American Journal of Analytical Chemistry 2014 5 995-1009 Published Online October 2014 in SciRes httpwwwscirporgjournalajac httpdxdoiorg104236ajac2014514106

How to cite this paper Abdelkader SB Yahia FBH and Khattech I (2014) Preparation Characterization and Statistical Studies of the Physicochemical Results of Series of ldquoBrdquo Carbonated Calcium Hydroxyapatites Containing Mg2+ and 2

3CO minus American Journal of Analytical Chemistry 5 995-1009 httpdxdoiorg104236ajac2014514106

Preparation Characterization and Statistical Studies of the Physicochemical Results of Series of ldquoBrdquo Carbonated Calcium Hydroxyapatites Containing Mg2+ and 2

3CO minus

S Ben Abdelkader F Bel Hadj Yahia I Khattech Applied Thermodynamics Laboratory Chemistry Department Faculty of Sciences Tunis Tunisia Email faouziarockh1Gmailcom Received 21 August 2014 revised 6 October 2014 accepted 21 October 2014

Copyright copy 2014 by authors and Scientific Research Publishing Inc This work is licensed under the Creative Commons Attribution International License (CC BY) httpcreativecommonsorglicensesby40

Abstract In this study series of hydroxyapatites containing Mg2+ and 2

3CO minus are prepared by the precipita-tion method with independently varying concentrations of 2

3CO minus and Mg2+ All the compounds are characterized by infrared spectra (IR) powder X-ray diffraction (PXRD) and elemental analy-sis The physical analysis results show that the prepared compounds are pure B-type carbonate apatite The presence of Mg2+ and 2

3CO minus in the apatite cause the following effects on its physical properties a decrease in a-dimension but no changes in c-dimension and a decrease in crystallin-ity as shown in XDR patterns and IR spectra The results of the chemical analysis allow us to pre-dict the predominant substitution mechanisms for the 2

3CO minus and the Mg2+ incorporations in the calcium hydroxyapatites and to calculate their relative contributions x y and z

2 3 Ca 24 3Ca 2PO V 2CO+ minus minus+ harr + (II) 2+2 OH2 Ca 2 OH Mg 2 V+ minussdot + sdot harr + sdot 2 (IV) 3 2

4 3PO CO OHminus minus minusharr + (V) Statistical studies of the results ldquomultiple linear regression analysis of variance (ANOVA) and t-test of the regression coefficientsrdquo allow us to determine and to test the mathematical model proposed Finally the present study makes it possible to write the general formula for these com-pounds

Keywords B-Type Carbonated Calcium Hydroxyapatite Containing Magnesium ldquoBrdquo CO3Mg-HAps Substitution Mechanism(s) Multiple Linear Regression F-Test (ANOVA) t-Test of the Coefficients

S B Abdelkader et al

996

1 Introduction A number of studies have reported that the incorporation of magnesium in hydroxyapatites Ca10(PO4)6(OH)2 is limited [1]-[3] Previously it has been shown that the magnesium can disturb the crystallization of apatites when its concentration in the solution is sufficient to be a major competitor for calcium [4] But when the molar ratio of MgCa is higher than 01 another phase is observed the whitlockite [3] [5]-[7] The co-substitution of a second ionic species like the carbonate ion can increase the insertion of magnesium in the lattice and prevent the decomposition while stabilizing the structure [8] [9]

On the other hand it is now well established that the biological minerals are best described as carbonated apa-tites rather than as a hydroxyapatite [2] [10]-[12] The carbonate presents at 3 - 6 in biological apatites mostly substitutes for the phosphate ion in the crystal structure and has a significant influence on the incorpora-tion of other foreign ions into the apatite lattice Magnesium is one of the most abundant trace ions present in the biological hard tissues and in dental enamel its content approximately being 01 - 04 In dentin the magne-sium content is up to 11 while in bone it is found at 06 [13]-[15] Thusly Magnesium has been the subject of many studies To understand the role of magnesium on biological apatites the works using synthetic carbo-nated apatites are very helpful

Previous studies suggest that the magnesium is incorporated into or onto the carbonated apatites during their formation [16]-[24] Some of these works demonstrate the role of the carbonate concentration the pH of prepa-ration and the magnesium content incorporated into the apatites at similar quantities to those found in biological apatites [16] [17] [23] Other works report the effect of the magnesium on the parameters of the lattice of apa-tites prepared by precipitation or high-temperature synthesis [4] [25] Legeros et al [22] noted an increase in the dissolution rates of carbonate-containing apatites when the magnesium was incorporated Some studies have investigated the phasersquos composition after heat-treatment of the magnesiumcarbonate co-substituted in the hy-droxyapatite [16] [23] [24]

Despite numerous investigations the mechanism(s) by which the carbonate and the magnesium are incorpo-rated in the apatite lattice are not yet known Indications are found in the literature about the mechanisms by which 2

3CO minus and alkalimetal M+ are incorporated in the apatite lattice [26]-[28] In these works De Maeyer and Verbeeck suggest that six fundamental substitution mechanisms can contribute theoretically for these subs-titutions

2 3 Ca 2 OH4 3Ca PO OH V CO V+ minus minus minus+ + harr + + (I)

2 3 Ca 24 3Ca 2PO V 2CO+ minus minus+ harr + (II)

2 3 24 3Ca PO M CO+ minus + minus+ harr + (III)

2 OHCa OH M V+ minus ++ harr + (IV) 3 24 3PO CO OHminus minus minusharr + (V)

2 OH32OH CO Vminus minusharr + (VI)

where VX stands for a vacancy in the X-sublattice The contributions of each of these mechanisms should be es-timated on the basis of a thorough physicochemical studies of the samples

The present study tries to find the mechanism(s) which contribute to the incorporation of magnesium and carbonate in the apatites lattice For this purpose series of ldquoBrdquo carbonated calcium hydroxyapatites containing magnesium are prepared by the precipitation method In the first series the concentration of the 2

3CO minus solution is Cc = 000 M while the Mg2+ concentration CMg is 000 17 68 and 136 mM For the second the same pro-cedure is remade with Cc = 0025 M in the hydrolysis solution and for the third Cc is equal to 005 M The chemical and physical characteristics of the samples prepared are determined and an attempt is made to deduce the fundamental substitution mechanisms which determine their stoichiometry Finally statistical studies of the experiment results allow us to find the relationship between the different variables and to verify the proposed mechanisms by which 2

3CO minus and Mg2+ are incorporated in the apatite lattice

S B Abdelkader et al

997

2 Methods and Materials 21 Preparation of ldquoBrdquo Type Carbonated Hydroxyapatites Containing Magnesium The method of preparation used in this work is inspired from the method used in reference [23] but it is slightly modified The apaties are prepared by dropping 200 mL of a phosphate solution (NH4)2HPO4 (018 M) into 200 mL of a calcium solution Ca(NO3)4H2O (044 M) under reflux at 87˚C To the calcium solution is added 20 mL of a magnesium solution Mg(NO3)26H2O containing different concentrations CMg (000 17 68 and 136) mM The same procedure is remade by adding to the phosphate solution 5 mL of a carbonate solution NH4HCO3 (1 M) A third set of preparations is performed by adding 10 mL from the above carbonate solution The pH is maintained at 90 during the precipitation by adding an ammonia concentrated solution (28 weight) The pre-cipitation is carried out over 3 h Then the system is refluxed for an additional duration of 2 h The samples are filtered thoroughly washed with hot distilled water and dried overnight at 120˚C

22 Physical Analysis The powdered samples are identified by X-ray diffraction and by infra red spectroscopy Infrared spectra of the samples dispersed in KBr tablets are recorded using a Shimadzu Fourier transform infrared spectrophotometer in the range of 4000 - 400 cmminus1 Then the samples are analyzed by X-ray diffraction (XRD) using a Philips dif-fractometer using Cu Kα radiation The samples are scanned in the 2θ range of 20˚ - 60˚ The ldquoa and crdquo parame-ters of the lattice of the hexagonal unit cell are calculated using ldquowincellrdquo refinement program

23 Chemical Analysis The samples are analyzed for Ca PO4 CO3 and Mg The calcium content of the precipitates is determined by a complexometric titration with the ethylenediaminetetraacetic acid [29] the magnesium by atomic absorption the carbonate content is determined by coulometrically method and the phosphorus content by spectrophotome-trie of the phosphomolybdate complex [30]

3 Results 31 Results of Physical Analysis The IR Spectra of some representative samples (Mg4 Mg8 and Mg12) are shown in Figure 1 The spectra contain the characteristic bands of the phosphate group 3

4PO minus in the ranges 960 - 1100 and 570 - 610 cmminus1 Two broad bands around 1635 cmminus1 and 3400 cmminus1 confirm that the samples contain a significant amount of water On the

Figure 1 IR spectra of some representative samples

S B Abdelkader et al

998

spectra of the samples (Mg8 and Mg12) are displayed typical absorption bands of 23CO minus at ~873 and ~1420

cmminus1 and between 1450 and 1500 cmminus1 characterizing the vibration of the 23CO minus on 3

4PO minus lattice sites (B- type 2

3CO minus ) [31] From Figure 1 we can clearly see that the intensity of these absorptions increases with the increase of the carbonate content On the other hand the IR spectra of the compounds (Mg4 Mg8 and Mg12) show that the magnesium incorporated in the apatites causes the loss of resolution of the 3

4PO minus absorptions bands suggesting a decrease in the crystallinity [31]

The X-ray diffraction patterns of some representative samples are shown in Figure 2 The X-ray diffraction powder patterns of the compounds show only one crystal phase The peaks are sharp well resolved and charac-teristic of the hexagonal apatite phase No extraneous peaks attributable to other phases than apatite could be found in the diffractograms The increase of the level of 2

3CO minus substitution produces a loss of the resolution of the 112 peak and a decrease in the intensity of the 300 202 and 002 peaks

The Table 1 contains the values of the lattice parameters ldquoardquo and ldquocrdquo obtained for the different compounds From this table we can see that simultaneous incorporation of two elements ldquoCO3 and Mgrdquo results in an de-

crease of the ldquoardquo parameter This contraction is attributed to the simultaneous effects of the 23CO minus and Mg2+

substitutions [2]

Figure 2 X-ray diffraction patterns of some representative samples

Table 1 ldquoardquo and ldquocrdquo Lattice parameters of ldquoBrdquo CO3Mg-Haps

Samples CCM CMgmM aAring cAring ca

Mg1 0000 000 9439 plusmn 0004 6911 plusmn 0004 0732

Mg2 0000 017 9428 plusmn 0004 6901 plusmn 0003 0732

Mg3 0000 068 9412 plusmn 0006 6869 plusmn 0004 0730

Mg4 0000 136 9409 plusmn 0007 6862 plusmn 0005 0729

Mg5 0025 000 9410 plusmn 0005 6910 plusmn 0005 0735

Mg6 0025 017 9412 plusmn 0008 6911 plusmn 0005 0734

Mg7 0025 068 9386 plusmn 0008 6889 plusmn 0005 0734

Mg8 0025 136 9375 plusmn 0006 6896 plusmn 0004 0735

Mg9 0050 000 9369 plusmn 0007 6895 plusmn 0004 0736

Mg10 0050 017 9360 plusmn 0010 6899 plusmn 0007 0737

Mg11 0050 068 9343 plusmn 0014 6879 plusmn 0010 0736

Mg12 0050 136 9333 plusmn 0009 6872 plusmn 0004 0736

S B Abdelkader et al

999

32 Chemical Results The results of the chemical analysis of the samples in Weight are summarized in Table 2 This table also gives the hydroxide content of the samples calculated on the basis of the electroneutrality condition and the total mass balance sum obtained from the equation

4PO P 3 Ca P M M CO Mg OHsum = + sdot + + + (1)

With MX the atomic or ionic mass of X sum value is lower than 100 indicating that the samples of the present study still contain some water after drying at 120˚C

The results of the chemical and physical analysis (Table 2) allow us to calculate the number of each ion X per unit cell nx according to the following equation

( )3

XX P 3 CO

X 6nM P M CO M

=+

(2)

The results of these calculations are summarized in Table 3 The errors in Table 4 are estimated by the means of error propagation theory

Table 2 Chemical Composition (weight percent) and Total Mass Balance sum of the hydroxyapatites obtained by precipita-tion in solutions containing Cc (M) CO3 and CMg (mM) Mg

Sample CCM CMgmM Ca P CO3 Mg OHminus sum

Mg1 0000 000 3697 1683 099 001 319 9273

Mg2 0000 017 3709 1683 158 022 325 9371

Mg3 0000 068 3475 1683 000 082 299 9013

Mg4 0000 136 3435 1722 000 150 296 9158

Mg5 0025 000 3732 1584 445 001 315 9347

Mg6 0025 017 3673 1594 416 021 293 9287

Mg7 0025 068 3673 1633 465 084 289 9515

Mg8 0025 136 3514 1623 534 155 231 9407

Mg9 0050 000 3791 1485 822 001 315 9480

Mg10 0050 017 3762 1425 878 021 385 9413

Mg11 0050 068 3574 1475 871 080 229 9274

Mg12 0050 136 3564 1534 911 152 202 9530

Table 3 Unit cell compositions of NaCO3 Aaps calculated on the basis of the chemical composition and using Equation (2)

Sample CCM CMgmM nCa nP 3COn nMg nOH nMgnP 3COn nP

Mg1 0000 000 991 plusmn 021 582 plusmn 012 0170 plusmn 0007 000 200 008 000

Mg2 0000 017 977 plusmn 020 572 plusmn 011 0270 plusmn 0009 0090 plusmn 0003 200 013 000

Mg3 0000 068 960 plusmn 021 600 plusmn 013 000 0370 plusmn 0021 194 003 006

Mg4 0000 136 927 plusmn 018 600 plusmn 011 000 0660 plusmn 0055 188 006 012

Mg5 0025 000 957 plusmn 021 524 plusmn 011 076 plusmn 002 000 190 042 010

Mg6 0025 017 944 plusmn 020 529 plusmn 010 071 plusmn 002 0090 plusmn 0026 177 047 023

Mg7 0025 068 912 plusmn 018 523 plusmn 010 077 plusmn 002 0340 plusmn 0019 169 053 031

Mg8 0025 136 860 plusmn 018 513 plusmn 010 087 plusmn 002 0620 plusmn 0056 133 077 067

Mg9 0050 000 923 plusmn 019 466 plusmn 011 133 plusmn 003 000 180 076 020

Mg10 0050 017 892 plusmn 020 436 plusmn 009 164 plusmn 004 0080 plusmn 0025 164 099 036

Mg11 0050 068 863 plusmn 019 459 plusmn 010 140 plusmn 003 0320 plusmn 0017 130 105 070

Mg12 0050 136 827 plusmn 017 459 plusmn 009 141 plusmn 003 0580 plusmn 0049 110 115 090

S B Abdelkader et al

1000

Table 4 Multiple linear regression analysis of Yi = 3COn nP the molar ratio (Table 3) as a function of the concentration of

carbonate CcM and magnesium CMgmM in the solution (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 00156 Var(X1) = 00004 Var(X2) = 27997 Covariance Cov(X1 Y) = 00025 Cov(X2 Y) = minus00205 Cov(X1 X2) = 0

Coefficients of correlation R(X1 Y) = 0978 R(X2 Y) = minus0031 R(X1 X2) = 0 Residual mean squareStandard error s2 = 000087

R square R2 = 0958 Observations n = 12 Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 00165b = - - - - -

X1 1 5985b = ( )1 0418bσ = ( )1 1431T b = 1710minus7 5040 693

X2 2 000073b = minus ( )2 00016bσ = ( )2 0452T b = minus 0662 minus00043 00029

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 0179 00897 10254 474 Deviations 9 00078 00078

Total 11 0187 0017

4 Statistical Analysis of the Physicochemical Results 41 Influence of the Experimental Conditions on the Composition

of the Synthetic Apatites To know the influence of the experimental conditions on the incorporation of 2

3CO minus and Mg2+ in the lattice of these synthetic apatites we graph Yi =

3COn nP and nMgnP the molar ratios contents of the samples against Xi the concentration of 2

3CO minus Cc or the concentration of Mg2+ CMg in the solution Figure 3 and Figure 4 From the Figure 3(a) and Figure 4(a) it is seen that

3COn nP the molar ratio increases with the increase of the concentration of 2

3CO minus in the solution (CcM) Contrariwise it varies slightly with the concentration of the Mg2+ ions in the solution and vice versa for nMgnP (Figure 3(b) and Figure 4(b))

To estimate the simultaneous influence of the experimental conditions on 3COn nP and nMgnP the molar ratios

we construct a mathematical model of Yi = 3COn nP or nMgnP on two variables X1i = Cc and X2i = CMg

The mathematical model is described by the equation

0 1 1i 2 2iY X Xi b b b e= + + + (3)

The method of least squares (OLS) allows us to establish the predicted equation

i 0 1 1i 2 2iˆ ˆ ˆY b b X b X= + + (4)

that is most suitable to the data On the other hand this method allows us to calculate the estimated standard er-

rors of the coefficients ( ) ( )1 2ˆ ˆb and bσ σ the individual confidence interval at 95 level R2 the standardized

statistic and to test the null hypothesis H0 bj = 0 and its significances level The analysis of the variance for the linear regression or the F test allows us to ensure that at least one of the X-variables contributes to the regression The theoretical basis of these calculations is given in references [32]-[34] The calculations are summarized in Table 4 and Table 5

42 Influence of the Incorporation of 23CO minus and Mg2+ on the Variation of Ca2+ and OHminus

the Molar Ions Contents of the Synthetic Apatites In attempts to disentangle and to measure the effects of the insertion of 2

3CO minus and Mg2+ ions on the molar con-

S B Abdelkader et al

1001

(a) (b)

Figure 3 (a) 3COn nP molar ratio of the solid versus CMgmM for the samples prepared at different CcM (b) nMgnP

molar ratio of the solid versus CMgmM for the samples prepared at different CcM

(a) (b)

Figure 4 (a) 3COn nP molar ratio versus CcM for the samples prepared at different CMgmM (b) nMgnP molar ratio

versus CcM for the samples prepared at different CMgmM

Table 5 Multiple linear regression analysis of Yi = nMgnP molar ratio (Table 3) as a function of the concentration of carbo-nate CcM and magnesium CMgmM in the solution (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 00022 Var(X1) = 000042 Var(X2) = 27998 Covariance Cov(X1 Y) = 563 times 10minus5 Cov(X2 Y) = 0245 Cov(X1 X2) = 0

Coefficients of correlation R(X1 Y) = 00593 R(X2 Y) = 0995 R(X1 X2) = 0 Standard error s2 = 171 times 10minus5

R square R2 = 0994 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 00013b = minus - - - - -

X1 1 0134b = ( )1 0058bσ = ( )1 229T b = 0048 00023 0266

X2 2 00087b = ( )2 000022bσ = ( )2 3879T b = 000 00082 00092

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 00258 0012 75519 474 Deviations 9 0000154 17 times 10minus5

Total 11 00259 00024

S B Abdelkader et al

1002

tent of Ca2+ of the solid we use the multiple linear regression on two X-variables where X1 = 23nCO minus and X2 =

nMg2+ and Y is the estimate molar content of Ca2+ or OHminus (data Table 3) The results of these calculations are given in Table 6 and Table 7

Table 6 Multiple linear regression analysis of the estimated Yi = nCa2+ on X1i = nMg2+ X2i = 2

3nCO minus calcium carbonate and magnesium respectively molar contents of the solid ldquoBrdquo Mg-CO3 Haps (a) Regression statistic (c) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0059 Var(X2) = 0307 Covariance Cov(X1 Y) = minus0071 Cov(X2 Y) = minus0194 Cov(X1 X2) = minus0013

Coefficients of correlation R(X1 Y) = minus0607 R(X2 Y) = minus0722 R(X1 X2) = minus0101 Residual mean squareStandard error s2 = 00033

R square R2 = 0989 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 1009b = - - - - -

X1 1 1373b = minus ( )1 0068bσ = ( )1 2017T b = minus 110minus8 minus1527 minus1219

X2 2 0693b = minus ( )2 0030bσ = ( )2 2325T b = minus 000 minus0760 minus0625

(c)

Source of variation DF Sum of squares Mean square Fcal Ftab (5 2 9) [32]

Regression 2 279 1397 42812 474

Deviations 9 0029 00033

Total 11 2823 0257

Table 7 Multiple linear regression analysis of the estimated Yi = OHminus on X1 i = nMg2+ and X2i = 2

3nCO minus the molar con-tents of the solid ldquoBrdquo Mg-CO3 Haps (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0059 Var(X2) = 0307 Covariance Cov(X1 Y) = minus0038 Cov(X2 Y) = minus0111 Cov(X1 X2) = minus0013

Coefficients of correlation R(X1 Y) = minus0551 R(X2 Y) = minus0697 R(X1 X2) = minus0101 Residual mean squareStandard error s2 = 00137

R square R2 = 0875 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 219b = - - - - -

X1 1 0740b = minus ( )1 0139bσ = ( )1 532T b = minus 000048 minus1054 minus0426

X2 2 0393b = minus ( )2 0061bσ = ( )2 645T b = minus 000012 minus0531 minus0255

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 0862 0431 3161 474 Deviations 9 0123 00136

Total 11 0985 00896

S B Abdelkader et al

1003

43 The Determination of the Relationship between Y = ca Crystallographic Parameters Ratio and

3COn pn the Molar Ratio To estimate the influence of the incorporation of carbonate on the lattice parameters ldquoardquo and ldquocrdquo in presence of magnesium we plot ca crystallographic parameters ratio (Table 1) as a function of molar ratio

3COn nP (Table 3) for 0 le nMg le 174 mM (Figure 5)

Given that the shape of the curve obtained in Figure 5 is a polynomial we construct a multiple linear regres-sion on Yi = ca as a function of three X-variables where X1 =

3COn nP X2 = (3COn nP)2 and X3 = (

3COn nP)3 (data Table 2) and Y is the estimate ratio of the hexagonal lattice dimensions (data Table 1) The mathematical model equation is

0 1 1 2 2 3 3Y b b X b X b X= + + + + ε (5) Least square [33] allows calculating the regression and correlation coefficients regression of the predicted

equation

0 1 1 2 2 3 3ˆ ˆ ˆ ˆY b b X b X b X= + + + (6)

These estimated rgression coefficients 0 1 2 3ˆ ˆ ˆ ˆb b b and b are calculated from the values of correlation coeffici-

ents variance and covariance according to the method of Scherrer [33] This method allows us to test the utility of the model or the F-test according to

( )( )

2

2

n m 1 RF

m 1 Rminus minus sdot

=sdot minus

(7)

where n is sample size m is number of parameters and (n minus m minus 1) is degree of freedom On the other hand this method allows us to calculate the standard errors of the coefficients 0 1 2 3

ˆ ˆ ˆ ˆb b b and b and to conduct t-tests on the brsquos (to discover which variable(s) is related to estimate Y ) and to calculate the individual confidence interval at 95 level The results of these calculations are given in Table 8

The analysis of variance (ANOVA) shows that F-test = 20189 is higher than criterion F(5 3 8) = 407

Table 8 Multiple linear regression analysis of Yi= ca ratio of the lattice parameters of Mg-CO3 HAps (Table 1) on X1i = 3COn nP X2i = (

3COn nP)2 X3i = (3COn nP)3 the molar ratio Table 3 (a) Statistic regression (b) Coefficients

(a)

Variances Covariances Correlation coefficients

var(Y) 597 times 10minus6 Cov(X1 Y) 284 times 10minus4 1X YR 0931

v(X1) 156 times 10minus3 Cov(X2 Y) 908 times 10minus5 2X YR 0811

v(X2) 21 times 10minus3 Cov(X3 Y) 294 times 10minus5 3X YR 0731

v(X3) 27 times 10minus4 Cov(X1 X2) 55 times 10minus3 2 1X XR 0960

Cov(X1 X3) 186 times 10minus3 2 3X XR 0986

Cov(X2 X3) 744 times 10minus4 3 1X XR 0906

(b)

Standard error of the regression sr = 218 R square R2 = 098 Observations n = 12 Degree of freedom ν = 8

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 0730b = - - - - -

X1 1 0052b = ( ) 71 6793 10bσ minus= times ( )1 6331t b = 000 005217 005218

X2 2 0168b = minus ( ) 62 1852 10bσ minus= times ( )2 12384t b = minus 000 minus016851 minus016850

X3 3 0212b = ( ) 63 515 10bσ minus= times ( )3 9349t b = 000 021221 021223

S B Abdelkader et al

1004

Figure 5 ca parameters ratio as a function of

3COn nP the molar ratio for the apatites prepared at different values of CMg

5 Determination of the General Formula of the Unit Cell of the Synthetic ldquoBrdquo CO3Mg-HAps

The relative composition (Table 3) and the results of the physical analysis demonstrate that the samples are pure ldquoBrdquo type carbonated apatites containing Mg2+ ions Thus mechanisms I II III and V could be account in the incorporation of 2

3CO minus on 34PO minus and Mg2+ ions are incorporated in the apatite lattice according to mechan-

isms III and or IV Moreover the study carried out previously (paragraph 41) show that the 2

3CO minus ions are incorporated in the apatite lattice independently of the concentration of Mg2+ ions solution This result confirms that mechanism IV does and mechanism III does not contribute to the incorporation of Mg2+ in the apatites

Many works [27] [28] have demonstrated that mechanism I andor II are the main mechanisms for the incor-poration of CO3 Otherwise according the reference [27] the contribution of mechanism I seems to be hardly influenced by the alkali metal which is not our case Therefore we consider that mechanism II contribute to the insertion of CO3 ions in the lattice of the solid

Table 7 show that the variation of nOHminus depends on the increase of 23CO minus and Mg2+ So it may be said in

the present study that the mechanism V could account Then the fundamental substitution mechanisms for the incorporation of 2

3CO minus and Mg2+ in the HAp lattice are

2 3 Ca 24 3Ca 2PO V 2CO+ minus minus+ harr + (II)

2 2 OH2Ca 2OH Mg 2V+ minus ++ harr + 2(IV) 3 24 3PO CO OHminus minus minusharr + (V)

where VOH stands for a vacancy in the OHminus sub lattice If x y and z are the contributions of mechanisms II 2IV and V respectively thus

2nCa 10 x 2 y+ = minus minus sdot (8) 34nPO 6 2 x zminus = minus sdot minus (9)

2nMg y+ = (10) 23nCO 2 x zminus = sdot + (11)

and nOH 2 2 y zminus = minus sdot + (12)

and the generic formula has the following expression

( ) ( )( ) ( )( ) ( )( )y 4 310 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OHminus minus sdot minus minus minus ++

S B Abdelkader et al

1005

The values of x y and z the contribution of mechanisms II 2IV and V respectively are calculated from the data (Table 3) and the following equations Then statistical studies are conducted to verify the accuracy of the proposed formula The results of these calculations are summarized in Tables 9-11

Table 9 The values of x y and z the contributions of the mechanisms II 2IV and V respectively calculated from Equations (10)-(14)

