In situ compressibility of carbonated hydroxyapatite in tooth dentine measured under hydrostatic pressure by high energy X-ray diffraction

journal of the mechanical behavior of biomedical materials 50 (2015) 171–179

Available online at www.sciencedirect.com

www.elsevier.com/locate/jmbbm

Research Paper

In situ compressibility of carbonated hydroxyapatite in tooth dentine measured under hydrostatic pressure by high energy X-ray diffraction Jean-Baptiste Foriena, Claudia Fleckb, Christina Krywkac, Emil Zolotoyabkod, Paul Zaslanskya,n a

Julius Wolff Institute, Charité – Universitätsmedizin, 13353 Berlin, Germany Materials Engineering, Technische Universität Berlin, 10623 Berlin, Germany c Helmholtz-Zentrum Geesthacht, 21502 Geesthacht, Germany d Department of Materials Science and Engineering, Technion – Israel Institute of Technology, 32000 Haifa, Israel b

art i cle i nfo

ab st rac t

Article history:

Tooth dentine and other bone-like materials contain carbonated hydroxyapatite nano-

Received 5 April 2015

particles within a network of collagen fibrils. It is widely assumed that the elastic

Received in revised form

properties of biogenic hydroxyapatites are identical to those of geological apatite. By

1 June 2015

applying hydrostatic pressure and by in situ measurements of the a- and c- lattice

Accepted 2 June 2015

parameters using high energy X-ray diffraction, we characterize the anisotropic deform-

Available online 12 June 2015

ability of the mineral in the crowns and roots of teeth. The collected data allowed us to

Keywords:

calculate the bulk modulus and to derive precise estimates of Young's moduli and

X-ray diffraction

Poisson's ratios of the biogenic mineral particles. The results show that the dentine

Apatite

apatite particles are about 20% less stiff than geological and synthetic apatites and that the

Hydrostatic pressure Bulk modulus Elastic properties Anisotropy

mineral has an average bulk modulus K¼ 82.7 GPa. A 5% anisotropy is observed in the derived values of Young's moduli, with E11 E 91 GPa and E33 E 96 GPa, indicating that the nanoparticles are only slightly stiffer along their long axis. Poisson's ratio spans νE 0.30– 0.35, as expected. Our findings suggest that the carbonated nanoparticles of biogenic apatite are significantly softer than previously thought and that their elastic properties can be considered to be nearly isotropic. & 2015 Elsevier Ltd. All rights reserved.

1.

Introduction

Carbonated hydroxyapatite (cHAp) mineral, known as dahllite, is the mineral found in bone and is one of the main constituents of dentine, the bulk-forming bio-composite material in n

Corresponding author. E-mail address: [email protected] (P. Zaslansky).

http://dx.doi.org/10.1016/j.jmbbm.2015.06.005 1751-6161/& 2015 Elsevier Ltd. All rights reserved.

teeth. Dahllite mineral deposits surrounding dental tubules form “peritubular dentine” (Fig. 1a and b) columns running through the tissue. The rest of the matrix of dentine is stiffened by cHAp particles (Johansen and Parks, 1960), surrounding the organic mesh, essentially composed of collagen

172

journal of the mechanical behavior of biomedical materials 50 (2015) 171 –179

Fig. 1 – Dentine microstructure: (a) Scanning electron microscope micrograph of a polished section of dentine obtained from the crown part of a tooth exhibiting tubules surrounded by a thick layer of mineral (peritubular dentine, white rings) and suspended in the mineralized collagen matrix (intertubular dentine: light grey). (b) Schematic representation of typical dentine microstructure (not drawn to scale).Tubules lined with mineral (light green) are surrounded by a matrix of mineralized collagen fibres (dark green). These in turn are made of clusters of mineral-containing nanofibrils, with the long plate axis corresponding to the collagen axis (c). The mineral particles in dentine (d) are compositionally similar to the mineral of bone, carbonated hydroxyapatite (dahllite); the c-axis of the apatite crystals corresponds to the long particle axis. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

