ArchsoralBid Vol.22.pp.399to403.PergamonPress1977. Printed in Great Brieah
SOME
ELECTROCHEMICAL CHARACTERISTICS HUMAN TOOTH ENAMEL J.
W.
E. VAN DIJK, N. E. WATERS,
F. C. M.
OF
J. M. P. M. BORGGREVEN and
DRIE%%NS
Dental School, Catholic University of Nijmegen, “Heyendael”, Nijmegen, The Netherlands and Department of Physical Sciences, The Royal Dental Hospital, School for Dental Surgery
University of London, Leicester Square, London WC2, England Summary--The fixed charge of enamel and the ratio of the diffusion coefficients of cations to anions of some common electrolytes within enamel were calculated from steady state electromotive forces developed in concentration cells across enamel caps of whole human teeth, sections of human enamel or synthetic hydroxyapatite membranes. The electrolytes were KCl, KF, NaCI, C&l,, KH2POh and K2HP0,. The results indicate that enamel behaves like an ion exchange membrane of which the fixed charge depends on the electrolytic environment and the sequence of exposure to various electrolytes.
INTRODUCTION
i the parameter for species i. The symbols are:
In initial smooth surface dental caries, the inorganic phase of enamel dissolves partially and the ions must pass a relatively intact surface layer of enamel by diffusion (Moreno, 1974; Silverstone, 1973). Thus for understanding the mechanism of carious breakdown, it is important to know what the transport properties of enamel are and how they can be influenced by chemical agents and physical influences. Amberson, Williams and Klein (1926) and Klein (1932) measured the electromotive force (e.m.f.) of concentration cells in which enamel served as a membrane. They proved that enamel behaves as an ion-selective porous membrane with a fixed charge on the pore-walls. It seems that this charge can be changed by the action of electrolytes like CaCl,. Waters (1968, 1971, 1975) confirmed these findings, giving more information about the influence of some common electrolytes on the chemical properties of enamel. These studies are qualitative in the sense that they express the selectivity of enamel in terms of membrane potential only. We now present a mathematical procedure which enables the calculation of the two parameters that are characteristic of the selectivity of a charge membrane, namely the ratio of the diffusion coefficients of the cations and anions within the membrane and its effective fixed charge.
MATE:RIALS AND METHODS
Physical and mthmutical
methods
J D
c
a In(a)
-
ax
F R aE
the flux the diffusion coefficient the concentration the gradient of the logarithm of the activity Faraday’s constant gas constant the gradient of the electrical potential
SX
z
the charge of the ion.
We used single electrolytes and therefore there are only two flux equations (I), one for the cation and one for the anion. An exact solution of these differential equations is not possible because of several unknown parameters, including the concentration dependence of the activity of the ions within a pore. It was necessary to conform to the following assumptions which are inherent in the TMS theory: 1. All activities are set equal to concentration. 2. The diffusion coefficients of the ions do not change with concentration nor with the position within the membrane. 3. There is no convection within the membrane. 4. The membrane has a constant composition throughout its entire thickness. In addition to the two differential equations and the assumptions, there are two conditions: 5. There is no electrical current XziJi = 0
The basis of the calculations is the Teorell-MeyerSievers theory (TMS theory) (Teorell, 1953; Helfferich, 1962; Schlogl, 1954) which describes the transport of ions through a porous membrane having fixed charges within its structure. The theory uses the Nernst-Planck flux equation in the form (Helfferich, 1962):
where ?? the fixed charge of the membrane
a In(@) __
w the sign of the fixed charge.
Ji =
-jiici
ax
(1)
(21
and 6. Within the membrane there is electro-neutrality ZZiEi + wx = 0
(3)
Using the two flux equations (1) and the assumptions and conditions, we can get an explicit expression for
A bar denotes a parameter inside the membrane and 399
400
J. W. E. van Dijk et al
1965). The calculations have been carried out on the IBM370/158 computer of the Catholic University of Nijmegen using PL/l programs. In the Appendix, the principle of the calculation of the parameters 0+/oand WX as well as their standard errors is explained.
the total membrane potential E,; E, = E + E:, + Eb’ B, -Di Z+D+ - z-i( ‘)( z+D+(r’)i+ - z_B_(r’)‘xln c cl’ z+D+(r”)l+ - z_b_(r”)r- >
E, = y
I,
+lnY i r’
)I
EXPERIMENT
(4)
The single or double marks denote a parameter at the left and right interface respectively, a+ or athe parameters for a cation or an anion. r denotes the Donnan distribution coefficient for the membrane/solution boundaries and Ed the corresponding Donnan potential. Figure 1 shows a scheme of the various variables. The Donnan distribution coefficients give a relation between the concentration of an ion inside and outside the charged membrane at the interface. C+ = r”. c+ and?_
= ?m.c_.
(5)
By substituting (5) and (3) we get an equation of higher degree from which the distribution coefficients can be calculated if the fixed charge is known: z+c+f4
+ z-c-i-
+ OX = 0.