Sample x y z

Mg1 0085 000 000

Mg2 0045 009 018

Mg3 minus034 037 068

Mg4 minus060 066 120

Mg5 043 000 minus01

Mg6 038 009 minus005

Mg7 020 034 037

Mg8 015 062 057

Mg9 076 000 minus020

Mg10 092 008 minus020

Mg11 073 032 minus006

Mg12 057 058 026

Table 10 Multiple linear regression analysis of the estimated Yi = nCa2+ the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = x- and X2i = y the contribution of mechanisms II and 2IV (Table 9) (a) Regression statistic (b) Coeffi-cients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0187 Var(X2) =0059

Covariance Cov(X1 Y) = minus0093 Cov(X2 Y) = minus0071 Cov(X1 X2) = minus0046

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 188 times 10minus5

R square R2 = 0999

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 999b = - - - - -

X1 1 0995b = minus ( )1 00032bσ = ( )1 30816T b = minus 000 minus1002 minus0987

X2 2 1999b = minus ( )2 00057bσ = ( )2 34779T b = minus 000 minus2012 minus1986

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32]

Regression 2 2862 1412 7414888 474

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

S B Abdelkader et al

1006

Table 11 Multiple linear regression analysis of the estimated Yi = nOHminus the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = z and X2i = z the contribution of mechanisms II and V (Table 9) (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0164 Var(X2) = 0059

Covariance Cov(X1 Y) = 00055 Cov(X2 Y) = minus0038 Cov(X1 X2) = 0079

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 274 times 10minus32

R square R2 = 100

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 200b = - - - - -

X1 1 100b = ( ) 161 1997 10bσ minus= times ( ) 15

1 500 10T b += times 000 1 1

X2 2 200b = minus ( ) 162 3338 10bσ minus= times ( ) 15

2 599 10T b += minus times 000 minus2 minus2

(c)

Source of variation DF Sum of squares Mean square Fcal

Regression 2 2862 1412 7414888

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

6 Discussion From Table 1 we can see that simultaneous incorporation of two elements ldquoCO3 and Mgrdquo results in an de-crease of the ldquoardquo parameter This contraction is attributed to the simultaneous effects of the 2

3CO minus and Mg2+ substitutions [4]

In Table 2 and Table 3 it is seen that the concentration of the 23CO minus ions in the solution Cc does not affect

the quantities of Mg2+ ions inserted in the solid Because regardless of the concentration of the 23CO minus ions in

the solution Cc the Mg2+ ions contents of the samples increase proportionally with the increase of the concentra-tion of Mg2+ ions in solution CMg This result is in agreement with reference [4] For the same concentration of Mg2+ions in the solution CMg the variation of 3

4PO minus and 23CO minus contents of the solid do not seem to be corre-

lated with the concentration of the Mg2+ ions in the solution while the Ca2+content depends on the concentra-tions of 2

3CO minus Cc and Mg2+ CMg in the solution Figure 3 and Figure 4 show that nCO3nP the molar ratio increases with the increasing of the concentration

of 23CO minus in solution (CcM) Contrariwise it varies slightly with the concentration of Mg2+ ions in solution and

vice versa for nMgnP The statistical treatment of the experimental data Table 4 and Table 5 allows us to es-tablish the estimated equations between these variables at 95 levels

( )3 cnCO nP 00165 5986 0945 C M= + plusmn sdot (13)

and ( ) ( )c MgnMg nP 00013 0135 0132 C M 00087 00005 C mM= minus + plusmn sdot + plusmn sdot (14)

Equations (9) and (10) show that the concentration of the 23CO minus ions in the hydrolysis solution Cc affects the

quantities of 23CO minus and Mg2+ ions incorporated in the solid but the concentration of Mg2+ ions in solution CMg

does not affects the quantities of 23CO minus ions in the solid These results are in agreement with those found in the

reference [27]

S B Abdelkader et al

1007

To know the relationship between the variation of Ca2+ and OHminus with the increasing of 23CO minus and Mg2+

content in the solid statistical studies are conducted The results of multiple linear regression Table 6 and Table 7 show that the estimated equations on these variables are represented at 95 level by

( ) ( )2 23

2Mg CO

nCa 1009 1373 0153 n 0698 0067 n+ minus+ = minus plusmn sdot minus plusmn sdot (15)

( ) ( )2 23Mg CO

nOH 219 0740 0313 n 0393 0138 n+ minusminus = minus plusmn minus plusmn sdot (16)

From the intercepts of the following equations it can seen that within experimental error a carbonate and magnesium-free apatite (nCO3 = 0 nMg = 0) contains 10 Ca2+ and 2 OHminus ions per unit cell Equations (11) and (12) These results are in agreement with those in literature [24]-[26] [34] [35]

As shown in Figure 5 there is a correlation of the unit cell parameters of the apatites with their chemical compositions Indeed the changes in the unit cell parameter ldquoardquo of the compounds are attributed to the additive effects of the substitution in the lattice of either carbonate and magnesium [2] [34] [35] The solid line of best fit for these series of compounds in Figure 5 extrapolates to a ratio ca very close to that in hydroxyapatite This result is similary to these obtained previously [34] [35] The application of multiple linear regression to Yi = ca

on ( )3 3

2

1i CO P 2i CO PX n n X n n= = ( )3

3

3i CO Pand X n n= allows us to establish the predicted equation at

95 level

( ) ( ) ( )

( ) ( )

26 6i CO3 P CO3 P

35CO3 P

Y c a 073 0052 156 10 n n 0168 426 10 n n

0212 118 10 n n

minus minus

minus

= = + plusmn times sdot + plusmn times sdot

+ plusmn times sdot (17)

To verify the general formula proposed We apply the multiple linear regression to Yi = nCa2+ on X1i = x and X2i = y (the contributions of the mechanisms II and IV) Similar treatment is realized for Yi = nOHminus on X1i = z and X2i = y (the contributions of the mechanisms V and IV) Table 10 and Table 11 The results of these calcu-lations show that the predicted equations at 95 level are

( ) ( )2nCa 999 0995 0007 x 1999 0013 y+ = minus plusmn minus plusmn sdot (18)

( ) ( )16 16nOH 200 100 452 10 z 200 754 10 yminus minus minus= + plusmn times sdot minus plusmn times sdot (19)

7 Conclusion The theoretical calculations of the present study indicate unambiguously that the mechanisms II III and V con-tribute to the incorporation of Mg and Ca in the lattice of apatite This corroborates in more definite way our as-sumptions obtained from the experimental data and allows us to propose for these compounds the general for-mula

( ) ( )( ) ( )( ) ( )( )y 4 3 210 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OH nH Ominus minus sdot minus minus minus ++

sdot

References [1] Kreidler ER and Hummel FA (1970) The Crystal Chemistry of Apatite Structure Fields of Fluor- and Chlorapatite

American Mineralogist 55 170 [2] Legeros RZ (1984) Tooth Enamel IV In Fearnhead RW and Sugas S Eds Elsevier Amsterdam 32-36 [3] Ben Abdelkader S Khattech I Rey C and Jemal M (2001) Syntheacutese Caracteacuterisation et Thermochimie drsquoApatites

Calco-Magneacutesiennes Hydroxyleacutees et Fluoreacutees Thermochimica Acta 376 25-36 httpdxdoiorg101016S0040-6031(01)00565-2

[4] Terpstra RA and Driessens FCM (1986) Magnesium in Tooth Enamel and Synthetic Apatites Calcified Tissue In-ternational 39 348-354 httpdxdoiorg101007BF02555203

[5] Hayek E and Newsely H (1958) Uumlber die Existenz von Tricalciumphosphat in waumlBriger Loumlsung Mn Chem 89 88 [6] Rowles SL (1968) Crystallographic Study of Biological Apatites Bulletin de la Socieacuteteacute Chimique de France 1797 [7] Hamad M and Heughebaert JC (1987) Study of Apatite and Whitlockite Formation at 100˚C in the System CaO-

MgO-P2O5-H2O J Chim Phys 84 985

S B Abdelkader et al

997

2 Methods and Materials 21 Preparation of ldquoBrdquo Type Carbonated Hydroxyapatites Containing Magnesium The method of preparation used in this work is inspired from the method used in reference [23] but it is slightly modified The apaties are prepared by dropping 200 mL of a phosphate solution (NH4)2HPO4 (018 M) into 200 mL of a calcium solution Ca(NO3)4H2O (044 M) under reflux at 87˚C To the calcium solution is added 20 mL of a magnesium solution Mg(NO3)26H2O containing different concentrations CMg (000 17 68 and 136) mM The same procedure is remade by adding to the phosphate solution 5 mL of a carbonate solution NH4HCO3 (1 M) A third set of preparations is performed by adding 10 mL from the above carbonate solution The pH is maintained at 90 during the precipitation by adding an ammonia concentrated solution (28 weight) The pre-cipitation is carried out over 3 h Then the system is refluxed for an additional duration of 2 h The samples are filtered thoroughly washed with hot distilled water and dried overnight at 120˚C

22 Physical Analysis The powdered samples are identified by X-ray diffraction and by infra red spectroscopy Infrared spectra of the samples dispersed in KBr tablets are recorded using a Shimadzu Fourier transform infrared spectrophotometer in the range of 4000 - 400 cmminus1 Then the samples are analyzed by X-ray diffraction (XRD) using a Philips dif-fractometer using Cu Kα radiation The samples are scanned in the 2θ range of 20˚ - 60˚ The ldquoa and crdquo parame-ters of the lattice of the hexagonal unit cell are calculated using ldquowincellrdquo refinement program

23 Chemical Analysis The samples are analyzed for Ca PO4 CO3 and Mg The calcium content of the precipitates is determined by a complexometric titration with the ethylenediaminetetraacetic acid [29] the magnesium by atomic absorption the carbonate content is determined by coulometrically method and the phosphorus content by spectrophotome-trie of the phosphomolybdate complex [30]

3 Results 31 Results of Physical Analysis The IR Spectra of some representative samples (Mg4 Mg8 and Mg12) are shown in Figure 1 The spectra contain the characteristic bands of the phosphate group 3

4PO minus in the ranges 960 - 1100 and 570 - 610 cmminus1 Two broad bands around 1635 cmminus1 and 3400 cmminus1 confirm that the samples contain a significant amount of water On the

Figure 1 IR spectra of some representative samples

S B Abdelkader et al

998

spectra of the samples (Mg8 and Mg12) are displayed typical absorption bands of 23CO minus at ~873 and ~1420

cmminus1 and between 1450 and 1500 cmminus1 characterizing the vibration of the 23CO minus on 3

4PO minus lattice sites (B- type 2

3CO minus ) [31] From Figure 1 we can clearly see that the intensity of these absorptions increases with the increase of the carbonate content On the other hand the IR spectra of the compounds (Mg4 Mg8 and Mg12) show that the magnesium incorporated in the apatites causes the loss of resolution of the 3

4PO minus absorptions bands suggesting a decrease in the crystallinity [31]

The X-ray diffraction patterns of some representative samples are shown in Figure 2 The X-ray diffraction powder patterns of the compounds show only one crystal phase The peaks are sharp well resolved and charac-teristic of the hexagonal apatite phase No extraneous peaks attributable to other phases than apatite could be found in the diffractograms The increase of the level of 2

3CO minus substitution produces a loss of the resolution of the 112 peak and a decrease in the intensity of the 300 202 and 002 peaks

The Table 1 contains the values of the lattice parameters ldquoardquo and ldquocrdquo obtained for the different compounds From this table we can see that simultaneous incorporation of two elements ldquoCO3 and Mgrdquo results in an de-

crease of the ldquoardquo parameter This contraction is attributed to the simultaneous effects of the 23CO minus and Mg2+

substitutions [2]

Figure 2 X-ray diffraction patterns of some representative samples

Table 1 ldquoardquo and ldquocrdquo Lattice parameters of ldquoBrdquo CO3Mg-Haps

Samples CCM CMgmM aAring cAring ca

Mg1 0000 000 9439 plusmn 0004 6911 plusmn 0004 0732

Mg2 0000 017 9428 plusmn 0004 6901 plusmn 0003 0732

Mg3 0000 068 9412 plusmn 0006 6869 plusmn 0004 0730

Mg4 0000 136 9409 plusmn 0007 6862 plusmn 0005 0729

Mg5 0025 000 9410 plusmn 0005 6910 plusmn 0005 0735

Mg6 0025 017 9412 plusmn 0008 6911 plusmn 0005 0734

Mg7 0025 068 9386 plusmn 0008 6889 plusmn 0005 0734

Mg8 0025 136 9375 plusmn 0006 6896 plusmn 0004 0735

Mg9 0050 000 9369 plusmn 0007 6895 plusmn 0004 0736

Mg10 0050 017 9360 plusmn 0010 6899 plusmn 0007 0737

Mg11 0050 068 9343 plusmn 0014 6879 plusmn 0010 0736

Mg12 0050 136 9333 plusmn 0009 6872 plusmn 0004 0736

S B Abdelkader et al

999

32 Chemical Results The results of the chemical analysis of the samples in Weight are summarized in Table 2 This table also gives the hydroxide content of the samples calculated on the basis of the electroneutrality condition and the total mass balance sum obtained from the equation

4PO P 3 Ca P M M CO Mg OHsum = + sdot + + + (1)

With MX the atomic or ionic mass of X sum value is lower than 100 indicating that the samples of the present study still contain some water after drying at 120˚C

The results of the chemical and physical analysis (Table 2) allow us to calculate the number of each ion X per unit cell nx according to the following equation

( )3

XX P 3 CO

X 6nM P M CO M

=+

(2)

The results of these calculations are summarized in Table 3 The errors in Table 4 are estimated by the means of error propagation theory

Table 2 Chemical Composition (weight percent) and Total Mass Balance sum of the hydroxyapatites obtained by precipita-tion in solutions containing Cc (M) CO3 and CMg (mM) Mg

Sample CCM CMgmM Ca P CO3 Mg OHminus sum

Mg1 0000 000 3697 1683 099 001 319 9273

Mg2 0000 017 3709 1683 158 022 325 9371

Mg3 0000 068 3475 1683 000 082 299 9013

Mg4 0000 136 3435 1722 000 150 296 9158

Mg5 0025 000 3732 1584 445 001 315 9347

Mg6 0025 017 3673 1594 416 021 293 9287

Mg7 0025 068 3673 1633 465 084 289 9515

Mg8 0025 136 3514 1623 534 155 231 9407

Mg9 0050 000 3791 1485 822 001 315 9480

Mg10 0050 017 3762 1425 878 021 385 9413

Mg11 0050 068 3574 1475 871 080 229 9274

Mg12 0050 136 3564 1534 911 152 202 9530

Table 3 Unit cell compositions of NaCO3 Aaps calculated on the basis of the chemical composition and using Equation (2)

Sample CCM CMgmM nCa nP 3COn nMg nOH nMgnP 3COn nP

Mg1 0000 000 991 plusmn 021 582 plusmn 012 0170 plusmn 0007 000 200 008 000

Mg2 0000 017 977 plusmn 020 572 plusmn 011 0270 plusmn 0009 0090 plusmn 0003 200 013 000

Mg3 0000 068 960 plusmn 021 600 plusmn 013 000 0370 plusmn 0021 194 003 006

Mg4 0000 136 927 plusmn 018 600 plusmn 011 000 0660 plusmn 0055 188 006 012

Mg5 0025 000 957 plusmn 021 524 plusmn 011 076 plusmn 002 000 190 042 010

Mg6 0025 017 944 plusmn 020 529 plusmn 010 071 plusmn 002 0090 plusmn 0026 177 047 023

Mg7 0025 068 912 plusmn 018 523 plusmn 010 077 plusmn 002 0340 plusmn 0019 169 053 031

Mg8 0025 136 860 plusmn 018 513 plusmn 010 087 plusmn 002 0620 plusmn 0056 133 077 067

Mg9 0050 000 923 plusmn 019 466 plusmn 011 133 plusmn 003 000 180 076 020

Mg10 0050 017 892 plusmn 020 436 plusmn 009 164 plusmn 004 0080 plusmn 0025 164 099 036

Mg11 0050 068 863 plusmn 019 459 plusmn 010 140 plusmn 003 0320 plusmn 0017 130 105 070

Mg12 0050 136 827 plusmn 017 459 plusmn 009 141 plusmn 003 0580 plusmn 0049 110 115 090

S B Abdelkader et al

1000

Table 4 Multiple linear regression analysis of Yi = 3COn nP the molar ratio (Table 3) as a function of the concentration of

carbonate CcM and magnesium CMgmM in the solution (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 00156 Var(X1) = 00004 Var(X2) = 27997 Covariance Cov(X1 Y) = 00025 Cov(X2 Y) = minus00205 Cov(X1 X2) = 0

Coefficients of correlation R(X1 Y) = 0978 R(X2 Y) = minus0031 R(X1 X2) = 0 Residual mean squareStandard error s2 = 000087

R square R2 = 0958 Observations n = 12 Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 00165b = - - - - -

X1 1 5985b = ( )1 0418bσ = ( )1 1431T b = 1710minus7 5040 693

X2 2 000073b = minus ( )2 00016bσ = ( )2 0452T b = minus 0662 minus00043 00029

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 0179 00897 10254 474 Deviations 9 00078 00078

Total 11 0187 0017

4 Statistical Analysis of the Physicochemical Results 41 Influence of the Experimental Conditions on the Composition

of the Synthetic Apatites To know the influence of the experimental conditions on the incorporation of 2

3CO minus and Mg2+ in the lattice of these synthetic apatites we graph Yi =

3COn nP and nMgnP the molar ratios contents of the samples against Xi the concentration of 2

3CO minus Cc or the concentration of Mg2+ CMg in the solution Figure 3 and Figure 4 From the Figure 3(a) and Figure 4(a) it is seen that

3COn nP the molar ratio increases with the increase of the concentration of 2

3CO minus in the solution (CcM) Contrariwise it varies slightly with the concentration of the Mg2+ ions in the solution and vice versa for nMgnP (Figure 3(b) and Figure 4(b))

To estimate the simultaneous influence of the experimental conditions on 3COn nP and nMgnP the molar ratios

we construct a mathematical model of Yi = 3COn nP or nMgnP on two variables X1i = Cc and X2i = CMg

The mathematical model is described by the equation

0 1 1i 2 2iY X Xi b b b e= + + + (3)

The method of least squares (OLS) allows us to establish the predicted equation

i 0 1 1i 2 2iˆ ˆ ˆY b b X b X= + + (4)

that is most suitable to the data On the other hand this method allows us to calculate the estimated standard er-

rors of the coefficients ( ) ( )1 2ˆ ˆb and bσ σ the individual confidence interval at 95 level R2 the standardized

statistic and to test the null hypothesis H0 bj = 0 and its significances level The analysis of the variance for the linear regression or the F test allows us to ensure that at least one of the X-variables contributes to the regression The theoretical basis of these calculations is given in references [32]-[34] The calculations are summarized in Table 4 and Table 5

42 Influence of the Incorporation of 23CO minus and Mg2+ on the Variation of Ca2+ and OHminus

the Molar Ions Contents of the Synthetic Apatites In attempts to disentangle and to measure the effects of the insertion of 2

3CO minus and Mg2+ ions on the molar con-

S B Abdelkader et al

1001

(a) (b)

Figure 3 (a) 3COn nP molar ratio of the solid versus CMgmM for the samples prepared at different CcM (b) nMgnP

molar ratio of the solid versus CMgmM for the samples prepared at different CcM

(a) (b)

Figure 4 (a) 3COn nP molar ratio versus CcM for the samples prepared at different CMgmM (b) nMgnP molar ratio

versus CcM for the samples prepared at different CMgmM

Table 5 Multiple linear regression analysis of Yi = nMgnP molar ratio (Table 3) as a function of the concentration of carbo-nate CcM and magnesium CMgmM in the solution (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 00022 Var(X1) = 000042 Var(X2) = 27998 Covariance Cov(X1 Y) = 563 times 10minus5 Cov(X2 Y) = 0245 Cov(X1 X2) = 0

Coefficients of correlation R(X1 Y) = 00593 R(X2 Y) = 0995 R(X1 X2) = 0 Standard error s2 = 171 times 10minus5

R square R2 = 0994 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 00013b = minus - - - - -

X1 1 0134b = ( )1 0058bσ = ( )1 229T b = 0048 00023 0266

X2 2 00087b = ( )2 000022bσ = ( )2 3879T b = 000 00082 00092

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 00258 0012 75519 474 Deviations 9 0000154 17 times 10minus5

Total 11 00259 00024

S B Abdelkader et al

1002

tent of Ca2+ of the solid we use the multiple linear regression on two X-variables where X1 = 23nCO minus and X2 =

nMg2+ and Y is the estimate molar content of Ca2+ or OHminus (data Table 3) The results of these calculations are given in Table 6 and Table 7

Table 6 Multiple linear regression analysis of the estimated Yi = nCa2+ on X1i = nMg2+ X2i = 2

3nCO minus calcium carbonate and magnesium respectively molar contents of the solid ldquoBrdquo Mg-CO3 Haps (a) Regression statistic (c) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0059 Var(X2) = 0307 Covariance Cov(X1 Y) = minus0071 Cov(X2 Y) = minus0194 Cov(X1 X2) = minus0013

Coefficients of correlation R(X1 Y) = minus0607 R(X2 Y) = minus0722 R(X1 X2) = minus0101 Residual mean squareStandard error s2 = 00033

R square R2 = 0989 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 1009b = - - - - -

X1 1 1373b = minus ( )1 0068bσ = ( )1 2017T b = minus 110minus8 minus1527 minus1219

X2 2 0693b = minus ( )2 0030bσ = ( )2 2325T b = minus 000 minus0760 minus0625

(c)

Source of variation DF Sum of squares Mean square Fcal Ftab (5 2 9) [32]

Regression 2 279 1397 42812 474

Deviations 9 0029 00033

Total 11 2823 0257

Table 7 Multiple linear regression analysis of the estimated Yi = OHminus on X1 i = nMg2+ and X2i = 2

3nCO minus the molar con-tents of the solid ldquoBrdquo Mg-CO3 Haps (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0059 Var(X2) = 0307 Covariance Cov(X1 Y) = minus0038 Cov(X2 Y) = minus0111 Cov(X1 X2) = minus0013

Coefficients of correlation R(X1 Y) = minus0551 R(X2 Y) = minus0697 R(X1 X2) = minus0101 Residual mean squareStandard error s2 = 00137

R square R2 = 0875 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 219b = - - - - -

X1 1 0740b = minus ( )1 0139bσ = ( )1 532T b = minus 000048 minus1054 minus0426

X2 2 0393b = minus ( )2 0061bσ = ( )2 645T b = minus 000012 minus0531 minus0255

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 0862 0431 3161 474 Deviations 9 0123 00136

Total 11 0985 00896

S B Abdelkader et al

1003

43 The Determination of the Relationship between Y = ca Crystallographic Parameters Ratio and

3COn pn the Molar Ratio To estimate the influence of the incorporation of carbonate on the lattice parameters ldquoardquo and ldquocrdquo in presence of magnesium we plot ca crystallographic parameters ratio (Table 1) as a function of molar ratio

3COn nP (Table 3) for 0 le nMg le 174 mM (Figure 5)

Given that the shape of the curve obtained in Figure 5 is a polynomial we construct a multiple linear regres-sion on Yi = ca as a function of three X-variables where X1 =

3COn nP X2 = (3COn nP)2 and X3 = (

3COn nP)3 (data Table 2) and Y is the estimate ratio of the hexagonal lattice dimensions (data Table 1) The mathematical model equation is

0 1 1 2 2 3 3Y b b X b X b X= + + + + ε (5) Least square [33] allows calculating the regression and correlation coefficients regression of the predicted

equation

0 1 1 2 2 3 3ˆ ˆ ˆ ˆY b b X b X b X= + + + (6)

These estimated rgression coefficients 0 1 2 3ˆ ˆ ˆ ˆb b b and b are calculated from the values of correlation coeffici-

ents variance and covariance according to the method of Scherrer [33] This method allows us to test the utility of the model or the F-test according to

( )( )

2

2

n m 1 RF

m 1 Rminus minus sdot

=sdot minus

(7)

where n is sample size m is number of parameters and (n minus m minus 1) is degree of freedom On the other hand this method allows us to calculate the standard errors of the coefficients 0 1 2 3

ˆ ˆ ˆ ˆb b b and b and to conduct t-tests on the brsquos (to discover which variable(s) is related to estimate Y ) and to calculate the individual confidence interval at 95 level The results of these calculations are given in Table 8

The analysis of variance (ANOVA) shows that F-test = 20189 is higher than criterion F(5 3 8) = 407

Table 8 Multiple linear regression analysis of Yi= ca ratio of the lattice parameters of Mg-CO3 HAps (Table 1) on X1i = 3COn nP X2i = (

3COn nP)2 X3i = (3COn nP)3 the molar ratio Table 3 (a) Statistic regression (b) Coefficients

(a)

Variances Covariances Correlation coefficients

var(Y) 597 times 10minus6 Cov(X1 Y) 284 times 10minus4 1X YR 0931

v(X1) 156 times 10minus3 Cov(X2 Y) 908 times 10minus5 2X YR 0811

v(X2) 21 times 10minus3 Cov(X3 Y) 294 times 10minus5 3X YR 0731

v(X3) 27 times 10minus4 Cov(X1 X2) 55 times 10minus3 2 1X XR 0960

Cov(X1 X3) 186 times 10minus3 2 3X XR 0986

Cov(X2 X3) 744 times 10minus4 3 1X XR 0906

(b)

Standard error of the regression sr = 218 R square R2 = 098 Observations n = 12 Degree of freedom ν = 8

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 0730b = - - - - -

X1 1 0052b = ( ) 71 6793 10bσ minus= times ( )1 6331t b = 000 005217 005218

X2 2 0168b = minus ( ) 62 1852 10bσ minus= times ( )2 12384t b = minus 000 minus016851 minus016850

X3 3 0212b = ( ) 63 515 10bσ minus= times ( )3 9349t b = 000 021221 021223

S B Abdelkader et al

1004

Figure 5 ca parameters ratio as a function of

3COn nP the molar ratio for the apatites prepared at different values of CMg

5 Determination of the General Formula of the Unit Cell of the Synthetic ldquoBrdquo CO3Mg-HAps

The relative composition (Table 3) and the results of the physical analysis demonstrate that the samples are pure ldquoBrdquo type carbonated apatites containing Mg2+ ions Thus mechanisms I II III and V could be account in the incorporation of 2

3CO minus on 34PO minus and Mg2+ ions are incorporated in the apatite lattice according to mechan-

isms III and or IV Moreover the study carried out previously (paragraph 41) show that the 2

3CO minus ions are incorporated in the apatite lattice independently of the concentration of Mg2+ ions solution This result confirms that mechanism IV does and mechanism III does not contribute to the incorporation of Mg2+ in the apatites

Many works [27] [28] have demonstrated that mechanism I andor II are the main mechanisms for the incor-poration of CO3 Otherwise according the reference [27] the contribution of mechanism I seems to be hardly influenced by the alkali metal which is not our case Therefore we consider that mechanism II contribute to the insertion of CO3 ions in the lattice of the solid

Table 7 show that the variation of nOHminus depends on the increase of 23CO minus and Mg2+ So it may be said in

the present study that the mechanism V could account Then the fundamental substitution mechanisms for the incorporation of 2

3CO minus and Mg2+ in the HAp lattice are

2 3 Ca 24 3Ca 2PO V 2CO+ minus minus+ harr + (II)

