ARTICLE IN PRESS

Biomaterials 25 (2004) 1151–1158

Dry mechanochemical synthesis of hydroxyapatites from DCPD and CaO: influence of instrumental parameters on the reaction kinetics Carolina Mochalesa, Hassane El Briak-BenAbdeslamb, Maria Pau Ginebraa, Alain Terolb, Josep A. Planella, Philippe Boudevilleb,* a

Research Center in Biomedical Engineering, Biomaterials Division, Department of Materials Science and Metallurgy, Technological University of Catalonia (UPC), Avda. Diagonal 647, Barcelona E-08028, Spain b Laboratoire de Chimie G!en!erale et Min!erale, Facult!e de Pharmacie, Universite de Montpellier I, 15 Av. Charles Flahault, BP 14491, 34093 Montpellier, Cedex 5, France Received 4 June 2003; accepted 7 August 2003

Abstract Mechanochemistry is a possible route to synthesize calcium deficient hydroxyapatite (CDHA) with an expected molar calcium-tophosphate (Ca/P) ratio 70.01. To optimize the experimental conditions of CDHA preparation from dicalcium phosphate dihydrate (DCPD) and calcium oxide by dry mechanosynthesis reaction, we performed the kinetic study varying some experimental parameters. This kinetic study was carried out with two different planetary ball mills (Retsch or Fritsch Instuments). Results obtained with the two mills led to the same conclusions although the values of the rate constants of DCPD disappearance and times for complete reaction were very different. Certainly, the origin of these differences was from the mills used, thus we investigated the influence of instrumental parameters such as the mass and the surface area of the balls or the rotation velocity on the mechanochemical reaction kinetics of DCPD with CaO. Results show that the DCPD reaction rate constant and the inverse of the time for complete disappearance of CaO both vary linearly with (i) the square of the rotation velocity, (ii) the square of eccentricity of the vial on the rotating disc and (iii) the product of the mass by the surface area of the balls. These observations comply with theoretical models developed for mechanical alloying. The consideration of these four parameters allows the transposition of experimental conditions from one mill to another or the comparison between results obtained with different planetary ball mills. These instrumental parameters have to be well described in papers concerning mechanochemistry or when grinding is an important stage in a process. r 2003 Elsevier Ltd. All rights reserved. Keywords: Mechanosynthesis; Kinetic study; Dicalcium phosphate dihydrate; Calcium oxide; Hydroxyapatite; Planetary ball mill

1. Introduction Numerous biomaterials have been developed as alternatives to autogenous and allogenous bone grafts. Among these synthetic bone substitutes, calcium phosphate ceramics have been used successfully in orthopedics, dentistry and maxillofacial surgery [1–4]. These ceramics are composed either of pure beta-tricalcium phosphate (b-TCP) with a calcium-to-phosphate (Ca/P) ratio of 1.5, or of stoichiometric hydroxyapatite (HA) with Ca/P=1.67, or of a mixture of b-TCP and HA (biphasic calcium phosphate ceramic BCP) with *Corresponding author. Tel.: +33-4-67-54-80-78; fax: +33-4-67-5480-82. E-mail address: [email protected] (P. Boudeville). 0142-9612/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.biomaterials.2003.08.002

1.5oCa/Po1.67. They are generally prepared either by solid-state reaction at high temperature between a calcium salt and a phosphate salt or by heating a poorly crystallized HA or calcium deficient hydroxyapatite (CDHA). In previous papers, we showed that dry mechanochemistry is an alternative route to prepare CDHA [5–7] with an improved precision in the Ca/P ratio. In mechanosynthesis, the reaction is activated by mechanical milling: the reactants are crushed between balls or ball and wall. They absorb a part of the energy provided by the collisions which allows their reaction. However, one of the drawbacks attributed to this technique is the lack of reproducibility and the difficulty to compare results obtained by different authors with different mills [8]. Certainly, the origin of these differences arises from

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the different characteristics of the mills used, and in this sense it is necessary to investigate the influence of the different instrumental factors on the mechanochemical reaction kinetics. Therefore, the aim of this work is to evaluate the effect of different instrumental parameters, such as the mass and the surface area of the balls, or the rotation velocity on the reaction kinetics. This can allow the formulation of general laws applicable to different milling equipments and protocols.