(6)
If the e.m.f. of a concentration cell is measured at two or more pairs of concentrations, c’ and c”, the two independent and unknown parameters 0+/oand WX can be calculated using the equations (4) and (6) in an iterative numerical procedure. In fact the e.m.f. at more than two concentration pairs has been measured. Values for 0+/D- and ox have then been calculated that give the smallest sum of squares of the difference between the measured e.m.f. and the e.m.f. calculated by substituting the values of D+/n_ and OX in (6) and (4). This non-linear least-squares problem is solved by using a numerical procedure for minimizing a non-linear sum of squares (Powell,
All calculations refer to e.m.f. data from experiments published in earlier papers (Waters, 1971, 1972, 1975). In all experiments the e.m.f. was measured on concentration cells of the type: ref. electr./lsolution
1lmembranelsolution
2llref. electr.
Mercuryalomel half cells were used as reference electrodes and were connected to the solutions by saturated KCl-agar salt bridges. The potential developed across the cell was measured with a high-impedance potentiometric recorder. The technique including that for the preparation of the human enamel caps and synthetic membranes has been described (Chick and Waters, 1963; Waters, 1971, 1970). All electrolyte solutions, except those with and K2HP0, were buffered with KH,PO, 0.002 mol/l barbituric acid and adjusted to pH 7.4 with 2 mol/l KOH. The influence of the ions from Em
mV
4c
30
2c
10
bulk
membrane
bulk
0
- ,I ‘cl
C’.
Em
-10
-20
E.0
-30
-40 0.2 x=0
Fig. 1. Graph lyte and the
x=6
of the concentrations of a I.1 valent electroe.m.f. in a negatively charged membrane (Ed are the Donnan potentials).
0.4
0.6
0.6
1.0
1.2
1.4
1.6
log f
Fig. 2. Plot of the e.m.f. versus log c’jc” measured on tooth Tl9 with c’ = O.I,mol/l (+KCl; A-NaCI; q--Call,). Standard errors S, [see equation (A,)] were 0.4 mV-KCl. 0.9 mV-NaCl and 0.8 mV-CaCl,.
401
Electrochemical characteristics of tooth enamel Table 1. Fixed charge in mequiv/l of human dental enamel and synthetic hydroxyapatite
in various electrolytes
Electrolyte Membrane T12 T15 T19 T21 T23 T24 T31 T32 T34 T35 T36 T38 ClE C4E C5E C6E ClHA C2HA C3HA CllHA C12HA C13HA C14HA
KC1 --4 -15 --9 -34 -15 --6
(1) (4) (1) (24) (1) (2)
--3 (2) -16 (3) --3 (2) --7 (2) -7 (3) -- 0.9 (0.3) --7 (1) -21 (6) -- 1.7 (0.6) -22 (9) -- 3.1 (0.6) -27 (8) -- 1.2 (0.5) -- 1.6 (0.7)
KCI* -25 (16)
-34(11) - 24 (6) -13(l)
NaCl -5 -32 -10 -16 -25
CaCl,
(1) (7) (2) (2) (20)
0 -10 -4 -4 -9
K,HPO,t
(5) (4) (1) (4) (2) -13(5) -37(14) -17(4) -15(7) -26(13) -53 (45)
- 20 (6) -3 (1) -25 (7)
KH,POIS
- 1.5 (0.5) -29 (6) - 1.8 (0.2) - 1.2 (0.4)
-8 (1) -5 (7) -17 (3) - 5.7 (0.3)
- 1.7 (0.7)
+23
(5)
- 2.0 (0.5)
+21
(2)
- 3.6 (0.5) -4 (0.4)
+18 +18
(4) (4)
- 16(12) -22 (19) - 23 (27) -22(18) -31 (37) -27(16)
- 24 (8)
The membrane:; denoted by T are whole human teeth, those denoted by C-E are sections of human teeth and those denoted bv C-HA are sections of synthetic hydroxyapatite; figures within parentheses represent the calculated standard eirors (s,~). See equation*(A5). * After treatment with 1 mol/l KF. t PO:- and H,PO; concentration neglected. $ PO:- and HPO:- concentration neglected. the buffer on the e.m.f. was neglected. The solutions of KH2P04 and K2HP04 were at their natural pH. In the concentration range used, they are about 4.5 and 9.3 respectively. In some experiments, the, human enamel or synthetic hydroxyapatite was treated with a 1 mol/KF solution at its natural pH of 7.6 for 24 h. A series of e.m.f. measurements with KC1 solutions was carried out before and after this treatment. RESULTS
In Fig. 2 the measured e.m.f.s of a typical experiment are plotted. The data for sE are the standard errors in the e.m.f. according to the least squares solution as explained in the Appendix. In Table 1, the calculated fixed charges with their standard errors are tabulated in mequiv/l. For the calculations the K2HP04 and KH[2POI solutions were supposed to ions and K’ and contain only K+ and HPO:H2PO; ions respectively. In fact, at their natural pH the concentration of the other phosphate ions is less than 1 per cent. In Table 2, the corresponding results for the ratio of the diffusion coefficients of cations and anions are tabulated. DISCUSSION The assumptions used in deriving the formulas (4) and (6) for the membrane potential are less stringent than they seem. Eloth concentrations and diffusion