2 2 OH2Ca 2OH Mg 2V+ minus ++ harr + 2(IV) 3 24 3PO CO OHminus minus minusharr + (V)

where VOH stands for a vacancy in the OHminus sub lattice If x y and z are the contributions of mechanisms II 2IV and V respectively thus

2nCa 10 x 2 y+ = minus minus sdot (8) 34nPO 6 2 x zminus = minus sdot minus (9)

2nMg y+ = (10) 23nCO 2 x zminus = sdot + (11)

and nOH 2 2 y zminus = minus sdot + (12)

and the generic formula has the following expression

( ) ( )( ) ( )( ) ( )( )y 4 310 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OHminus minus sdot minus minus minus ++

S B Abdelkader et al

1005

The values of x y and z the contribution of mechanisms II 2IV and V respectively are calculated from the data (Table 3) and the following equations Then statistical studies are conducted to verify the accuracy of the proposed formula The results of these calculations are summarized in Tables 9-11

Table 9 The values of x y and z the contributions of the mechanisms II 2IV and V respectively calculated from Equations (10)-(14)

Sample x y z

Mg1 0085 000 000

Mg2 0045 009 018

Mg3 minus034 037 068

Mg4 minus060 066 120

Mg5 043 000 minus01

Mg6 038 009 minus005

Mg7 020 034 037

Mg8 015 062 057

Mg9 076 000 minus020

Mg10 092 008 minus020

Mg11 073 032 minus006

Mg12 057 058 026

Table 10 Multiple linear regression analysis of the estimated Yi = nCa2+ the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = x- and X2i = y the contribution of mechanisms II and 2IV (Table 9) (a) Regression statistic (b) Coeffi-cients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0187 Var(X2) =0059

Covariance Cov(X1 Y) = minus0093 Cov(X2 Y) = minus0071 Cov(X1 X2) = minus0046

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 188 times 10minus5

R square R2 = 0999

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 999b = - - - - -

X1 1 0995b = minus ( )1 00032bσ = ( )1 30816T b = minus 000 minus1002 minus0987

X2 2 1999b = minus ( )2 00057bσ = ( )2 34779T b = minus 000 minus2012 minus1986

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32]

Regression 2 2862 1412 7414888 474

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

S B Abdelkader et al

1006

Table 11 Multiple linear regression analysis of the estimated Yi = nOHminus the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = z and X2i = z the contribution of mechanisms II and V (Table 9) (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0164 Var(X2) = 0059

Covariance Cov(X1 Y) = 00055 Cov(X2 Y) = minus0038 Cov(X1 X2) = 0079

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 274 times 10minus32

R square R2 = 100

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 200b = - - - - -

X1 1 100b = ( ) 161 1997 10bσ minus= times ( ) 15

1 500 10T b += times 000 1 1

X2 2 200b = minus ( ) 162 3338 10bσ minus= times ( ) 15

2 599 10T b += minus times 000 minus2 minus2

(c)

Source of variation DF Sum of squares Mean square Fcal

Regression 2 2862 1412 7414888

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

6 Discussion From Table 1 we can see that simultaneous incorporation of two elements ldquoCO3 and Mgrdquo results in an de-crease of the ldquoardquo parameter This contraction is attributed to the simultaneous effects of the 2

3CO minus and Mg2+ substitutions [4]

In Table 2 and Table 3 it is seen that the concentration of the 23CO minus ions in the solution Cc does not affect

the quantities of Mg2+ ions inserted in the solid Because regardless of the concentration of the 23CO minus ions in

the solution Cc the Mg2+ ions contents of the samples increase proportionally with the increase of the concentra-tion of Mg2+ ions in solution CMg This result is in agreement with reference [4] For the same concentration of Mg2+ions in the solution CMg the variation of 3

4PO minus and 23CO minus contents of the solid do not seem to be corre-

lated with the concentration of the Mg2+ ions in the solution while the Ca2+content depends on the concentra-tions of 2

3CO minus Cc and Mg2+ CMg in the solution Figure 3 and Figure 4 show that nCO3nP the molar ratio increases with the increasing of the concentration

of 23CO minus in solution (CcM) Contrariwise it varies slightly with the concentration of Mg2+ ions in solution and

vice versa for nMgnP The statistical treatment of the experimental data Table 4 and Table 5 allows us to es-tablish the estimated equations between these variables at 95 levels

( )3 cnCO nP 00165 5986 0945 C M= + plusmn sdot (13)

and ( ) ( )c MgnMg nP 00013 0135 0132 C M 00087 00005 C mM= minus + plusmn sdot + plusmn sdot (14)

Equations (9) and (10) show that the concentration of the 23CO minus ions in the hydrolysis solution Cc affects the

quantities of 23CO minus and Mg2+ ions incorporated in the solid but the concentration of Mg2+ ions in solution CMg

does not affects the quantities of 23CO minus ions in the solid These results are in agreement with those found in the

reference [27]

S B Abdelkader et al

1007

To know the relationship between the variation of Ca2+ and OHminus with the increasing of 23CO minus and Mg2+

content in the solid statistical studies are conducted The results of multiple linear regression Table 6 and Table 7 show that the estimated equations on these variables are represented at 95 level by

( ) ( )2 23

2Mg CO

nCa 1009 1373 0153 n 0698 0067 n+ minus+ = minus plusmn sdot minus plusmn sdot (15)

( ) ( )2 23Mg CO

nOH 219 0740 0313 n 0393 0138 n+ minusminus = minus plusmn minus plusmn sdot (16)

From the intercepts of the following equations it can seen that within experimental error a carbonate and magnesium-free apatite (nCO3 = 0 nMg = 0) contains 10 Ca2+ and 2 OHminus ions per unit cell Equations (11) and (12) These results are in agreement with those in literature [24]-[26] [34] [35]

As shown in Figure 5 there is a correlation of the unit cell parameters of the apatites with their chemical compositions Indeed the changes in the unit cell parameter ldquoardquo of the compounds are attributed to the additive effects of the substitution in the lattice of either carbonate and magnesium [2] [34] [35] The solid line of best fit for these series of compounds in Figure 5 extrapolates to a ratio ca very close to that in hydroxyapatite This result is similary to these obtained previously [34] [35] The application of multiple linear regression to Yi = ca

on ( )3 3

2

1i CO P 2i CO PX n n X n n= = ( )3

3

3i CO Pand X n n= allows us to establish the predicted equation at

95 level

( ) ( ) ( )

( ) ( )

26 6i CO3 P CO3 P

35CO3 P

Y c a 073 0052 156 10 n n 0168 426 10 n n

0212 118 10 n n

minus minus

minus

= = + plusmn times sdot + plusmn times sdot

+ plusmn times sdot (17)

To verify the general formula proposed We apply the multiple linear regression to Yi = nCa2+ on X1i = x and X2i = y (the contributions of the mechanisms II and IV) Similar treatment is realized for Yi = nOHminus on X1i = z and X2i = y (the contributions of the mechanisms V and IV) Table 10 and Table 11 The results of these calcu-lations show that the predicted equations at 95 level are

( ) ( )2nCa 999 0995 0007 x 1999 0013 y+ = minus plusmn minus plusmn sdot (18)

( ) ( )16 16nOH 200 100 452 10 z 200 754 10 yminus minus minus= + plusmn times sdot minus plusmn times sdot (19)

7 Conclusion The theoretical calculations of the present study indicate unambiguously that the mechanisms II III and V con-tribute to the incorporation of Mg and Ca in the lattice of apatite This corroborates in more definite way our as-sumptions obtained from the experimental data and allows us to propose for these compounds the general for-mula

( ) ( )( ) ( )( ) ( )( )y 4 3 210 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OH nH Ominus minus sdot minus minus minus ++

sdot

References [1] Kreidler ER and Hummel FA (1970) The Crystal Chemistry of Apatite Structure Fields of Fluor- and Chlorapatite

American Mineralogist 55 170 [2] Legeros RZ (1984) Tooth Enamel IV In Fearnhead RW and Sugas S Eds Elsevier Amsterdam 32-36 [3] Ben Abdelkader S Khattech I Rey C and Jemal M (2001) Syntheacutese Caracteacuterisation et Thermochimie drsquoApatites

Calco-Magneacutesiennes Hydroxyleacutees et Fluoreacutees Thermochimica Acta 376 25-36 httpdxdoiorg101016S0040-6031(01)00565-2

[4] Terpstra RA and Driessens FCM (1986) Magnesium in Tooth Enamel and Synthetic Apatites Calcified Tissue In-ternational 39 348-354 httpdxdoiorg101007BF02555203

[5] Hayek E and Newsely H (1958) Uumlber die Existenz von Tricalciumphosphat in waumlBriger Loumlsung Mn Chem 89 88 [6] Rowles SL (1968) Crystallographic Study of Biological Apatites Bulletin de la Socieacuteteacute Chimique de France 1797 [7] Hamad M and Heughebaert JC (1987) Study of Apatite and Whitlockite Formation at 100˚C in the System CaO-

MgO-P2O5-H2O J Chim Phys 84 985

S B Abdelkader et al

998

spectra of the samples (Mg8 and Mg12) are displayed typical absorption bands of 23CO minus at ~873 and ~1420

cmminus1 and between 1450 and 1500 cmminus1 characterizing the vibration of the 23CO minus on 3

4PO minus lattice sites (B- type 2

3CO minus ) [31] From Figure 1 we can clearly see that the intensity of these absorptions increases with the increase of the carbonate content On the other hand the IR spectra of the compounds (Mg4 Mg8 and Mg12) show that the magnesium incorporated in the apatites causes the loss of resolution of the 3

4PO minus absorptions bands suggesting a decrease in the crystallinity [31]

The X-ray diffraction patterns of some representative samples are shown in Figure 2 The X-ray diffraction powder patterns of the compounds show only one crystal phase The peaks are sharp well resolved and charac-teristic of the hexagonal apatite phase No extraneous peaks attributable to other phases than apatite could be found in the diffractograms The increase of the level of 2

3CO minus substitution produces a loss of the resolution of the 112 peak and a decrease in the intensity of the 300 202 and 002 peaks

The Table 1 contains the values of the lattice parameters ldquoardquo and ldquocrdquo obtained for the different compounds From this table we can see that simultaneous incorporation of two elements ldquoCO3 and Mgrdquo results in an de-

crease of the ldquoardquo parameter This contraction is attributed to the simultaneous effects of the 23CO minus and Mg2+

substitutions [2]

Figure 2 X-ray diffraction patterns of some representative samples

Table 1 ldquoardquo and ldquocrdquo Lattice parameters of ldquoBrdquo CO3Mg-Haps

Samples CCM CMgmM aAring cAring ca

Mg1 0000 000 9439 plusmn 0004 6911 plusmn 0004 0732

Mg2 0000 017 9428 plusmn 0004 6901 plusmn 0003 0732

Mg3 0000 068 9412 plusmn 0006 6869 plusmn 0004 0730

Mg4 0000 136 9409 plusmn 0007 6862 plusmn 0005 0729

Mg5 0025 000 9410 plusmn 0005 6910 plusmn 0005 0735

Mg6 0025 017 9412 plusmn 0008 6911 plusmn 0005 0734

Mg7 0025 068 9386 plusmn 0008 6889 plusmn 0005 0734

Mg8 0025 136 9375 plusmn 0006 6896 plusmn 0004 0735

Mg9 0050 000 9369 plusmn 0007 6895 plusmn 0004 0736

Mg10 0050 017 9360 plusmn 0010 6899 plusmn 0007 0737

Mg11 0050 068 9343 plusmn 0014 6879 plusmn 0010 0736

Mg12 0050 136 9333 plusmn 0009 6872 plusmn 0004 0736

S B Abdelkader et al

999

32 Chemical Results The results of the chemical analysis of the samples in Weight are summarized in Table 2 This table also gives the hydroxide content of the samples calculated on the basis of the electroneutrality condition and the total mass balance sum obtained from the equation

4PO P 3 Ca P M M CO Mg OHsum = + sdot + + + (1)

With MX the atomic or ionic mass of X sum value is lower than 100 indicating that the samples of the present study still contain some water after drying at 120˚C

The results of the chemical and physical analysis (Table 2) allow us to calculate the number of each ion X per unit cell nx according to the following equation

( )3

XX P 3 CO

X 6nM P M CO M

=+

(2)

The results of these calculations are summarized in Table 3 The errors in Table 4 are estimated by the means of error propagation theory

Table 2 Chemical Composition (weight percent) and Total Mass Balance sum of the hydroxyapatites obtained by precipita-tion in solutions containing Cc (M) CO3 and CMg (mM) Mg

Sample CCM CMgmM Ca P CO3 Mg OHminus sum

Mg1 0000 000 3697 1683 099 001 319 9273

Mg2 0000 017 3709 1683 158 022 325 9371

Mg3 0000 068 3475 1683 000 082 299 9013

Mg4 0000 136 3435 1722 000 150 296 9158

Mg5 0025 000 3732 1584 445 001 315 9347

Mg6 0025 017 3673 1594 416 021 293 9287

Mg7 0025 068 3673 1633 465 084 289 9515

Mg8 0025 136 3514 1623 534 155 231 9407

Mg9 0050 000 3791 1485 822 001 315 9480

Mg10 0050 017 3762 1425 878 021 385 9413

Mg11 0050 068 3574 1475 871 080 229 9274

Mg12 0050 136 3564 1534 911 152 202 9530

Table 3 Unit cell compositions of NaCO3 Aaps calculated on the basis of the chemical composition and using Equation (2)

Sample CCM CMgmM nCa nP 3COn nMg nOH nMgnP 3COn nP

Mg1 0000 000 991 plusmn 021 582 plusmn 012 0170 plusmn 0007 000 200 008 000

Mg2 0000 017 977 plusmn 020 572 plusmn 011 0270 plusmn 0009 0090 plusmn 0003 200 013 000

Mg3 0000 068 960 plusmn 021 600 plusmn 013 000 0370 plusmn 0021 194 003 006

Mg4 0000 136 927 plusmn 018 600 plusmn 011 000 0660 plusmn 0055 188 006 012

Mg5 0025 000 957 plusmn 021 524 plusmn 011 076 plusmn 002 000 190 042 010

Mg6 0025 017 944 plusmn 020 529 plusmn 010 071 plusmn 002 0090 plusmn 0026 177 047 023

Mg7 0025 068 912 plusmn 018 523 plusmn 010 077 plusmn 002 0340 plusmn 0019 169 053 031

Mg8 0025 136 860 plusmn 018 513 plusmn 010 087 plusmn 002 0620 plusmn 0056 133 077 067

Mg9 0050 000 923 plusmn 019 466 plusmn 011 133 plusmn 003 000 180 076 020

Mg10 0050 017 892 plusmn 020 436 plusmn 009 164 plusmn 004 0080 plusmn 0025 164 099 036

Mg11 0050 068 863 plusmn 019 459 plusmn 010 140 plusmn 003 0320 plusmn 0017 130 105 070

Mg12 0050 136 827 plusmn 017 459 plusmn 009 141 plusmn 003 0580 plusmn 0049 110 115 090

S B Abdelkader et al

1000

Table 4 Multiple linear regression analysis of Yi = 3COn nP the molar ratio (Table 3) as a function of the concentration of

carbonate CcM and magnesium CMgmM in the solution (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 00156 Var(X1) = 00004 Var(X2) = 27997 Covariance Cov(X1 Y) = 00025 Cov(X2 Y) = minus00205 Cov(X1 X2) = 0

Coefficients of correlation R(X1 Y) = 0978 R(X2 Y) = minus0031 R(X1 X2) = 0 Residual mean squareStandard error s2 = 000087

R square R2 = 0958 Observations n = 12 Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 00165b = - - - - -

X1 1 5985b = ( )1 0418bσ = ( )1 1431T b = 1710minus7 5040 693

X2 2 000073b = minus ( )2 00016bσ = ( )2 0452T b = minus 0662 minus00043 00029

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 0179 00897 10254 474 Deviations 9 00078 00078

Total 11 0187 0017

4 Statistical Analysis of the Physicochemical Results 41 Influence of the Experimental Conditions on the Composition

of the Synthetic Apatites To know the influence of the experimental conditions on the incorporation of 2

3CO minus and Mg2+ in the lattice of these synthetic apatites we graph Yi =

3COn nP and nMgnP the molar ratios contents of the samples against Xi the concentration of 2

3CO minus Cc or the concentration of Mg2+ CMg in the solution Figure 3 and Figure 4 From the Figure 3(a) and Figure 4(a) it is seen that

3COn nP the molar ratio increases with the increase of the concentration of 2

3CO minus in the solution (CcM) Contrariwise it varies slightly with the concentration of the Mg2+ ions in the solution and vice versa for nMgnP (Figure 3(b) and Figure 4(b))

To estimate the simultaneous influence of the experimental conditions on 3COn nP and nMgnP the molar ratios

we construct a mathematical model of Yi = 3COn nP or nMgnP on two variables X1i = Cc and X2i = CMg

The mathematical model is described by the equation

0 1 1i 2 2iY X Xi b b b e= + + + (3)

The method of least squares (OLS) allows us to establish the predicted equation

i 0 1 1i 2 2iˆ ˆ ˆY b b X b X= + + (4)

that is most suitable to the data On the other hand this method allows us to calculate the estimated standard er-

rors of the coefficients ( ) ( )1 2ˆ ˆb and bσ σ the individual confidence interval at 95 level R2 the standardized

statistic and to test the null hypothesis H0 bj = 0 and its significances level The analysis of the variance for the linear regression or the F test allows us to ensure that at least one of the X-variables contributes to the regression The theoretical basis of these calculations is given in references [32]-[34] The calculations are summarized in Table 4 and Table 5

42 Influence of the Incorporation of 23CO minus and Mg2+ on the Variation of Ca2+ and OHminus

the Molar Ions Contents of the Synthetic Apatites In attempts to disentangle and to measure the effects of the insertion of 2

3CO minus and Mg2+ ions on the molar con-

S B Abdelkader et al

1001

(a) (b)

Figure 3 (a) 3COn nP molar ratio of the solid versus CMgmM for the samples prepared at different CcM (b) nMgnP

molar ratio of the solid versus CMgmM for the samples prepared at different CcM

(a) (b)

Figure 4 (a) 3COn nP molar ratio versus CcM for the samples prepared at different CMgmM (b) nMgnP molar ratio

versus CcM for the samples prepared at different CMgmM

Table 5 Multiple linear regression analysis of Yi = nMgnP molar ratio (Table 3) as a function of the concentration of carbo-nate CcM and magnesium CMgmM in the solution (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 00022 Var(X1) = 000042 Var(X2) = 27998 Covariance Cov(X1 Y) = 563 times 10minus5 Cov(X2 Y) = 0245 Cov(X1 X2) = 0

Coefficients of correlation R(X1 Y) = 00593 R(X2 Y) = 0995 R(X1 X2) = 0 Standard error s2 = 171 times 10minus5

R square R2 = 0994 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 00013b = minus - - - - -

X1 1 0134b = ( )1 0058bσ = ( )1 229T b = 0048 00023 0266

X2 2 00087b = ( )2 000022bσ = ( )2 3879T b = 000 00082 00092

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 00258 0012 75519 474 Deviations 9 0000154 17 times 10minus5

Total 11 00259 00024

S B Abdelkader et al

1002

tent of Ca2+ of the solid we use the multiple linear regression on two X-variables where X1 = 23nCO minus and X2 =

nMg2+ and Y is the estimate molar content of Ca2+ or OHminus (data Table 3) The results of these calculations are given in Table 6 and Table 7

Table 6 Multiple linear regression analysis of the estimated Yi = nCa2+ on X1i = nMg2+ X2i = 2

3nCO minus calcium carbonate and magnesium respectively molar contents of the solid ldquoBrdquo Mg-CO3 Haps (a) Regression statistic (c) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0059 Var(X2) = 0307 Covariance Cov(X1 Y) = minus0071 Cov(X2 Y) = minus0194 Cov(X1 X2) = minus0013

Coefficients of correlation R(X1 Y) = minus0607 R(X2 Y) = minus0722 R(X1 X2) = minus0101 Residual mean squareStandard error s2 = 00033

R square R2 = 0989 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 1009b = - - - - -

X1 1 1373b = minus ( )1 0068bσ = ( )1 2017T b = minus 110minus8 minus1527 minus1219

X2 2 0693b = minus ( )2 0030bσ = ( )2 2325T b = minus 000 minus0760 minus0625

(c)

Source of variation DF Sum of squares Mean square Fcal Ftab (5 2 9) [32]

Regression 2 279 1397 42812 474

Deviations 9 0029 00033

Total 11 2823 0257

Table 7 Multiple linear regression analysis of the estimated Yi = OHminus on X1 i = nMg2+ and X2i = 2

3nCO minus the molar con-tents of the solid ldquoBrdquo Mg-CO3 Haps (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0059 Var(X2) = 0307 Covariance Cov(X1 Y) = minus0038 Cov(X2 Y) = minus0111 Cov(X1 X2) = minus0013

Coefficients of correlation R(X1 Y) = minus0551 R(X2 Y) = minus0697 R(X1 X2) = minus0101 Residual mean squareStandard error s2 = 00137

R square R2 = 0875 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 219b = - - - - -

X1 1 0740b = minus ( )1 0139bσ = ( )1 532T b = minus 000048 minus1054 minus0426

X2 2 0393b = minus ( )2 0061bσ = ( )2 645T b = minus 000012 minus0531 minus0255

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 0862 0431 3161 474 Deviations 9 0123 00136

Total 11 0985 00896

S B Abdelkader et al

1003

43 The Determination of the Relationship between Y = ca Crystallographic Parameters Ratio and

3COn pn the Molar Ratio To estimate the influence of the incorporation of carbonate on the lattice parameters ldquoardquo and ldquocrdquo in presence of magnesium we plot ca crystallographic parameters ratio (Table 1) as a function of molar ratio

3COn nP (Table 3) for 0 le nMg le 174 mM (Figure 5)

Given that the shape of the curve obtained in Figure 5 is a polynomial we construct a multiple linear regres-sion on Yi = ca as a function of three X-variables where X1 =

3COn nP X2 = (3COn nP)2 and X3 = (

3COn nP)3 (data Table 2) and Y is the estimate ratio of the hexagonal lattice dimensions (data Table 1) The mathematical model equation is

0 1 1 2 2 3 3Y b b X b X b X= + + + + ε (5) Least square [33] allows calculating the regression and correlation coefficients regression of the predicted

equation

0 1 1 2 2 3 3ˆ ˆ ˆ ˆY b b X b X b X= + + + (6)

These estimated rgression coefficients 0 1 2 3ˆ ˆ ˆ ˆb b b and b are calculated from the values of correlation coeffici-

ents variance and covariance according to the method of Scherrer [33] This method allows us to test the utility of the model or the F-test according to

( )( )

2

2

n m 1 RF

m 1 Rminus minus sdot

=sdot minus

(7)

where n is sample size m is number of parameters and (n minus m minus 1) is degree of freedom On the other hand this method allows us to calculate the standard errors of the coefficients 0 1 2 3

ˆ ˆ ˆ ˆb b b and b and to conduct t-tests on the brsquos (to discover which variable(s) is related to estimate Y ) and to calculate the individual confidence interval at 95 level The results of these calculations are given in Table 8

The analysis of variance (ANOVA) shows that F-test = 20189 is higher than criterion F(5 3 8) = 407

Table 8 Multiple linear regression analysis of Yi= ca ratio of the lattice parameters of Mg-CO3 HAps (Table 1) on X1i = 3COn nP X2i = (

3COn nP)2 X3i = (3COn nP)3 the molar ratio Table 3 (a) Statistic regression (b) Coefficients

(a)

Variances Covariances Correlation coefficients

var(Y) 597 times 10minus6 Cov(X1 Y) 284 times 10minus4 1X YR 0931

v(X1) 156 times 10minus3 Cov(X2 Y) 908 times 10minus5 2X YR 0811

v(X2) 21 times 10minus3 Cov(X3 Y) 294 times 10minus5 3X YR 0731

v(X3) 27 times 10minus4 Cov(X1 X2) 55 times 10minus3 2 1X XR 0960

Cov(X1 X3) 186 times 10minus3 2 3X XR 0986

Cov(X2 X3) 744 times 10minus4 3 1X XR 0906

(b)

Standard error of the regression sr = 218 R square R2 = 098 Observations n = 12 Degree of freedom ν = 8

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 0730b = - - - - -

X1 1 0052b = ( ) 71 6793 10bσ minus= times ( )1 6331t b = 000 005217 005218

X2 2 0168b = minus ( ) 62 1852 10bσ minus= times ( )2 12384t b = minus 000 minus016851 minus016850

X3 3 0212b = ( ) 63 515 10bσ minus= times ( )3 9349t b = 000 021221 021223

S B Abdelkader et al

1004

Figure 5 ca parameters ratio as a function of

3COn nP the molar ratio for the apatites prepared at different values of CMg

5 Determination of the General Formula of the Unit Cell of the Synthetic ldquoBrdquo CO3Mg-HAps

The relative composition (Table 3) and the results of the physical analysis demonstrate that the samples are pure ldquoBrdquo type carbonated apatites containing Mg2+ ions Thus mechanisms I II III and V could be account in the incorporation of 2

3CO minus on 34PO minus and Mg2+ ions are incorporated in the apatite lattice according to mechan-

isms III and or IV Moreover the study carried out previously (paragraph 41) show that the 2

3CO minus ions are incorporated in the apatite lattice independently of the concentration of Mg2+ ions solution This result confirms that mechanism IV does and mechanism III does not contribute to the incorporation of Mg2+ in the apatites

Many works [27] [28] have demonstrated that mechanism I andor II are the main mechanisms for the incor-poration of CO3 Otherwise according the reference [27] the contribution of mechanism I seems to be hardly influenced by the alkali metal which is not our case Therefore we consider that mechanism II contribute to the insertion of CO3 ions in the lattice of the solid

Table 7 show that the variation of nOHminus depends on the increase of 23CO minus and Mg2+ So it may be said in

the present study that the mechanism V could account Then the fundamental substitution mechanisms for the incorporation of 2

3CO minus and Mg2+ in the HAp lattice are

2 3 Ca 24 3Ca 2PO V 2CO+ minus minus+ harr + (II)

2 2 OH2Ca 2OH Mg 2V+ minus ++ harr + 2(IV) 3 24 3PO CO OHminus minus minusharr + (V)

where VOH stands for a vacancy in the OHminus sub lattice If x y and z are the contributions of mechanisms II 2IV and V respectively thus

2nCa 10 x 2 y+ = minus minus sdot (8) 34nPO 6 2 x zminus = minus sdot minus (9)

2nMg y+ = (10) 23nCO 2 x zminus = sdot + (11)

and nOH 2 2 y zminus = minus sdot + (12)

and the generic formula has the following expression

( ) ( )( ) ( )( ) ( )( )y 4 310 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OHminus minus sdot minus minus minus ++

S B Abdelkader et al

1005

The values of x y and z the contribution of mechanisms II 2IV and V respectively are calculated from the data (Table 3) and the following equations Then statistical studies are conducted to verify the accuracy of the proposed formula The results of these calculations are summarized in Tables 9-11

Table 9 The values of x y and z the contributions of the mechanisms II 2IV and V respectively calculated from Equations (10)-(14)

