633, 1952. 7. PIGMAN, W., and SOGNNAES, R.F.: Histologic Studies of Carious-Like Lesions Produced in the Artificial Mouth, Oral Surg 8: 530-538, 1955. 8. McDoUGALL, W.A., and ADKINS, K.S.: A Method for the In Vitro Study of the Demineralization and Remineralization of the Subsurface Enamel, Aust Dent J 11:20-26, 1966. 9. LEVICH1, V.G.: Physicochemical Hydrodynamics, Englewood Cliffs, NJ: Prentice-Hall, 1962. 10. FoGG, D.N., and WILKINSON, N.T.: Colorimetric Determination of Phosphorus, Analyst (Lond) 83:403-404, 1958. 11. RIDDIFORD, A.C.: The Rotating Risk System, in DELAHAY, P. (ed): Advances in Electrochemistry and Electrochemical Engineering, Vol IV, New York: Interscience, 1966, pp 47-116. 12. WASLEY, G.D.: Advances in Technique, Baltimore: Williams & Wilkins Co., 1973. 13. BATES, R.G.: Determination of pH Theory and Practice, New York: John Wiley & Sons, p 123. 14. CHUGHTAL, A.; MARSHALL, R.; and NANCOLLAS, G.H.: Complexes in Calcium Phosphate Solutions, J Phys Chenm 72:208-211, 1968. 15. JONES, A.L.; LINGE, H.R.; and WILSON, I.R.: The Dissolution of Silver Chromate
I Dent
Res
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1977
into Aqueous Solutions, J Crystal Growth 12:201-208, 1972. 16. WILLARD, H.H.; MERRITT, L.L. JR.; and DEAN, J.A.: Instrumental Methods of Analysis, 4th ed, New York: Van Nostrand, 1965, Chap. 22. 17. MOORE, W.J.: Physical Chemistry, 3rd ed., Englewood Cliffs, NJ: Prentice-Hall, 1963. 18. DAVIES, C.W.: Ion Association, London: Butterworth's, 1962. 19. LAITINEN, H.A.: Chemical Analysis, New
York: McGraw-Hill, 1960. 20. MORENO, E.C.; GREGORY, T.M.; and BROWN, W.E.: Preparation and Solubility of Hydroxyapatite, J Res NBS 72A: 773-782, 1968. 21. LAMER, V.K.: Solubility Behavior of Hydroxyapatite, J Phys Chem 66:973-978, 1962. 22. NEUMAN, W.F., and NEUMAN, M.W.: The Chemical Dynamics of Bone Material, Chicago: University of Chicago Press, 1958. 23. MORENO, E.C.; BROWN, W.E.; and OsBORNE, G.: Stability of Dicalcium Phosphate Dihydrate in Aqueous Solutions and Solubility of Octocalcium Phosphate, Soil Sci Soc Am Proc 24:99-102, 1960. 24. LEVINSKAS, G.J., and NEUMAN, W.F.: The Solubility of Bone Mineral: I. Solubility Studies of Synthetic Hydroxyapatite, J Phys Chem 59:164-168, 1955. 25. ROOTARE, H.M.; DEITZ, V.R.; and CARPENTER, F.G.: Solubility Product Phenomena in Hydroxyapatite Water Systems, J Colloid Sci 17:179-206, 1962. 26. POSNER, A.S.: Crystal Chemistry of Bone Mineral, Physiological Reviews 49: 760-792, 1969. 27. YOUNG, F.; FAWZI, M.G.; DEDHIYA, M.G.; Wu, M.S.; and HIGUCHI, W.I.: Dual Mechanism for Dental Enamel Dissolution in Acid Buffers, J Dent Res 53 (Special Issue): Abstract No. 576, 1974. 28. FAWZI, M.; SONOBE, T.; HIGUCHI, W.I.;
and HEFFERREN, J.J.: Proposed Mechanism of Hydroxypatite Dissolution Under Partial Saturation: Synchronized Crystal Dissolution, J Dent Res 54: (Special Issue A): 157, No. 449, 1975. 29. BROWN, W.E.; PATEL, P.R.; and CHOW, L.C.: Formation of CaHPO4 2H20 from Enamel Mineral and Its Relationship to Caries Mechanism, J Dent Res 54:475-481, 1975. 30. ROBINSON, R.A., and STOKES, R.H.: Electrolyte Solutions, 2nd ed, London: Butterworth's, 1970. 31. NERNST, W.: Zur Kinetik der in Losung Befindlichen Korper, Z Physik Chem 2:613637, 1888.
