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Biomaterials 25 (2004) 3453–3462

Effect of the particle size on the micro and nanostructural features of a calcium phosphate cement: a kinetic analysis M.P. Ginebra*, F.C.M. Driessens, J.A. Planell Research Centre in Biomedical Engineering, Biomaterials Division, Department of Materials Science and Metallurgy, Technical University of Catalonia (UPC), Av. Diagonal 647, E08028 Barcelona, Spain Received 21 February 2003; accepted 12 October 2003

Abstract The aim of this work is to investigate the possibility of controlling the final micro and nanostructural features of a calcium phosphate cement by modifying the particle size of the starting powder, and to study the effect of this parameter on the kinetics of the setting reaction. The development of calcium phosphate materials with tailored structures at the micro and nanoscale levels could allow the modulation of some specific responses in biologic phenomena such as protein adsorption and cell adhesion, which strongly depend on the nano-sized roughness of the interface. It is shown that the higher specific surface, produced by the reduction of the particle size of the powder, strongly accelerates the hydrolysis of the a-TCP into calcium-deficient hydroxyapatite. The higher degree of supersaturation attained in the solution favours the nucleation of smaller crystals. Thus, by increasing the specific surface of the starting powder in a factor of 5, the size of the precipitated crystals is strongly reduced, and the specific surface of the set cement increases by a factor of 2. The reduction of the particle size produces a substantial decrease of the setting time and accelerates the hardening of the cement without significantly affecting the final strength attained. The mechanical strength achieved by the cement cannot be univocally related to the degree of reaction, without considering the microstructural features. r 2003 Elsevier Ltd. All rights reserved. Keywords: Calcium phosphate cement; Bone tissue engineering; Microstructure; Mechanical properties

1. Introduction Recent studies have put forward the significant role that both microstructure and nanostructure of surfaces and interfaces play in biological phenomena such as protein adsorption and bone cell adhesion in different materials [1], and specifically in hydroxyapatite ceramics [2]. These findings suggest that the capability to process materials with tailored structures at the micro and nanoscale level provides the possibility to elicit specific responses from surrounding cells and tissues. This is indeed relevant to optimise the in vivo response of materials, but it is even more crucial to design adequate substrates and scaffolds for bone tissue engineering, where the cell response can be more sensitive to the surface material properties. The applications of calcium phosphate cements as materials for bone regeneration have been increasing in *Corresponding author. Fax: +34-93-4016706. E-mail address: [email protected] (M.P. Ginebra). 0142-9612/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.biomaterials.2003.10.049

different fields such as orthopaedic surgery, dentistry, maxillofacial surgery and reconstructive surgery [3,4] since they were discovered by Brown and Chow in 1983 [5,6]. Moreover, they are also being used as candidate materials for the development of scaffolds for bone tissue engineering [7,8], as an alternative to the traditional porous-sintered ceramic scaffolds. This approach takes advantage of the fact that the final product is a low temperature precipitated hydroxyapatite, more reactive than the sintered one, and more similar to the biological apatites. In this context, the possibility to control the morphology of the interface of the calcium phosphate cements both at a microstructural and nanostructural level is a crucial issue. The chemical reactions that take place during the setting of a calcium phosphate cement depend on its chemical composition. However, in general, it can be stated that the setting and hardening of the cement paste is the result of dissolution and precipitation processes, and the particle size of the powder phase is a key factor

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that must be considered when the properties of the cement are optimised to comply with the clinical requirements of a given application. Indeed, the effect of the particle size of the cement powder on the setting and hardening properties has been extensively studied in other hydraulic cements, such as Portland cement [9–12]. As a matter of fact, the wellknown fast setting Portland cements have as one of their characteristics the small particle size of their powder [9]. In the field of calcium phosphate cements, some authors have studied the influence of the particle size of the reactants on the final mechanical properties of some cements, such as dicalcium phosphate (DCP)- and tetracalcium phosphate (TTCP)-based cements [13] or dicalcium phosphate dihydrate (DCPD)- and TTCPbased cements [14]. Also, the effect of this parameter on the pH, workability and setting times of a DCP-atricalcium phosphate cement has been studied [15]. However, little attention has been paid to the effect of this processing parameter on the final micro and nanostructural features of the cement and on the kinetic aspects of the setting reaction. The aim of this work is to verify the possibility to control the final micro and nanostructure of a calcium phosphate cement through the particle size of the initial powder phase. The cement studied has a-tricalcium phosphate (a-TCP) as the only active reactant. In previous studies, it has been shown that the setting and hardening of this cementing system is the result of the hydrolysis of a-TCP according to the following reaction: 3Ca3 ðPO4 Þ2 þH2 O-Ca9 ðHPO4 ÞðPO4 Þ5 OH:

ð1Þ

The progressive dissolution of the a-TCP particles and the simultaneous formation of an entangled network of precipitated calcium-deficient hydroxyapatite (CDHA) crystals determine the setting and hardening of the cement [16,17]. Many of the commercial calcium phosphate cements developed in the last years have aTCP as the main reactant [3]. In this work, the particle size of the powder phase is introduced as a variable, in order to determine its influence on the final structure of the cement at the micro and nanoscale range. The reaction kinetics and the underlying physico-chemical mechanisms responsible for the final structure and properties of the cement are investigated.

1300 C for 15 h and then quenching to room temperature in air an appropriate mixture of CaHPO4 (Merck 02144) and CaCO3 (Merck 2076). Two cement powders with different particle sizes were studied. To obtain them the a-TCP was milled for two different lengths of time (30 and 360 min) in an agate ball mill (Pulverisette 6, Fritsch GmbH) at two different rotating speeds (525 and 650 rpm, respectively). In both cements, a 2 wt% of precipitated hydroxyapatite (Merck 2143) was added as a seed in the powder phase. The liquid phase consisted of an aqueous solution 2.5 wt% of disodium hydrogen phosphate, Na2HPO4 (Merck 6586). The liquid-to-powder ratio used was 0.32 ml/g. 2.2. Powder characterisation The specific surface of the two TCP powders used was measured by nitrogen adsorption. The particle size distribution was measured by means of laser diffraction. The powder had been previously dispersed in ethanol in an ultrasonic bath. Hereafter, the fine powder will be coded as Powder F, and the coarse powder as Powder C, and the cements prepared by mixing each powder with the liquid phase as Cement F and Cement C, respectively. 2.3. Setting times and cohesion time Powder and liquid were mixed in a mortar for about 1 min, and the initial and final setting times of the cement paste were determined with Gillmore needles according to the C266-ASTM standard [18]. The cohesion time, defined as the minimum time after which the cement did not suffer disintegration when immersed in Ringer’s solution, was determined by visual inspection [19]. 2.4. Cement hardening To evaluate the strength development, cylindrical specimens with a diameter of 6 mm and a height of 12 mm were moulded. After 15 min, they were immersed in Ringer’s solution at 37 C for periods of 1, 2, 4, 8, 16, 32, 64 and 360 h. Eight samples were prepared for each cement and period. After the immersion, the samples were polished, removed from the mould and tested under compression at a cross-head speed or 1 mm/min in a Universal Electromechanical Testing Machine.

2. Materials and methods 2.5. Reaction kinetics 2.1. Cement preparation The cement powder consisted of 85% a-TCP and 15% b-TCP, according to quantitative X-ray diffraction (XRD) analysis [16]. It was prepared by heating in air at

Immediately after they had been tested in compression, the specimens were quenched in acetone and dried to stop the reaction. Two of the specimens were kept for microstructural characterisation, and the others were

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crushed in order to perform XRD. The extent of conversion or degree of reaction in function of the reaction time was calculated as described in a previous paper from XRD results, on the basis of the peak intensity variation by means of the external standard method [16]. 2.6. Microstructural development and specific surface The microstructural development was investigated by scanning electron microscopy (SEM) on the broken surfaces of the two specimens previously tested in compression and kept to this end. In these specimens, the reaction was stopped after different reaction times (0, 1, 2, 4, 8, 16, 32, 64 and 360 h) as described above. The fracture surfaces were gold-covered previous to SEM observation. The specific surface of the two cements after 360 h in Ringer’s solution was measured by nitrogen adsorption.