Sample x y z

Mg1 0085 000 000

Mg2 0045 009 018

Mg3 minus034 037 068

Mg4 minus060 066 120

Mg5 043 000 minus01

Mg6 038 009 minus005

Mg7 020 034 037

Mg8 015 062 057

Mg9 076 000 minus020

Mg10 092 008 minus020

Mg11 073 032 minus006

Mg12 057 058 026

Table 10 Multiple linear regression analysis of the estimated Yi = nCa2+ the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = x- and X2i = y the contribution of mechanisms II and 2IV (Table 9) (a) Regression statistic (b) Coeffi-cients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0187 Var(X2) =0059

Covariance Cov(X1 Y) = minus0093 Cov(X2 Y) = minus0071 Cov(X1 X2) = minus0046

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 188 times 10minus5

R square R2 = 0999

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 999b = - - - - -

X1 1 0995b = minus ( )1 00032bσ = ( )1 30816T b = minus 000 minus1002 minus0987

X2 2 1999b = minus ( )2 00057bσ = ( )2 34779T b = minus 000 minus2012 minus1986

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32]

Regression 2 2862 1412 7414888 474

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

S B Abdelkader et al

1006

Table 11 Multiple linear regression analysis of the estimated Yi = nOHminus the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = z and X2i = z the contribution of mechanisms II and V (Table 9) (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0164 Var(X2) = 0059

Covariance Cov(X1 Y) = 00055 Cov(X2 Y) = minus0038 Cov(X1 X2) = 0079

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 274 times 10minus32

R square R2 = 100

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 200b = - - - - -

X1 1 100b = ( ) 161 1997 10bσ minus= times ( ) 15

1 500 10T b += times 000 1 1

X2 2 200b = minus ( ) 162 3338 10bσ minus= times ( ) 15

2 599 10T b += minus times 000 minus2 minus2

(c)

Source of variation DF Sum of squares Mean square Fcal

Regression 2 2862 1412 7414888

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

6 Discussion From Table 1 we can see that simultaneous incorporation of two elements ldquoCO3 and Mgrdquo results in an de-crease of the ldquoardquo parameter This contraction is attributed to the simultaneous effects of the 2

3CO minus and Mg2+ substitutions [4]

In Table 2 and Table 3 it is seen that the concentration of the 23CO minus ions in the solution Cc does not affect

the quantities of Mg2+ ions inserted in the solid Because regardless of the concentration of the 23CO minus ions in

the solution Cc the Mg2+ ions contents of the samples increase proportionally with the increase of the concentra-tion of Mg2+ ions in solution CMg This result is in agreement with reference [4] For the same concentration of Mg2+ions in the solution CMg the variation of 3

4PO minus and 23CO minus contents of the solid do not seem to be corre-

lated with the concentration of the Mg2+ ions in the solution while the Ca2+content depends on the concentra-tions of 2

3CO minus Cc and Mg2+ CMg in the solution Figure 3 and Figure 4 show that nCO3nP the molar ratio increases with the increasing of the concentration

of 23CO minus in solution (CcM) Contrariwise it varies slightly with the concentration of Mg2+ ions in solution and

vice versa for nMgnP The statistical treatment of the experimental data Table 4 and Table 5 allows us to es-tablish the estimated equations between these variables at 95 levels

( )3 cnCO nP 00165 5986 0945 C M= + plusmn sdot (13)

and ( ) ( )c MgnMg nP 00013 0135 0132 C M 00087 00005 C mM= minus + plusmn sdot + plusmn sdot (14)

Equations (9) and (10) show that the concentration of the 23CO minus ions in the hydrolysis solution Cc affects the

quantities of 23CO minus and Mg2+ ions incorporated in the solid but the concentration of Mg2+ ions in solution CMg

does not affects the quantities of 23CO minus ions in the solid These results are in agreement with those found in the

reference [27]

S B Abdelkader et al

1007

To know the relationship between the variation of Ca2+ and OHminus with the increasing of 23CO minus and Mg2+

content in the solid statistical studies are conducted The results of multiple linear regression Table 6 and Table 7 show that the estimated equations on these variables are represented at 95 level by

( ) ( )2 23

2Mg CO

nCa 1009 1373 0153 n 0698 0067 n+ minus+ = minus plusmn sdot minus plusmn sdot (15)

( ) ( )2 23Mg CO

nOH 219 0740 0313 n 0393 0138 n+ minusminus = minus plusmn minus plusmn sdot (16)

From the intercepts of the following equations it can seen that within experimental error a carbonate and magnesium-free apatite (nCO3 = 0 nMg = 0) contains 10 Ca2+ and 2 OHminus ions per unit cell Equations (11) and (12) These results are in agreement with those in literature [24]-[26] [34] [35]

As shown in Figure 5 there is a correlation of the unit cell parameters of the apatites with their chemical compositions Indeed the changes in the unit cell parameter ldquoardquo of the compounds are attributed to the additive effects of the substitution in the lattice of either carbonate and magnesium [2] [34] [35] The solid line of best fit for these series of compounds in Figure 5 extrapolates to a ratio ca very close to that in hydroxyapatite This result is similary to these obtained previously [34] [35] The application of multiple linear regression to Yi = ca

on ( )3 3

2

1i CO P 2i CO PX n n X n n= = ( )3

3

3i CO Pand X n n= allows us to establish the predicted equation at

95 level

( ) ( ) ( )

( ) ( )

26 6i CO3 P CO3 P

35CO3 P

Y c a 073 0052 156 10 n n 0168 426 10 n n

0212 118 10 n n

minus minus

minus

= = + plusmn times sdot + plusmn times sdot

+ plusmn times sdot (17)

To verify the general formula proposed We apply the multiple linear regression to Yi = nCa2+ on X1i = x and X2i = y (the contributions of the mechanisms II and IV) Similar treatment is realized for Yi = nOHminus on X1i = z and X2i = y (the contributions of the mechanisms V and IV) Table 10 and Table 11 The results of these calcu-lations show that the predicted equations at 95 level are

( ) ( )2nCa 999 0995 0007 x 1999 0013 y+ = minus plusmn minus plusmn sdot (18)

( ) ( )16 16nOH 200 100 452 10 z 200 754 10 yminus minus minus= + plusmn times sdot minus plusmn times sdot (19)

7 Conclusion The theoretical calculations of the present study indicate unambiguously that the mechanisms II III and V con-tribute to the incorporation of Mg and Ca in the lattice of apatite This corroborates in more definite way our as-sumptions obtained from the experimental data and allows us to propose for these compounds the general for-mula

( ) ( )( ) ( )( ) ( )( )y 4 3 210 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OH nH Ominus minus sdot minus minus minus ++

sdot

References [1] Kreidler ER and Hummel FA (1970) The Crystal Chemistry of Apatite Structure Fields of Fluor- and Chlorapatite

American Mineralogist 55 170 [2] Legeros RZ (1984) Tooth Enamel IV In Fearnhead RW and Sugas S Eds Elsevier Amsterdam 32-36 [3] Ben Abdelkader S Khattech I Rey C and Jemal M (2001) Syntheacutese Caracteacuterisation et Thermochimie drsquoApatites

Calco-Magneacutesiennes Hydroxyleacutees et Fluoreacutees Thermochimica Acta 376 25-36 httpdxdoiorg101016S0040-6031(01)00565-2

[4] Terpstra RA and Driessens FCM (1986) Magnesium in Tooth Enamel and Synthetic Apatites Calcified Tissue In-ternational 39 348-354 httpdxdoiorg101007BF02555203

[5] Hayek E and Newsely H (1958) Uumlber die Existenz von Tricalciumphosphat in waumlBriger Loumlsung Mn Chem 89 88 [6] Rowles SL (1968) Crystallographic Study of Biological Apatites Bulletin de la Socieacuteteacute Chimique de France 1797 [7] Hamad M and Heughebaert JC (1987) Study of Apatite and Whitlockite Formation at 100˚C in the System CaO-

MgO-P2O5-H2O J Chim Phys 84 985

S B Abdelkader et al

999

32 Chemical Results The results of the chemical analysis of the samples in Weight are summarized in Table 2 This table also gives the hydroxide content of the samples calculated on the basis of the electroneutrality condition and the total mass balance sum obtained from the equation

4PO P 3 Ca P M M CO Mg OHsum = + sdot + + + (1)

With MX the atomic or ionic mass of X sum value is lower than 100 indicating that the samples of the present study still contain some water after drying at 120˚C

The results of the chemical and physical analysis (Table 2) allow us to calculate the number of each ion X per unit cell nx according to the following equation

( )3

XX P 3 CO

X 6nM P M CO M

=+

(2)

The results of these calculations are summarized in Table 3 The errors in Table 4 are estimated by the means of error propagation theory

Table 2 Chemical Composition (weight percent) and Total Mass Balance sum of the hydroxyapatites obtained by precipita-tion in solutions containing Cc (M) CO3 and CMg (mM) Mg

Sample CCM CMgmM Ca P CO3 Mg OHminus sum

Mg1 0000 000 3697 1683 099 001 319 9273

Mg2 0000 017 3709 1683 158 022 325 9371

Mg3 0000 068 3475 1683 000 082 299 9013

Mg4 0000 136 3435 1722 000 150 296 9158

Mg5 0025 000 3732 1584 445 001 315 9347

Mg6 0025 017 3673 1594 416 021 293 9287

Mg7 0025 068 3673 1633 465 084 289 9515

Mg8 0025 136 3514 1623 534 155 231 9407

Mg9 0050 000 3791 1485 822 001 315 9480

Mg10 0050 017 3762 1425 878 021 385 9413

Mg11 0050 068 3574 1475 871 080 229 9274

Mg12 0050 136 3564 1534 911 152 202 9530

Table 3 Unit cell compositions of NaCO3 Aaps calculated on the basis of the chemical composition and using Equation (2)

Sample CCM CMgmM nCa nP 3COn nMg nOH nMgnP 3COn nP

Mg1 0000 000 991 plusmn 021 582 plusmn 012 0170 plusmn 0007 000 200 008 000

Mg2 0000 017 977 plusmn 020 572 plusmn 011 0270 plusmn 0009 0090 plusmn 0003 200 013 000

Mg3 0000 068 960 plusmn 021 600 plusmn 013 000 0370 plusmn 0021 194 003 006

Mg4 0000 136 927 plusmn 018 600 plusmn 011 000 0660 plusmn 0055 188 006 012

Mg5 0025 000 957 plusmn 021 524 plusmn 011 076 plusmn 002 000 190 042 010

Mg6 0025 017 944 plusmn 020 529 plusmn 010 071 plusmn 002 0090 plusmn 0026 177 047 023

Mg7 0025 068 912 plusmn 018 523 plusmn 010 077 plusmn 002 0340 plusmn 0019 169 053 031

Mg8 0025 136 860 plusmn 018 513 plusmn 010 087 plusmn 002 0620 plusmn 0056 133 077 067

Mg9 0050 000 923 plusmn 019 466 plusmn 011 133 plusmn 003 000 180 076 020

Mg10 0050 017 892 plusmn 020 436 plusmn 009 164 plusmn 004 0080 plusmn 0025 164 099 036

Mg11 0050 068 863 plusmn 019 459 plusmn 010 140 plusmn 003 0320 plusmn 0017 130 105 070

Mg12 0050 136 827 plusmn 017 459 plusmn 009 141 plusmn 003 0580 plusmn 0049 110 115 090

S B Abdelkader et al

1000

Table 4 Multiple linear regression analysis of Yi = 3COn nP the molar ratio (Table 3) as a function of the concentration of

carbonate CcM and magnesium CMgmM in the solution (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 00156 Var(X1) = 00004 Var(X2) = 27997 Covariance Cov(X1 Y) = 00025 Cov(X2 Y) = minus00205 Cov(X1 X2) = 0

Coefficients of correlation R(X1 Y) = 0978 R(X2 Y) = minus0031 R(X1 X2) = 0 Residual mean squareStandard error s2 = 000087

R square R2 = 0958 Observations n = 12 Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 00165b = - - - - -

X1 1 5985b = ( )1 0418bσ = ( )1 1431T b = 1710minus7 5040 693

X2 2 000073b = minus ( )2 00016bσ = ( )2 0452T b = minus 0662 minus00043 00029

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 0179 00897 10254 474 Deviations 9 00078 00078

Total 11 0187 0017

4 Statistical Analysis of the Physicochemical Results 41 Influence of the Experimental Conditions on the Composition

of the Synthetic Apatites To know the influence of the experimental conditions on the incorporation of 2

3CO minus and Mg2+ in the lattice of these synthetic apatites we graph Yi =

3COn nP and nMgnP the molar ratios contents of the samples against Xi the concentration of 2

3CO minus Cc or the concentration of Mg2+ CMg in the solution Figure 3 and Figure 4 From the Figure 3(a) and Figure 4(a) it is seen that

3COn nP the molar ratio increases with the increase of the concentration of 2

3CO minus in the solution (CcM) Contrariwise it varies slightly with the concentration of the Mg2+ ions in the solution and vice versa for nMgnP (Figure 3(b) and Figure 4(b))

To estimate the simultaneous influence of the experimental conditions on 3COn nP and nMgnP the molar ratios

we construct a mathematical model of Yi = 3COn nP or nMgnP on two variables X1i = Cc and X2i = CMg

The mathematical model is described by the equation

0 1 1i 2 2iY X Xi b b b e= + + + (3)

The method of least squares (OLS) allows us to establish the predicted equation

i 0 1 1i 2 2iˆ ˆ ˆY b b X b X= + + (4)

that is most suitable to the data On the other hand this method allows us to calculate the estimated standard er-

rors of the coefficients ( ) ( )1 2ˆ ˆb and bσ σ the individual confidence interval at 95 level R2 the standardized

statistic and to test the null hypothesis H0 bj = 0 and its significances level The analysis of the variance for the linear regression or the F test allows us to ensure that at least one of the X-variables contributes to the regression The theoretical basis of these calculations is given in references [32]-[34] The calculations are summarized in Table 4 and Table 5

42 Influence of the Incorporation of 23CO minus and Mg2+ on the Variation of Ca2+ and OHminus

the Molar Ions Contents of the Synthetic Apatites In attempts to disentangle and to measure the effects of the insertion of 2

3CO minus and Mg2+ ions on the molar con-

S B Abdelkader et al

1001

(a) (b)

Figure 3 (a) 3COn nP molar ratio of the solid versus CMgmM for the samples prepared at different CcM (b) nMgnP

molar ratio of the solid versus CMgmM for the samples prepared at different CcM

(a) (b)

Figure 4 (a) 3COn nP molar ratio versus CcM for the samples prepared at different CMgmM (b) nMgnP molar ratio

versus CcM for the samples prepared at different CMgmM

Table 5 Multiple linear regression analysis of Yi = nMgnP molar ratio (Table 3) as a function of the concentration of carbo-nate CcM and magnesium CMgmM in the solution (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 00022 Var(X1) = 000042 Var(X2) = 27998 Covariance Cov(X1 Y) = 563 times 10minus5 Cov(X2 Y) = 0245 Cov(X1 X2) = 0

Coefficients of correlation R(X1 Y) = 00593 R(X2 Y) = 0995 R(X1 X2) = 0 Standard error s2 = 171 times 10minus5

R square R2 = 0994 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 00013b = minus - - - - -

X1 1 0134b = ( )1 0058bσ = ( )1 229T b = 0048 00023 0266

X2 2 00087b = ( )2 000022bσ = ( )2 3879T b = 000 00082 00092

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 00258 0012 75519 474 Deviations 9 0000154 17 times 10minus5

Total 11 00259 00024

S B Abdelkader et al

1002

tent of Ca2+ of the solid we use the multiple linear regression on two X-variables where X1 = 23nCO minus and X2 =

nMg2+ and Y is the estimate molar content of Ca2+ or OHminus (data Table 3) The results of these calculations are given in Table 6 and Table 7

Table 6 Multiple linear regression analysis of the estimated Yi = nCa2+ on X1i = nMg2+ X2i = 2

3nCO minus calcium carbonate and magnesium respectively molar contents of the solid ldquoBrdquo Mg-CO3 Haps (a) Regression statistic (c) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0059 Var(X2) = 0307 Covariance Cov(X1 Y) = minus0071 Cov(X2 Y) = minus0194 Cov(X1 X2) = minus0013

Coefficients of correlation R(X1 Y) = minus0607 R(X2 Y) = minus0722 R(X1 X2) = minus0101 Residual mean squareStandard error s2 = 00033

R square R2 = 0989 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 1009b = - - - - -

X1 1 1373b = minus ( )1 0068bσ = ( )1 2017T b = minus 110minus8 minus1527 minus1219

X2 2 0693b = minus ( )2 0030bσ = ( )2 2325T b = minus 000 minus0760 minus0625

(c)

Source of variation DF Sum of squares Mean square Fcal Ftab (5 2 9) [32]

Regression 2 279 1397 42812 474

Deviations 9 0029 00033

Total 11 2823 0257

Table 7 Multiple linear regression analysis of the estimated Yi = OHminus on X1 i = nMg2+ and X2i = 2

3nCO minus the molar con-tents of the solid ldquoBrdquo Mg-CO3 Haps (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0059 Var(X2) = 0307 Covariance Cov(X1 Y) = minus0038 Cov(X2 Y) = minus0111 Cov(X1 X2) = minus0013

Coefficients of correlation R(X1 Y) = minus0551 R(X2 Y) = minus0697 R(X1 X2) = minus0101 Residual mean squareStandard error s2 = 00137

R square R2 = 0875 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 219b = - - - - -

X1 1 0740b = minus ( )1 0139bσ = ( )1 532T b = minus 000048 minus1054 minus0426

X2 2 0393b = minus ( )2 0061bσ = ( )2 645T b = minus 000012 minus0531 minus0255

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 0862 0431 3161 474 Deviations 9 0123 00136

Total 11 0985 00896

S B Abdelkader et al

1003

43 The Determination of the Relationship between Y = ca Crystallographic Parameters Ratio and

3COn pn the Molar Ratio To estimate the influence of the incorporation of carbonate on the lattice parameters ldquoardquo and ldquocrdquo in presence of magnesium we plot ca crystallographic parameters ratio (Table 1) as a function of molar ratio

3COn nP (Table 3) for 0 le nMg le 174 mM (Figure 5)

Given that the shape of the curve obtained in Figure 5 is a polynomial we construct a multiple linear regres-sion on Yi = ca as a function of three X-variables where X1 =

3COn nP X2 = (3COn nP)2 and X3 = (

3COn nP)3 (data Table 2) and Y is the estimate ratio of the hexagonal lattice dimensions (data Table 1) The mathematical model equation is

0 1 1 2 2 3 3Y b b X b X b X= + + + + ε (5) Least square [33] allows calculating the regression and correlation coefficients regression of the predicted

equation

0 1 1 2 2 3 3ˆ ˆ ˆ ˆY b b X b X b X= + + + (6)

These estimated rgression coefficients 0 1 2 3ˆ ˆ ˆ ˆb b b and b are calculated from the values of correlation coeffici-

ents variance and covariance according to the method of Scherrer [33] This method allows us to test the utility of the model or the F-test according to

( )( )

2

2

n m 1 RF

m 1 Rminus minus sdot

=sdot minus

(7)

where n is sample size m is number of parameters and (n minus m minus 1) is degree of freedom On the other hand this method allows us to calculate the standard errors of the coefficients 0 1 2 3

ˆ ˆ ˆ ˆb b b and b and to conduct t-tests on the brsquos (to discover which variable(s) is related to estimate Y ) and to calculate the individual confidence interval at 95 level The results of these calculations are given in Table 8

The analysis of variance (ANOVA) shows that F-test = 20189 is higher than criterion F(5 3 8) = 407

Table 8 Multiple linear regression analysis of Yi= ca ratio of the lattice parameters of Mg-CO3 HAps (Table 1) on X1i = 3COn nP X2i = (

3COn nP)2 X3i = (3COn nP)3 the molar ratio Table 3 (a) Statistic regression (b) Coefficients

(a)

Variances Covariances Correlation coefficients

var(Y) 597 times 10minus6 Cov(X1 Y) 284 times 10minus4 1X YR 0931

v(X1) 156 times 10minus3 Cov(X2 Y) 908 times 10minus5 2X YR 0811

v(X2) 21 times 10minus3 Cov(X3 Y) 294 times 10minus5 3X YR 0731

v(X3) 27 times 10minus4 Cov(X1 X2) 55 times 10minus3 2 1X XR 0960

Cov(X1 X3) 186 times 10minus3 2 3X XR 0986

Cov(X2 X3) 744 times 10minus4 3 1X XR 0906

(b)

Standard error of the regression sr = 218 R square R2 = 098 Observations n = 12 Degree of freedom ν = 8

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 0730b = - - - - -

X1 1 0052b = ( ) 71 6793 10bσ minus= times ( )1 6331t b = 000 005217 005218

X2 2 0168b = minus ( ) 62 1852 10bσ minus= times ( )2 12384t b = minus 000 minus016851 minus016850

X3 3 0212b = ( ) 63 515 10bσ minus= times ( )3 9349t b = 000 021221 021223

S B Abdelkader et al

1004

Figure 5 ca parameters ratio as a function of

3COn nP the molar ratio for the apatites prepared at different values of CMg

5 Determination of the General Formula of the Unit Cell of the Synthetic ldquoBrdquo CO3Mg-HAps

The relative composition (Table 3) and the results of the physical analysis demonstrate that the samples are pure ldquoBrdquo type carbonated apatites containing Mg2+ ions Thus mechanisms I II III and V could be account in the incorporation of 2

3CO minus on 34PO minus and Mg2+ ions are incorporated in the apatite lattice according to mechan-

isms III and or IV Moreover the study carried out previously (paragraph 41) show that the 2

3CO minus ions are incorporated in the apatite lattice independently of the concentration of Mg2+ ions solution This result confirms that mechanism IV does and mechanism III does not contribute to the incorporation of Mg2+ in the apatites

Many works [27] [28] have demonstrated that mechanism I andor II are the main mechanisms for the incor-poration of CO3 Otherwise according the reference [27] the contribution of mechanism I seems to be hardly influenced by the alkali metal which is not our case Therefore we consider that mechanism II contribute to the insertion of CO3 ions in the lattice of the solid

Table 7 show that the variation of nOHminus depends on the increase of 23CO minus and Mg2+ So it may be said in

the present study that the mechanism V could account Then the fundamental substitution mechanisms for the incorporation of 2

3CO minus and Mg2+ in the HAp lattice are

2 3 Ca 24 3Ca 2PO V 2CO+ minus minus+ harr + (II)

2 2 OH2Ca 2OH Mg 2V+ minus ++ harr + 2(IV) 3 24 3PO CO OHminus minus minusharr + (V)

where VOH stands for a vacancy in the OHminus sub lattice If x y and z are the contributions of mechanisms II 2IV and V respectively thus

2nCa 10 x 2 y+ = minus minus sdot (8) 34nPO 6 2 x zminus = minus sdot minus (9)

2nMg y+ = (10) 23nCO 2 x zminus = sdot + (11)

and nOH 2 2 y zminus = minus sdot + (12)

and the generic formula has the following expression

( ) ( )( ) ( )( ) ( )( )y 4 310 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OHminus minus sdot minus minus minus ++

S B Abdelkader et al

1005

The values of x y and z the contribution of mechanisms II 2IV and V respectively are calculated from the data (Table 3) and the following equations Then statistical studies are conducted to verify the accuracy of the proposed formula The results of these calculations are summarized in Tables 9-11

Table 9 The values of x y and z the contributions of the mechanisms II 2IV and V respectively calculated from Equations (10)-(14)

Sample x y z

Mg1 0085 000 000

Mg2 0045 009 018

Mg3 minus034 037 068

Mg4 minus060 066 120

Mg5 043 000 minus01

Mg6 038 009 minus005

Mg7 020 034 037

Mg8 015 062 057

Mg9 076 000 minus020

Mg10 092 008 minus020

Mg11 073 032 minus006

Mg12 057 058 026

Table 10 Multiple linear regression analysis of the estimated Yi = nCa2+ the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = x- and X2i = y the contribution of mechanisms II and 2IV (Table 9) (a) Regression statistic (b) Coeffi-cients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0187 Var(X2) =0059

Covariance Cov(X1 Y) = minus0093 Cov(X2 Y) = minus0071 Cov(X1 X2) = minus0046

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 188 times 10minus5

R square R2 = 0999

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 999b = - - - - -

X1 1 0995b = minus ( )1 00032bσ = ( )1 30816T b = minus 000 minus1002 minus0987

X2 2 1999b = minus ( )2 00057bσ = ( )2 34779T b = minus 000 minus2012 minus1986

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32]

Regression 2 2862 1412 7414888 474

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

S B Abdelkader et al

1006

Table 11 Multiple linear regression analysis of the estimated Yi = nOHminus the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = z and X2i = z the contribution of mechanisms II and V (Table 9) (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0164 Var(X2) = 0059

Covariance Cov(X1 Y) = 00055 Cov(X2 Y) = minus0038 Cov(X1 X2) = 0079

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 274 times 10minus32

R square R2 = 100

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 200b = - - - - -

X1 1 100b = ( ) 161 1997 10bσ minus= times ( ) 15

1 500 10T b += times 000 1 1

X2 2 200b = minus ( ) 162 3338 10bσ minus= times ( ) 15

2 599 10T b += minus times 000 minus2 minus2

(c)

Source of variation DF Sum of squares Mean square Fcal

Regression 2 2862 1412 7414888

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

6 Discussion From Table 1 we can see that simultaneous incorporation of two elements ldquoCO3 and Mgrdquo results in an de-crease of the ldquoardquo parameter This contraction is attributed to the simultaneous effects of the 2

3CO minus and Mg2+ substitutions [4]

In Table 2 and Table 3 it is seen that the concentration of the 23CO minus ions in the solution Cc does not affect

the quantities of Mg2+ ions inserted in the solid Because regardless of the concentration of the 23CO minus ions in

the solution Cc the Mg2+ ions contents of the samples increase proportionally with the increase of the concentra-tion of Mg2+ ions in solution CMg This result is in agreement with reference [4] For the same concentration of Mg2+ions in the solution CMg the variation of 3

4PO minus and 23CO minus contents of the solid do not seem to be corre-

lated with the concentration of the Mg2+ ions in the solution while the Ca2+content depends on the concentra-tions of 2

3CO minus Cc and Mg2+ CMg in the solution Figure 3 and Figure 4 show that nCO3nP the molar ratio increases with the increasing of the concentration

of 23CO minus in solution (CcM) Contrariwise it varies slightly with the concentration of Mg2+ ions in solution and

vice versa for nMgnP The statistical treatment of the experimental data Table 4 and Table 5 allows us to es-tablish the estimated equations between these variables at 95 levels

( )3 cnCO nP 00165 5986 0945 C M= + plusmn sdot (13)

and ( ) ( )c MgnMg nP 00013 0135 0132 C M 00087 00005 C mM= minus + plusmn sdot + plusmn sdot (14)

Equations (9) and (10) show that the concentration of the 23CO minus ions in the hydrolysis solution Cc affects the

quantities of 23CO minus and Mg2+ ions incorporated in the solid but the concentration of Mg2+ ions in solution CMg

does not affects the quantities of 23CO minus ions in the solid These results are in agreement with those found in the

reference [27]