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Department of Chenmistry, State University of New York at Buffalo, Buffalo, New York 14214,.
UTSA A rotating disk method has been used to study the dissolution of bovine enamel. A diffusion model adequately describes the reaction for a wide range of conditions, and the rate is also hydrogen ion dependent. The solubility product of bovine enamel, 5.9 ± 1.8 X 10-59 moles9 1-9 at 25 C is derived by a method that minimizes the possibility of phase changes.
S;nce dental caries is a disease that involves the dissolution of dental enamel by bacterially produced acid, many studies have been made of the rates and mechanisms of the reaction.1-8 Because knowledge of the processes occurring in the mouth is the ultimate goal, a number of these studies have attempted to reproduce conditions found in the mouth.68 However, the complex nature of the fluid dynamics and coDditions of the oral environment makes the interpretation of suchi results difficult. In this study, conditions have been optimized for the measurement of dissolution rates accessible to treatment by fundamental theories of solution chemistry. Such rate laws may be used to interpret experiments designed to more exactly reproduce conditions found in nature or to quantitate the effects of natural or synthetic inhibitors. The exceptionally stable and reproducible mass transport conditions found at the surface of a rotating disk allow calculations of the hydrodynamic boundary layer dimensions to be made using the Levich theory9 for fluid flow at a rotating disk. In the present work, the dependence of the rate of dissolution of bovine enamel on the concentrations of hydrogen, calcium, and phosphate ions has been studied under a variety of hydrodynamic conditions. Materials and Methods Analytical grade chemicals, water prepaled by deionization followed by double distillation tinder nitrogen, and Grade A glassware
Ct.
Received for publication January 12, 1976. Accepted for publication July 9, 1976. * Dowex 50 ion exchanger, Dowex, Phiuipsburg, NJ. t Perkin Elmer Model 503, Perkin Elmer, Norwalk, +
524
';Orthocryl," Stratford-Cookson Co., Yeadon, Pa.
were used throughout. Solution calcium concentrations were determined by passing aliquots through a cation exchange resin* in the hydrogen form and by titrating the eluate with standard base. For solutions containing more than one cation, calcium analysis was made using an atomic absorption spectrometer.t Phosphate was determined spectrophotometrically as the phasphomolybdate.'0 The bovine teeth used in this study included both fully matured permanent incisors from animals 7 to 10 years old and erupted deciduous incisors. The selected bovine incisors were centrally set in an inert, self-setting, acrylic resinj and machined to the shape shown in Figure 1. This. shape has been shownIJ to meet the requirements of the hydrodvnamic boundary layer calculations of Levich.9 The exposed enamel surface (about 1 cM2) was carefully polished with 6-micron and finally with 1-micron diamond dust. The enamel surfaces were stored in Hanks' balanced salt solution'2 before and after polish-ing to avoid changes in the organic matrix. Before use in a rate experiment, they were washed twice in an ultrasonic cleaner with distilled water and thoroughly rinsed with distilled water between changes of bulk solution. The surfacer areas were determined as the geometric area by photographic reproduction and enlargement. Dissolution reactions were carried out in the presence of 0.02 M potassium nitrate using a thermostatically controlled (25.00 + 0.05 C) Pyrex glass cell of about 150 ml capacity which was purged with nitrogen presaturated by pass-ing through a 0.02 M potassium nitrate solu-
exposed
bovacrylic disk
expseti
exposed
enamel
etin of
~~~~~~~bovinetooth,
FIG 1. Rotating disk.
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VFol. 56 No. 5
ENAMEL DISSOLUTION
525
of acid
ml
_
A I
0
I_
I
2 4 6 8 10 12 14 time in min FIG 2.-Plots of acid uptake as function of tiimie. Ionic concenitration products = 1.77 X 10- 9. Circle, 903 rpnm: triangle. 401 rpm.