the setting of the cement was accelerated in a factor between 2 (I-T) and 5 (F-T), and a fast setting cement was obtained. 3.3. Compressive strength The strength development of both cements is shown in Fig. 4. In both cases, the evolution of the compressive strength with the reaction time was fitted to an exponential function. The value for Cs (compressive strength at saturation) was slightly higher for the Fine Cement, that is 41 (71.8) vs. 34.6 (71.0) MPa, the value for the Coarse Cement. A more significant difference was found in the hardening rate of the two cements, as reflected by the time constant t obtained for the different setting reactions, which indicates the time when the cement reaches a 63% of the compressive strength at saturation (Cs ). Thus, whereas the Coarse Cement had a value of t ¼ 28:2 (72.2) h, this value was

100 90 80 70 60 50 40 30 20 10 0 0.1

12

powder C

10 8 6 4 2

1

10

100

0 1000 12

powder F

10 8 6 4 2

1

10 size (µm)

100

diff volume (%)

Both cements gave a homogeneous paste, with acceptable wettability of the powder by the liquid. However, as could be expected, the paste was thicker for the cement that had a smaller particle size of the two (Cement F). The results obtained for the cohesion time (C-T), the initial setting time (I-T) and the final setting time (F-T) are given in Fig. 3. It can be observed that a reduction of the particle size produced a substantial decrease of both the setting times and the cohesion time. With a procedure as simple as a longer grinding of the powder,

cum volume (%)

3.2. Workability and setting parameters

100 90 80 70 60 50 40 30 20 10 0 0.1

diff volume (%)

The particle-size distribution of the two starting powders, coarse (C) and fine (F), as obtained by laser diffraction is shown in Fig. 1. The main parameters that characterise the particle-size distribution and the specific surface area are shown in Table 1. The powder morphology as observed by SEM is shown in Fig. 2.

cum volume (%)

3. Results 3.1. Powder characterisation

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0 1000

Fig. 1. Particle-size distribution of the two starting powders: coarse (Powder C) and fine (Powder F).

Table 1 Some particle-size parameters and specific surface of the two a-TCP powders used a-TCP

D(10) (mm)a

Median Size, D(50) (mm)

D(90) (mm)b

Particle size volume average, Mv (mm)

Particle size number average, Mn (mm)

Specific surface (m2/g)

Powder C Powder F

2.12 1.39

10.88 2.22

25.12 11.69

12.95 5.06

2.07 1.55

0.5470.04 2.7370.08

a b

D(10) means 10% of the powder particles are smaller than this value. D(90) means 90% of the powder particles are smaller than this value.

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Compressive strength (MPa)

45 40 35 30 25 20

C(t)= Cs(1-exp (-t/τ))

15 10

Cement F: Cs=40.1(1.8)MPa; τ=6.1(0.9)h Cement C: Cs=34.6(1.0)MPa: τ=28.2(2.2)h

5 0

(*standard deviation between brackets)

0

50

100

150

200

250

300

350

400

time (h) Fig. 4. Evolution of the compressive strength (C) of the two cements studied with time.

Fig. 2. Morphology of the two starting powders as observed by SEM: (a) Powder C; (b) Powder F.

CementC CementF

Time (min)

40 30 20 10 0 C-T

I-T

F-T

Fig. 3. Cohesion time (C-T), initial setting time (I-T) and final setting time (F-T) for the cements with coarse (cement C) and fine (cement F) particle size of the powder phase.

reduced to t ¼ 6:1 (70.9) h for the Fine Cement, which is a reduction in a factor of 4.5 by reducing the particle size of the cement.