ωv

vial ωd

rd rv

Sun disc

2. Materials and methods 2.1. Chemicals and milling equipments

Fig. 1. Diagram of the planetary ball mill with the representation of parameters given in Table 1 and used in several equations.

Chemicals were analytical reagent grade purchased from Aldrich (CaO) and Fluka (dicalcium phosphate dihydrate—DCPD). Commercial CaO was heated at 900
C for 2 h to remove H2O and CO2 and stored in a vacuum desiccator. The CaO median particle size d50 was around 7 mm (d10 2d90 ¼ 2240 mm; calculated specific surface area=4.3 m2 g1, Mastersizer, Malvern Instruments). DCPD was used as received, its median particle size was 8 mm (1.6–27 mm; calculated specific surface area, 3.5 m2 g1). Two planetary ball mills were used: Retsch Instrument (R) and Fritsch Instrument (F). Their technical specifications are given in Table 1. In this table the term eccentricity (rd ) corresponds to the distance between the center of the vial and the center of the rotating disc on which the vial is fixed, as represented in Fig. 1. 2.2. General grinding procedure DCPD and calcium oxide, each weighted to obtain 15 g of mixtures with the desired Ca/P ratio (from 1.5 to 1.7), were placed in the 500-ml vial with several balls. To

Table 1 Specifications of the two mills, vials and balls used in this work Retsch

study the effect of two different parameters on the reaction kinetics, two protocols were defined: 2.2.1. Effect of the mass and surface area of the balls The Retsch mill was used with a rotation speed of 350 rotations per minute (rpm) and without change in the rotation direction. The Ca/P ratio was adjusted to 1.6. The number and the diameter of the balls were adjusted to obtain different total masses and surface areas, trying to get in some experiments the greatest mass for the lowest surface area and inversely. Before all experiments, the balls were heated at 110
C for 1 h, and after cooling their mass and diameter were measured accurately. 2.2.2. Effect of the mill rotation velocity The powder was milled at different rotating speeds, in the two mills. For the R mill five balls were used (mean diameter 2.6 cm, total mass 135 g and surface area 110.8 cm2 for k ¼ f ðod Þ or 2.5 cm, 119 g and 98.2 cm2, respectively, for 1=tf ¼ f ðod Þ), and for the F mill four balls (3.0 cm in diameter, total mass 150 g and surface area 116 cm2). 2.3. Sample evolution characterization

Fritsch

Vial Material Radius rv (cm) Volume (ml)

Porcelain 5 500

Agate 5 500

Balls Material Mean diameter (cm) Mean weight (g) Mean surface area (cm2)

Porcelain 2.6/2.0 24/9.5 22/12.6

Agate 3.0 37 28.3

Rotation velocity od (rpm) Eccentricity rd (cm)

160–380 3.65

220–680 1.5

At different intervals depending on the transformation rate, 50–100 mg of powder were taken for XRD (Automatic diffractometer Philips PW3830 using an ( with a nickel filter) or anti-cathode Cu (Ka 1,5405 A) DSC analysis (DSC 6 Perkin-Elmer using indium and zinc as calorimetric and thermometric references), in order to determine the reaction kinetics and to evaluate the changes in the powder composition. When the sample analysis was delayed, samples were washed with acetone to stop an eventual continuation of the reaction because of the presence of water in low amount coming from the DCPD [7]. The relative amount of DCPD in

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2.5

1.0

k

-1

(h )

1.5

DCPD

2.0

0.5 0.0 0

20

40

60

80

100

120

140 160

Total mass M of balls (g)

(a)

2.5

1.0

k

-1

(h )

1.5

DCPD

2.0

0.5 0.0 0

20

40

60

80

100

120

140

Total surface area Sa of balls (cm²)

(b)

2.5

1.0

-1

(h )