coefficients occur as ratios in equation (4) so that deviations from thermodynamic ideality which work in the same direction for cations and anions cancel out approximately. The third assumption, that there is no convection, is not expected to introduce a major error because an important convection term is only expected in systems which have a high water permeability and a large membrane charge (Schliigl, 1954) neither of which is the case for enamel. The last assumption, that the enamel membrane has a constant composition and structure throughout its entire thickness is not valid (Weatherell and Robinson, 1974). This means that the results of our calculations are an approximate mean value for the ratio of the diffusion coefficients as well as for the fixed charge. A good criterion for the fit of the theoretical model to the experimental values is the standard error in the e.m.f.s, calculated from the sum of squares of the difference between measured and calculated e.m.f.s (Appendix). These standard errors for the experiments on tooth 19 are mentioned in Fig. 2. The average value of this standard error for all experiments (Tables 1 and 2) is 1.6 mV, which can reasonably be explained from the errors in the e.m.f. measurements as such. This indicates that, in the range of concentrations used, the TMS theory offers an acceptable model for the transport of ions through enamel. The relative standard error (Tables 1 and 2) in the fixed charge is in general much higher than that in
402
Table
J. W. E. van Dijk et al. 2. The ratio of the diffusion coefficients of the cation and anion of various electrolytes enamel and synthetic hydroxyapatite and those in the bulk solutions
in human
Electrolyte Membrane T12 Tl5 T19 T21 T23 T24 T31 T32 T34 T35 T36 T38 ClE C4E C5E C6E ClHA C2HA C3HA Cl 1HA C12HA Cl3HA Cl4HA water
KC1 1.7 4.4 2.4 0.9 2.63 3.2
(0.2) (0.5) (0.1) (0.5) (0.08) (0.3)
3.1 (0.4) 4.0 (0.4) 1.8 (0.2) 2.1 (0.2) 6.0 (0.6) 1.42 (0.04) 3.3 (0.1) 2.2 (0.3) 1.90 (0.08) 1.6 (0.4) 1.33 (0.06) 1.9 (0.4) 1.40 (0.05) 1.49 (0.08) 0.9%
KCl* 1.3 (0.6)
1.9 (0.5) 2.6 (0.4) 3.6 (0.1)
CaCl,
NaCl 1.12 2.5 1.32 0.90 1.4
(0.08) (0.4) (0.08) (0.07) (0.7)
0.5 0.58 0.29 0.46 0.38
K,HPObl’
(0.1) (0.07) (0.01) (0.08) (0.02) 5.6 4 6.1 7.9 8 3
1.37 2.6 0.87 1.49 4.1 (0.6) 2.7 (0.2) 4.5 (0.7)
(0.06) (0.4) (0.02) (0.05)
0.38 0.9 0.11 0.30
KH,POhS
(0.5) (1) (0.4) (0.8) (2) (2)
2.4 (0.8) 2 (1) 3 (2) 2.2 (0.9) 2 (2) 21 (12)
(0.01) (0.1) (0.01) (0.01)
1.49 (0.08)
0.10 (0.01)
1.06 (0.05)
0.17 (0.01)
0.92 (0.02) 0.94 (0.03) 0.66$
0.23 (0.02) 0.18 (0.01) 0.399
3.6 (0.7)
0.96s
3.2011
2.23T
The membranes denoted by T are whole human teeth, those denoted by C-E are sections of human teeth and those denoted by C-HA are sections of synthetic hydroxyapatite; figures within parentheses represent the calculated standard errors. (e+/n_). See equation (A6). * After treatment with 1 mol/l KF. t PO; and H,PO; concentration neglected. $ PO; and HPO; concentration neglected. 5 From Robinson and Stokes (1970) // From Tatarinov (1960). C From Mason (1949). the ratio of the diffusion doefficients, because a small change in the ratio of the diffusion coefficients has a far more pronounced effect on the e.m.f. than a small change in the tied charge. Although the standard errors are large, some trends in the fixed charge seem to appear (p-values about 7.5 per cent). The fixed charge of both human dental enamel and of synthetic hydroxyapatite was negative in all electrolyte solutions used. The fixed charge of enamel was the same in KC1 and NaCl solutions. Also the ratio of the diffusion coefficients of K and Na did not differ significantly from that in water. For whole teeth, the fixed charge in CaCl, solutions was less negative than that of the same samples in the other electrolyte solutions and it tended to be more negative in the phosphate solutions than in the KC1 or NaCl solutions. These findings are in agreement with the results of zeta potential measurements on fluorapatite as a function of the kind and the concentration of electrolytes (Somasundaran and Agar, 1972). The difference in behaviour of dental enamel and synthetic hydroxyapatite towards/ a 1 mol/KF solution is remarkable. The concentration of the fixed charge of human enamel tended to become more negative, whereas that of synthetic hydroxyapatite remained constant. The opposite was observed for the ratio of the diffusion coefficients of K+ and Cl- ; this remained constant for dental enamel and seemed to rise in synthetic hydroxyapatite. The influence of
CaCl, solutions on the fixed charge of synthetic hydroxyapatite, however, was more pronounced than that on dental enamel. Relatively large positive values were found, whereas in dental enamel the fixed charge became less negative. The effects of the changes of the fixed charge which can be induced by a fluoride treatment on the fluxes and the concentrations inside the enamel in uivo are not easily predictable. To enable estimates of such effects to be made, calculations were carried out on a system which contained the 7 ions that are the major components of saliva and the intercellular fluid. These calculations indicated that, if the fixed charge is changed by a factor two, the fluxes of the univalent ions and their concentrations inside the enamel are altered only by a few per cent. The flux and concentrations of Ca2+ ions within the enamel, however, doubles or trebles. Accordingly the transport of Ca* + ions through enamel in the oral system wduld be noncongruent (Schlijgl, 1964). In other words, Ca2+ is transported from saliva to the internal compartments although its concentration in that compartment is higher. than that in saliva. This is relevant to the mechanism of remineralization. The most important conclusion of our study is that the fixed charge of tooth enamel, which is mostly negative, depends on the composition of its environment. Fluoride and phosphate solutions make the fixed charge more negative and calcium solutions