S B Abdelkader et al

1007

To know the relationship between the variation of Ca2+ and OHminus with the increasing of 23CO minus and Mg2+

content in the solid statistical studies are conducted The results of multiple linear regression Table 6 and Table 7 show that the estimated equations on these variables are represented at 95 level by

( ) ( )2 23

2Mg CO

nCa 1009 1373 0153 n 0698 0067 n+ minus+ = minus plusmn sdot minus plusmn sdot (15)

( ) ( )2 23Mg CO

nOH 219 0740 0313 n 0393 0138 n+ minusminus = minus plusmn minus plusmn sdot (16)

From the intercepts of the following equations it can seen that within experimental error a carbonate and magnesium-free apatite (nCO3 = 0 nMg = 0) contains 10 Ca2+ and 2 OHminus ions per unit cell Equations (11) and (12) These results are in agreement with those in literature [24]-[26] [34] [35]

As shown in Figure 5 there is a correlation of the unit cell parameters of the apatites with their chemical compositions Indeed the changes in the unit cell parameter ldquoardquo of the compounds are attributed to the additive effects of the substitution in the lattice of either carbonate and magnesium [2] [34] [35] The solid line of best fit for these series of compounds in Figure 5 extrapolates to a ratio ca very close to that in hydroxyapatite This result is similary to these obtained previously [34] [35] The application of multiple linear regression to Yi = ca

on ( )3 3

2

1i CO P 2i CO PX n n X n n= = ( )3

3

3i CO Pand X n n= allows us to establish the predicted equation at

95 level

( ) ( ) ( )

( ) ( )

26 6i CO3 P CO3 P

35CO3 P

Y c a 073 0052 156 10 n n 0168 426 10 n n

0212 118 10 n n

minus minus

minus

= = + plusmn times sdot + plusmn times sdot

+ plusmn times sdot (17)

To verify the general formula proposed We apply the multiple linear regression to Yi = nCa2+ on X1i = x and X2i = y (the contributions of the mechanisms II and IV) Similar treatment is realized for Yi = nOHminus on X1i = z and X2i = y (the contributions of the mechanisms V and IV) Table 10 and Table 11 The results of these calcu-lations show that the predicted equations at 95 level are

( ) ( )2nCa 999 0995 0007 x 1999 0013 y+ = minus plusmn minus plusmn sdot (18)

( ) ( )16 16nOH 200 100 452 10 z 200 754 10 yminus minus minus= + plusmn times sdot minus plusmn times sdot (19)

7 Conclusion The theoretical calculations of the present study indicate unambiguously that the mechanisms II III and V con-tribute to the incorporation of Mg and Ca in the lattice of apatite This corroborates in more definite way our as-sumptions obtained from the experimental data and allows us to propose for these compounds the general for-mula

( ) ( )( ) ( )( ) ( )( )y 4 3 210 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OH nH Ominus minus sdot minus minus minus ++

sdot

References [1] Kreidler ER and Hummel FA (1970) The Crystal Chemistry of Apatite Structure Fields of Fluor- and Chlorapatite

American Mineralogist 55 170 [2] Legeros RZ (1984) Tooth Enamel IV In Fearnhead RW and Sugas S Eds Elsevier Amsterdam 32-36 [3] Ben Abdelkader S Khattech I Rey C and Jemal M (2001) Syntheacutese Caracteacuterisation et Thermochimie drsquoApatites

Calco-Magneacutesiennes Hydroxyleacutees et Fluoreacutees Thermochimica Acta 376 25-36 httpdxdoiorg101016S0040-6031(01)00565-2

[4] Terpstra RA and Driessens FCM (1986) Magnesium in Tooth Enamel and Synthetic Apatites Calcified Tissue In-ternational 39 348-354 httpdxdoiorg101007BF02555203

[5] Hayek E and Newsely H (1958) Uumlber die Existenz von Tricalciumphosphat in waumlBriger Loumlsung Mn Chem 89 88 [6] Rowles SL (1968) Crystallographic Study of Biological Apatites Bulletin de la Socieacuteteacute Chimique de France 1797 [7] Hamad M and Heughebaert JC (1987) Study of Apatite and Whitlockite Formation at 100˚C in the System CaO-

MgO-P2O5-H2O J Chim Phys 84 985

S B Abdelkader et al

1000

Table 4 Multiple linear regression analysis of Yi = 3COn nP the molar ratio (Table 3) as a function of the concentration of

carbonate CcM and magnesium CMgmM in the solution (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 00156 Var(X1) = 00004 Var(X2) = 27997 Covariance Cov(X1 Y) = 00025 Cov(X2 Y) = minus00205 Cov(X1 X2) = 0

Coefficients of correlation R(X1 Y) = 0978 R(X2 Y) = minus0031 R(X1 X2) = 0 Residual mean squareStandard error s2 = 000087

R square R2 = 0958 Observations n = 12 Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 00165b = - - - - -

X1 1 5985b = ( )1 0418bσ = ( )1 1431T b = 1710minus7 5040 693

X2 2 000073b = minus ( )2 00016bσ = ( )2 0452T b = minus 0662 minus00043 00029

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 0179 00897 10254 474 Deviations 9 00078 00078

Total 11 0187 0017

4 Statistical Analysis of the Physicochemical Results 41 Influence of the Experimental Conditions on the Composition

of the Synthetic Apatites To know the influence of the experimental conditions on the incorporation of 2

3CO minus and Mg2+ in the lattice of these synthetic apatites we graph Yi =

3COn nP and nMgnP the molar ratios contents of the samples against Xi the concentration of 2

3CO minus Cc or the concentration of Mg2+ CMg in the solution Figure 3 and Figure 4 From the Figure 3(a) and Figure 4(a) it is seen that

3COn nP the molar ratio increases with the increase of the concentration of 2

3CO minus in the solution (CcM) Contrariwise it varies slightly with the concentration of the Mg2+ ions in the solution and vice versa for nMgnP (Figure 3(b) and Figure 4(b))

To estimate the simultaneous influence of the experimental conditions on 3COn nP and nMgnP the molar ratios

we construct a mathematical model of Yi = 3COn nP or nMgnP on two variables X1i = Cc and X2i = CMg

The mathematical model is described by the equation

0 1 1i 2 2iY X Xi b b b e= + + + (3)

The method of least squares (OLS) allows us to establish the predicted equation

i 0 1 1i 2 2iˆ ˆ ˆY b b X b X= + + (4)

that is most suitable to the data On the other hand this method allows us to calculate the estimated standard er-

rors of the coefficients ( ) ( )1 2ˆ ˆb and bσ σ the individual confidence interval at 95 level R2 the standardized

statistic and to test the null hypothesis H0 bj = 0 and its significances level The analysis of the variance for the linear regression or the F test allows us to ensure that at least one of the X-variables contributes to the regression The theoretical basis of these calculations is given in references [32]-[34] The calculations are summarized in Table 4 and Table 5

42 Influence of the Incorporation of 23CO minus and Mg2+ on the Variation of Ca2+ and OHminus

the Molar Ions Contents of the Synthetic Apatites In attempts to disentangle and to measure the effects of the insertion of 2

3CO minus and Mg2+ ions on the molar con-

S B Abdelkader et al

1001

(a) (b)

Figure 3 (a) 3COn nP molar ratio of the solid versus CMgmM for the samples prepared at different CcM (b) nMgnP

molar ratio of the solid versus CMgmM for the samples prepared at different CcM

(a) (b)

Figure 4 (a) 3COn nP molar ratio versus CcM for the samples prepared at different CMgmM (b) nMgnP molar ratio

versus CcM for the samples prepared at different CMgmM

Table 5 Multiple linear regression analysis of Yi = nMgnP molar ratio (Table 3) as a function of the concentration of carbo-nate CcM and magnesium CMgmM in the solution (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 00022 Var(X1) = 000042 Var(X2) = 27998 Covariance Cov(X1 Y) = 563 times 10minus5 Cov(X2 Y) = 0245 Cov(X1 X2) = 0

Coefficients of correlation R(X1 Y) = 00593 R(X2 Y) = 0995 R(X1 X2) = 0 Standard error s2 = 171 times 10minus5

R square R2 = 0994 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 00013b = minus - - - - -

X1 1 0134b = ( )1 0058bσ = ( )1 229T b = 0048 00023 0266

X2 2 00087b = ( )2 000022bσ = ( )2 3879T b = 000 00082 00092

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 00258 0012 75519 474 Deviations 9 0000154 17 times 10minus5

Total 11 00259 00024

S B Abdelkader et al

1002

tent of Ca2+ of the solid we use the multiple linear regression on two X-variables where X1 = 23nCO minus and X2 =

nMg2+ and Y is the estimate molar content of Ca2+ or OHminus (data Table 3) The results of these calculations are given in Table 6 and Table 7

Table 6 Multiple linear regression analysis of the estimated Yi = nCa2+ on X1i = nMg2+ X2i = 2

3nCO minus calcium carbonate and magnesium respectively molar contents of the solid ldquoBrdquo Mg-CO3 Haps (a) Regression statistic (c) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0059 Var(X2) = 0307 Covariance Cov(X1 Y) = minus0071 Cov(X2 Y) = minus0194 Cov(X1 X2) = minus0013

Coefficients of correlation R(X1 Y) = minus0607 R(X2 Y) = minus0722 R(X1 X2) = minus0101 Residual mean squareStandard error s2 = 00033

R square R2 = 0989 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 1009b = - - - - -

X1 1 1373b = minus ( )1 0068bσ = ( )1 2017T b = minus 110minus8 minus1527 minus1219

X2 2 0693b = minus ( )2 0030bσ = ( )2 2325T b = minus 000 minus0760 minus0625

(c)

Source of variation DF Sum of squares Mean square Fcal Ftab (5 2 9) [32]

Regression 2 279 1397 42812 474

Deviations 9 0029 00033

Total 11 2823 0257

Table 7 Multiple linear regression analysis of the estimated Yi = OHminus on X1 i = nMg2+ and X2i = 2

3nCO minus the molar con-tents of the solid ldquoBrdquo Mg-CO3 Haps (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0059 Var(X2) = 0307 Covariance Cov(X1 Y) = minus0038 Cov(X2 Y) = minus0111 Cov(X1 X2) = minus0013

Coefficients of correlation R(X1 Y) = minus0551 R(X2 Y) = minus0697 R(X1 X2) = minus0101 Residual mean squareStandard error s2 = 00137

R square R2 = 0875 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 219b = - - - - -

X1 1 0740b = minus ( )1 0139bσ = ( )1 532T b = minus 000048 minus1054 minus0426

X2 2 0393b = minus ( )2 0061bσ = ( )2 645T b = minus 000012 minus0531 minus0255

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 0862 0431 3161 474 Deviations 9 0123 00136

Total 11 0985 00896

S B Abdelkader et al

1003

43 The Determination of the Relationship between Y = ca Crystallographic Parameters Ratio and

3COn pn the Molar Ratio To estimate the influence of the incorporation of carbonate on the lattice parameters ldquoardquo and ldquocrdquo in presence of magnesium we plot ca crystallographic parameters ratio (Table 1) as a function of molar ratio

3COn nP (Table 3) for 0 le nMg le 174 mM (Figure 5)

Given that the shape of the curve obtained in Figure 5 is a polynomial we construct a multiple linear regres-sion on Yi = ca as a function of three X-variables where X1 =

3COn nP X2 = (3COn nP)2 and X3 = (

3COn nP)3 (data Table 2) and Y is the estimate ratio of the hexagonal lattice dimensions (data Table 1) The mathematical model equation is

0 1 1 2 2 3 3Y b b X b X b X= + + + + ε (5) Least square [33] allows calculating the regression and correlation coefficients regression of the predicted

equation

0 1 1 2 2 3 3ˆ ˆ ˆ ˆY b b X b X b X= + + + (6)

These estimated rgression coefficients 0 1 2 3ˆ ˆ ˆ ˆb b b and b are calculated from the values of correlation coeffici-

ents variance and covariance according to the method of Scherrer [33] This method allows us to test the utility of the model or the F-test according to

( )( )

2

2

n m 1 RF

m 1 Rminus minus sdot

=sdot minus

(7)

where n is sample size m is number of parameters and (n minus m minus 1) is degree of freedom On the other hand this method allows us to calculate the standard errors of the coefficients 0 1 2 3

ˆ ˆ ˆ ˆb b b and b and to conduct t-tests on the brsquos (to discover which variable(s) is related to estimate Y ) and to calculate the individual confidence interval at 95 level The results of these calculations are given in Table 8

The analysis of variance (ANOVA) shows that F-test = 20189 is higher than criterion F(5 3 8) = 407

Table 8 Multiple linear regression analysis of Yi= ca ratio of the lattice parameters of Mg-CO3 HAps (Table 1) on X1i = 3COn nP X2i = (

3COn nP)2 X3i = (3COn nP)3 the molar ratio Table 3 (a) Statistic regression (b) Coefficients

(a)

Variances Covariances Correlation coefficients

var(Y) 597 times 10minus6 Cov(X1 Y) 284 times 10minus4 1X YR 0931

v(X1) 156 times 10minus3 Cov(X2 Y) 908 times 10minus5 2X YR 0811

v(X2) 21 times 10minus3 Cov(X3 Y) 294 times 10minus5 3X YR 0731

v(X3) 27 times 10minus4 Cov(X1 X2) 55 times 10minus3 2 1X XR 0960

Cov(X1 X3) 186 times 10minus3 2 3X XR 0986

Cov(X2 X3) 744 times 10minus4 3 1X XR 0906

(b)

Standard error of the regression sr = 218 R square R2 = 098 Observations n = 12 Degree of freedom ν = 8

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 0730b = - - - - -

X1 1 0052b = ( ) 71 6793 10bσ minus= times ( )1 6331t b = 000 005217 005218

X2 2 0168b = minus ( ) 62 1852 10bσ minus= times ( )2 12384t b = minus 000 minus016851 minus016850

X3 3 0212b = ( ) 63 515 10bσ minus= times ( )3 9349t b = 000 021221 021223

S B Abdelkader et al

1004

Figure 5 ca parameters ratio as a function of

3COn nP the molar ratio for the apatites prepared at different values of CMg

5 Determination of the General Formula of the Unit Cell of the Synthetic ldquoBrdquo CO3Mg-HAps

The relative composition (Table 3) and the results of the physical analysis demonstrate that the samples are pure ldquoBrdquo type carbonated apatites containing Mg2+ ions Thus mechanisms I II III and V could be account in the incorporation of 2

3CO minus on 34PO minus and Mg2+ ions are incorporated in the apatite lattice according to mechan-

isms III and or IV Moreover the study carried out previously (paragraph 41) show that the 2

3CO minus ions are incorporated in the apatite lattice independently of the concentration of Mg2+ ions solution This result confirms that mechanism IV does and mechanism III does not contribute to the incorporation of Mg2+ in the apatites

Many works [27] [28] have demonstrated that mechanism I andor II are the main mechanisms for the incor-poration of CO3 Otherwise according the reference [27] the contribution of mechanism I seems to be hardly influenced by the alkali metal which is not our case Therefore we consider that mechanism II contribute to the insertion of CO3 ions in the lattice of the solid

Table 7 show that the variation of nOHminus depends on the increase of 23CO minus and Mg2+ So it may be said in

the present study that the mechanism V could account Then the fundamental substitution mechanisms for the incorporation of 2

3CO minus and Mg2+ in the HAp lattice are

2 3 Ca 24 3Ca 2PO V 2CO+ minus minus+ harr + (II)

2 2 OH2Ca 2OH Mg 2V+ minus ++ harr + 2(IV) 3 24 3PO CO OHminus minus minusharr + (V)

where VOH stands for a vacancy in the OHminus sub lattice If x y and z are the contributions of mechanisms II 2IV and V respectively thus

2nCa 10 x 2 y+ = minus minus sdot (8) 34nPO 6 2 x zminus = minus sdot minus (9)

2nMg y+ = (10) 23nCO 2 x zminus = sdot + (11)

and nOH 2 2 y zminus = minus sdot + (12)

and the generic formula has the following expression

( ) ( )( ) ( )( ) ( )( )y 4 310 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OHminus minus sdot minus minus minus ++

S B Abdelkader et al

1005

The values of x y and z the contribution of mechanisms II 2IV and V respectively are calculated from the data (Table 3) and the following equations Then statistical studies are conducted to verify the accuracy of the proposed formula The results of these calculations are summarized in Tables 9-11

Table 9 The values of x y and z the contributions of the mechanisms II 2IV and V respectively calculated from Equations (10)-(14)

Sample x y z

Mg1 0085 000 000

Mg2 0045 009 018

Mg3 minus034 037 068

Mg4 minus060 066 120

Mg5 043 000 minus01

Mg6 038 009 minus005

Mg7 020 034 037

Mg8 015 062 057

Mg9 076 000 minus020

Mg10 092 008 minus020

Mg11 073 032 minus006

Mg12 057 058 026

Table 10 Multiple linear regression analysis of the estimated Yi = nCa2+ the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = x- and X2i = y the contribution of mechanisms II and 2IV (Table 9) (a) Regression statistic (b) Coeffi-cients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0187 Var(X2) =0059

Covariance Cov(X1 Y) = minus0093 Cov(X2 Y) = minus0071 Cov(X1 X2) = minus0046

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 188 times 10minus5

R square R2 = 0999

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 999b = - - - - -

X1 1 0995b = minus ( )1 00032bσ = ( )1 30816T b = minus 000 minus1002 minus0987

X2 2 1999b = minus ( )2 00057bσ = ( )2 34779T b = minus 000 minus2012 minus1986

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32]

Regression 2 2862 1412 7414888 474

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

S B Abdelkader et al

1006

Table 11 Multiple linear regression analysis of the estimated Yi = nOHminus the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = z and X2i = z the contribution of mechanisms II and V (Table 9) (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0164 Var(X2) = 0059

Covariance Cov(X1 Y) = 00055 Cov(X2 Y) = minus0038 Cov(X1 X2) = 0079

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 274 times 10minus32

R square R2 = 100

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 200b = - - - - -

X1 1 100b = ( ) 161 1997 10bσ minus= times ( ) 15

1 500 10T b += times 000 1 1

X2 2 200b = minus ( ) 162 3338 10bσ minus= times ( ) 15

2 599 10T b += minus times 000 minus2 minus2

(c)

Source of variation DF Sum of squares Mean square Fcal

Regression 2 2862 1412 7414888

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

6 Discussion From Table 1 we can see that simultaneous incorporation of two elements ldquoCO3 and Mgrdquo results in an de-crease of the ldquoardquo parameter This contraction is attributed to the simultaneous effects of the 2

3CO minus and Mg2+ substitutions [4]

In Table 2 and Table 3 it is seen that the concentration of the 23CO minus ions in the solution Cc does not affect

the quantities of Mg2+ ions inserted in the solid Because regardless of the concentration of the 23CO minus ions in

the solution Cc the Mg2+ ions contents of the samples increase proportionally with the increase of the concentra-tion of Mg2+ ions in solution CMg This result is in agreement with reference [4] For the same concentration of Mg2+ions in the solution CMg the variation of 3

4PO minus and 23CO minus contents of the solid do not seem to be corre-

lated with the concentration of the Mg2+ ions in the solution while the Ca2+content depends on the concentra-tions of 2

3CO minus Cc and Mg2+ CMg in the solution Figure 3 and Figure 4 show that nCO3nP the molar ratio increases with the increasing of the concentration

of 23CO minus in solution (CcM) Contrariwise it varies slightly with the concentration of Mg2+ ions in solution and

vice versa for nMgnP The statistical treatment of the experimental data Table 4 and Table 5 allows us to es-tablish the estimated equations between these variables at 95 levels

( )3 cnCO nP 00165 5986 0945 C M= + plusmn sdot (13)

and ( ) ( )c MgnMg nP 00013 0135 0132 C M 00087 00005 C mM= minus + plusmn sdot + plusmn sdot (14)

Equations (9) and (10) show that the concentration of the 23CO minus ions in the hydrolysis solution Cc affects the

quantities of 23CO minus and Mg2+ ions incorporated in the solid but the concentration of Mg2+ ions in solution CMg

does not affects the quantities of 23CO minus ions in the solid These results are in agreement with those found in the

reference [27]

S B Abdelkader et al

1007

To know the relationship between the variation of Ca2+ and OHminus with the increasing of 23CO minus and Mg2+

content in the solid statistical studies are conducted The results of multiple linear regression Table 6 and Table 7 show that the estimated equations on these variables are represented at 95 level by

( ) ( )2 23

2Mg CO

nCa 1009 1373 0153 n 0698 0067 n+ minus+ = minus plusmn sdot minus plusmn sdot (15)

( ) ( )2 23Mg CO

nOH 219 0740 0313 n 0393 0138 n+ minusminus = minus plusmn minus plusmn sdot (16)

From the intercepts of the following equations it can seen that within experimental error a carbonate and magnesium-free apatite (nCO3 = 0 nMg = 0) contains 10 Ca2+ and 2 OHminus ions per unit cell Equations (11) and (12) These results are in agreement with those in literature [24]-[26] [34] [35]

As shown in Figure 5 there is a correlation of the unit cell parameters of the apatites with their chemical compositions Indeed the changes in the unit cell parameter ldquoardquo of the compounds are attributed to the additive effects of the substitution in the lattice of either carbonate and magnesium [2] [34] [35] The solid line of best fit for these series of compounds in Figure 5 extrapolates to a ratio ca very close to that in hydroxyapatite This result is similary to these obtained previously [34] [35] The application of multiple linear regression to Yi = ca

on ( )3 3

2

1i CO P 2i CO PX n n X n n= = ( )3

3

3i CO Pand X n n= allows us to establish the predicted equation at

95 level

( ) ( ) ( )

( ) ( )

26 6i CO3 P CO3 P

35CO3 P

Y c a 073 0052 156 10 n n 0168 426 10 n n

0212 118 10 n n

minus minus

minus

= = + plusmn times sdot + plusmn times sdot

+ plusmn times sdot (17)

To verify the general formula proposed We apply the multiple linear regression to Yi = nCa2+ on X1i = x and X2i = y (the contributions of the mechanisms II and IV) Similar treatment is realized for Yi = nOHminus on X1i = z and X2i = y (the contributions of the mechanisms V and IV) Table 10 and Table 11 The results of these calcu-lations show that the predicted equations at 95 level are

( ) ( )2nCa 999 0995 0007 x 1999 0013 y+ = minus plusmn minus plusmn sdot (18)

( ) ( )16 16nOH 200 100 452 10 z 200 754 10 yminus minus minus= + plusmn times sdot minus plusmn times sdot (19)

7 Conclusion The theoretical calculations of the present study indicate unambiguously that the mechanisms II III and V con-tribute to the incorporation of Mg and Ca in the lattice of apatite This corroborates in more definite way our as-sumptions obtained from the experimental data and allows us to propose for these compounds the general for-mula

( ) ( )( ) ( )( ) ( )( )y 4 3 210 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OH nH Ominus minus sdot minus minus minus ++

sdot

References [1] Kreidler ER and Hummel FA (1970) The Crystal Chemistry of Apatite Structure Fields of Fluor- and Chlorapatite

American Mineralogist 55 170 [2] Legeros RZ (1984) Tooth Enamel IV In Fearnhead RW and Sugas S Eds Elsevier Amsterdam 32-36 [3] Ben Abdelkader S Khattech I Rey C and Jemal M (2001) Syntheacutese Caracteacuterisation et Thermochimie drsquoApatites

Calco-Magneacutesiennes Hydroxyleacutees et Fluoreacutees Thermochimica Acta 376 25-36 httpdxdoiorg101016S0040-6031(01)00565-2

[4] Terpstra RA and Driessens FCM (1986) Magnesium in Tooth Enamel and Synthetic Apatites Calcified Tissue In-ternational 39 348-354 httpdxdoiorg101007BF02555203

[5] Hayek E and Newsely H (1958) Uumlber die Existenz von Tricalciumphosphat in waumlBriger Loumlsung Mn Chem 89 88 [6] Rowles SL (1968) Crystallographic Study of Biological Apatites Bulletin de la Socieacuteteacute Chimique de France 1797 [7] Hamad M and Heughebaert JC (1987) Study of Apatite and Whitlockite Formation at 100˚C in the System CaO-

MgO-P2O5-H2O J Chim Phys 84 985

S B Abdelkader et al

1001

(a) (b)

Figure 3 (a) 3COn nP molar ratio of the solid versus CMgmM for the samples prepared at different CcM (b) nMgnP

molar ratio of the solid versus CMgmM for the samples prepared at different CcM

(a) (b)

Figure 4 (a) 3COn nP molar ratio versus CcM for the samples prepared at different CMgmM (b) nMgnP molar ratio

versus CcM for the samples prepared at different CMgmM

Table 5 Multiple linear regression analysis of Yi = nMgnP molar ratio (Table 3) as a function of the concentration of carbo-nate CcM and magnesium CMgmM in the solution (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 00022 Var(X1) = 000042 Var(X2) = 27998 Covariance Cov(X1 Y) = 563 times 10minus5 Cov(X2 Y) = 0245 Cov(X1 X2) = 0

Coefficients of correlation R(X1 Y) = 00593 R(X2 Y) = 0995 R(X1 X2) = 0 Standard error s2 = 171 times 10minus5

R square R2 = 0994 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 00013b = minus - - - - -

X1 1 0134b = ( )1 0058bσ = ( )1 229T b = 0048 00023 0266

X2 2 00087b = ( )2 000022bσ = ( )2 3879T b = 000 00082 00092

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 00258 0012 75519 474 Deviations 9 0000154 17 times 10minus5

Total 11 00259 00024

S B Abdelkader et al

1002

tent of Ca2+ of the solid we use the multiple linear regression on two X-variables where X1 = 23nCO minus and X2 =

nMg2+ and Y is the estimate molar content of Ca2+ or OHminus (data Table 3) The results of these calculations are given in Table 6 and Table 7

Table 6 Multiple linear regression analysis of the estimated Yi = nCa2+ on X1i = nMg2+ X2i = 2

3nCO minus calcium carbonate and magnesium respectively molar contents of the solid ldquoBrdquo Mg-CO3 Haps (a) Regression statistic (c) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0059 Var(X2) = 0307 Covariance Cov(X1 Y) = minus0071 Cov(X2 Y) = minus0194 Cov(X1 X2) = minus0013

Coefficients of correlation R(X1 Y) = minus0607 R(X2 Y) = minus0722 R(X1 X2) = minus0101 Residual mean squareStandard error s2 = 00033

R square R2 = 0989 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 1009b = - - - - -

X1 1 1373b = minus ( )1 0068bσ = ( )1 2017T b = minus 110minus8 minus1527 minus1219

X2 2 0693b = minus ( )2 0030bσ = ( )2 2325T b = minus 000 minus0760 minus0625

(c)

Source of variation DF Sum of squares Mean square Fcal Ftab (5 2 9) [32]

Regression 2 279 1397 42812 474

Deviations 9 0029 00033

Total 11 2823 0257

Table 7 Multiple linear regression analysis of the estimated Yi = OHminus on X1 i = nMg2+ and X2i = 2

3nCO minus the molar con-tents of the solid ldquoBrdquo Mg-CO3 Haps (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0059 Var(X2) = 0307 Covariance Cov(X1 Y) = minus0038 Cov(X2 Y) = minus0111 Cov(X1 X2) = minus0013