1.5

DCPD

2.0

k

the powder at each interval was determined either by XRD (area of a DCPD diffraction peak) or by DSC (enthalpy of the endotherm corresponding to the dehydration of DCPD) [7,9]. By XRD, the area of the DCPD X-ray diffraction peak at 5.80
y (plane 0.2.0) was calculated on patterns recorded from 5
y to 6.5
y by 0.01
y (20 acquisitions, acquisition delay of 500 ms) after baseline subtraction. By DSC, all experiments were performed under nitrogen flow (30 ml min1) on 3–4 mg samples in aluminum crucibles and a heating rate of 20
C min1. The enthalpy (expressed in J g1) of the DCPD endothermic peak at 170–210
C was determined on thermograms recorded using two temperature programs depending on the sample preparation [7]. The DCPD disappearance rate constant was determined by plotting the natural logarithm of the DCPD content versus the time of grinding. The slope of the straight line obtained represents the rate constant k expressed in h1. In the previous work [7], it was shown that the DCPD consumption did not indicate the end of the reaction but an additional time was required, which increased exponentially with the Ca/P ratio, to consume the CaO and complete the reaction. Therefore, the final reaction time (tf ) was determined by a phenolphthalein test that indicated, when it was negative, the complete disappearance of calcium oxide or calcium hydroxide [7]. To have a reasonable final reaction time even under unfavorable grinding conditions (tf increasing exponentially with the Ca/P ratio) [7], the Ca/P of the samples studied was fixed into 1.6.

1153

0.5 0.0 0 (c)

3. Results

5000

10000

15000

20000

Mass x surface area of balls (g cm²)

3.1. Effect of ball mass and surface area

Fig. 2. Variation in the DCPD disappearance rate constant k with (a) the total mass M; (b) the total surface area of the balls Sa, and (c) the product M Sa: Retsch mill, rotation speed 350 rpm, Ca/P=1.60 and 15 g of powder. Full lines represent the correlation lines.

The rate constant of DCPD disappearance varies with the total mass and the total surface area of the balls (Figs. 2(a)–(c) and Eqs. (1)–(3)) and the best fit was obtained by plotting the rate constant versus the product of the total mass (M) and total surface area (Sa) of the balls (Fig. 2c and Eq. (3)) (units are given in parentheses)

(Figs. 3(a)–(c), Eqs. (4)–(6)). As for the DCPD disappearance, the best fit was obtained by plotting 1=tf versus the product of the total mass (M) and total surface area (Sa) of the balls (Fig. 3(c) and Eq. (6))

k ðh1 Þ ¼ 1:64  102 ðg hÞ1 M  0:324 ðh1 Þ; ðr ¼ 0:977Þ;

1=tf ðh1 Þ ¼ 3:48  103 ðcm2 hÞ1 Sa  0:157 ðh1 Þ; ðr ¼ 0:898Þ; ð5Þ

ð2Þ

1=tf ðh1 Þ ¼ 1:85  105 ðg cm2 hÞ1 M Sa  0:0027 ðh1 Þ;

k ðh1 Þ ¼ 1:24  104 ðg cm2 hÞ1 M Sa  0:0071 ðh1 Þ;

ðr ¼ 0:998Þ:

ð4Þ

ð1Þ

k ðh1 Þ ¼ 2:26  102 ðcm hÞ1 Sa  0:864 ðh1 Þ; ðr ¼ 0:827Þ;

1=tf ðh1 Þ ¼ 2:46  103 ðg hÞ1 M  0:054 ðh1 Þ; ðr ¼ 0:981Þ;

ð3Þ

The inverse of the final reaction time tf (equivalent in dimensions to a rate constant) is also affected by the total mass and the total surface area of the balls

ðr ¼ 0:998Þ:

ð6Þ

3.2. Effect of the mill rotating speed The DCPD disappearance rate constants determined for different rotation velocities with the two mills were

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0.30

2.5 2.0 1.5 1.0

-1

(h )

0.20

DCPD

-1

1 / tf (h )

0.25

0.15 0.10

k

0.05

0.5

0.00 0

20

40

60

80

100 120 140 160 0.0

Total mass M of balls (g)

(a)

0

100000

0.30

-1

1 / tf (h )

0.25

200000

300000

ωd ² (rpm²)

400000

500000

Fig. 4. Variation in the DCPD disappearance rate constant k with the square of the rotation velocity o2d obtained by grinding DCPD+CaO with the Retsch mill (E) and the Fritsch mill, (’). Mass of powder ground 15 g. Full lines represent the correlation lines.