403
Electrochemical characteristics of tooth enamel
make it less negative, which is contrary to the inference drawn by Waters (1971) who, by analogy with the behaviour of synthetic anion-exchange membranes concluded that, in the presence of strong calcium solutions, enamel acquires an effective positive fixed charge. The general conclusion, however, that enamel normally i,s-viuo has a preferential selectivity for calcium ions is supported by our study. REFERENCES
Amberson W. R., Williams R. W. and Klein H. (1926) Electromotive phenomena in teeth and bones. Am. J. med. sci. 171, 926-927. Helfferich F. 1962. Ion Exchange. McGraw-Hill, New : York. Klein H. 1932. Physico-chemical studies on the structure of dental enamel. J. dent. Res. 12, 79-98. Mason C. M. and Culvern J. B. 1949. Electrical conductivity of orthophosphoric acid and of sodium and potassium dihydrogen phosphates at 25°C. J. Am. &em. Sot. 71, 2387-2393. Moreno E. C. and Zahradnik R. T. 1974. Chemistry of enamel subsurface demineralisation. J. dent. Res. 53, 226-236. Powell M. J. D. 1965. A method for minimizing a sum of squares of non-linear functions without calculating derivatives. Computer J. 7, 303-307. Robinson R. A. and Stokes R. H. 1970. Electrolyte So/ulions. Butterworth, London. Schlijgl R. 1954. Elek.trodiffusion in freier LGsung und geladencn Membranen. Z. Phys. Chem. NF 1, 305-339. Schlijgl R. 1964. Sto@wqwrt durch Membranen. Dr. Dietrich Steinkopf Verlag, Darmstadt. Silverstone L. M. 1973. Structure of carious enamel, including the early lesion. Oral Sci. Rev. 3, 100-160. Somasundaran P. and Agar G. E. 1972. Further streaming potential studies on apatite in inorganic electrolytes.
APPENDIX
Mathematical basis of the computer program calculating the fixed charge and ratio of the difision coefficientsfrom e.m.f: measurements
Our purpose was to-find those values for the two parameters WX and D./D_, that give the closest approach of the theoretical e.m.f.s of the equation (4) to the measured e.m.f.s. The best fit of a model to the experimental measurements can be found by applying a least-squares technique, i.e. by determining the minimum value of the function f(wX,D+/D_)
-8f(wX.D+/D-) __ZZ adi
o
267-269.
Waters N. E. 1971. The selectivity of human dental enamel to ionic transport. Archs oral Biol. 16, 305-322. Waters N. E. 1972. The electrochemical behaviour of human dental enamel after a topical fluoride treatment. Co/c. Tiss. Res. 108. 314-322. Waters N. E. 1975. ‘Electrochemistry of human enamel: selectivity to potassium in solutions containing calcium or phosphate ions. Archs oral Biol. 20, 195-201. Weatherell J. A. and Robinson C. 1973. The inorganic composition of teeth. In: Biological Mineralization (Edited by Zipkin I), pp. 43-74. John Wiley, New York.
and
8f(wX,D+lD-) ~ ab+/D-
= 0.
(A2)
The result is a set of two equations with two unknown parameters which, in general, can be solved. In this case, however, the minimum can be determined only by a numerical optimization procedure. A great variety of iterative numerical optimization procedure is available. The method of Powell (Powell, 1965) suited this problem because the partial derivatives need only to be calculated in the first iteration, and it converges in about 10 steps. Let Q be the minimum value of the least squares function (Al), than the standard error in the e.m.f. is give by: SE =
Q n-2
J(-1
-
-
The standard errors in the two parameters ox and D+D_ can be obtained by calculating the variance-covariance matrix (this is generated in Powell’s method without any additional operations). If f is the matrix with the elements
aEi
305-369.
Waters N. E. 1968. Electrochemical properties of human dental enamel. Nature, Land. 219, 62-63. Waters N. E. 1970. A cell for membrane and diffusion studies across en.lmel sections. Archs oral Biol. 15,
(Al)
where E^ is the expected value for the c.m.f., that is the one that can be calculated with the formula (4) for some w% and fi+/D_. E is the measured e.m.f. i = 1, 2, 3,. ,n is the number of a measurement, n the number of e.m.f. measurements = the number of different concentration pairs used. The minimum of the function (Al) is obtained if the two partial derivatives are set equal to zero
Trans. AIME 252, 348-352.