tion. The acrylic disk xwith the polished tooth placed on the end of a Teflon-covx red spindle that xwas rotated by a synchlronious alternating current motor§ coupled to a pully assermnly that proxideod fixed rotationi sp)eds in the ranige of 400 to 2,500 i(pmii. A lMetrohm plI1 stat¶f with a sensitivity of 0.01 plH ullnit was used to (ontrol the pH Iy the adclitioni of 2.0 X 1t)- M HNlOu in 0.02 Al KNO5 froml a di,ital burette. The COLt of the reaction cotulc! be conveniently followed Ib observation of the r-ate of acic uiptake. T'lhe glass andI referenc electrodes xwere standardized in NBS phthalate anci phosphate bUffers" and hecked after each experimilenit; drifts in lfbUffer adjustmiienit Settinlgs of the instrtmnieint we eI alxw ay less thani 0.01 pH uinit. In txpical experiment blank acrv li( disk wvas firsit placed int the cell solution to sial)ilize the system. At this tim(', any nec essarx large pLI adjtistmlenits made with 0.10 ol 0.001 N KOH HNO. to m1inimi/(%t' v'olilule' changes; fiiial pH adjulstlltetit x-as mi,ade Il the pH-stat. The enamel SOuI-Ce WxaS tlenII place( in the c('II solutioni and the rate of eactioti was ohserved usinii a variety of totational s.peeds. The solution wxas thein remioxved for aIalNsis; a slightly imiore concenititated (calciumi phosphate solution in 0.02 Al potassiLlIli nitrate xas substituited,ainid the procedure repeated tintil the was
se
e
re
s
a
a
were
or
§ Type NSY-12 motor, Bodine Electric Co., Chicago, I 1.
; Combititrator 3D, Brinkmann InstruLments, Wesbtury. NY.
final soltution on(centration xwas close to calcittm phosphate satttration that the rtates beame too smalla for,accttrate m(easuement. Thsls mtethod of soltitioni additiont tl loxxed the rai'es of dissoluitioni to obt.tined over vide ranoIO of stthsattirtatiotis xw ithottt material lx chh:tn,itt c
so
he
the tootl
Lexich
xcii
fac e
theotxy.
h! droxvapatite
a
a
ouditio1
r0euictr((l
hv
the
(Calcultatiots of the amoutnt (f (I I.\Lr)remoxved Indicated tilht
\x0otlcl r(c(d((I bx nio 1mot e than -I X a complete Stele (i.Soucixh e\p(eriliet'its. Since a sill(otil Strfaxe is rTitica. disks tcre repol shed betxeenutse i'f there xW as the surfa-e
0-' ('11m
in
evidence of sutrfac e rtot-henitng. Soltitus of calcium phosphate xwith stoichtioltett i calciuIIt-phosphatte molar r atios of 5: 3 as req(luir de for HIAP as xwelIl as ratios of 1: 1 and :.5 xxe( uts(d in number of exp)erimuents andl the dleplendence of the rate of clissolutioiu On the hlt-
anyx
a
droeni iGol concentrationi xw
experiments at 5.30, anid 5.60.
as
detetrmined
pH valuCes of 4.4-0,
Results
anid
1.70,
bx
5.00.
DiscuissioIn
Typical plots of acid tiptake x erssus timlie in Figure 2. The slopcs of these lines yield the rate of dissolutioni R-) expressed - at eaCl as Imioles c:tlx itlm and phosphate concentration, aiid tihe liitial CtUrvatures reflect the timne lag for the pH stat to teachI a steadx state. Figutirc 3 shoxxws the variation of dlissolare shoxxin
sec
tion
in
rate
xith talc
iuLm1
phosplhate
conicenti-ation
solution.