0.20 0.15 0.10 0.05

0.5

0.00 0

20

40

60

80

100 120 140 160 0.4

Total surface area Sa of balls (cm²)

1 / tf (h )

(b)

-1

0.30

-1

1 / tf (h )

0.25 0.20 0.15

0.3 0.2 0.1

0.10 0.0

0.05

0

0.00 0 (c)

4000

8000

12000

Fig. 3. Variation in the inverse of the final reaction time 1=tf with (a) the total mass M; (b) the total surface area of the balls Sa, and (c) the product M Sa: Retsch mill, rotation speed 350 rpm, Ca/P=1.60 and 15 g of powder. Full lines represent the correlation lines.

plotted against the mill rotation velocity. For the two mills, a straight line was only obtained by plotting the rate constants with the square of the rotation velocity o2d (Fig. 4). Equations of these straight lines are given: Eq. (7) for results from the Retsch mill and Eq. (8) from the Fritsch mill

300000

400000

500000

Fig. 5. Variation in the inverse of the final reaction time 1=tf with the square of the rotation velocity o2d obtained by grinding DCPD+CaO with the Retsch mill (~) and the Fritsch mill, (’). Mass of powder ground 15 g and Ca/P=1.6. Full lines represent the correlation lines.

Eqs. (9) and (10)). R mill :

1=tf ðh1 Þ ¼ 27:6  107 ðrpm2 hÞ1 o2d  0:0325 ðh1 Þ;

ðr ¼ 0:999Þ; ð9Þ

F mill : 1=tf ðh1 Þ ¼ 9:4  107 ðrpm2 hÞ1 o2d  0:0070 ðh1 Þ;

R mill : k ðh1 Þ ¼ 19:15  106 ðrpm2 hÞ1 o2d ðr ¼ 0:986Þ;

200000

ω d² (rpm²)

16000

Mass x Surface area (g cm²)

 0:369 ðh1 Þ;

100000

ðr ¼ 0:998Þ: ð10Þ

ð7Þ 4. Discussion

F mill : k ðh1 Þ ¼ 3:60  106 ðrpm2 hÞ1 o2d  0:091 ðh1 Þ;

ðr ¼ 0:970Þ:

ð8Þ

The inverse of the final reaction time (corresponding to a Ca/P ratio of 1.6 in the mixture) also varies linearly with the square of the rotation velocity (Fig. 5 and

In mechanosynthesis, reactants are crushed between balls or ball and vial wall. The energy provided by these collisions allows both the fragmentation and/or aggregation of the powder particles and their eventual reaction. Some works have attempted to model the behavior of balls and the energy exchanges during these shocks.