Tatarinov B. P. and Fursenko V. F. 1960. Electrical conductivity of dilute solutions of Na*HPO+ and KH2POI. Zb. fiz. i
SOME
ELECTROCHEMICAL CHARACTERISTICS HUMAN TOOTH ENAMEL J.
W.
E. VAN DIJK, N. E. WATERS,
F. C. M.
OF
J. M. P. M. BORGGREVEN and
DRIE%%NS
Dental School, Catholic University of Nijmegen, “Heyendael”, Nijmegen, The Netherlands and Department of Physical Sciences, The Royal Dental Hospital, School for Dental Surgery
University of London, Leicester Square, London WC2, England Summary--The fixed charge of enamel and the ratio of the diffusion coefficients of cations to anions of some common electrolytes within enamel were calculated from steady state electromotive forces developed in concentration cells across enamel caps of whole human teeth, sections of human enamel or synthetic hydroxyapatite membranes. The electrolytes were KCl, KF, NaCI, C&l,, KH2POh and K2HP0,. The results indicate that enamel behaves like an ion exchange membrane of which the fixed charge depends on the electrolytic environment and the sequence of exposure to various electrolytes.
INTRODUCTION
i the parameter for species i. The symbols are:
In initial smooth surface dental caries, the inorganic phase of enamel dissolves partially and the ions must pass a relatively intact surface layer of enamel by diffusion (Moreno, 1974; Silverstone, 1973). Thus for understanding the mechanism of carious breakdown, it is important to know what the transport properties of enamel are and how they can be influenced by chemical agents and physical influences. Amberson, Williams and Klein (1926) and Klein (1932) measured the electromotive force (e.m.f.) of concentration cells in which enamel served as a membrane. They proved that enamel behaves as an ion-selective porous membrane with a fixed charge on the pore-walls. It seems that this charge can be changed by the action of electrolytes like CaCl,. Waters (1968, 1971, 1975) confirmed these findings, giving more information about the influence of some common electrolytes on the chemical properties of enamel. These studies are qualitative in the sense that they express the selectivity of enamel in terms of membrane potential only. We now present a mathematical procedure which enables the calculation of the two parameters that are characteristic of the selectivity of a charge membrane, namely the ratio of the diffusion coefficients of the cations and anions within the membrane and its effective fixed charge.
MATE:RIALS AND METHODS
Physical and mthmutical
methods
J D
c
a In(a)
-
ax
F R aE
the flux the diffusion coefficient the concentration the gradient of the logarithm of the activity Faraday’s constant gas constant the gradient of the electrical potential
SX
z
the charge of the ion.
We used single electrolytes and therefore there are only two flux equations (I), one for the cation and one for the anion. An exact solution of these differential equations is not possible because of several unknown parameters, including the concentration dependence of the activity of the ions within a pore. It was necessary to conform to the following assumptions which are inherent in the TMS theory: 1. All activities are set equal to concentration. 2. The diffusion coefficients of the ions do not change with concentration nor with the position within the membrane. 3. There is no convection within the membrane. 4. The membrane has a constant composition throughout its entire thickness. In addition to the two differential equations and the assumptions, there are two conditions: 5. There is no electrical current XziJi = 0
The basis of the calculations is the Teorell-MeyerSievers theory (TMS theory) (Teorell, 1953; Helfferich, 1962; Schlogl, 1954) which describes the transport of ions through a porous membrane having fixed charges within its structure. The theory uses the Nernst-Planck flux equation in the form (Helfferich, 1962):
where ?? the fixed charge of the membrane
a In(@) __
w the sign of the fixed charge.
Ji =
-jiici
ax
(1)
(21
and 6. Within the membrane there is electro-neutrality ZZiEi + wx = 0
(3)
Using the two flux equations (1) and the assumptions and conditions, we can get an explicit expression for
A bar denotes a parameter inside the membrane and 399
400
J. W. E. van Dijk et al
1965). The calculations have been carried out on the IBM370/158 computer of the Catholic University of Nijmegen using PL/l programs. In the Appendix, the principle of the calculation of the parameters 0+/oand WX as well as their standard errors is explained.
the total membrane potential E,; E, = E + E:, + Eb’ B, -Di Z+D+ - z-i( ‘)( z+D+(r’)i+ - z_B_(r’)‘xln c cl’ z+D+(r”)l+ - z_b_(r”)r- >
E, = y
I,
+lnY i r’
)I
EXPERIMENT
(4)
The single or double marks denote a parameter at the left and right interface respectively, a+ or athe parameters for a cation or an anion. r denotes the Donnan distribution coefficient for the membrane/solution boundaries and Ed the corresponding Donnan potential. Figure 1 shows a scheme of the various variables. The Donnan distribution coefficients give a relation between the concentration of an ion inside and outside the charged membrane at the interface. C+ = r”. c+ and?_
= ?m.c_.
(5)
By substituting (5) and (3) we get an equation of higher degree from which the distribution coefficients can be calculated if the fixed charge is known: z+c+f4
+ z-c-i-
+ OX = 0.