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526
WHITE & NANCOLLAS
Quantitative represent ation of ioin transport processes involvititu spe ies stuch as calc imlli, hydrogen, and phosphate ioiis is coImlplicatedl 1)oth bx the e';SteneC of h1om0o1en0onLs eri1illbrialt iln the diffusion laver and bx- the differencltS in ionic charges anld dijfti,ion oefficiel-tS which (an lead t) harge as well as co( enttrationg aralients in the dliffn sion lax er. 1hle lassical methods of Nernst based oni Fick's first lawN are inapl)lical te if c harge giradients atre pieseni I omvever, the addition of a relatix (eI la-rge ount of ani inert' salt minimi/es the dcvelopmnunt of c harge gradients and unrlde- ( ertaimi conditioisl.t. permits a tIllthemati(cal representation of the conceuntration gradients hx the Lex\iIh thcorx f(r fluitii floxs ait a rttating tlisk. Unider the ptesensl expeitmental conditions, the uise of 0.02 .A KNO as a stlp)p)ortilg ele( trolx te has )eenl shoxswn to sumpprses charge gradients, and experinents durimug this study lretonhfime(
these findlings. Ac( ordiln, to the Lexit t theorv tlle dtepentlence of R-j, tIhe steadx state dissoltition r-ate, on tlec stirrin!g rate is of the form' R(= tt, r1 wxhere co is the ang-ular velocitv anld 0 is a comllplex integral, dlepeudent on the solution anld
J Dent Res Mlay 1977surfac e cou entration of ea li ion in the Nernst bounldar lav er, bttt niot ani explicit fuinctioni of W. IF r laminar flows, a x=w-hereas for tur< a < 1.1 Plots of the dissoltibtllet floHtw tionI ratt, R, explressed as imilliliters of acid added 1)t-r t inn1te-, against I 1 at several uincletnst-t.t'aios for a single (lisk, shoxsn in Figt1 . onftirsI that Iuder the coniditionis of thefl e experiments both- the conditions of diffiisioit contitol and latmtinar flows are met. It has been replsrtetl that larger caltltil
at
higlher sin speeds and
ttlncentratioins Imtpor taut a-
Itn tilits of sut ftwe Contrtol arc .llcotutecr1e. If thet riac tion waas ompletelx sturfat ontrolledI, the tate wxould not be an e\l)litit ftiutction tf thie thic kiess of the Nernst boundaiv laver andt Fi2cItt xi - twtld shotl a series of hom,ono Iital i, S. If tuitbtl entt floss xw as tate deterltttittin,' as thle anl(Ular walocits xas incr teased, Figtjtr+e 4 woldd shosw a sr i's tf ur'vs t conc(\ax (: Upwatrd. .itim l is rept (sent ative of the tresultis of all the txl)erimnttts made at the v artolis ioict elilt 1ratiotns of t alt iut anid phlosphatt aindi at thlec vartious totatioI trat(e itn the Ipresent xvttk. Sill e (litftl.sioll SeeIlls to be tiate controllinlg ritdeAr contlitiots itt \vbith hlar(e g-radiemnts hase beetn stlppl-essed. it is apl)Olpopriate to U-se-
0.3
Rd ml/min acid
% HAP saturation
FIG 3. Variation of dissolution rate with perceittage HAP satuiration in soluition. Ttiangle, 1.603 rpm: circle, 903 rpm: plts sign, 401 rpm. Downloaded from jdr.sagepub.com at Bobst Library, New York University on May 9, 2015 For personal use only. No other uses without permission.
ENAMEL DISSOLUTION
Vol. 56 No. 5
527
Rd ml/min acid
0.20
10 '2 W
in (radians/sec)
'/2
FIG 4. Plots of dissolution rate against w')2 at variouse calcium phosphate concentrations expressed as percentage HAP saturation, pH- 5.00.
Fick's lawv to represent the nieasured rate of dissolution. Such an equtation could be derived in terms of the separate diffusion of each ionic species if the ionic diffusion is not charge coupled; however, in the present iiistance these diffusion processes are coupled both by stoichic-inetry and by solubility limllitatioins. For diffusion control, the concentrations at the enamiiel surface are coupled by the solubility product constant, K,P, since the linit of the Kspi mulst !)e approached for diffu.sioni control. In adclition, for the stead) state, the ratios at wshich specific ions leave the diffusion layer mllust be in the correct stoichiometric ratios for the substance dissolving. In terms of these coonsiderations, an enmpirical equation F2] can be written relating the rate of diffnIsion of hydroxyapatite in "moles" sec-' cm-, R1, to solution concentrations: 16,17 Rd =V
d (HAP)
.-SDHAP
103 di [2] (HAP ) {(HAPo) In equation F2] V is the volume of the cell solution in liters; S, the area of enaimiel exposed in centimters'; DII AP the "overall" diffusion coefficienit of HAP in centinmeters2 plr seconid; S., the thickness in centimeters of the Nernst diffusion layer; (HAP), .he nmolar activity of aquated HAP in the bulk solution given by -
).
equation 3; and (HAP0), the value of the activitv a" the enamllel surface corresponding to satturationi.