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These models have been exclusively developed for metals in the so-called mechanical alloying (MA). The different aspects of MA have been exhaustively reviewed by Suryanarayama in a recent paper [10]. Although reaction mechanisms involved in MA are different from the mechanisms of chemical reactions induced by mechanical activation, some of the conclusions of these studies can be used to understand the differences we obtained in our results. For MA modeling, different approaches have been taken: mechanistic, atomistic, thermodynamic or kinetic models that have been reviewed and discussed by Khina and Froes [11]. To treat our results we used only data from the mechanistic models. These models consider cold welding of powders in a ball mill (‘‘local’’ models) and the ball dynamics in a milling device (‘‘global’’ models) [12–21]. In the local models, collisions between bodies are normally conceived as being either perfectly elastic in which there is no loss in kinetic energy, or imperfectly elastic, in which energy is dissipated. Without powder, collisions can be considered as purely elastic and treated by the Hertzian impact theory. In the presence of powder, a part of the energy exchanged is absorbed by the powder for packing it down but this absorbed part is much less than the elastic energy (around 1%) [13]. For Magini et al. [14], the energy transferred in each collision is DE ¼ Ka Ek ; where Ek is the kinetic energy of the ball and Ka a factor that is equal to 0 for perfect elastic collisions (no energy transfer) and equal to 1 for perfect inelastic collisions. They showed that if the balls are covered with powder, the collisions are almost perfectly inelastic, so that Ka E1 [14] and it is generally stated that the kinetic shock energy is released in totality into the powder [14,16–18]. Therefore, it can be expected that the instrumental parameters which can modify the kinetic energy of the balls will have an effect on the kinetics of the reaction. 4.1. Effect of ball mass and surface area The correlations obtained for k and 1=tf with the mass M and the surface area Sa of the balls show that these two parameters are interdependent in their influence on the kinetic data. Indeed, the best fits were obtained by correlating them with the product of the two parameters M Sa (Figs. 2c and 3c and Eqs. (3) and (6)). Moreover, it is reasonable to suppose that if the mass and consequently the surface area of the balls is equal to zero, whatever the rotation velocity, the reaction does not occur, and thus k and 1=tf ¼ 0: This signifies that the intercept in Figs. 2 and 3 should pass through the origin of the coordinates system. This is only true when the results are represented in function of the product M Sa (Figs. 2(c) and 3(c)). The influence of the mass of the balls can be related to the ball kinetic energy, which is Ek ¼ 12 mv2 ; where m is

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its mass and v its linear velocity at the collision time. The influence of the surface area of the balls has not been intrinsically taken into account in the different models developed. However, this parameter has a real influence as shown by correlations (3) and (6) and Figs. 2(c) and 3(c). This result is more delicate to explain but confirms a suggestion we did in a previous paper [5] that the amount of powder crushed per unit time was in relation with the total surface available, and therefore to the surface area of the balls. This point will be addressed again at the end of the discussion. 4.2. Effect of the mill rotating speed Figs. 4 and 5 show that the kinetic parameters (both k and 1=tf ), determined using the two mills, are proportional to the square of the mill rotating speed. However, the slopes are very different (Eqs. (7)–(10)). Note that the intercepts of the straight lines k or 1=tf ¼ f ðo2d Þ are negative (Eqs. (7)–(10) and Figs. 4 and 5). Because k and 1=tf cannot be negative, a possible explanation could be that, for a given mass and surface area of the balls, a minimum rotation velocity would be required to activate the mechanochemical reaction. The calculation of this minimum rotation velocity from these equations, taking k and 1=tf ¼ 0; gives respectively, 138 and 108 rpm for the Retsch mill and 159 and 86 for the Fritsch mill. However, these values have to be relativized because the intercept of a regression is generally more sensitive to a spurious point than its slope. The straight lines obtained by correlating the kinetic parameters with the square of the rotation velocity and also the great difference in their slopes can be explained in terms of the kinetic energy of the balls. Indeed, from mathematical analyses of milling mechanics in a planetary ball mill (global models) [14–18], equations that describe the position of a ball, its velocity, a detachment criterion, the collision frequency, and the shock power were established taking into account instrumental parameters. In this present work, we will use the equation developed by Chattopadhyay et al. [18] for the impact velocity v of a ball (Eq. (11)) v2 ¼ o2d r2d þ ðod  ov Þ2 r2v þ 2od ðod  ov Þ rd rv cos f1 ;

ð11Þ

where od is the rotation speed of the disc in the anticlockwise direction, ov the rotation speed of the vial in the clockwise direction, rd the distance between the center of the sun disc and the center of the vial (eccentricity), rv the vial radius (Fig. 1) and cos f1 =S=rv ðod  ov Þ2 =rd o2d where S is the detachment parameter (0pSp1). The mills used in this work do not allow for varying the rotation speed of the vial independent of the disc rotation speed. The rotation of the disc and of the vial are coupled and od ¼ ov

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whatever the od value was. In this way, Eq. (11) can be simplified (Eq. (12)) v2 ¼ o2d r2d :