(6)
If the e.m.f. of a concentration cell is measured at two or more pairs of concentrations, c’ and c”, the two independent and unknown parameters 0+/oand WX can be calculated using the equations (4) and (6) in an iterative numerical procedure. In fact the e.m.f. at more than two concentration pairs has been measured. Values for 0+/D- and ox have then been calculated that give the smallest sum of squares of the difference between the measured e.m.f. and the e.m.f. calculated by substituting the values of D+/n_ and OX in (6) and (4). This non-linear least-squares problem is solved by using a numerical procedure for minimizing a non-linear sum of squares (Powell,
All calculations refer to e.m.f. data from experiments published in earlier papers (Waters, 1971, 1972, 1975). In all experiments the e.m.f. was measured on concentration cells of the type: ref. electr./lsolution
1lmembranelsolution
2llref. electr.
Mercuryalomel half cells were used as reference electrodes and were connected to the solutions by saturated KCl-agar salt bridges. The potential developed across the cell was measured with a high-impedance potentiometric recorder. The technique including that for the preparation of the human enamel caps and synthetic membranes has been described (Chick and Waters, 1963; Waters, 1971, 1970). All electrolyte solutions, except those with and K2HP0, were buffered with KH,PO, 0.002 mol/l barbituric acid and adjusted to pH 7.4 with 2 mol/l KOH. The influence of the ions from Em
mV
4c
30
2c
10
bulk
membrane
bulk
0
- ,I ‘cl
C’.
Em
-10
-20
E.0
-30
-40 0.2 x=0
Fig. 1. Graph lyte and the
x=6
of the concentrations of a I.1 valent electroe.m.f. in a negatively charged membrane (Ed are the Donnan potentials).
0.4
0.6
0.6
1.0
1.2
1.4
1.6
log f
Fig. 2. Plot of the e.m.f. versus log c’jc” measured on tooth Tl9 with c’ = O.I,mol/l (+KCl; A-NaCI; q--Call,). Standard errors S, [see equation (A,)] were 0.4 mV-KCl. 0.9 mV-NaCl and 0.8 mV-CaCl,.
401
Electrochemical characteristics of tooth enamel Table 1. Fixed charge in mequiv/l of human dental enamel and synthetic hydroxyapatite
in various electrolytes
Electrolyte Membrane T12 T15 T19 T21 T23 T24 T31 T32 T34 T35 T36 T38 ClE C4E C5E C6E ClHA C2HA C3HA CllHA C12HA C13HA C14HA
KC1 --4 -15 --9 -34 -15 --6
(1) (4) (1) (24) (1) (2)
--3 (2) -16 (3) --3 (2) --7 (2) -7 (3) -- 0.9 (0.3) --7 (1) -21 (6) -- 1.7 (0.6) -22 (9) -- 3.1 (0.6) -27 (8) -- 1.2 (0.5) -- 1.6 (0.7)
KCI* -25 (16)
-34(11) - 24 (6) -13(l)
NaCl -5 -32 -10 -16 -25
CaCl,
(1) (7) (2) (2) (20)
0 -10 -4 -4 -9
K,HPO,t
(5) (4) (1) (4) (2) -13(5) -37(14) -17(4) -15(7) -26(13) -53 (45)
- 20 (6) -3 (1) -25 (7)
KH,POIS
- 1.5 (0.5) -29 (6) - 1.8 (0.2) - 1.2 (0.4)
-8 (1) -5 (7) -17 (3) - 5.7 (0.3)
- 1.7 (0.7)
+23
(5)
- 2.0 (0.5)
+21
(2)
- 3.6 (0.5) -4 (0.4)
+18 +18
(4) (4)
- 16(12) -22 (19) - 23 (27) -22(18) -31 (37) -27(16)
- 24 (8)
The membrane:; denoted by T are whole human teeth, those denoted by C-E are sections of human teeth and those denoted bv C-HA are sections of synthetic hydroxyapatite; figures within parentheses represent the calculated standard eirors (s,~). See equation*(A5). * After treatment with 1 mol/l KF. t PO:- and H,PO; concentration neglected. $ PO:- and HPO:- concentration neglected. the buffer on the e.m.f. was neglected. The solutions of KH2P04 and K2HP04 were at their natural pH. In the concentration range used, they are about 4.5 and 9.3 respectively. In some experiments, the, human enamel or synthetic hydroxyapatite was treated with a 1 mol/KF solution at its natural pH of 7.6 for 24 h. A series of e.m.f. measurements with KC1 solutions was carried out before and after this treatment. RESULTS
In Fig. 2 the measured e.m.f.s of a typical experiment are plotted. The data for sE are the standard errors in the e.m.f. according to the least squares solution as explained in the Appendix. In Table 1, the calculated fixed charges with their standard errors are tabulated in mequiv/l. For the calculations the K2HP04 and KH[2POI solutions were supposed to ions and K’ and contain only K+ and HPO:H2PO; ions respectively. In fact, at their natural pH the concentration of the other phosphate ions is less than 1 per cent. In Table 2, the corresponding results for the ratio of the diffusion coefficients of cations and anions are tabulated. DISCUSSION The assumptions used in deriving the formulas (4) and (6) for the membrane potential are less stringent than they seem. Eloth concentrations and diffusion