3.4. Reaction kinetics The phase evolution during the setting reaction in the two cements studied is shown in Fig. 5, where the XRD

diagrams obtained after different reaction times are shown. The XRD diagrams for t ¼ 0 h correspond to the unreacted powder phase of the two cements. Both could be identified as a-TCP with approximately a 15 wt% bTCP formed during the quenching. A widening of the peaks was clearly noticed in the powder phase of Cement F, due to the crystallite size reduction caused by the more severe grinding. The evolution of the XRD diagrams with time showed that the cement setting was the consequence of a hydrolysis reaction, according to Eq. (1). Indeed, the a-TCP was progressively transformed into CDHA, while b-TCP remained unreacted, in agreement with the results published in previous studies [16]. The XRD diagrams showed that the disappearance of the a-TCP, and the simultaneous formation of CDHA was much faster in the Fine Cement than in the coarse one. Thus, in the Fine Cement, the presence of an apatitic phase could be clearly detected already after 2 h, and after 8 h it was already the predominant phase. In contrast, in the Coarse Cement, CDHA was hardly detected after 2 h and after 8 h its presence was scarce. Although the kinetics of the reaction was different, the transformation of the a-TCP into CDHA was complete in the two cements studied. Moreover, it was observed that the CDHA formed in Cement F had a lower crystallinity than that formed in Cement C, as indicated by the lower intensity and widening of the XRD peaks. The average size of the diffraction domains responsible for the Bragg reflection of the (0 0 2) planes was determined using the well-known Scherrer formula [20]. The values obtained for the (0 0 2) diffraction plane were 40 and 54 nm for the Fine and Coarse cements, respectively. The quantification of the amounts of the different phases present, calculated on the basis of the peak intensity variation using the external standard method,

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Cement F

Cement C α-TCP β-TCP CDHA

t = 0h

α-TCP β-TCP CDHA

t = 0h

t = 2h

I (cts)

t = 2h

I (cts)

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t = 8h t = 8h

t = 64h t = 64h

t = 360h t = 360h

20

(a)

30

40

20

50

2θ

30

40

(b)

50

2θ

Fig. 5. XRD diagrams of the two cements studied after different reaction times.

1.0 0.8 0.6

R=Rs(1-exp(-t/τ))

R

allowed to represent the degree of reaction R; as a function of the reaction time, as shown in Fig. 6. An exponential function was fitted, being the degree of reaction at saturation (Rs ) equal to 1 in both cases, since the transformation of a-TCP into CDHA was complete. The time constant t was four times smaller for the Fine Cement (5.570.1 h) than for the Coarse one (20.570.6 h). Therefore, as put in evidence previously by the faster cement hardening, the reduction of the particle size had a strong accelerating effect in the hydrolysis of the cement powder.

0.4 0.2

Cement C; Rs = 1; τ = 20.5(0.6)h Cement F; Rs = 1; τ = 5.5(0.1)h

0.0 -50

0

50

100

150

200

250

300

350

400

time (h)

3.5. Microstructural evolution and specific surface

Fig. 6. Degree of reaction (R) as a function of the reaction time for the two cements studied.

The microstructure of the Cement C and Cement F after 2, 8, 64 and 360 h of setting as observed by SEM is shown in Figs. 7 and 8, respectively. It can be observed that the a-TCP particles are much smaller in Fig. 8 than in Fig. 7. After 2 h of reaction (Figs. 7a and 8a), small crystals surrounding the initial powder could be observed. After 8 h (Figs. 7b and 8b), the amount of precipitated crystals increased in both cements, and two differential features were detected between the two cements: the size of the precipitated crystals was clearly smaller in the Fine Cement, and also their topological distribution was different. In the Coarse Cement, the precipitated crystals grew forming shells which surrounded the a-TCP particles, and some voids were located between these shells. In contrast, in the Fine Cement, a denser net of crystals filled the space between a-TCP particles, but there was some free space between the reactant particles and the precipitated crystals. This certainly will be of

relevance when the mechanisms controlling the reaction are discussed. After 64 h, the Coarse Cement presented a more crystalline aspect (Fig. 7c). Some unreacted particles were still detected. In contrast, the Fine Cement was nearly transformed into CDHA, and a-TCP particles were not detected (Fig. 8c). After complete reaction (360 h, Figs. 7d and 8d), it could be clearly observed that the crystallite size of the precipitated CDHA was much smaller for the Fine Cement than for the Coarse Cement. A micrograph of the Fine Cement at higher magnification is shown in Fig. 9. Another significant difference observed between the two cements was that, in the Fine Cement, the space occupied by the a-TCP particles was not completely filled by the small and compactly packed precipitated crystals, giving rise to small voids or pores all over the cement. This did not happen in the Coarse Cement, due

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Fig. 7. Evolution of the microstructure with the time of reaction for the Coarse Cement (Cement C) as observed by SEM: (a) after 2 h of reaction; (b) 8 h of reaction; (c) 64 h of reaction; (d) 360 h of reaction.