ð12Þ

It could seem surprising that the ball velocity appears independent of the vial radius and velocity but this was confirmed by simulations carried out by Abdellaoui and Gaffet (see Fig. 1 of Ref. [17]) and Chattopadhyay et al. (see Fig. 3 of Ref. [18]) showing that the kinetic energy of the ball is a parabolic function of only the disc rotation velocity and is scarcely dependent on vial parameters. Note that our two vials had the same radius (Table 1). Inversely, the shock frequency varies linearly both with od and ov (Fig. 2, Ref. [17] and Fig. 2, Ref. [18]). The results shown in Figs. 4 and 5 can be justified considering that (i) the reaction kinetics parameters k or 1=tf are in relation with the energy exchanged during shocks, (ii) this exchanged energy is proportional to the kinetic energy of the ball at the collision time, but not equal because collisions can be from frontal to tangential (impact mode or friction mode [20,21] with a statistical distribution of these modes with the time of grinding), (iii) for a given mass of the balls, the kinetic energy is a quadratic function of the ball velocity and therefore is a quadratic function of the rotation speed of the disc and of the eccentricity rd of the vial on the rotating disc (Eq. (12)). Effectively, the kinetic parameters k and 1=tf vary linearly with the square of the rotation velocity (Eqs. (7) and (8)) and if Eq. (12) is true, there would also be a quadratic function of the eccentricity. Thus we corrected the kinetic parameters for the total mass and surface area of the balls and for the square of the rotation velocity (k=ðM Sa o2d Þ or ð1=tf Þ=ðM Sa o2d Þ) and we correlated these normalized parameters with the square of the eccentricity (Fig. 6 and Eq. (13)). In this correlation we added the point (0, 0) because, from Eq. (12), if rd ¼ 0; v ¼ 0 and consequently the ball kinetic energy Ek ¼ 0: Thus the reaction does not take place and k ¼ 0: The point (0; 0) well integrates the correlation k=ðM Sa o2d Þ ¼ 7:86  1011 r2d þ 2:5  1011 ; ðr ¼ 0:993Þ:

ð13Þ

The correlation ð1=tf Þ/(M Sa o2d )=f ðr2d Þ was slightly less good but gave results in the same order of magnitude. Thus we can conclude that the kinetic parameters are in direct relation with the square of the eccentricity. This dependence of the kinetic parameters with the square of the eccentricity explains the difference observed in the slopes of the correlations k or 1=tf ¼ f ðo2d Þ obtained with the two mills. Correcting these slopes for the total mass and surface area of balls, we obtained an experimental ratio Rexp of the k ¼ f ðo2d Þ corrected slopes of 6.18 (Eq. 14) that well complies with

1.2E-09 1.0E-09

k / (M*Sa* ω d²)

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8.0E-10 6.0E-10 4.0E-10 2.0E-10 0.0E+00 0

2

4

6

8

10

12

14

rd² (cm²) Fig. 6. Variation in the normalized DCPD disappearance rate constant (k/(M Sa o2d )) with the square of the eccentricity r2d : Full line represents the correlation line taking into account the particular point (0, 0). Mass of powder ground 15 g.

the theoretical ratio of the square of the eccentricity of the two vials: Rtheo ¼ r2d Retsch =r2d Fritsch ¼ 5:9: Rexp ¼

19:15  106  150  116 ¼ 6:18: 3:6  106  135  110:8

ð14Þ

The same calculation applied to the slopes of the correlations 1=tf ¼ f ðo2d Þ gives Rexp ¼ 4:5: The agreement is less good than the previous ratio but remains in the same order. On a theoretical standpoint, the agreement between these experimental results and the theoretical models corroborates both the validity of our kinetic approach of the DCPD+CaO reaction mechanochemically activated and the equations developed for MA mechanistic global models. Although these models have been generally developed considering only one ball, the transposition seems adequate when several balls are used. But, the exact influence of the surface area of the balls remains without real explanation and raises more questions than answers. For Maurice and Courtney [13]: ‘‘powder particle aggregates are much smaller than the colliding bodies they are trapped between, thus it is reasonable to view the surfaces of the colliding bodies as having infinite curvature relative to the powder. The collisions with powder may then be viewed as an upset forging process between two parallel planes (Fig. 7). The length of the porous powder cylinder entrapped between two colliding bodies is twice the Hertz radius rh which is the consequence of the ball deformation during the collision. Indeed, during the short impact time laps the ball is compressed for a distance dmax and the spherical surface becomes flat. The radius of this flat surface is the Hertz radius rh that depends on many factors [13] such as the Young’s modulus and the density of the ball material, the precollision relative velocity and the radius of the ball. The radius of contact between the colliding surfaces at maximum compression is a point of importance