coefficients occur as ratios in equation (4) so that deviations from thermodynamic ideality which work in the same direction for cations and anions cancel out approximately. The third assumption, that there is no convection, is not expected to introduce a major error because an important convection term is only expected in systems which have a high water permeability and a large membrane charge (Schliigl, 1954) neither of which is the case for enamel. The last assumption, that the enamel membrane has a constant composition and structure throughout its entire thickness is not valid (Weatherell and Robinson, 1974). This means that the results of our calculations are an approximate mean value for the ratio of the diffusion coefficients as well as for the fixed charge. A good criterion for the fit of the theoretical model to the experimental values is the standard error in the e.m.f.s, calculated from the sum of squares of the difference between measured and calculated e.m.f.s (Appendix). These standard errors for the experiments on tooth 19 are mentioned in Fig. 2. The average value of this standard error for all experiments (Tables 1 and 2) is 1.6 mV, which can reasonably be explained from the errors in the e.m.f. measurements as such. This indicates that, in the range of concentrations used, the TMS theory offers an acceptable model for the transport of ions through enamel. The relative standard error (Tables 1 and 2) in the fixed charge is in general much higher than that in
402
Table
J. W. E. van Dijk et al. 2. The ratio of the diffusion coefficients of the cation and anion of various electrolytes enamel and synthetic hydroxyapatite and those in the bulk solutions
in human
Electrolyte Membrane T12 Tl5 T19 T21 T23 T24 T31 T32 T34 T35 T36 T38 ClE C4E C5E C6E ClHA C2HA C3HA Cl 1HA C12HA Cl3HA Cl4HA water
KC1 1.7 4.4 2.4 0.9 2.63 3.2
(0.2) (0.5) (0.1) (0.5) (0.08) (0.3)
3.1 (0.4) 4.0 (0.4) 1.8 (0.2) 2.1 (0.2) 6.0 (0.6) 1.42 (0.04) 3.3 (0.1) 2.2 (0.3) 1.90 (0.08) 1.6 (0.4) 1.33 (0.06) 1.9 (0.4) 1.40 (0.05) 1.49 (0.08) 0.9%
KCl* 1.3 (0.6)
1.9 (0.5) 2.6 (0.4) 3.6 (0.1)
CaCl,
NaCl 1.12 2.5 1.32 0.90 1.4
(0.08) (0.4) (0.08) (0.07) (0.7)
0.5 0.58 0.29 0.46 0.38
K,HPObl’
(0.1) (0.07) (0.01) (0.08) (0.02) 5.6 4 6.1 7.9 8 3
1.37 2.6 0.87 1.49 4.1 (0.6) 2.7 (0.2) 4.5 (0.7)
(0.06) (0.4) (0.02) (0.05)
0.38 0.9 0.11 0.30
KH,POhS
(0.5) (1) (0.4) (0.8) (2) (2)
2.4 (0.8) 2 (1) 3 (2) 2.2 (0.9) 2 (2) 21 (12)
(0.01) (0.1) (0.01) (0.01)
1.49 (0.08)
0.10 (0.01)
1.06 (0.05)
0.17 (0.01)
0.92 (0.02) 0.94 (0.03) 0.66$
0.23 (0.02) 0.18 (0.01) 0.399
3.6 (0.7)
0.96s
3.2011
2.23T
The membranes denoted by T are whole human teeth, those denoted by C-E are sections of human teeth and those denoted by C-HA are sections of synthetic hydroxyapatite; figures within parentheses represent the calculated standard errors. (e+/n_). See equation (A6). * After treatment with 1 mol/l KF. t PO; and H,PO; concentration neglected. $ PO; and HPO; concentration neglected. 5 From Robinson and Stokes (1970) // From Tatarinov (1960). C From Mason (1949). the ratio of the diffusion doefficients, because a small change in the ratio of the diffusion coefficients has a far more pronounced effect on the e.m.f. than a small change in the tied charge. Although the standard errors are large, some trends in the fixed charge seem to appear (p-values about 7.5 per cent). The fixed charge of both human dental enamel and of synthetic hydroxyapatite was negative in all electrolyte solutions used. The fixed charge of enamel was the same in KC1 and NaCl solutions. Also the ratio of the diffusion coefficients of K and Na did not differ significantly from that in water. For whole teeth, the fixed charge in CaCl, solutions was less negative than that of the same samples in the other electrolyte solutions and it tended to be more negative in the phosphate solutions than in the KC1 or NaCl solutions. These findings are in agreement with the results of zeta potential measurements on fluorapatite as a function of the kind and the concentration of electrolytes (Somasundaran and Agar, 1972). The difference in behaviour of dental enamel and synthetic hydroxyapatite towards/ a 1 mol/KF solution is remarkable. The concentration of the fixed charge of human enamel tended to become more negative, whereas that of synthetic hydroxyapatite remained constant. The opposite was observed for the ratio of the diffusion coefficients of K+ and Cl- ; this remained constant for dental enamel and seemed to rise in synthetic hydroxyapatite. The influence of
CaCl, solutions on the fixed charge of synthetic hydroxyapatite, however, was more pronounced than that on dental enamel. Relatively large positive values were found, whereas in dental enamel the fixed charge became less negative. The effects of the changes of the fixed charge which can be induced by a fluoride treatment on the fluxes and the concentrations inside the enamel in uivo are not easily predictable. To enable estimates of such effects to be made, calculations were carried out on a system which contained the 7 ions that are the major components of saliva and the intercellular fluid. These calculations indicated that, if the fixed charge is changed by a factor two, the fluxes of the univalent ions and their concentrations inside the enamel are altered only by a few per cent. The flux and concentrations of Ca2+ ions within the enamel, however, doubles or trebles. Accordingly the transport of Ca* + ions through enamel in the oral system wduld be noncongruent (Schlijgl, 1964). In other words, Ca2+ is transported from saliva to the internal compartments although its concentration in that compartment is higher. than that in saliva. This is relevant to the mechanism of remineralization. The most important conclusion of our study is that the fixed charge of tooth enamel, which is mostly negative, depends on the composition of its environment. Fluoride and phosphate solutions make the fixed charge more negative and calcium solutions