Coefficients of correlation R(X1 Y) = minus0551 R(X2 Y) = minus0697 R(X1 X2) = minus0101 Residual mean squareStandard error s2 = 00137

R square R2 = 0875 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 219b = - - - - -

X1 1 0740b = minus ( )1 0139bσ = ( )1 532T b = minus 000048 minus1054 minus0426

X2 2 0393b = minus ( )2 0061bσ = ( )2 645T b = minus 000012 minus0531 minus0255

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 0862 0431 3161 474 Deviations 9 0123 00136

Total 11 0985 00896

S B Abdelkader et al

1003

43 The Determination of the Relationship between Y = ca Crystallographic Parameters Ratio and

3COn pn the Molar Ratio To estimate the influence of the incorporation of carbonate on the lattice parameters ldquoardquo and ldquocrdquo in presence of magnesium we plot ca crystallographic parameters ratio (Table 1) as a function of molar ratio

3COn nP (Table 3) for 0 le nMg le 174 mM (Figure 5)

Given that the shape of the curve obtained in Figure 5 is a polynomial we construct a multiple linear regres-sion on Yi = ca as a function of three X-variables where X1 =

3COn nP X2 = (3COn nP)2 and X3 = (

3COn nP)3 (data Table 2) and Y is the estimate ratio of the hexagonal lattice dimensions (data Table 1) The mathematical model equation is

0 1 1 2 2 3 3Y b b X b X b X= + + + + ε (5) Least square [33] allows calculating the regression and correlation coefficients regression of the predicted

equation

0 1 1 2 2 3 3ˆ ˆ ˆ ˆY b b X b X b X= + + + (6)

These estimated rgression coefficients 0 1 2 3ˆ ˆ ˆ ˆb b b and b are calculated from the values of correlation coeffici-

ents variance and covariance according to the method of Scherrer [33] This method allows us to test the utility of the model or the F-test according to

( )( )

2

2

n m 1 RF

m 1 Rminus minus sdot

=sdot minus

(7)

where n is sample size m is number of parameters and (n minus m minus 1) is degree of freedom On the other hand this method allows us to calculate the standard errors of the coefficients 0 1 2 3

ˆ ˆ ˆ ˆb b b and b and to conduct t-tests on the brsquos (to discover which variable(s) is related to estimate Y ) and to calculate the individual confidence interval at 95 level The results of these calculations are given in Table 8

The analysis of variance (ANOVA) shows that F-test = 20189 is higher than criterion F(5 3 8) = 407

Table 8 Multiple linear regression analysis of Yi= ca ratio of the lattice parameters of Mg-CO3 HAps (Table 1) on X1i = 3COn nP X2i = (

3COn nP)2 X3i = (3COn nP)3 the molar ratio Table 3 (a) Statistic regression (b) Coefficients

(a)

Variances Covariances Correlation coefficients

var(Y) 597 times 10minus6 Cov(X1 Y) 284 times 10minus4 1X YR 0931

v(X1) 156 times 10minus3 Cov(X2 Y) 908 times 10minus5 2X YR 0811

v(X2) 21 times 10minus3 Cov(X3 Y) 294 times 10minus5 3X YR 0731

v(X3) 27 times 10minus4 Cov(X1 X2) 55 times 10minus3 2 1X XR 0960

Cov(X1 X3) 186 times 10minus3 2 3X XR 0986

Cov(X2 X3) 744 times 10minus4 3 1X XR 0906

(b)

Standard error of the regression sr = 218 R square R2 = 098 Observations n = 12 Degree of freedom ν = 8

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 0730b = - - - - -

X1 1 0052b = ( ) 71 6793 10bσ minus= times ( )1 6331t b = 000 005217 005218

X2 2 0168b = minus ( ) 62 1852 10bσ minus= times ( )2 12384t b = minus 000 minus016851 minus016850

X3 3 0212b = ( ) 63 515 10bσ minus= times ( )3 9349t b = 000 021221 021223

S B Abdelkader et al

1004

Figure 5 ca parameters ratio as a function of

3COn nP the molar ratio for the apatites prepared at different values of CMg

5 Determination of the General Formula of the Unit Cell of the Synthetic ldquoBrdquo CO3Mg-HAps

The relative composition (Table 3) and the results of the physical analysis demonstrate that the samples are pure ldquoBrdquo type carbonated apatites containing Mg2+ ions Thus mechanisms I II III and V could be account in the incorporation of 2

3CO minus on 34PO minus and Mg2+ ions are incorporated in the apatite lattice according to mechan-

isms III and or IV Moreover the study carried out previously (paragraph 41) show that the 2

3CO minus ions are incorporated in the apatite lattice independently of the concentration of Mg2+ ions solution This result confirms that mechanism IV does and mechanism III does not contribute to the incorporation of Mg2+ in the apatites

Many works [27] [28] have demonstrated that mechanism I andor II are the main mechanisms for the incor-poration of CO3 Otherwise according the reference [27] the contribution of mechanism I seems to be hardly influenced by the alkali metal which is not our case Therefore we consider that mechanism II contribute to the insertion of CO3 ions in the lattice of the solid

Table 7 show that the variation of nOHminus depends on the increase of 23CO minus and Mg2+ So it may be said in

the present study that the mechanism V could account Then the fundamental substitution mechanisms for the incorporation of 2

3CO minus and Mg2+ in the HAp lattice are

2 3 Ca 24 3Ca 2PO V 2CO+ minus minus+ harr + (II)

2 2 OH2Ca 2OH Mg 2V+ minus ++ harr + 2(IV) 3 24 3PO CO OHminus minus minusharr + (V)

where VOH stands for a vacancy in the OHminus sub lattice If x y and z are the contributions of mechanisms II 2IV and V respectively thus

2nCa 10 x 2 y+ = minus minus sdot (8) 34nPO 6 2 x zminus = minus sdot minus (9)

2nMg y+ = (10) 23nCO 2 x zminus = sdot + (11)

and nOH 2 2 y zminus = minus sdot + (12)

and the generic formula has the following expression

( ) ( )( ) ( )( ) ( )( )y 4 310 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OHminus minus sdot minus minus minus ++

S B Abdelkader et al

1005

The values of x y and z the contribution of mechanisms II 2IV and V respectively are calculated from the data (Table 3) and the following equations Then statistical studies are conducted to verify the accuracy of the proposed formula The results of these calculations are summarized in Tables 9-11

Table 9 The values of x y and z the contributions of the mechanisms II 2IV and V respectively calculated from Equations (10)-(14)

Sample x y z

Mg1 0085 000 000

Mg2 0045 009 018

Mg3 minus034 037 068

Mg4 minus060 066 120

Mg5 043 000 minus01

Mg6 038 009 minus005

Mg7 020 034 037

Mg8 015 062 057

Mg9 076 000 minus020

Mg10 092 008 minus020

Mg11 073 032 minus006

Mg12 057 058 026

Table 10 Multiple linear regression analysis of the estimated Yi = nCa2+ the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = x- and X2i = y the contribution of mechanisms II and 2IV (Table 9) (a) Regression statistic (b) Coeffi-cients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0187 Var(X2) =0059

Covariance Cov(X1 Y) = minus0093 Cov(X2 Y) = minus0071 Cov(X1 X2) = minus0046

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 188 times 10minus5

R square R2 = 0999

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 999b = - - - - -

X1 1 0995b = minus ( )1 00032bσ = ( )1 30816T b = minus 000 minus1002 minus0987

X2 2 1999b = minus ( )2 00057bσ = ( )2 34779T b = minus 000 minus2012 minus1986

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32]

Regression 2 2862 1412 7414888 474

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

S B Abdelkader et al

1006

Table 11 Multiple linear regression analysis of the estimated Yi = nOHminus the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = z and X2i = z the contribution of mechanisms II and V (Table 9) (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0164 Var(X2) = 0059

Covariance Cov(X1 Y) = 00055 Cov(X2 Y) = minus0038 Cov(X1 X2) = 0079

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 274 times 10minus32

R square R2 = 100

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 200b = - - - - -

X1 1 100b = ( ) 161 1997 10bσ minus= times ( ) 15

1 500 10T b += times 000 1 1

X2 2 200b = minus ( ) 162 3338 10bσ minus= times ( ) 15

2 599 10T b += minus times 000 minus2 minus2

(c)

Source of variation DF Sum of squares Mean square Fcal

Regression 2 2862 1412 7414888

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

6 Discussion From Table 1 we can see that simultaneous incorporation of two elements ldquoCO3 and Mgrdquo results in an de-crease of the ldquoardquo parameter This contraction is attributed to the simultaneous effects of the 2

3CO minus and Mg2+ substitutions [4]

In Table 2 and Table 3 it is seen that the concentration of the 23CO minus ions in the solution Cc does not affect

the quantities of Mg2+ ions inserted in the solid Because regardless of the concentration of the 23CO minus ions in

the solution Cc the Mg2+ ions contents of the samples increase proportionally with the increase of the concentra-tion of Mg2+ ions in solution CMg This result is in agreement with reference [4] For the same concentration of Mg2+ions in the solution CMg the variation of 3

4PO minus and 23CO minus contents of the solid do not seem to be corre-

lated with the concentration of the Mg2+ ions in the solution while the Ca2+content depends on the concentra-tions of 2

3CO minus Cc and Mg2+ CMg in the solution Figure 3 and Figure 4 show that nCO3nP the molar ratio increases with the increasing of the concentration

of 23CO minus in solution (CcM) Contrariwise it varies slightly with the concentration of Mg2+ ions in solution and

vice versa for nMgnP The statistical treatment of the experimental data Table 4 and Table 5 allows us to es-tablish the estimated equations between these variables at 95 levels

( )3 cnCO nP 00165 5986 0945 C M= + plusmn sdot (13)

and ( ) ( )c MgnMg nP 00013 0135 0132 C M 00087 00005 C mM= minus + plusmn sdot + plusmn sdot (14)

Equations (9) and (10) show that the concentration of the 23CO minus ions in the hydrolysis solution Cc affects the

quantities of 23CO minus and Mg2+ ions incorporated in the solid but the concentration of Mg2+ ions in solution CMg

does not affects the quantities of 23CO minus ions in the solid These results are in agreement with those found in the

reference [27]

S B Abdelkader et al

1007

To know the relationship between the variation of Ca2+ and OHminus with the increasing of 23CO minus and Mg2+

content in the solid statistical studies are conducted The results of multiple linear regression Table 6 and Table 7 show that the estimated equations on these variables are represented at 95 level by

( ) ( )2 23

2Mg CO

nCa 1009 1373 0153 n 0698 0067 n+ minus+ = minus plusmn sdot minus plusmn sdot (15)

( ) ( )2 23Mg CO

nOH 219 0740 0313 n 0393 0138 n+ minusminus = minus plusmn minus plusmn sdot (16)

From the intercepts of the following equations it can seen that within experimental error a carbonate and magnesium-free apatite (nCO3 = 0 nMg = 0) contains 10 Ca2+ and 2 OHminus ions per unit cell Equations (11) and (12) These results are in agreement with those in literature [24]-[26] [34] [35]

As shown in Figure 5 there is a correlation of the unit cell parameters of the apatites with their chemical compositions Indeed the changes in the unit cell parameter ldquoardquo of the compounds are attributed to the additive effects of the substitution in the lattice of either carbonate and magnesium [2] [34] [35] The solid line of best fit for these series of compounds in Figure 5 extrapolates to a ratio ca very close to that in hydroxyapatite This result is similary to these obtained previously [34] [35] The application of multiple linear regression to Yi = ca

on ( )3 3

2

1i CO P 2i CO PX n n X n n= = ( )3

3

3i CO Pand X n n= allows us to establish the predicted equation at

95 level

( ) ( ) ( )

( ) ( )

26 6i CO3 P CO3 P

35CO3 P

Y c a 073 0052 156 10 n n 0168 426 10 n n

0212 118 10 n n

minus minus

minus

= = + plusmn times sdot + plusmn times sdot

+ plusmn times sdot (17)

To verify the general formula proposed We apply the multiple linear regression to Yi = nCa2+ on X1i = x and X2i = y (the contributions of the mechanisms II and IV) Similar treatment is realized for Yi = nOHminus on X1i = z and X2i = y (the contributions of the mechanisms V and IV) Table 10 and Table 11 The results of these calcu-lations show that the predicted equations at 95 level are

( ) ( )2nCa 999 0995 0007 x 1999 0013 y+ = minus plusmn minus plusmn sdot (18)

( ) ( )16 16nOH 200 100 452 10 z 200 754 10 yminus minus minus= + plusmn times sdot minus plusmn times sdot (19)

7 Conclusion The theoretical calculations of the present study indicate unambiguously that the mechanisms II III and V con-tribute to the incorporation of Mg and Ca in the lattice of apatite This corroborates in more definite way our as-sumptions obtained from the experimental data and allows us to propose for these compounds the general for-mula

( ) ( )( ) ( )( ) ( )( )y 4 3 210 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OH nH Ominus minus sdot minus minus minus ++

sdot

References [1] Kreidler ER and Hummel FA (1970) The Crystal Chemistry of Apatite Structure Fields of Fluor- and Chlorapatite

American Mineralogist 55 170 [2] Legeros RZ (1984) Tooth Enamel IV In Fearnhead RW and Sugas S Eds Elsevier Amsterdam 32-36 [3] Ben Abdelkader S Khattech I Rey C and Jemal M (2001) Syntheacutese Caracteacuterisation et Thermochimie drsquoApatites

Calco-Magneacutesiennes Hydroxyleacutees et Fluoreacutees Thermochimica Acta 376 25-36 httpdxdoiorg101016S0040-6031(01)00565-2

[4] Terpstra RA and Driessens FCM (1986) Magnesium in Tooth Enamel and Synthetic Apatites Calcified Tissue In-ternational 39 348-354 httpdxdoiorg101007BF02555203

[5] Hayek E and Newsely H (1958) Uumlber die Existenz von Tricalciumphosphat in waumlBriger Loumlsung Mn Chem 89 88 [6] Rowles SL (1968) Crystallographic Study of Biological Apatites Bulletin de la Socieacuteteacute Chimique de France 1797 [7] Hamad M and Heughebaert JC (1987) Study of Apatite and Whitlockite Formation at 100˚C in the System CaO-

MgO-P2O5-H2O J Chim Phys 84 985

S B Abdelkader et al

1002

tent of Ca2+ of the solid we use the multiple linear regression on two X-variables where X1 = 23nCO minus and X2 =

nMg2+ and Y is the estimate molar content of Ca2+ or OHminus (data Table 3) The results of these calculations are given in Table 6 and Table 7

Table 6 Multiple linear regression analysis of the estimated Yi = nCa2+ on X1i = nMg2+ X2i = 2

3nCO minus calcium carbonate and magnesium respectively molar contents of the solid ldquoBrdquo Mg-CO3 Haps (a) Regression statistic (c) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0059 Var(X2) = 0307 Covariance Cov(X1 Y) = minus0071 Cov(X2 Y) = minus0194 Cov(X1 X2) = minus0013

Coefficients of correlation R(X1 Y) = minus0607 R(X2 Y) = minus0722 R(X1 X2) = minus0101 Residual mean squareStandard error s2 = 00033

R square R2 = 0989 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 1009b = - - - - -

X1 1 1373b = minus ( )1 0068bσ = ( )1 2017T b = minus 110minus8 minus1527 minus1219

X2 2 0693b = minus ( )2 0030bσ = ( )2 2325T b = minus 000 minus0760 minus0625

(c)

Source of variation DF Sum of squares Mean square Fcal Ftab (5 2 9) [32]

Regression 2 279 1397 42812 474

Deviations 9 0029 00033

Total 11 2823 0257

Table 7 Multiple linear regression analysis of the estimated Yi = OHminus on X1 i = nMg2+ and X2i = 2

3nCO minus the molar con-tents of the solid ldquoBrdquo Mg-CO3 Haps (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0059 Var(X2) = 0307 Covariance Cov(X1 Y) = minus0038 Cov(X2 Y) = minus0111 Cov(X1 X2) = minus0013

Coefficients of correlation R(X1 Y) = minus0551 R(X2 Y) = minus0697 R(X1 X2) = minus0101 Residual mean squareStandard error s2 = 00137

R square R2 = 0875 Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 219b = - - - - -

X1 1 0740b = minus ( )1 0139bσ = ( )1 532T b = minus 000048 minus1054 minus0426

X2 2 0393b = minus ( )2 0061bσ = ( )2 645T b = minus 000012 minus0531 minus0255

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32] Regression 2 0862 0431 3161 474 Deviations 9 0123 00136

Total 11 0985 00896

S B Abdelkader et al

1003

43 The Determination of the Relationship between Y = ca Crystallographic Parameters Ratio and

3COn pn the Molar Ratio To estimate the influence of the incorporation of carbonate on the lattice parameters ldquoardquo and ldquocrdquo in presence of magnesium we plot ca crystallographic parameters ratio (Table 1) as a function of molar ratio

3COn nP (Table 3) for 0 le nMg le 174 mM (Figure 5)

Given that the shape of the curve obtained in Figure 5 is a polynomial we construct a multiple linear regres-sion on Yi = ca as a function of three X-variables where X1 =

3COn nP X2 = (3COn nP)2 and X3 = (

3COn nP)3 (data Table 2) and Y is the estimate ratio of the hexagonal lattice dimensions (data Table 1) The mathematical model equation is

0 1 1 2 2 3 3Y b b X b X b X= + + + + ε (5) Least square [33] allows calculating the regression and correlation coefficients regression of the predicted

equation

0 1 1 2 2 3 3ˆ ˆ ˆ ˆY b b X b X b X= + + + (6)

These estimated rgression coefficients 0 1 2 3ˆ ˆ ˆ ˆb b b and b are calculated from the values of correlation coeffici-

ents variance and covariance according to the method of Scherrer [33] This method allows us to test the utility of the model or the F-test according to

( )( )

2

2

n m 1 RF

m 1 Rminus minus sdot

=sdot minus

(7)

where n is sample size m is number of parameters and (n minus m minus 1) is degree of freedom On the other hand this method allows us to calculate the standard errors of the coefficients 0 1 2 3

ˆ ˆ ˆ ˆb b b and b and to conduct t-tests on the brsquos (to discover which variable(s) is related to estimate Y ) and to calculate the individual confidence interval at 95 level The results of these calculations are given in Table 8

The analysis of variance (ANOVA) shows that F-test = 20189 is higher than criterion F(5 3 8) = 407

Table 8 Multiple linear regression analysis of Yi= ca ratio of the lattice parameters of Mg-CO3 HAps (Table 1) on X1i = 3COn nP X2i = (

3COn nP)2 X3i = (3COn nP)3 the molar ratio Table 3 (a) Statistic regression (b) Coefficients

(a)

Variances Covariances Correlation coefficients

var(Y) 597 times 10minus6 Cov(X1 Y) 284 times 10minus4 1X YR 0931

v(X1) 156 times 10minus3 Cov(X2 Y) 908 times 10minus5 2X YR 0811

v(X2) 21 times 10minus3 Cov(X3 Y) 294 times 10minus5 3X YR 0731

v(X3) 27 times 10minus4 Cov(X1 X2) 55 times 10minus3 2 1X XR 0960

Cov(X1 X3) 186 times 10minus3 2 3X XR 0986

Cov(X2 X3) 744 times 10minus4 3 1X XR 0906

(b)

Standard error of the regression sr = 218 R square R2 = 098 Observations n = 12 Degree of freedom ν = 8

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 0730b = - - - - -

X1 1 0052b = ( ) 71 6793 10bσ minus= times ( )1 6331t b = 000 005217 005218

X2 2 0168b = minus ( ) 62 1852 10bσ minus= times ( )2 12384t b = minus 000 minus016851 minus016850

X3 3 0212b = ( ) 63 515 10bσ minus= times ( )3 9349t b = 000 021221 021223

S B Abdelkader et al

1004

Figure 5 ca parameters ratio as a function of

3COn nP the molar ratio for the apatites prepared at different values of CMg

5 Determination of the General Formula of the Unit Cell of the Synthetic ldquoBrdquo CO3Mg-HAps

The relative composition (Table 3) and the results of the physical analysis demonstrate that the samples are pure ldquoBrdquo type carbonated apatites containing Mg2+ ions Thus mechanisms I II III and V could be account in the incorporation of 2

3CO minus on 34PO minus and Mg2+ ions are incorporated in the apatite lattice according to mechan-

isms III and or IV Moreover the study carried out previously (paragraph 41) show that the 2

3CO minus ions are incorporated in the apatite lattice independently of the concentration of Mg2+ ions solution This result confirms that mechanism IV does and mechanism III does not contribute to the incorporation of Mg2+ in the apatites

Many works [27] [28] have demonstrated that mechanism I andor II are the main mechanisms for the incor-poration of CO3 Otherwise according the reference [27] the contribution of mechanism I seems to be hardly influenced by the alkali metal which is not our case Therefore we consider that mechanism II contribute to the insertion of CO3 ions in the lattice of the solid

Table 7 show that the variation of nOHminus depends on the increase of 23CO minus and Mg2+ So it may be said in

the present study that the mechanism V could account Then the fundamental substitution mechanisms for the incorporation of 2

3CO minus and Mg2+ in the HAp lattice are

2 3 Ca 24 3Ca 2PO V 2CO+ minus minus+ harr + (II)

2 2 OH2Ca 2OH Mg 2V+ minus ++ harr + 2(IV) 3 24 3PO CO OHminus minus minusharr + (V)

where VOH stands for a vacancy in the OHminus sub lattice If x y and z are the contributions of mechanisms II 2IV and V respectively thus

2nCa 10 x 2 y+ = minus minus sdot (8) 34nPO 6 2 x zminus = minus sdot minus (9)

2nMg y+ = (10) 23nCO 2 x zminus = sdot + (11)

and nOH 2 2 y zminus = minus sdot + (12)

and the generic formula has the following expression

( ) ( )( ) ( )( ) ( )( )y 4 310 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OHminus minus sdot minus minus minus ++

S B Abdelkader et al

1005

The values of x y and z the contribution of mechanisms II 2IV and V respectively are calculated from the data (Table 3) and the following equations Then statistical studies are conducted to verify the accuracy of the proposed formula The results of these calculations are summarized in Tables 9-11

Table 9 The values of x y and z the contributions of the mechanisms II 2IV and V respectively calculated from Equations (10)-(14)

Sample x y z

Mg1 0085 000 000

Mg2 0045 009 018

Mg3 minus034 037 068

Mg4 minus060 066 120

Mg5 043 000 minus01

Mg6 038 009 minus005

Mg7 020 034 037

Mg8 015 062 057

Mg9 076 000 minus020

Mg10 092 008 minus020

Mg11 073 032 minus006

Mg12 057 058 026

Table 10 Multiple linear regression analysis of the estimated Yi = nCa2+ the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = x- and X2i = y the contribution of mechanisms II and 2IV (Table 9) (a) Regression statistic (b) Coeffi-cients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0187 Var(X2) =0059

Covariance Cov(X1 Y) = minus0093 Cov(X2 Y) = minus0071 Cov(X1 X2) = minus0046

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 188 times 10minus5

R square R2 = 0999

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 999b = - - - - -

X1 1 0995b = minus ( )1 00032bσ = ( )1 30816T b = minus 000 minus1002 minus0987

X2 2 1999b = minus ( )2 00057bσ = ( )2 34779T b = minus 000 minus2012 minus1986

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32]

Regression 2 2862 1412 7414888 474

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

S B Abdelkader et al

1006

Table 11 Multiple linear regression analysis of the estimated Yi = nOHminus the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = z and X2i = z the contribution of mechanisms II and V (Table 9) (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0164 Var(X2) = 0059

Covariance Cov(X1 Y) = 00055 Cov(X2 Y) = minus0038 Cov(X1 X2) = 0079

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 274 times 10minus32

R square R2 = 100

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 200b = - - - - -

X1 1 100b = ( ) 161 1997 10bσ minus= times ( ) 15

1 500 10T b += times 000 1 1

X2 2 200b = minus ( ) 162 3338 10bσ minus= times ( ) 15

2 599 10T b += minus times 000 minus2 minus2

(c)

Source of variation DF Sum of squares Mean square Fcal

Regression 2 2862 1412 7414888

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

6 Discussion From Table 1 we can see that simultaneous incorporation of two elements ldquoCO3 and Mgrdquo results in an de-crease of the ldquoardquo parameter This contraction is attributed to the simultaneous effects of the 2

3CO minus and Mg2+ substitutions [4]

In Table 2 and Table 3 it is seen that the concentration of the 23CO minus ions in the solution Cc does not affect

the quantities of Mg2+ ions inserted in the solid Because regardless of the concentration of the 23CO minus ions in

the solution Cc the Mg2+ ions contents of the samples increase proportionally with the increase of the concentra-tion of Mg2+ ions in solution CMg This result is in agreement with reference [4] For the same concentration of Mg2+ions in the solution CMg the variation of 3

4PO minus and 23CO minus contents of the solid do not seem to be corre-

lated with the concentration of the Mg2+ ions in the solution while the Ca2+content depends on the concentra-tions of 2

3CO minus Cc and Mg2+ CMg in the solution Figure 3 and Figure 4 show that nCO3nP the molar ratio increases with the increasing of the concentration

of 23CO minus in solution (CcM) Contrariwise it varies slightly with the concentration of Mg2+ ions in solution and

vice versa for nMgnP The statistical treatment of the experimental data Table 4 and Table 5 allows us to es-tablish the estimated equations between these variables at 95 levels

( )3 cnCO nP 00165 5986 0945 C M= + plusmn sdot (13)

and ( ) ( )c MgnMg nP 00013 0135 0132 C M 00087 00005 C mM= minus + plusmn sdot + plusmn sdot (14)

Equations (9) and (10) show that the concentration of the 23CO minus ions in the hydrolysis solution Cc affects the

quantities of 23CO minus and Mg2+ ions incorporated in the solid but the concentration of Mg2+ ions in solution CMg

does not affects the quantities of 23CO minus ions in the solid These results are in agreement with those found in the

reference [27]

S B Abdelkader et al

1007

To know the relationship between the variation of Ca2+ and OHminus with the increasing of 23CO minus and Mg2+

content in the solid statistical studies are conducted The results of multiple linear regression Table 6 and Table 7 show that the estimated equations on these variables are represented at 95 level by

( ) ( )2 23

2Mg CO

nCa 1009 1373 0153 n 0698 0067 n+ minus+ = minus plusmn sdot minus plusmn sdot (15)

( ) ( )2 23Mg CO

nOH 219 0740 0313 n 0393 0138 n+ minusminus = minus plusmn minus plusmn sdot (16)

From the intercepts of the following equations it can seen that within experimental error a carbonate and magnesium-free apatite (nCO3 = 0 nMg = 0) contains 10 Ca2+ and 2 OHminus ions per unit cell Equations (11) and (12) These results are in agreement with those in literature [24]-[26] [34] [35]