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ball

rh h0

1157

of the whole parameters on the kinetics of the DCPDCaO reaction to optimize the mechanochemical synthesis of apatites, we do not look further, at this stage, in the comprehension of the phenomena occurring in mechanochemistry.

5. Conclusion

ball Fig. 7. Representation of the porous powder cylinder entrapped between the colliding surfaces during a collision event (redrawn from Fig. 4 of Ref. [13]).

because it partly defines the volume of materials impacted per collision ðQ ¼ h0 pr2h Þ [13,14] thus that can react (Fig. 7) and, because rh is dependent on the ball radius [13], the volume of materials impacted is consequently in direct relation with the surface area of the ball’’. Another interesting point developed in this local model is that the height h0 of the powder cylinder entrapped between balls or ball and wall is inversely proportional to the charge ratio [12] (CR=mass of the balls/mass of powder=M=mp ). Thus, for the same mass and surface area of balls, h0 and consequently the powder cylinder volume entrapped are proportional to the mass of powder. For a given ball kinetic energy, the energy released to a volume unit of powder that will allow its reaction will be inversely proportional to the mass mp of powder ground and the kinetics constants will vary in the same way. In the previous paper [7], we showed that k and 1=tf effectively varied linearly with the inverse of the mass of powder ground: mp k and mp =tf =constant. These experimental results reported in the previous paper seem also in agreement with the theoretical models developed. In a first view, these theoretical considerations could explain the influence of the surface area of one ball but not necessarily of several balls. Indeed, if we consider a total mass M of balls, this mass can be obtained by taking, for example, four balls with a great diameter or 10 balls with a lower diameter. In this way the total surface area is increased (and consequently the volume of powder entrapped that can react) but the mass of each ball is decreased and consequently the ball kinetic energy is lowered. Moreover, by increasing the number of balls, the collision frequency is increased but, because at the collision time the ball is stopped, its mean velocity is certainly lowered, and therefore its kinetic energy. How are these different parameters linked? There is certainly a subtle compensation between them because we have obtained excellent linear correlations between the kinetic data and the product M Sa: But because the aim of our works was the quantification of the influence

These experiments have allowed the quantification of the changes in instrumental parameters on the kinetic constants of the mechanochemical synthesis of CDHA from DCPD and CaO. The kinetic parameters (k and 1=tf ) vary linearly with (i) the product of the total mass and surface area of the balls (M Sa), (ii) the square of the rotation velocity ðo2d Þ of the disc supporting the vial and (iii) the square of the eccentricity ðr2d Þ of the vial on the rotating disc. On a practical standpoint, these results show clearly that (i) under given experimental and instrumental conditions, mechanosynthesis is a reproducible technique, (ii) comparison or transposition is easy if all the experimental and instrumental parameters are well known, these parameters have to be well defined in the material section of papers treating on mechanosynthesis and, (iii) the kinetic study of a mechanochemically induced reaction is easily feasible and the results allow the optimization of synthesis conditions. This last point is of importance to minimize the pollution of the powder by the mill material [10,22]. On an academic point of view, mechanistic models developed to describe MA seem adequate for mechanochemical synthesis. Although these models were generally developed by considering only one ball, the transposition seems correct even when several balls are used. A kinetic constant is a good parameter to describe a reaction, and its variations can be used to develop and validate theoretical models. In this idea the reaction of DCPD with CaO, that is reproducible and relatively fast, could be an interesting tool.

Acknowledgements Authors thank B. Pauvert and G. Des for their technical assistance.

References [1] De Groot K. Bioceramics of calcium phosphate. Boca Raton, FL: CRC Press; 1983. [2] Daculsi G, LeGeros RZ, Nery E, Lynch K, Kerebel B. Formation of biphasic calcium phosphate ceramics in vitro: ultrastructural and physiological characterization. J Biomed Mater Res 1989;23: 883–94.