Fig. 8. Evolution of the microstructure with the time of reaction for the Fine Cement (Cement F) as observed by SEM: (a) after 2 h of reaction; (b) 8 h of reaction; (c) 64 h of reaction; (d) 360 h of reaction.

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Fig. 9. Final microstructure of the Fine Cement, at a higher magnification.

probably to the larger size and low packing density of the precipitated crystals. The specific surface of the Coarse and Fine set Cements, after 360 h reaction was 15.9670.03 and 30.3970.05 m2/g, respectively.

4. Discussion In general, the results obtained showed that the particle size is a key factor, which modifies significantly the final properties of the cement, and that specially affects the kinetics of the chemical reaction and the mechanical consolidation of the cement. Therefore, it is a parameter that can be very useful in order to adjust the behaviour of the cement to the specific clinical requirements of each different application. In many cases, the optimisation of the particle size could avoid the need to incorporate other additives to the calcium phosphate cements, for instance polymeric substances, which in many cases can worsen the mechanical properties of the cement and in some instances can compromise the excellent biocompatibility of the calcium phosphate cements. In this study, the powder phase of the cement was grinded following two different milling protocols. It has to be noted than, as could be expected in powders grinded in the same mill, the dispersion of the particle sizes obtained in the two cases was similar. As the effect of the more severe milling, the number of particles with a size under 3 mm was drastically increased. One of the consequences of this reduction in the particle size was the fact that the paste was thicker, and therefore less injectable, for Cement F than for Cement C. This was explained by the higher specific surface of the Fine Cement, and the higher amount of water adsorbed on the solid particles.

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As it was expected, the reduction of the particle size resulted in a cement with a much faster setting (Fig. 3). This behaviour can be explained as the result of the different interactions occurring in the cement paste. Indeed, it has to be kept in mind that, in general, both the chemical reaction that is taking place, and the physical attractive forces between the cement particles contribute to the cement setting. In the case of the Fine Cement, due to the higher specific surface of the reactants, the chemical reaction took place at a much higher rate at the initial stages. Additionally, the smaller the particles the stronger the electrostatic attractive forces were [21]. The same arguments can explain also the fact that the cohesion time was reduced when the fineness of the cement powder increased. Moreover, the reduction of the cohesion time is in agreement with the Kozeny–Carman equation, which relates the permeability coefficient of a porous solid with the inverse of the square of its specific surface [22]. The hardening of the cement was also accelerated when the particle size of the cement powder was reduced. Indeed, by increasing the specific surface of the powder by a factor of 5, the strengthening rate of the cement at the initial stages (CN =t) increased also by a factor of 5. However, the final compressive strength at saturation was only slightly higher for the Fine Cement than for the Coarse one. Physically, it is accepted that in calcium phosphate cements, the cement hardening is due to the entanglement of the crystals of the product phase, and therefore the morphology and size of these crystals should condition the final strength reached by the cement. As reported in the previous section, both the specific surface measurements and the microstructural study showed that the CDHA crystals formed in the Fine Cement were significantly smaller than those formed in the Coarse Cement. This can be explained by the higher degree of supersaturation attained by the dissolution of the powder with higher specific surface. Indeed, a higher supersaturation in the solution favours the formation of crystalline nuclei, giving rise to the precipitation of more crystals. When the supersaturation is lower, less nuclei precipitate and crystal growth is favoured. This result means that the surface properties (roughness, specific surface, porosity) of the interface between the material and the biologic environment (in vivo or in vitro) can be controlled by modifying the particle size of the initial powder used in the cement. And, as it has been shown in recent investigations, these surface properties, at the micro and nanostructural level can have a very strong influence on the protein adsorption and therefore on the osteoblast adhesion [1,2]. The reduction in crystal size was also assessed by calculating the crystallite dimensions by the Scherrer formula on the XRD patterns. However, the crystallite dimensions obtained in this way were much smaller than