(HAP)
AVithin
{[Ca"+] f,/5)
9
t{P043 ] f3/3}'9
{[OH. f1 } /!. equatioin [3], squaire brackets
[-I
denote coincentrationis and braces, activities; fz is the actixity coefficient of an ion of charge z. The latter was calculated using the extended form of the Debye Jijickel equation proposed by Davies.18 In all the experiments, the ionic strenigth wvas effecti-.,ely constant, 0.02 AS, mraintained b)y the relatively large potassiuim nitrate concentration. The corresponding value of f, xwas 0.85 + 0.01.
Phosphate activities were calculated by stanidard alpha" fractions-'3 using activity coefficienits. No corrections were made for the small quanitities (less ltan 2% of analytical calcium) of CaH.PPO4+ and CaHPO4 present.'4 Values of the phosphoric acid protonation constants w-ere pK,- 2.16; pK, = 7.13; and pK, = 12.30.19 WVith a pH of 5.00, the stoichiometry of the dissolution is represented by equation 4 as follows: Ca5 (PO4) :OH + 6.976H+ - 5Ca-+ + H.O + 0.012 (3) HPO4 2- +0.987 (3) [4] HPO-4 + 0.001 (3)H3PO4. It caln be seen in Figure 5 that eqtiation
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WHITE & NANCOLLAS
528
I Dent
Res
May
1977
[2] satisfactorily represents the data within experimental accuracy with the exception of a small nonzero intercept. This probably indi6 cates an error in the value, 3.7 + 0.5 X 10-5& XX0 used for Ksp which was taken as the value for -Rd + HAP synthetic HAP reported by Moreno, Gregory, moles per cm-40 and Brown.20 There has been considerable debate over the solubility product of both < i+ ° 2 HAP20-25 and enamel,26 and a recent model proposes more than one Ksp value27-28 for enamel. A corrected solubility product for 0 2 4 6 8 10 bovine enamei, which would require the lines (HAPo) (HAP) mules per liter X107 in Figure 5 to pass through the origin was calFIG 6.-Plots of rate of dissolution against culated at 5.9 + 1.8 X 10-59. As can be seen in - (HAP) according to equation [2] using (HAP0) Figure 6, the use of this value gave satisfactory = 10-59 for bovine enamel. convergence of the data with little change of Ksp 5.9 X -the straicrht lines from those ofJX6%Fimre 5. It is In this equation, v is the kinematic viscosity in %1LW notew ,orthy that the same value of the solucentimeters2in per second and Dj is the diffusion bilitv product was obtained with disks prepared coefficient centimeters2 per second of the jth using different teeth. In contrast, Brown, Patel, Diffusion coefficients were calculated and C hoW.29 in a study of human tooth enamel species. .. from conductances30 s equivalent n p except for the powd( i phosphate ion value, which was estimated e varied found tha solubility 10-.6 cm2 sec-1) from the mobility of simimulative amount of dissolution. TThe aad- la(8Xrvln the cuimulative vanta£ge of the present method is that the solurvalen is. 1\alues used species 1~~ ~ ~ ~ were Do V iOnl D03 bility product can be calculated from~~e the 1- for re-5.13X = 7.94X cm2 other sec1 and sults oif experiments in which there is effectively a of th soiufc. Thr.sltl no etc ;bin hi 10-6 cm2 sec-1. Nernst31 shown that for surface phase i-odifica- electrolvtes of type CA, thehas proper average diffor surface chanc(e, ttherefore, for tion dluring the dissolution experiments. The fusion coefficient is given by 2DCDA/(Dc + error limits indicated in this report are those DA), where DC and DA are the diffusion cocorres' ponding to one SDj SNof equation [2] can efficients of the individual species C and A. can and neglecting Using a similar relationship onetheLDjofequahtieony be dettonding trom from Levich theory as112modiinteraction terms, the average diffusion cofied b)y Riddiford'0 and is given by equation 5 : trCintrs h a-rg ifso o n i efficient for HAP calculated be to a may first (Ddo13 -1g2 [ s =N 1 80.5 approximiation as 1.4X 10-5 cm2 sec-l from E0.8934 + 0.316 p co equation 6: (Dj)0.36 [51 2DOH (2 DCaDpO/[Dca + Dp0j]) V *J D =6] -
.1
I.
sporouct woltit amolunilityprofdisstion the
os
phasemi lfica-
fore,
~ermined
A / / 0/ o ^//0
6 -Rd moles HAP per cm-sec
X
+
101,
A
+
+0
2
6 2 4 8 6 i0 (HAP.) - (HAP) moles per iter Xil7 FI,G 5.-PlotS of rate of dissolution against 0
0
*ration
subsatu (HAP) - (HAP) according to equatic)n [2] using K5p = 3.7 X 10 58 for bovine enamell.