403
Electrochemical characteristics of tooth enamel
make it less negative, which is contrary to the inference drawn by Waters (1971) who, by analogy with the behaviour of synthetic anion-exchange membranes concluded that, in the presence of strong calcium solutions, enamel acquires an effective positive fixed charge. The general conclusion, however, that enamel normally i,s-viuo has a preferential selectivity for calcium ions is supported by our study. REFERENCES
Amberson W. R., Williams R. W. and Klein H. (1926) Electromotive phenomena in teeth and bones. Am. J. med. sci. 171, 926-927. Helfferich F. 1962. Ion Exchange. McGraw-Hill, New : York. Klein H. 1932. Physico-chemical studies on the structure of dental enamel. J. dent. Res. 12, 79-98. Mason C. M. and Culvern J. B. 1949. Electrical conductivity of orthophosphoric acid and of sodium and potassium dihydrogen phosphates at 25°C. J. Am. &em. Sot. 71, 2387-2393. Moreno E. C. and Zahradnik R. T. 1974. Chemistry of enamel subsurface demineralisation. J. dent. Res. 53, 226-236. Powell M. J. D. 1965. A method for minimizing a sum of squares of non-linear functions without calculating derivatives. Computer J. 7, 303-307. Robinson R. A. and Stokes R. H. 1970. Electrolyte So/ulions. Butterworth, London. Schlijgl R. 1954. Elek.trodiffusion in freier LGsung und geladencn Membranen. Z. Phys. Chem. NF 1, 305-339. Schlijgl R. 1964. Sto@wqwrt durch Membranen. Dr. Dietrich Steinkopf Verlag, Darmstadt. Silverstone L. M. 1973. Structure of carious enamel, including the early lesion. Oral Sci. Rev. 3, 100-160. Somasundaran P. and Agar G. E. 1972. Further streaming potential studies on apatite in inorganic electrolytes.
APPENDIX
Mathematical basis of the computer program calculating the fixed charge and ratio of the difision coefficientsfrom e.m.f: measurements
Our purpose was to-find those values for the two parameters WX and D./D_, that give the closest approach of the theoretical e.m.f.s of the equation (4) to the measured e.m.f.s. The best fit of a model to the experimental measurements can be found by applying a least-squares technique, i.e. by determining the minimum value of the function f(wX,D+/D_)
-8f(wX.D+/D-) __ZZ adi
o
267-269.
Waters N. E. 1971. The selectivity of human dental enamel to ionic transport. Archs oral Biol. 16, 305-322. Waters N. E. 1972. The electrochemical behaviour of human dental enamel after a topical fluoride treatment. Co/c. Tiss. Res. 108. 314-322. Waters N. E. 1975. ‘Electrochemistry of human enamel: selectivity to potassium in solutions containing calcium or phosphate ions. Archs oral Biol. 20, 195-201. Weatherell J. A. and Robinson C. 1973. The inorganic composition of teeth. In: Biological Mineralization (Edited by Zipkin I), pp. 43-74. John Wiley, New York.
and
8f(wX,D+lD-) ~ ab+/D-
= 0.
(A2)
The result is a set of two equations with two unknown parameters which, in general, can be solved. In this case, however, the minimum can be determined only by a numerical optimization procedure. A great variety of iterative numerical optimization procedure is available. The method of Powell (Powell, 1965) suited this problem because the partial derivatives need only to be calculated in the first iteration, and it converges in about 10 steps. Let Q be the minimum value of the least squares function (Al), than the standard error in the e.m.f. is give by: SE =
Q n-2
J(-1
-
-
The standard errors in the two parameters ox and D+D_ can be obtained by calculating the variance-covariance matrix (this is generated in Powell’s method without any additional operations). If f is the matrix with the elements
aEi
305-369.
Waters N. E. 1968. Electrochemical properties of human dental enamel. Nature, Land. 219, 62-63. Waters N. E. 1970. A cell for membrane and diffusion studies across en.lmel sections. Archs oral Biol. 15,
(Al)
where E^ is the expected value for the c.m.f., that is the one that can be calculated with the formula (4) for some w% and fi+/D_. E is the measured e.m.f. i = 1, 2, 3,. ,n is the number of a measurement, n the number of e.m.f. measurements = the number of different concentration pairs used. The minimum of the function (Al) is obtained if the two partial derivatives are set equal to zero
Trans. AIME 252, 348-352.
Tatarinov B. P. and Fursenko V. F. 1960. Electrical conductivity of dilute solutions of Na*HPO+ and KH2POI. Zb. fiz. i