As shown in Figure 5 there is a correlation of the unit cell parameters of the apatites with their chemical compositions Indeed the changes in the unit cell parameter ldquoardquo of the compounds are attributed to the additive effects of the substitution in the lattice of either carbonate and magnesium [2] [34] [35] The solid line of best fit for these series of compounds in Figure 5 extrapolates to a ratio ca very close to that in hydroxyapatite This result is similary to these obtained previously [34] [35] The application of multiple linear regression to Yi = ca

on ( )3 3

2

1i CO P 2i CO PX n n X n n= = ( )3

3

3i CO Pand X n n= allows us to establish the predicted equation at

95 level

( ) ( ) ( )

( ) ( )

26 6i CO3 P CO3 P

35CO3 P

Y c a 073 0052 156 10 n n 0168 426 10 n n

0212 118 10 n n

minus minus

minus

= = + plusmn times sdot + plusmn times sdot

+ plusmn times sdot (17)

To verify the general formula proposed We apply the multiple linear regression to Yi = nCa2+ on X1i = x and X2i = y (the contributions of the mechanisms II and IV) Similar treatment is realized for Yi = nOHminus on X1i = z and X2i = y (the contributions of the mechanisms V and IV) Table 10 and Table 11 The results of these calcu-lations show that the predicted equations at 95 level are

( ) ( )2nCa 999 0995 0007 x 1999 0013 y+ = minus plusmn minus plusmn sdot (18)

( ) ( )16 16nOH 200 100 452 10 z 200 754 10 yminus minus minus= + plusmn times sdot minus plusmn times sdot (19)

7 Conclusion The theoretical calculations of the present study indicate unambiguously that the mechanisms II III and V con-tribute to the incorporation of Mg and Ca in the lattice of apatite This corroborates in more definite way our as-sumptions obtained from the experimental data and allows us to propose for these compounds the general for-mula

( ) ( )( ) ( )( ) ( )( )y 4 3 210 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OH nH Ominus minus sdot minus minus minus ++

sdot

References [1] Kreidler ER and Hummel FA (1970) The Crystal Chemistry of Apatite Structure Fields of Fluor- and Chlorapatite

American Mineralogist 55 170 [2] Legeros RZ (1984) Tooth Enamel IV In Fearnhead RW and Sugas S Eds Elsevier Amsterdam 32-36 [3] Ben Abdelkader S Khattech I Rey C and Jemal M (2001) Syntheacutese Caracteacuterisation et Thermochimie drsquoApatites

Calco-Magneacutesiennes Hydroxyleacutees et Fluoreacutees Thermochimica Acta 376 25-36 httpdxdoiorg101016S0040-6031(01)00565-2

[4] Terpstra RA and Driessens FCM (1986) Magnesium in Tooth Enamel and Synthetic Apatites Calcified Tissue In-ternational 39 348-354 httpdxdoiorg101007BF02555203

[5] Hayek E and Newsely H (1958) Uumlber die Existenz von Tricalciumphosphat in waumlBriger Loumlsung Mn Chem 89 88 [6] Rowles SL (1968) Crystallographic Study of Biological Apatites Bulletin de la Socieacuteteacute Chimique de France 1797 [7] Hamad M and Heughebaert JC (1987) Study of Apatite and Whitlockite Formation at 100˚C in the System CaO-

MgO-P2O5-H2O J Chim Phys 84 985

S B Abdelkader et al

1003

43 The Determination of the Relationship between Y = ca Crystallographic Parameters Ratio and

3COn pn the Molar Ratio To estimate the influence of the incorporation of carbonate on the lattice parameters ldquoardquo and ldquocrdquo in presence of magnesium we plot ca crystallographic parameters ratio (Table 1) as a function of molar ratio

3COn nP (Table 3) for 0 le nMg le 174 mM (Figure 5)

Given that the shape of the curve obtained in Figure 5 is a polynomial we construct a multiple linear regres-sion on Yi = ca as a function of three X-variables where X1 =

3COn nP X2 = (3COn nP)2 and X3 = (

3COn nP)3 (data Table 2) and Y is the estimate ratio of the hexagonal lattice dimensions (data Table 1) The mathematical model equation is

0 1 1 2 2 3 3Y b b X b X b X= + + + + ε (5) Least square [33] allows calculating the regression and correlation coefficients regression of the predicted

equation

0 1 1 2 2 3 3ˆ ˆ ˆ ˆY b b X b X b X= + + + (6)

These estimated rgression coefficients 0 1 2 3ˆ ˆ ˆ ˆb b b and b are calculated from the values of correlation coeffici-

ents variance and covariance according to the method of Scherrer [33] This method allows us to test the utility of the model or the F-test according to

( )( )

2

2

n m 1 RF

m 1 Rminus minus sdot

=sdot minus

(7)

where n is sample size m is number of parameters and (n minus m minus 1) is degree of freedom On the other hand this method allows us to calculate the standard errors of the coefficients 0 1 2 3

ˆ ˆ ˆ ˆb b b and b and to conduct t-tests on the brsquos (to discover which variable(s) is related to estimate Y ) and to calculate the individual confidence interval at 95 level The results of these calculations are given in Table 8

The analysis of variance (ANOVA) shows that F-test = 20189 is higher than criterion F(5 3 8) = 407

Table 8 Multiple linear regression analysis of Yi= ca ratio of the lattice parameters of Mg-CO3 HAps (Table 1) on X1i = 3COn nP X2i = (

3COn nP)2 X3i = (3COn nP)3 the molar ratio Table 3 (a) Statistic regression (b) Coefficients

(a)

Variances Covariances Correlation coefficients

var(Y) 597 times 10minus6 Cov(X1 Y) 284 times 10minus4 1X YR 0931

v(X1) 156 times 10minus3 Cov(X2 Y) 908 times 10minus5 2X YR 0811

v(X2) 21 times 10minus3 Cov(X3 Y) 294 times 10minus5 3X YR 0731

v(X3) 27 times 10minus4 Cov(X1 X2) 55 times 10minus3 2 1X XR 0960

Cov(X1 X3) 186 times 10minus3 2 3X XR 0986

Cov(X2 X3) 744 times 10minus4 3 1X XR 0906

(b)

Standard error of the regression sr = 218 R square R2 = 098 Observations n = 12 Degree of freedom ν = 8

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 0730b = - - - - -

X1 1 0052b = ( ) 71 6793 10bσ minus= times ( )1 6331t b = 000 005217 005218

X2 2 0168b = minus ( ) 62 1852 10bσ minus= times ( )2 12384t b = minus 000 minus016851 minus016850

X3 3 0212b = ( ) 63 515 10bσ minus= times ( )3 9349t b = 000 021221 021223

S B Abdelkader et al

1004

Figure 5 ca parameters ratio as a function of

3COn nP the molar ratio for the apatites prepared at different values of CMg

5 Determination of the General Formula of the Unit Cell of the Synthetic ldquoBrdquo CO3Mg-HAps

The relative composition (Table 3) and the results of the physical analysis demonstrate that the samples are pure ldquoBrdquo type carbonated apatites containing Mg2+ ions Thus mechanisms I II III and V could be account in the incorporation of 2

3CO minus on 34PO minus and Mg2+ ions are incorporated in the apatite lattice according to mechan-

isms III and or IV Moreover the study carried out previously (paragraph 41) show that the 2

3CO minus ions are incorporated in the apatite lattice independently of the concentration of Mg2+ ions solution This result confirms that mechanism IV does and mechanism III does not contribute to the incorporation of Mg2+ in the apatites

Many works [27] [28] have demonstrated that mechanism I andor II are the main mechanisms for the incor-poration of CO3 Otherwise according the reference [27] the contribution of mechanism I seems to be hardly influenced by the alkali metal which is not our case Therefore we consider that mechanism II contribute to the insertion of CO3 ions in the lattice of the solid

Table 7 show that the variation of nOHminus depends on the increase of 23CO minus and Mg2+ So it may be said in

the present study that the mechanism V could account Then the fundamental substitution mechanisms for the incorporation of 2

3CO minus and Mg2+ in the HAp lattice are

2 3 Ca 24 3Ca 2PO V 2CO+ minus minus+ harr + (II)

2 2 OH2Ca 2OH Mg 2V+ minus ++ harr + 2(IV) 3 24 3PO CO OHminus minus minusharr + (V)

where VOH stands for a vacancy in the OHminus sub lattice If x y and z are the contributions of mechanisms II 2IV and V respectively thus

2nCa 10 x 2 y+ = minus minus sdot (8) 34nPO 6 2 x zminus = minus sdot minus (9)

2nMg y+ = (10) 23nCO 2 x zminus = sdot + (11)

and nOH 2 2 y zminus = minus sdot + (12)

and the generic formula has the following expression

( ) ( )( ) ( )( ) ( )( )y 4 310 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OHminus minus sdot minus minus minus ++

S B Abdelkader et al

1005

The values of x y and z the contribution of mechanisms II 2IV and V respectively are calculated from the data (Table 3) and the following equations Then statistical studies are conducted to verify the accuracy of the proposed formula The results of these calculations are summarized in Tables 9-11

Table 9 The values of x y and z the contributions of the mechanisms II 2IV and V respectively calculated from Equations (10)-(14)

Sample x y z

Mg1 0085 000 000

Mg2 0045 009 018

Mg3 minus034 037 068

Mg4 minus060 066 120

Mg5 043 000 minus01

Mg6 038 009 minus005

Mg7 020 034 037

Mg8 015 062 057

Mg9 076 000 minus020

Mg10 092 008 minus020

Mg11 073 032 minus006

Mg12 057 058 026

Table 10 Multiple linear regression analysis of the estimated Yi = nCa2+ the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = x- and X2i = y the contribution of mechanisms II and 2IV (Table 9) (a) Regression statistic (b) Coeffi-cients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0187 Var(X2) =0059

Covariance Cov(X1 Y) = minus0093 Cov(X2 Y) = minus0071 Cov(X1 X2) = minus0046

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 188 times 10minus5

R square R2 = 0999

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 999b = - - - - -

X1 1 0995b = minus ( )1 00032bσ = ( )1 30816T b = minus 000 minus1002 minus0987

X2 2 1999b = minus ( )2 00057bσ = ( )2 34779T b = minus 000 minus2012 minus1986

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32]

Regression 2 2862 1412 7414888 474

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

S B Abdelkader et al

1006

Table 11 Multiple linear regression analysis of the estimated Yi = nOHminus the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = z and X2i = z the contribution of mechanisms II and V (Table 9) (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0164 Var(X2) = 0059

Covariance Cov(X1 Y) = 00055 Cov(X2 Y) = minus0038 Cov(X1 X2) = 0079

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 274 times 10minus32

R square R2 = 100

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 200b = - - - - -

X1 1 100b = ( ) 161 1997 10bσ minus= times ( ) 15

1 500 10T b += times 000 1 1

X2 2 200b = minus ( ) 162 3338 10bσ minus= times ( ) 15

2 599 10T b += minus times 000 minus2 minus2

(c)

Source of variation DF Sum of squares Mean square Fcal

Regression 2 2862 1412 7414888

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

6 Discussion From Table 1 we can see that simultaneous incorporation of two elements ldquoCO3 and Mgrdquo results in an de-crease of the ldquoardquo parameter This contraction is attributed to the simultaneous effects of the 2

3CO minus and Mg2+ substitutions [4]

In Table 2 and Table 3 it is seen that the concentration of the 23CO minus ions in the solution Cc does not affect

the quantities of Mg2+ ions inserted in the solid Because regardless of the concentration of the 23CO minus ions in

the solution Cc the Mg2+ ions contents of the samples increase proportionally with the increase of the concentra-tion of Mg2+ ions in solution CMg This result is in agreement with reference [4] For the same concentration of Mg2+ions in the solution CMg the variation of 3

4PO minus and 23CO minus contents of the solid do not seem to be corre-

lated with the concentration of the Mg2+ ions in the solution while the Ca2+content depends on the concentra-tions of 2

3CO minus Cc and Mg2+ CMg in the solution Figure 3 and Figure 4 show that nCO3nP the molar ratio increases with the increasing of the concentration

of 23CO minus in solution (CcM) Contrariwise it varies slightly with the concentration of Mg2+ ions in solution and

vice versa for nMgnP The statistical treatment of the experimental data Table 4 and Table 5 allows us to es-tablish the estimated equations between these variables at 95 levels

( )3 cnCO nP 00165 5986 0945 C M= + plusmn sdot (13)

and ( ) ( )c MgnMg nP 00013 0135 0132 C M 00087 00005 C mM= minus + plusmn sdot + plusmn sdot (14)

Equations (9) and (10) show that the concentration of the 23CO minus ions in the hydrolysis solution Cc affects the

quantities of 23CO minus and Mg2+ ions incorporated in the solid but the concentration of Mg2+ ions in solution CMg

does not affects the quantities of 23CO minus ions in the solid These results are in agreement with those found in the

reference [27]

S B Abdelkader et al

1007

To know the relationship between the variation of Ca2+ and OHminus with the increasing of 23CO minus and Mg2+

content in the solid statistical studies are conducted The results of multiple linear regression Table 6 and Table 7 show that the estimated equations on these variables are represented at 95 level by

( ) ( )2 23

2Mg CO

nCa 1009 1373 0153 n 0698 0067 n+ minus+ = minus plusmn sdot minus plusmn sdot (15)

( ) ( )2 23Mg CO

nOH 219 0740 0313 n 0393 0138 n+ minusminus = minus plusmn minus plusmn sdot (16)

From the intercepts of the following equations it can seen that within experimental error a carbonate and magnesium-free apatite (nCO3 = 0 nMg = 0) contains 10 Ca2+ and 2 OHminus ions per unit cell Equations (11) and (12) These results are in agreement with those in literature [24]-[26] [34] [35]

As shown in Figure 5 there is a correlation of the unit cell parameters of the apatites with their chemical compositions Indeed the changes in the unit cell parameter ldquoardquo of the compounds are attributed to the additive effects of the substitution in the lattice of either carbonate and magnesium [2] [34] [35] The solid line of best fit for these series of compounds in Figure 5 extrapolates to a ratio ca very close to that in hydroxyapatite This result is similary to these obtained previously [34] [35] The application of multiple linear regression to Yi = ca

on ( )3 3

2

1i CO P 2i CO PX n n X n n= = ( )3

3

3i CO Pand X n n= allows us to establish the predicted equation at

95 level

( ) ( ) ( )

( ) ( )

26 6i CO3 P CO3 P

35CO3 P

Y c a 073 0052 156 10 n n 0168 426 10 n n

0212 118 10 n n

minus minus

minus

= = + plusmn times sdot + plusmn times sdot

+ plusmn times sdot (17)

To verify the general formula proposed We apply the multiple linear regression to Yi = nCa2+ on X1i = x and X2i = y (the contributions of the mechanisms II and IV) Similar treatment is realized for Yi = nOHminus on X1i = z and X2i = y (the contributions of the mechanisms V and IV) Table 10 and Table 11 The results of these calcu-lations show that the predicted equations at 95 level are

( ) ( )2nCa 999 0995 0007 x 1999 0013 y+ = minus plusmn minus plusmn sdot (18)

( ) ( )16 16nOH 200 100 452 10 z 200 754 10 yminus minus minus= + plusmn times sdot minus plusmn times sdot (19)

7 Conclusion The theoretical calculations of the present study indicate unambiguously that the mechanisms II III and V con-tribute to the incorporation of Mg and Ca in the lattice of apatite This corroborates in more definite way our as-sumptions obtained from the experimental data and allows us to propose for these compounds the general for-mula

( ) ( )( ) ( )( ) ( )( )y 4 3 210 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OH nH Ominus minus sdot minus minus minus ++

sdot

References [1] Kreidler ER and Hummel FA (1970) The Crystal Chemistry of Apatite Structure Fields of Fluor- and Chlorapatite

American Mineralogist 55 170 [2] Legeros RZ (1984) Tooth Enamel IV In Fearnhead RW and Sugas S Eds Elsevier Amsterdam 32-36 [3] Ben Abdelkader S Khattech I Rey C and Jemal M (2001) Syntheacutese Caracteacuterisation et Thermochimie drsquoApatites

Calco-Magneacutesiennes Hydroxyleacutees et Fluoreacutees Thermochimica Acta 376 25-36 httpdxdoiorg101016S0040-6031(01)00565-2

[4] Terpstra RA and Driessens FCM (1986) Magnesium in Tooth Enamel and Synthetic Apatites Calcified Tissue In-ternational 39 348-354 httpdxdoiorg101007BF02555203

[5] Hayek E and Newsely H (1958) Uumlber die Existenz von Tricalciumphosphat in waumlBriger Loumlsung Mn Chem 89 88 [6] Rowles SL (1968) Crystallographic Study of Biological Apatites Bulletin de la Socieacuteteacute Chimique de France 1797 [7] Hamad M and Heughebaert JC (1987) Study of Apatite and Whitlockite Formation at 100˚C in the System CaO-

MgO-P2O5-H2O J Chim Phys 84 985

S B Abdelkader et al

1004

Figure 5 ca parameters ratio as a function of

3COn nP the molar ratio for the apatites prepared at different values of CMg

5 Determination of the General Formula of the Unit Cell of the Synthetic ldquoBrdquo CO3Mg-HAps

The relative composition (Table 3) and the results of the physical analysis demonstrate that the samples are pure ldquoBrdquo type carbonated apatites containing Mg2+ ions Thus mechanisms I II III and V could be account in the incorporation of 2

3CO minus on 34PO minus and Mg2+ ions are incorporated in the apatite lattice according to mechan-

isms III and or IV Moreover the study carried out previously (paragraph 41) show that the 2

3CO minus ions are incorporated in the apatite lattice independently of the concentration of Mg2+ ions solution This result confirms that mechanism IV does and mechanism III does not contribute to the incorporation of Mg2+ in the apatites

Many works [27] [28] have demonstrated that mechanism I andor II are the main mechanisms for the incor-poration of CO3 Otherwise according the reference [27] the contribution of mechanism I seems to be hardly influenced by the alkali metal which is not our case Therefore we consider that mechanism II contribute to the insertion of CO3 ions in the lattice of the solid

Table 7 show that the variation of nOHminus depends on the increase of 23CO minus and Mg2+ So it may be said in

the present study that the mechanism V could account Then the fundamental substitution mechanisms for the incorporation of 2

3CO minus and Mg2+ in the HAp lattice are

2 3 Ca 24 3Ca 2PO V 2CO+ minus minus+ harr + (II)

2 2 OH2Ca 2OH Mg 2V+ minus ++ harr + 2(IV) 3 24 3PO CO OHminus minus minusharr + (V)

where VOH stands for a vacancy in the OHminus sub lattice If x y and z are the contributions of mechanisms II 2IV and V respectively thus

2nCa 10 x 2 y+ = minus minus sdot (8) 34nPO 6 2 x zminus = minus sdot minus (9)

2nMg y+ = (10) 23nCO 2 x zminus = sdot + (11)

and nOH 2 2 y zminus = minus sdot + (12)

and the generic formula has the following expression

( ) ( )( ) ( )( ) ( )( )y 4 310 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OHminus minus sdot minus minus minus ++

S B Abdelkader et al

1005

The values of x y and z the contribution of mechanisms II 2IV and V respectively are calculated from the data (Table 3) and the following equations Then statistical studies are conducted to verify the accuracy of the proposed formula The results of these calculations are summarized in Tables 9-11

Table 9 The values of x y and z the contributions of the mechanisms II 2IV and V respectively calculated from Equations (10)-(14)

Sample x y z

Mg1 0085 000 000

Mg2 0045 009 018

Mg3 minus034 037 068

Mg4 minus060 066 120

Mg5 043 000 minus01

Mg6 038 009 minus005

Mg7 020 034 037

Mg8 015 062 057

Mg9 076 000 minus020

Mg10 092 008 minus020

Mg11 073 032 minus006

Mg12 057 058 026

Table 10 Multiple linear regression analysis of the estimated Yi = nCa2+ the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = x- and X2i = y the contribution of mechanisms II and 2IV (Table 9) (a) Regression statistic (b) Coeffi-cients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0187 Var(X2) =0059

Covariance Cov(X1 Y) = minus0093 Cov(X2 Y) = minus0071 Cov(X1 X2) = minus0046

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 188 times 10minus5

R square R2 = 0999

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 999b = - - - - -

X1 1 0995b = minus ( )1 00032bσ = ( )1 30816T b = minus 000 minus1002 minus0987

X2 2 1999b = minus ( )2 00057bσ = ( )2 34779T b = minus 000 minus2012 minus1986

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32]

Regression 2 2862 1412 7414888 474

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

S B Abdelkader et al

1006

Table 11 Multiple linear regression analysis of the estimated Yi = nOHminus the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = z and X2i = z the contribution of mechanisms II and V (Table 9) (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0164 Var(X2) = 0059

Covariance Cov(X1 Y) = 00055 Cov(X2 Y) = minus0038 Cov(X1 X2) = 0079

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 274 times 10minus32

R square R2 = 100

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 200b = - - - - -

X1 1 100b = ( ) 161 1997 10bσ minus= times ( ) 15

1 500 10T b += times 000 1 1

X2 2 200b = minus ( ) 162 3338 10bσ minus= times ( ) 15

2 599 10T b += minus times 000 minus2 minus2

(c)

Source of variation DF Sum of squares Mean square Fcal

Regression 2 2862 1412 7414888

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

6 Discussion From Table 1 we can see that simultaneous incorporation of two elements ldquoCO3 and Mgrdquo results in an de-crease of the ldquoardquo parameter This contraction is attributed to the simultaneous effects of the 2

3CO minus and Mg2+ substitutions [4]

In Table 2 and Table 3 it is seen that the concentration of the 23CO minus ions in the solution Cc does not affect

the quantities of Mg2+ ions inserted in the solid Because regardless of the concentration of the 23CO minus ions in

the solution Cc the Mg2+ ions contents of the samples increase proportionally with the increase of the concentra-tion of Mg2+ ions in solution CMg This result is in agreement with reference [4] For the same concentration of Mg2+ions in the solution CMg the variation of 3

4PO minus and 23CO minus contents of the solid do not seem to be corre-

lated with the concentration of the Mg2+ ions in the solution while the Ca2+content depends on the concentra-tions of 2

3CO minus Cc and Mg2+ CMg in the solution Figure 3 and Figure 4 show that nCO3nP the molar ratio increases with the increasing of the concentration

of 23CO minus in solution (CcM) Contrariwise it varies slightly with the concentration of Mg2+ ions in solution and

vice versa for nMgnP The statistical treatment of the experimental data Table 4 and Table 5 allows us to es-tablish the estimated equations between these variables at 95 levels

( )3 cnCO nP 00165 5986 0945 C M= + plusmn sdot (13)

and ( ) ( )c MgnMg nP 00013 0135 0132 C M 00087 00005 C mM= minus + plusmn sdot + plusmn sdot (14)

Equations (9) and (10) show that the concentration of the 23CO minus ions in the hydrolysis solution Cc affects the

quantities of 23CO minus and Mg2+ ions incorporated in the solid but the concentration of Mg2+ ions in solution CMg

does not affects the quantities of 23CO minus ions in the solid These results are in agreement with those found in the

reference [27]

S B Abdelkader et al

1007

To know the relationship between the variation of Ca2+ and OHminus with the increasing of 23CO minus and Mg2+

content in the solid statistical studies are conducted The results of multiple linear regression Table 6 and Table 7 show that the estimated equations on these variables are represented at 95 level by

( ) ( )2 23

2Mg CO

nCa 1009 1373 0153 n 0698 0067 n+ minus+ = minus plusmn sdot minus plusmn sdot (15)

( ) ( )2 23Mg CO

nOH 219 0740 0313 n 0393 0138 n+ minusminus = minus plusmn minus plusmn sdot (16)

From the intercepts of the following equations it can seen that within experimental error a carbonate and magnesium-free apatite (nCO3 = 0 nMg = 0) contains 10 Ca2+ and 2 OHminus ions per unit cell Equations (11) and (12) These results are in agreement with those in literature [24]-[26] [34] [35]

As shown in Figure 5 there is a correlation of the unit cell parameters of the apatites with their chemical compositions Indeed the changes in the unit cell parameter ldquoardquo of the compounds are attributed to the additive effects of the substitution in the lattice of either carbonate and magnesium [2] [34] [35] The solid line of best fit for these series of compounds in Figure 5 extrapolates to a ratio ca very close to that in hydroxyapatite This result is similary to these obtained previously [34] [35] The application of multiple linear regression to Yi = ca

on ( )3 3

2

1i CO P 2i CO PX n n X n n= = ( )3

3

3i CO Pand X n n= allows us to establish the predicted equation at

95 level

( ) ( ) ( )

( ) ( )

26 6i CO3 P CO3 P

35CO3 P

Y c a 073 0052 156 10 n n 0168 426 10 n n

0212 118 10 n n

minus minus

minus

= = + plusmn times sdot + plusmn times sdot

+ plusmn times sdot (17)

To verify the general formula proposed We apply the multiple linear regression to Yi = nCa2+ on X1i = x and X2i = y (the contributions of the mechanisms II and IV) Similar treatment is realized for Yi = nOHminus on X1i = z and X2i = y (the contributions of the mechanisms V and IV) Table 10 and Table 11 The results of these calcu-lations show that the predicted equations at 95 level are

( ) ( )2nCa 999 0995 0007 x 1999 0013 y+ = minus plusmn minus plusmn sdot (18)

( ) ( )16 16nOH 200 100 452 10 z 200 754 10 yminus minus minus= + plusmn times sdot minus plusmn times sdot (19)

7 Conclusion The theoretical calculations of the present study indicate unambiguously that the mechanisms II III and V con-tribute to the incorporation of Mg and Ca in the lattice of apatite This corroborates in more definite way our as-sumptions obtained from the experimental data and allows us to propose for these compounds the general for-mula

( ) ( )( ) ( )( ) ( )( )y 4 3 210 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OH nH Ominus minus sdot minus minus minus ++

sdot

References [1] Kreidler ER and Hummel FA (1970) The Crystal Chemistry of Apatite Structure Fields of Fluor- and Chlorapatite

American Mineralogist 55 170 [2] Legeros RZ (1984) Tooth Enamel IV In Fearnhead RW and Sugas S Eds Elsevier Amsterdam 32-36 [3] Ben Abdelkader S Khattech I Rey C and Jemal M (2001) Syntheacutese Caracteacuterisation et Thermochimie drsquoApatites

Calco-Magneacutesiennes Hydroxyleacutees et Fluoreacutees Thermochimica Acta 376 25-36 httpdxdoiorg101016S0040-6031(01)00565-2

[4] Terpstra RA and Driessens FCM (1986) Magnesium in Tooth Enamel and Synthetic Apatites Calcified Tissue In-ternational 39 348-354 httpdxdoiorg101007BF02555203

[5] Hayek E and Newsely H (1958) Uumlber die Existenz von Tricalciumphosphat in waumlBriger Loumlsung Mn Chem 89 88 [6] Rowles SL (1968) Crystallographic Study of Biological Apatites Bulletin de la Socieacuteteacute Chimique de France 1797 [7] Hamad M and Heughebaert JC (1987) Study of Apatite and Whitlockite Formation at 100˚C in the System CaO-

MgO-P2O5-H2O J Chim Phys 84 985

S B Abdelkader et al

1005

The values of x y and z the contribution of mechanisms II 2IV and V respectively are calculated from the data (Table 3) and the following equations Then statistical studies are conducted to verify the accuracy of the proposed formula The results of these calculations are summarized in Tables 9-11