HAP DOH + 2D0aDpO4/[Dea + DPO4] Setting v l.OX 10-2 cm2 sec-1 in equation [6], the Nemst boundary layer may be solved for
different rotation rates and ihis in turn can be Lised in equation [2] to calculate a theoretical reaction rate. It is found that there is a dependence of Rd on the hydrogen ion activity that is not explained by the simple model of equation [2]. This is clearly seen in Figure 7 in which the ratio of experimental to calculated rates is plotted as a function of hydroaen ion activity. A straight line with slope, 2.112X 106 M-', and intercept of 1.044 fits the data over the whole experimental range with a correlation coefficient if 0.9996. There is clearly a theoretical justification for considering the dissolution as resulting from a coupled diffusion of hydrogen
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Vol. 56 No. 5
ENAMEL DISSOLUTION
529
100
Rdexp. Rdcalc.
(H+) X 10 FIG 7. Plots of ratio of experimental to calculated dissolution rates as function of hydrogen ion activity.
ions inward and dissolution products outward. The term SN must be considered for each ion and in the case of phosphate and hydroxide, the actual diffusion distance may be shortened by reactions resulting in the formation of water and the hydrogen phosphate ion, respectively. The effcctive distance from the surface in which these reactions are essentially complete will depend inversely on the rate of inward diffusion of hydrogen ions, a term first order in bulk hydrogen ion concentration. This model involving a decrease in the diffusion distance explains the observed dissolution rate, which is more rapid than diffusion control for a single species and less mobile than a hydrogen ion. This observation will be covered in greater detail in a subsequent publication. In the light of the results presented, the overall dissolution rate may be expressed by equation [7]: -SDHAP H S{ (HAPO)-(HAP) }[a (H+) Rd _=
+ b],
[7]
where the constants are a = 6.469X 106 and
b = 32.99. Equation [7] reduces to a form similar to equation [2] at constant hydrogen ion concentration.
Conclusions The results of studies of the dissolution rate for several samples of bovine dental enamel under reproducible, well-defined hydrodynamic conditions lead to a simple equation based on fundamental solution chemistry and diffusion theory that adequately describes the reaction over a wide range of conditions. At the present time, although the equation is still largely empirical, it contains a term related to a coupled diffusion model that forms the basis of a rigorous representation of the reaction. Conditions of pH, ionic concentration, and ionic strength have been used that are typical of those in regions of carious lesions, and a value of Ksp for bovine enamel has been determined by a method that minimizes the possibility of phase changes. This confirms that dissolution of bovine enamel can be inter-
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530
WHITE & NANCOLLAS
preted in terms of conventional thermodynamic methods in a manner similar to that for synthetic HAP.
References 1. LINGE, H.G., and NANCOLLAS, G.H.: A Rotating Disc Study of the Dissolution of Den-