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those deduced from the SEM images. Several reasons can explain this apparent disagreement: (i) in the cements studied there is a wide distribution of crystal sizes, and the smaller crystals cannot be observed by SEM at the magnifications employed. The size obtained by the Scherrer formula is an average value; (ii) the crystals that contribute to the peak widening are only those below 200 nm approximately. This indicates that, in the Fine Cement, the proportion of crystals below this size increase. Apparently, one would expect a higher strength the smaller the entangled crystals are, since in small crystals there are more contact points, and the porosity is lower. In this sense, Otsuka et al. reported an increase of the compressive strength of a calcium phosphate cement when the specific surface of the precipitated phase increased [14]. However, in our case, a more careful observation of the microstructures explained the apparent contradiction. In fact, in the Fine Cement, both the better packaging of the finer initial powder and the formation of smaller precipitated crystals should favour a more compact microstructure. However, the precipitated crystals were unable to fill the space occupied by the reactant particles of a-TCP, due to the fact that small crystals can fill the space more efficiently, leaving a smaller fraction of empty space between crystals, and some cavities appeared, in the places where the a-TCP particles were located (Fig. 8b–d). This gave as a result the appearance of many voids, as large as 1–10 mm, which had a weakening effect. In the case of the Coarse Cement, the precipitated crystals were larger and they filled less efficiently the space available, being less densely packed. Therefore, they were able to fill the space occupied by the reactants. Therefore, the microstructure was more homogeneous (Fig. 7c and d). This could explain the fact that the final strength did not increase when the particle size of the cement powder was reduced. Therefore, it can be stated that the reduction of the particle size accelerated the setting and hardening of the cement without affecting significantly the final strength attained. From a chemical point of view, the XRD analysis showed that the underlying mechanism that caused this effect was the acceleration of the hydrolysis of the a-TCP. Indeed, chemically, the mechanism of hardening consisted in the dissolution of the cement powder, and the precipitation of a product phase, in this case a calcium-deficient hydroxyapatite. Both mechanisms, dissolution and precipitation, were accelerated when the specific surface of the powder increased. Since the specific surface increased, the dissolution was favoured and the liquid phase around the solid particles became more supersaturated with respect to phosphate and calcium ions, favouring a more rapid precipitation of the product phase. The rate of the reaction at the initial stages (1/t) increased by a factor of 4 in the Fine

Cement F: K = 1.46; R0 = 0.33 Cement C: K = 1.17; R0 = 0.17

1.0 0.8

C/Cs

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C/Cs=K(R-R0)

0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

R

Fig. 10. Normalised strength (C=Cs ) vs. normalised degree of reaction (R=Rs ¼ R; since Rs ¼ 1) for the two cements studied.

Cement as compared to the Coarse one. This value was similar but somewhat lower to that found when the initial strengthening rate of the two cements was compared. Indeed, the comparison between Figs. 4 and 6 suggested that a relation could be established between the kinetics of the mechanical consolidation of the cement and the chemical reaction taking place. In Fig. 10, the normalised strength (C=Cs ) is represented vs. the normalised degree of reaction (R=Rs ; in our case is R because Rs ¼ 1). In order to find the best fitting equation to each group of experimental points, and to find the statistic significance between groups, a dummy variable multiple linear analysis was performed [23]. The fitted model was the following: C=Cs ¼ b0 þ b1 R þ b2 f þ b3 Rf ;

ð2Þ

where C=Cs was the normalised strength, R was the degree of reaction, bi are the fitted parameters and f is the dummy variable, which gets 0 for Cement F and 1 for Cement C. The fitted equation was the following: C=Cs ¼ 0:480 þ 1:46 R þ 0:276 f  0:291Rf :

ð3Þ

The statistics of the regression is summarised in Table 2. All parameters were significant (po0:05), and as consequence, it could be stated that, although in the two cements studied the strength acquisition depended linearly on the degree of reaction, both the slope and the intersect at the origin were statistically different. Therefore, for a given chemical composition of the cement, it is not possible to relate univocally the mechanical strength achieved by the cement to the degree of reaction of the same. Indeed, the particle size of the cement powder has a significant effect in this dependence, which has to be taken into account.