Table 9 The values of x y and z the contributions of the mechanisms II 2IV and V respectively calculated from Equations (10)-(14)

Sample x y z

Mg1 0085 000 000

Mg2 0045 009 018

Mg3 minus034 037 068

Mg4 minus060 066 120

Mg5 043 000 minus01

Mg6 038 009 minus005

Mg7 020 034 037

Mg8 015 062 057

Mg9 076 000 minus020

Mg10 092 008 minus020

Mg11 073 032 minus006

Mg12 057 058 026

Table 10 Multiple linear regression analysis of the estimated Yi = nCa2+ the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = x- and X2i = y the contribution of mechanisms II and 2IV (Table 9) (a) Regression statistic (b) Coeffi-cients (c) Analysis of variance

(a)

Variance Var(Y) = 0235 Var(X1) = 0187 Var(X2) =0059

Covariance Cov(X1 Y) = minus0093 Cov(X2 Y) = minus0071 Cov(X1 X2) = minus0046

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 188 times 10minus5

R square R2 = 0999

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 999b = - - - - -

X1 1 0995b = minus ( )1 00032bσ = ( )1 30816T b = minus 000 minus1002 minus0987

X2 2 1999b = minus ( )2 00057bσ = ( )2 34779T b = minus 000 minus2012 minus1986

(c)

Source of variation DF Sum of squares Mean square Fcal F (5 2 9) [32]

Regression 2 2862 1412 7414888 474

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

S B Abdelkader et al

1006

Table 11 Multiple linear regression analysis of the estimated Yi = nOHminus the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = z and X2i = z the contribution of mechanisms II and V (Table 9) (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0164 Var(X2) = 0059

Covariance Cov(X1 Y) = 00055 Cov(X2 Y) = minus0038 Cov(X1 X2) = 0079

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 274 times 10minus32

R square R2 = 100

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 200b = - - - - -

X1 1 100b = ( ) 161 1997 10bσ minus= times ( ) 15

1 500 10T b += times 000 1 1

X2 2 200b = minus ( ) 162 3338 10bσ minus= times ( ) 15

2 599 10T b += minus times 000 minus2 minus2

(c)

Source of variation DF Sum of squares Mean square Fcal

Regression 2 2862 1412 7414888

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

6 Discussion From Table 1 we can see that simultaneous incorporation of two elements ldquoCO3 and Mgrdquo results in an de-crease of the ldquoardquo parameter This contraction is attributed to the simultaneous effects of the 2

3CO minus and Mg2+ substitutions [4]

In Table 2 and Table 3 it is seen that the concentration of the 23CO minus ions in the solution Cc does not affect

the quantities of Mg2+ ions inserted in the solid Because regardless of the concentration of the 23CO minus ions in

the solution Cc the Mg2+ ions contents of the samples increase proportionally with the increase of the concentra-tion of Mg2+ ions in solution CMg This result is in agreement with reference [4] For the same concentration of Mg2+ions in the solution CMg the variation of 3

4PO minus and 23CO minus contents of the solid do not seem to be corre-

lated with the concentration of the Mg2+ ions in the solution while the Ca2+content depends on the concentra-tions of 2

3CO minus Cc and Mg2+ CMg in the solution Figure 3 and Figure 4 show that nCO3nP the molar ratio increases with the increasing of the concentration

of 23CO minus in solution (CcM) Contrariwise it varies slightly with the concentration of Mg2+ ions in solution and

vice versa for nMgnP The statistical treatment of the experimental data Table 4 and Table 5 allows us to es-tablish the estimated equations between these variables at 95 levels

( )3 cnCO nP 00165 5986 0945 C M= + plusmn sdot (13)

and ( ) ( )c MgnMg nP 00013 0135 0132 C M 00087 00005 C mM= minus + plusmn sdot + plusmn sdot (14)

Equations (9) and (10) show that the concentration of the 23CO minus ions in the hydrolysis solution Cc affects the

quantities of 23CO minus and Mg2+ ions incorporated in the solid but the concentration of Mg2+ ions in solution CMg

does not affects the quantities of 23CO minus ions in the solid These results are in agreement with those found in the

reference [27]

S B Abdelkader et al

1007

To know the relationship between the variation of Ca2+ and OHminus with the increasing of 23CO minus and Mg2+

content in the solid statistical studies are conducted The results of multiple linear regression Table 6 and Table 7 show that the estimated equations on these variables are represented at 95 level by

( ) ( )2 23

2Mg CO

nCa 1009 1373 0153 n 0698 0067 n+ minus+ = minus plusmn sdot minus plusmn sdot (15)

( ) ( )2 23Mg CO

nOH 219 0740 0313 n 0393 0138 n+ minusminus = minus plusmn minus plusmn sdot (16)

From the intercepts of the following equations it can seen that within experimental error a carbonate and magnesium-free apatite (nCO3 = 0 nMg = 0) contains 10 Ca2+ and 2 OHminus ions per unit cell Equations (11) and (12) These results are in agreement with those in literature [24]-[26] [34] [35]

As shown in Figure 5 there is a correlation of the unit cell parameters of the apatites with their chemical compositions Indeed the changes in the unit cell parameter ldquoardquo of the compounds are attributed to the additive effects of the substitution in the lattice of either carbonate and magnesium [2] [34] [35] The solid line of best fit for these series of compounds in Figure 5 extrapolates to a ratio ca very close to that in hydroxyapatite This result is similary to these obtained previously [34] [35] The application of multiple linear regression to Yi = ca

on ( )3 3

2

1i CO P 2i CO PX n n X n n= = ( )3

3

3i CO Pand X n n= allows us to establish the predicted equation at

95 level

( ) ( ) ( )

( ) ( )

26 6i CO3 P CO3 P

35CO3 P

Y c a 073 0052 156 10 n n 0168 426 10 n n

0212 118 10 n n

minus minus

minus

= = + plusmn times sdot + plusmn times sdot

+ plusmn times sdot (17)

To verify the general formula proposed We apply the multiple linear regression to Yi = nCa2+ on X1i = x and X2i = y (the contributions of the mechanisms II and IV) Similar treatment is realized for Yi = nOHminus on X1i = z and X2i = y (the contributions of the mechanisms V and IV) Table 10 and Table 11 The results of these calcu-lations show that the predicted equations at 95 level are

( ) ( )2nCa 999 0995 0007 x 1999 0013 y+ = minus plusmn minus plusmn sdot (18)

( ) ( )16 16nOH 200 100 452 10 z 200 754 10 yminus minus minus= + plusmn times sdot minus plusmn times sdot (19)

7 Conclusion The theoretical calculations of the present study indicate unambiguously that the mechanisms II III and V con-tribute to the incorporation of Mg and Ca in the lattice of apatite This corroborates in more definite way our as-sumptions obtained from the experimental data and allows us to propose for these compounds the general for-mula

( ) ( )( ) ( )( ) ( )( )y 4 3 210 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OH nH Ominus minus sdot minus minus minus ++

sdot

References [1] Kreidler ER and Hummel FA (1970) The Crystal Chemistry of Apatite Structure Fields of Fluor- and Chlorapatite

American Mineralogist 55 170 [2] Legeros RZ (1984) Tooth Enamel IV In Fearnhead RW and Sugas S Eds Elsevier Amsterdam 32-36 [3] Ben Abdelkader S Khattech I Rey C and Jemal M (2001) Syntheacutese Caracteacuterisation et Thermochimie drsquoApatites

Calco-Magneacutesiennes Hydroxyleacutees et Fluoreacutees Thermochimica Acta 376 25-36 httpdxdoiorg101016S0040-6031(01)00565-2

[4] Terpstra RA and Driessens FCM (1986) Magnesium in Tooth Enamel and Synthetic Apatites Calcified Tissue In-ternational 39 348-354 httpdxdoiorg101007BF02555203

[5] Hayek E and Newsely H (1958) Uumlber die Existenz von Tricalciumphosphat in waumlBriger Loumlsung Mn Chem 89 88 [6] Rowles SL (1968) Crystallographic Study of Biological Apatites Bulletin de la Socieacuteteacute Chimique de France 1797 [7] Hamad M and Heughebaert JC (1987) Study of Apatite and Whitlockite Formation at 100˚C in the System CaO-

MgO-P2O5-H2O J Chim Phys 84 985

S B Abdelkader et al

1006

Table 11 Multiple linear regression analysis of the estimated Yi = nOHminus the molar content of the solid ldquoBrdquo Mg-CO3 HAps (Table 3) on X1i = z and X2i = z the contribution of mechanisms II and V (Table 9) (a) Regression statistic (b) Coefficients (c) Analysis of variance

(a)

Variance Var(Y) = 0082 Var(X1) = 0164 Var(X2) = 0059

Covariance Cov(X1 Y) = 00055 Cov(X2 Y) = minus0038 Cov(X1 X2) = 0079

Coefficients of correlation R(X1 Y) = minus0443 R(X2 Y) = minus0607 R(X1 X2) = minus0443

Residual mean squareStandard error s2 = 274 times 10minus32

R square R2 = 100

Observations n = 12 and Degree of freedom ν = 9

(b)

Coefficients Standard error T statistic P-value Lower 95 Upper 95

Y-intercept 0 200b = - - - - -

X1 1 100b = ( ) 161 1997 10bσ minus= times ( ) 15

1 500 10T b += times 000 1 1

X2 2 200b = minus ( ) 162 3338 10bσ minus= times ( ) 15

2 599 10T b += minus times 000 minus2 minus2

(c)

Source of variation DF Sum of squares Mean square Fcal

Regression 2 2862 1412 7414888

Deviations 9 000017 1904 times 10minus5

Total 11 2823 02567

6 Discussion From Table 1 we can see that simultaneous incorporation of two elements ldquoCO3 and Mgrdquo results in an de-crease of the ldquoardquo parameter This contraction is attributed to the simultaneous effects of the 2

3CO minus and Mg2+ substitutions [4]

In Table 2 and Table 3 it is seen that the concentration of the 23CO minus ions in the solution Cc does not affect

the quantities of Mg2+ ions inserted in the solid Because regardless of the concentration of the 23CO minus ions in

the solution Cc the Mg2+ ions contents of the samples increase proportionally with the increase of the concentra-tion of Mg2+ ions in solution CMg This result is in agreement with reference [4] For the same concentration of Mg2+ions in the solution CMg the variation of 3

4PO minus and 23CO minus contents of the solid do not seem to be corre-

lated with the concentration of the Mg2+ ions in the solution while the Ca2+content depends on the concentra-tions of 2

3CO minus Cc and Mg2+ CMg in the solution Figure 3 and Figure 4 show that nCO3nP the molar ratio increases with the increasing of the concentration

of 23CO minus in solution (CcM) Contrariwise it varies slightly with the concentration of Mg2+ ions in solution and

vice versa for nMgnP The statistical treatment of the experimental data Table 4 and Table 5 allows us to es-tablish the estimated equations between these variables at 95 levels

( )3 cnCO nP 00165 5986 0945 C M= + plusmn sdot (13)

and ( ) ( )c MgnMg nP 00013 0135 0132 C M 00087 00005 C mM= minus + plusmn sdot + plusmn sdot (14)

Equations (9) and (10) show that the concentration of the 23CO minus ions in the hydrolysis solution Cc affects the

quantities of 23CO minus and Mg2+ ions incorporated in the solid but the concentration of Mg2+ ions in solution CMg

does not affects the quantities of 23CO minus ions in the solid These results are in agreement with those found in the

reference [27]

S B Abdelkader et al

1007

To know the relationship between the variation of Ca2+ and OHminus with the increasing of 23CO minus and Mg2+

content in the solid statistical studies are conducted The results of multiple linear regression Table 6 and Table 7 show that the estimated equations on these variables are represented at 95 level by

( ) ( )2 23

2Mg CO

nCa 1009 1373 0153 n 0698 0067 n+ minus+ = minus plusmn sdot minus plusmn sdot (15)

( ) ( )2 23Mg CO

nOH 219 0740 0313 n 0393 0138 n+ minusminus = minus plusmn minus plusmn sdot (16)

From the intercepts of the following equations it can seen that within experimental error a carbonate and magnesium-free apatite (nCO3 = 0 nMg = 0) contains 10 Ca2+ and 2 OHminus ions per unit cell Equations (11) and (12) These results are in agreement with those in literature [24]-[26] [34] [35]

As shown in Figure 5 there is a correlation of the unit cell parameters of the apatites with their chemical compositions Indeed the changes in the unit cell parameter ldquoardquo of the compounds are attributed to the additive effects of the substitution in the lattice of either carbonate and magnesium [2] [34] [35] The solid line of best fit for these series of compounds in Figure 5 extrapolates to a ratio ca very close to that in hydroxyapatite This result is similary to these obtained previously [34] [35] The application of multiple linear regression to Yi = ca

on ( )3 3

2

1i CO P 2i CO PX n n X n n= = ( )3

3

3i CO Pand X n n= allows us to establish the predicted equation at

95 level

( ) ( ) ( )

( ) ( )

26 6i CO3 P CO3 P

35CO3 P

Y c a 073 0052 156 10 n n 0168 426 10 n n

0212 118 10 n n

minus minus

minus

= = + plusmn times sdot + plusmn times sdot

+ plusmn times sdot (17)

To verify the general formula proposed We apply the multiple linear regression to Yi = nCa2+ on X1i = x and X2i = y (the contributions of the mechanisms II and IV) Similar treatment is realized for Yi = nOHminus on X1i = z and X2i = y (the contributions of the mechanisms V and IV) Table 10 and Table 11 The results of these calcu-lations show that the predicted equations at 95 level are

( ) ( )2nCa 999 0995 0007 x 1999 0013 y+ = minus plusmn minus plusmn sdot (18)

( ) ( )16 16nOH 200 100 452 10 z 200 754 10 yminus minus minus= + plusmn times sdot minus plusmn times sdot (19)

7 Conclusion The theoretical calculations of the present study indicate unambiguously that the mechanisms II III and V con-tribute to the incorporation of Mg and Ca in the lattice of apatite This corroborates in more definite way our as-sumptions obtained from the experimental data and allows us to propose for these compounds the general for-mula

( ) ( )( ) ( )( ) ( )( )y 4 3 210 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OH nH Ominus minus sdot minus minus minus ++

sdot

References [1] Kreidler ER and Hummel FA (1970) The Crystal Chemistry of Apatite Structure Fields of Fluor- and Chlorapatite

American Mineralogist 55 170 [2] Legeros RZ (1984) Tooth Enamel IV In Fearnhead RW and Sugas S Eds Elsevier Amsterdam 32-36 [3] Ben Abdelkader S Khattech I Rey C and Jemal M (2001) Syntheacutese Caracteacuterisation et Thermochimie drsquoApatites

Calco-Magneacutesiennes Hydroxyleacutees et Fluoreacutees Thermochimica Acta 376 25-36 httpdxdoiorg101016S0040-6031(01)00565-2

[4] Terpstra RA and Driessens FCM (1986) Magnesium in Tooth Enamel and Synthetic Apatites Calcified Tissue In-ternational 39 348-354 httpdxdoiorg101007BF02555203

[5] Hayek E and Newsely H (1958) Uumlber die Existenz von Tricalciumphosphat in waumlBriger Loumlsung Mn Chem 89 88 [6] Rowles SL (1968) Crystallographic Study of Biological Apatites Bulletin de la Socieacuteteacute Chimique de France 1797 [7] Hamad M and Heughebaert JC (1987) Study of Apatite and Whitlockite Formation at 100˚C in the System CaO-

MgO-P2O5-H2O J Chim Phys 84 985

S B Abdelkader et al

1007

To know the relationship between the variation of Ca2+ and OHminus with the increasing of 23CO minus and Mg2+

content in the solid statistical studies are conducted The results of multiple linear regression Table 6 and Table 7 show that the estimated equations on these variables are represented at 95 level by

( ) ( )2 23

2Mg CO

nCa 1009 1373 0153 n 0698 0067 n+ minus+ = minus plusmn sdot minus plusmn sdot (15)

( ) ( )2 23Mg CO

nOH 219 0740 0313 n 0393 0138 n+ minusminus = minus plusmn minus plusmn sdot (16)

From the intercepts of the following equations it can seen that within experimental error a carbonate and magnesium-free apatite (nCO3 = 0 nMg = 0) contains 10 Ca2+ and 2 OHminus ions per unit cell Equations (11) and (12) These results are in agreement with those in literature [24]-[26] [34] [35]

As shown in Figure 5 there is a correlation of the unit cell parameters of the apatites with their chemical compositions Indeed the changes in the unit cell parameter ldquoardquo of the compounds are attributed to the additive effects of the substitution in the lattice of either carbonate and magnesium [2] [34] [35] The solid line of best fit for these series of compounds in Figure 5 extrapolates to a ratio ca very close to that in hydroxyapatite This result is similary to these obtained previously [34] [35] The application of multiple linear regression to Yi = ca

on ( )3 3

2

1i CO P 2i CO PX n n X n n= = ( )3

3

3i CO Pand X n n= allows us to establish the predicted equation at

95 level

( ) ( ) ( )

( ) ( )

26 6i CO3 P CO3 P

35CO3 P

Y c a 073 0052 156 10 n n 0168 426 10 n n

0212 118 10 n n

minus minus

minus

= = + plusmn times sdot + plusmn times sdot

+ plusmn times sdot (17)

To verify the general formula proposed We apply the multiple linear regression to Yi = nCa2+ on X1i = x and X2i = y (the contributions of the mechanisms II and IV) Similar treatment is realized for Yi = nOHminus on X1i = z and X2i = y (the contributions of the mechanisms V and IV) Table 10 and Table 11 The results of these calcu-lations show that the predicted equations at 95 level are

( ) ( )2nCa 999 0995 0007 x 1999 0013 y+ = minus plusmn minus plusmn sdot (18)

( ) ( )16 16nOH 200 100 452 10 z 200 754 10 yminus minus minus= + plusmn times sdot minus plusmn times sdot (19)

7 Conclusion The theoretical calculations of the present study indicate unambiguously that the mechanisms II III and V con-tribute to the incorporation of Mg and Ca in the lattice of apatite This corroborates in more definite way our as-sumptions obtained from the experimental data and allows us to propose for these compounds the general for-mula

( ) ( )( ) ( )( ) ( )( )y 4 3 210 x 2 y 6 2x z 2 2y z2x zCa Mg PO CO OH nH Ominus minus sdot minus minus minus ++

sdot

References [1] Kreidler ER and Hummel FA (1970) The Crystal Chemistry of Apatite Structure Fields of Fluor- and Chlorapatite

American Mineralogist 55 170 [2] Legeros RZ (1984) Tooth Enamel IV In Fearnhead RW and Sugas S Eds Elsevier Amsterdam 32-36 [3] Ben Abdelkader S Khattech I Rey C and Jemal M (2001) Syntheacutese Caracteacuterisation et Thermochimie drsquoApatites

Calco-Magneacutesiennes Hydroxyleacutees et Fluoreacutees Thermochimica Acta 376 25-36 httpdxdoiorg101016S0040-6031(01)00565-2

[4] Terpstra RA and Driessens FCM (1986) Magnesium in Tooth Enamel and Synthetic Apatites Calcified Tissue In-ternational 39 348-354 httpdxdoiorg101007BF02555203

[5] Hayek E and Newsely H (1958) Uumlber die Existenz von Tricalciumphosphat in waumlBriger Loumlsung Mn Chem 89 88 [6] Rowles SL (1968) Crystallographic Study of Biological Apatites Bulletin de la Socieacuteteacute Chimique de France 1797 [7] Hamad M and Heughebaert JC (1987) Study of Apatite and Whitlockite Formation at 100˚C in the System CaO-

MgO-P2O5-H2O J Chim Phys 84 985

S B Abdelkader et al

1008

[8] Vignoles M Bonel G and Young RA (1987) Occurrence of Nitrogenous Species in Precipitated B-Type Carbo-nated Hydroxyapatites Calcified Tissue International 40 64-70 httpdxdoiorg101007BF02555707

[9] Vignoles M Bonel G Holcomb DW and Young RA (1988) Influence of Preparation Conditions on the Compo-sition of Type B Carbonated Hydroxyapatite and on the Localization of the Carbonate Ions Calcified Tissue Interna-tional 43 33

[10] Young RA and Spooner S (1969) Neutron Diffraction Studies of Human Tooth Enamel Archives of Oral Biology 15 47-63 httpdxdoiorg1010160003-9969(70)90144-5

[11] Bigi A Foresti E Gregorini R Ripamonti A Roveri N and Shah JS (1992) The Role of Magnesium on the Structure of Biological Apatites Calcified Tissue International 50 439-444 httpdxdoiorg101007BF00296775

[12] Holden JL Clement JG and Phakey PP (1995) Age and Temperature Related Changes to the Ultrastructure and Composition of Human Bone Mineral Journal of Bone and Mineral Research 10 1400-1409 httpdxdoiorg101002jbmr5650100918

[13] Robinson C Weatherell JA and Hallsworth AS (1981) Distribution of Magnesium in Mature Human Enamel Ca-ries Research 15 70-77 httpdxdoiorg101159000260502

[14] Steinfort J Driessens FCM Heijligers HJM and Bertseen W (1991) The Distribution of Magnesium in Devel-oping Rat Incisor Dentin Journal of Dental Research 70 187-191 httpdxdoiorg10117700220345910700030601

[15] Tsuboi S Nakagi H Ishiguro K Kondo K Mukai M Robinson C and Weatherell JA (1994) Magnesium Distribution in Human Bone Calcified Tissue International 54 34-37 httpdxdoiorg101007BF00316287

[16] Bigi A Marchetti F Ripamonti A Roveri N and Foresti E (1981) Magnesium and Strontium Interaction with Carbonate-Containing Hydroxyapatite in Aqueous Medium Journal of Inorganic Biochemistry 15 317-327 httpdxdoiorg101016S0162-0134(00)80235-4

[17] Featherstone JDB Mayer I Driessens FCM Verbeeck RMH and Heijligers HJ (1983) Synthetic Apatites Containing Na Mg and CO3 and Their Comparison with Tooth Enamel Mineral Calcified Tissue International 35 169-171 httpdxdoiorg101007BF02405026

[18] Apfelbaum F Mayer I and Featherstone JDB (1991) The Role of 24HPO minus and 2

3CO minus Ions in the Transformation of Synthetic Apatites to β-Ca3(PO4)2 Journal of Inorganic Biochemistry 38 1-8 httpdxdoiorg1010160162-0134(90)85001-D

[19] Aoba T Moreno EC and Shimoda S (1992) Competitive Adsorption of Magnesium and Calcium Ions onto Syn-thetic and Biological Apatites Calcified Tissue International 51 143-150 httpdxdoiorg101007BF00298503

[20] Okazaki M and Legeros RZ (1992) Crystallographic and Chemical Properties of Mg-Containing Apatites before and after Suspension in Solutions Magnesium Research 5 103-108

[21] Zhou JM Zhang XD Chen JY Zeng SX and De Groot K (1993) High Temperature Characteristics of Syn-thetic Hydroxyapatite Journal of Materials Science Materials in Medicine 4 83-85 httpdxdoiorg101007BF00122983

[22] Legeros RZ Kijkowska R Bautista C and Legeros JP (1995) Synergistic Effects of Magnesium and Carbonate on Properties of Biological and Synthetic Apatites Connective Tissue Research 333 203

[23] Mayer I Schlam R and Featherstone JDB (1997) Magnesium-Containing Carbonate Apatites Journal of Inor-ganic Biochemistry 66 1-6 httpdxdoiorg101016S0162-0134(96)00145-6

[24] Gibson IR and Bonfield W (2002) Preparation and Characterization of MagnesiumCarbonate Co-Substituted Hy-droxyapatites Journal of Materials Science Materials in Medicine 13 685-693 httpdxdoiorg101023A1015793927364

[25] Baravell SS Bigi A Ripamonti A Roveri N and Foresti E (1984) Thermal Behavior of Bone and Synthetic Hy-droxyapatites Submitted to Magnesium Interaction in Aqueous Medium Journal of Inorganic Biochemistry 20 1-12 httpdxdoiorg1010160162-0134(84)80001-X

[26] De Maeyer EAP and Verbeeck RMH (1993) Possible Substitution Mechanisms for Sodium and Carbonate in Cal-ciumhydroxyapatite Bulletin des Socieacuteteacutes Chimiques Belges 102 601-609 httpdxdoiorg101002bscb19931020907

[27] De Maeyer EAP Verbeeck RMH and Pieters IY (1996) Influence of the Solution Composition on the Stoichi-ometry of Na+- and of K+-Containing Carbonated Apatites Obtained by the Hydrolysis of Monetite Journal of Crystal Growth 169 539-547 httpdxdoiorg101016S0022-0248(96)00424-1

[28] De Maeyer EAP Verbeeck RMH and Pieters IY (1996) Effect of K+ on the Stoichiometry of Carbonated Hy-droxyapatite Obtained by the Hydrolysis of Monetite Inorganic Chemistry 35 857-863 httpdxdoiorg101021ic950916k

[29] Charlot G (1966) Les Meacutethodes de la Chimie Analytique Masson Paris

S B Abdelkader et al

1009

[30] Gee A and Deitz VR (1953) Determination of Phosphate by Differential Spectrophotometry Analytical Chemistry 25 1320-1324 httpdxdoiorg101021ac60081a006

[31] Legeros RZ (1991) Calcium Phosphates in Oral Biology and Medicine Monographs in Oral Science 15 89-95 [32] Snedecor GW and Cochran WG (1980) Statistical Methods 7th Edition The Iowa State University Press Ames [33] Borcard D (2009) Regression Multiple Universiteacute de Montreacuteal Montreacuteal

httpbiol09biolumontrealcaBIO2042Regr_multpdf [34] Bel Hadj Yahia F and Jemal M (2010) Synthesis Structural Analysis and Thermochemistry of B-Type Carbonate

Apatites Thermochimica Acta 505 22-32 httpdxdoiorg101016jtca201003017 [35] Rockh B Hadj Yahia F and Khattech I (2014) Statistical Studies of the Physicochemical Analytic Results of a Series

of Synthetic Calcium Hydroxyapatite Containing Carbonate and Sodium American Journal of Analytical Chemistry 5 343-357

• Preparation Characterization and Statistical Studies of the Physicochemical Results of Series of ldquoBrdquo Carbonated Calcium Hydroxyapatites Containing Mg2+ and
• Abstract
• Keywords
• 1 Introduction
• 2 Methods and Materials
• • 21 Preparation of ldquoBrdquo Type Carbonated Hydroxyapatites Containing Magnesium
  • 22 Physical Analysis
  • 23 Chemical Analysis
  • • 3 Results
    • • 31 Results of Physical Analysis
      • 32 Chemical Results
      • • 4 Statistical Analysis of the Physicochemical Results
        • • 41 Influence of the Experimental Conditions on the Composition of the Synthetic Apatites
          • 42 Influence of the Incorporation of and Mg2+ on the Variation of Ca2+ and OHminus the Molar Ions Contents of the Synthetic Apatites
          • 43 The Determination of the Relationship between Y = ca Crystallographic Parameters Ratio and the Molar Ratio
          • • 5 Determination of the General Formula of the Unit Cell of the Synthetic ldquoBrdquo CO3Mg-HAps
            • 6 Discussion
            • 7 Conclusion
            • References