tal Enamel, Calcif Tissue Res 12:193-208, 1973. 2. GRAY, J.A.: Kinetics of the Dissolution of Human Dental Enamel in Acid, J Dent Res 41:633-645, 1962. 3. HIGUCHI, W.I.; GRAY, J.A.; HEFFERREN, J.J.; and PATEL, P.R.: Mechanism of Enamel Dissolution in Acid Buffers; J Dent Res 44:330-341, 1965. 4. DEDHIYA, M.G.; YOUNG, F.; and HIGUCHI, W.I.: Mechanism of Hydroxapatite Dissolution: The Synergistic Effects of Solution Fluoride, Strontium, and Phosphate, J Phys Chem 78:1273-1279, 1974. 5. NILSON, K.G., and HIGUCHI, W.I.: Mechanism of Fluoride Uptake by Hydroxyapatite from Acidic Fluoride Solutions: I. Theoretical Considerations, I Dent Res 49:15411548, 1970. 6. PIGMAN, W.; ELLIOTT, H.C; and LAFFRE, R.O.: An Artificial Mouth for Caries Research, J Dent Res 31:627-633, 1952. 7. PIGMAN, W., and SOGNNAES, R.F.: Histologic Studies of Carious-Like Lesions Produced in the Artificial Mouth, Oral Surg 8: 530-538, 1955. 8. McDoUGALL, W.A., and ADKINS, K.S.: A Method for the In Vitro Study of the Demineralization and Remineralization of the Subsurface Enamel, Aust Dent J 11:20-26, 1966. 9. LEVICH1, V.G.: Physicochemical Hydrodynamics, Englewood Cliffs, NJ: Prentice-Hall, 1962. 10. FoGG, D.N., and WILKINSON, N.T.: Colorimetric Determination of Phosphorus, Analyst (Lond) 83:403-404, 1958. 11. RIDDIFORD, A.C.: The Rotating Risk System, in DELAHAY, P. (ed): Advances in Electrochemistry and Electrochemical Engineering, Vol IV, New York: Interscience, 1966, pp 47-116. 12. WASLEY, G.D.: Advances in Technique, Baltimore: Williams & Wilkins Co., 1973. 13. BATES, R.G.: Determination of pH Theory and Practice, New York: John Wiley & Sons, p 123. 14. CHUGHTAL, A.; MARSHALL, R.; and NANCOLLAS, G.H.: Complexes in Calcium Phosphate Solutions, J Phys Chenm 72:208-211, 1968. 15. JONES, A.L.; LINGE, H.R.; and WILSON, I.R.: The Dissolution of Silver Chromate
I Dent
Res
May
1977
into Aqueous Solutions, J Crystal Growth 12:201-208, 1972. 16. WILLARD, H.H.; MERRITT, L.L. JR.; and DEAN, J.A.: Instrumental Methods of Analysis, 4th ed, New York: Van Nostrand, 1965, Chap. 22. 17. MOORE, W.J.: Physical Chemistry, 3rd ed., Englewood Cliffs, NJ: Prentice-Hall, 1963. 18. DAVIES, C.W.: Ion Association, London: Butterworth's, 1962. 19. LAITINEN, H.A.: Chemical Analysis, New
York: McGraw-Hill, 1960. 20. MORENO, E.C.; GREGORY, T.M.; and BROWN, W.E.: Preparation and Solubility of Hydroxyapatite, J Res NBS 72A: 773-782, 1968. 21. LAMER, V.K.: Solubility Behavior of Hydroxyapatite, J Phys Chem 66:973-978, 1962. 22. NEUMAN, W.F., and NEUMAN, M.W.: The Chemical Dynamics of Bone Material, Chicago: University of Chicago Press, 1958. 23. MORENO, E.C.; BROWN, W.E.; and OsBORNE, G.: Stability of Dicalcium Phosphate Dihydrate in Aqueous Solutions and Solubility of Octocalcium Phosphate, Soil Sci Soc Am Proc 24:99-102, 1960. 24. LEVINSKAS, G.J., and NEUMAN, W.F.: The Solubility of Bone Mineral: I. Solubility Studies of Synthetic Hydroxyapatite, J Phys Chem 59:164-168, 1955. 25. ROOTARE, H.M.; DEITZ, V.R.; and CARPENTER, F.G.: Solubility Product Phenomena in Hydroxyapatite Water Systems, J Colloid Sci 17:179-206, 1962. 26. POSNER, A.S.: Crystal Chemistry of Bone Mineral, Physiological Reviews 49: 760-792, 1969. 27. YOUNG, F.; FAWZI, M.G.; DEDHIYA, M.G.; Wu, M.S.; and HIGUCHI, W.I.: Dual Mechanism for Dental Enamel Dissolution in Acid Buffers, J Dent Res 53 (Special Issue): Abstract No. 576, 1974. 28. FAWZI, M.; SONOBE, T.; HIGUCHI, W.I.;
and HEFFERREN, J.J.: Proposed Mechanism of Hydroxypatite Dissolution Under Partial Saturation: Synchronized Crystal Dissolution, J Dent Res 54: (Special Issue A): 157, No. 449, 1975. 29. BROWN, W.E.; PATEL, P.R.; and CHOW, L.C.: Formation of CaHPO4 2H20 from Enamel Mineral and Its Relationship to Caries Mechanism, J Dent Res 54:475-481, 1975. 30. ROBINSON, R.A., and STOKES, R.H.: Electrolyte Solutions, 2nd ed, London: Butterworth's, 1970. 31. NERNST, W.: Zur Kinetik der in Losung Befindlichen Korper, Z Physik Chem 2:613637, 1888.
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