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[3] Daculsi G, Patussi N, Martin S, Deudon C, LeGeros RZ, Raher S. Macroporous calcium phosphate ceramic for long bone surgery in humans and dogs, clinical and histological study. J Biomed Mater Res 1990;24:379–96. [4] Van Blitterswijk CA, Grote JJ, Kuijpers W, Daems WTh, De Groot K. Macroporous tissue ingrowth: a quantitative and qualitative study on hydroxyapatite ceramic. Biomaterials 1986;7:137–43. [5] Serraj S, Boudeville P, Pauvert B, Terol A. Effect on composition of dry mechanical grinding of calcium phosphate mixtures. J Biomed Mater Res 2001;55:566–75. [6] Boudeville P, Pauvert B, Ginebra MP, Fernandez E, Planell JA. Dry mechanochemical synthesis of apatites and other calcium phosphates. In: Giannini S, Moroni A, editors. Bioceramics, vol. 13. Switzerland: Trans Tech Publications; 2000. p. 115–18. [7] El Briak-BenAbdeslam H, Mochales C, Ginebra MP, Nurit J, Planell JA, Boudeville P. Dry mechanochemical synthesis of hydroxyapatites from dicalcium phosphate dihydate and calcium oxide: a kinetic study. J Biomed Mater Res, in press. [8] Kanazawa T. Inorganic phosphate materials. Materials science monographs, 52. Kodansha Ltd. Tokyo and Elsevier Science Publishers: Amsterdam; 1989. p. 213. [9] Boudeville P, Serraj S, Leloup JM, Margerit J, Pauvert B, Terol A. Physical properties and self-setting mechanism of calcium phosphate cements from calcium bis-dihydrogenphosphate and calcium oxide. J Mater Sci Mater Med 1999; 10:99–109. [10] Suryanarayama C. Mechanical alloying and milling. Prog Mater Sci 2001;46:1–184. [11] Khina BB, Froes FH. Modeling mechanical alloying: advances and challenges. JOM 1996;48:36–8.

[12] Courtney TH, Maurice D. Process modeling of mechanics of mechanical alloying. Scr Mater 1996;34:5–11. [13] Maurice DR, Gourtney TH. The physics of mechanical alloying: a first report. Metal Trans A 1990;21A:289–303. [14] Magini M, Iasonna A, Padella F. Ball milling: an experimental support to the energy transfer evaluated by the collision model. Scr Mater 1996;34:13–9. [15] Lee GG, Hashimoto H, Watanabe R. Development of particle morphology during dry ball milling of Cu powder. Mater Trans JIM 1995;36:548–54. [16] Iasonna A, Magini M. Power measurements during mechanical milling, an experimental way to investigate the energy transfer phenomena. Acta Mater 1996;44:1109–17. [17] Abdellaoui M, Gaffet E. The physics of mechanical alloying in a planetary ball mill: mathematical treatment. Acta Metall Mater 1995;43:1087–98. [18] Chattopadhyay PP, Manna I, Talapatra S, Pabi SK. A mathematical analysis of milling mechanics in a planetary ball mill. Mater Chem Phys 2001;68:85–94. [19] Basset D, Matteazzi P, Miami F. Measuring the impact velocities of balls in high energy mills. Mater Sci Eng A 1994;174:71–4. [20] Dallimore MP, McCormick PG. Dynamics of planetary ball milling: a comparison of computer simulated processing parameters with CuO/Ni displacement reaction milling kinetics. Mater Trans JIM 1996;37:1091–8. [21] Le Brun P, Froyen L, Delaey L. The modelling of mechanical alloying process in a planetary ball mill: comparison between theory and in situ observations. Mater Sci Eng A 1993;161:75–82. [22] Nakano T, Tokumura A, Umakoshi Y, Imazato S, Ehara A, Ebisu S. Control of hydroxyapatite cristallinity by mechanical grinding method. J Mater Sci Mater Med 2001;12:703–6.