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Parameter

Coefficient

St.dev.

t-Ratio

p-Value

b0 b1 b2 b3

0.480 1.460 0.276 0.291

0.07 0.09 0.09 0.12

6.47 15.31 3.15 2.45

0.000 0.000 0.012 0.037

10 CEMENT F CEMENT C

8

δ (µm)

Table 2 Statistics of the dummy variable multiple linear regression of C=Cs vs. R

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δ = 0.14t

6 4 2

1/2

δ = 0.43t

0

According to the above results, the fitted equations for each cement were: Cement F : C=Cs ¼ b0 þ b1 R ¼ 0:480 þ 1:46 R ¼ 1:46ðR  0:33Þ;

δ = 0.09t

0

10

Time (h) Fig. 11. Depth of reaction (d) as a function of the reaction time for the two cements studied.

ð4Þ

Cement C : C=Cs ¼ ðb0 þ b2 Þ þ ðb1 þ b3 ÞR ¼  0:204 þ 1:17 R ¼ 1:17ðR  0:17Þ:

ð5Þ

When the fitted equations were written as C=Cs ¼ kðR  R0 Þ; R0 represented the degree of reaction that the cement has to surpass to achieve a detectable mechanical strength. Below this value, the strength of the cement was negligible. It could be observed that the parameter R0 was much higher for the Fine Cement than for the Coarse one. Also the slope (k) was higher for the Fine Cement, indicating that once R0 was attained, the increase of the compressive strength with the degree of reaction was more marked than in the Coarse Cement. These differences can be attributed to the different microstructure of the precipitated phase in each case as explained previously (size, shape and distribution of the crystals formed). In a previous study, a physico-chemical model was put forward to explain the kinetics of the hydrolysis of a-TCP in a cementing system, analogous to the one studied in this work, but with a different particle-size distribution [17]. From the temporal evolution of the depth of reaction, calculated from the degree of reaction data and the particle-size distribution, two rate-limiting mechanisms were proposed: initially the surface area of the reactants and, subsequently, the diffusion of the liquid phase through the hydrated layer formed around the reactants. These two stages were related to the formation of a shell-like structure that separated the reactants during the reaction [17]. The same calculations were performed with the two cements studied in this work, in order to elucidate whether the same mechanisms could be identified. Fig. 11 represents the temporal evolution of the depth of reaction for the two cements studied. In the Coarse Cement, the two controlling mechanisms could be identified, in two reaction steps: in the first one the reaction was controlled by the available surface area for

dissolution of the reactants (d ¼ K1 t), and in the second one it was controlled by diffusion of the liquid through the hydrated layer (d ¼ K2 t1=2 ). The transition time was around 20 h. However, in the Fine Cement, only one mechanism was identified, since the dependence of the depth of reaction with time was linear throughout the whole reaction, as shown in Fig. 11. This would suggest that the reaction was controlled only by the available surface area for dissolution, and therefore that the layer of products formed was not efficient to separate the liquid from the solid reactant. In fact, as it has been mentioned previously, it could be observed in the SEM microstructures that the formation of precipitated CDHA took place not directly on the reactant particles, but in the volume between particles (Fig. 8b), due to the higher supersaturation. A shell-like structure was not observed, and the voids were formed around the a-TCP particles and not between the shells of precipitated crystals, as was the case in the Coarse Cement. Consequently, the liquid and solid reactants would be in contact along the whole reaction and the specific surface of reaction would be the only rate-limiting parameter.

5. Conclusion The particle-size distribution is a variable that controls in a very significant way the kinetics of the setting reaction of the cement. The crystallite size of the final product can be strongly reduced by increasing the specific surface of the starting powder. This allows developing calcium phosphate materials with tailored structures at the micro and nanoscale levels, with the aim to modulate some specific responses in biologic phenomena such as protein adsorption and cell adhesion, which strongly depend on the structure of the interface at the nanometric range.

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Acknowledgements The authors thank the Science and Technology Spanish Ministry for funding this work through project CICYT MAT2002-04297.

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