protein nanofibrils (Jantou-Morris et al., 2010; McNally et al., 2012). The mineral and fibres combine to form the mineralized collagen fibre matrix of dentine (“intertubular dentine”), which resembles the matrix found in other members of the bone family of materials (Weiner and Wagner, 1998). Similar to all particle-reinforced composite materials and specifically similar to bone (Almer and Stock, 2007, 2005), the intimately attached organic phase (mainly collagen type 1) and cHAp particles carry the stress and deform jointly, when loaded by an external force (Deymier-Black et al., 2012, 2010). In this manner, strain energy is distributed throughout the dentine structures. The mineral provides stiffness in compression and thus the elastic properties of cHAp are of paramount importance for the rigidity and structural performance of the tissue. This has been shown in numerous models of bone and dental tissues (Bar-On and Wagner, 2012; Gao et al., 2003; Hellmich et al., 2004; Jäger and Fratzl, 2000; Vercher-Martínez et al., 2015; Zuo and Wei, 2007). It is technically very challenging to measure and derive the mechanical properties of biogenic hydroxyapatites. Consequently much of what is known is based on measurements of geological or other large-scale apatite samples. Relevant measurements reported for both cHAp and fluorapatites are summarized in Table 1. Initially, ultrasonic testing methods were employed to measure the apatite elastic constants (Bhimasenachar, 1945; Gilmore and Katz, 1982; Katz and Ukraincik, 1971; Sha et al., 1994; Tofail et al., 2009; Yoon and Newham, 1969) requiring either large naturally-occurring samples, compacted pellets or sintered powders. Many of these were measured under high pressure so as to remove the effects of porosity on the acoustic signals (e.g. (Gilmore and Katz, 1982)). More recent in silico (simulation) approaches rely on calculating apatite elastic constants by atomic modelling using force fields (Menéndez-Proupin et al., 2011; Mostafa and Brown, 2007), ab initio techniques (MenéndezProupin et al., 2011; Ren et al., 2013; Snyders et al., 2007), theoretical tensile test modelling (Ching et al., 2009), and

density functional theory computations (Li et al., 2015). Other work provided information about the bulk modulus and the zero-pressure volume of the apatite unit-cell, using a combination of X-ray diffraction and high-pressure experiments (Brunet et al., 1999; Matsukage et al., 2004). All the above have generated a consensus about estimates of Young's modulus of apatite mineral, on the order of 115–125 GPa. By extrapolation, similar apatite moduli can even be estimated from correlations between indentation modulus and mineral content measurements reported in teeth (Angker et al., 2004), from which the mineral is predicted to have a stiffness of 132 GPa. However, a curious discrepancy emerges, when examining more direct mechanical measurements obtained using classic techniques listed in Table 2, including nanoindentation (Snyders et al., 2007; Viswanath et al., 2007), and micromechanical three-point bending tests (Teraoka et al., 1998). While indentation measurements on geologic or synthetic apatites report the same high modulus range as mentioned above, indentation along enamel rods known to be composed essentially of biogenic cHAp (90 95 vol% (TenCate, 1998)) revealed lower stiffness, even when indenting specifically along the known crystal texture corresponding to the stiffer cHAp c-axis. The lower 85–95 GPa range of reported indentation moduli (Ang et al., 2010; Craig et al., 1961; Habelitz et al., 2001; Xu et al., 1998; Yilmaz et al., 2015) is typically attributed to the composite nature of enamel, known to contain minute amounts of organic material (Ge et al., 2005), found in the socalled interrod-regions between adjacent mineral clusters. Thus, assuming that the dentine bone-like nanoparticles and enamel mineral are made of similar cHAp, it is possible that tooth mineral is significantly less stiff than synthetic or other naturally occurring cHAp. To address this question, we performed direct measurements of the compressibility of hydroxyapatite nanoplatelets of tooth dentine, which we use to gain insight into other elastic properties of biogenic cHAp. We use X-ray

journal of the mechanical behavior of biomedical materials 50 (2015) 171 –179

173

Table 1 – Values of bulk and Young's moduli reported in the literature for different apatites, and corresponding back calculated elastic properties based on these literature reports. Author

Type of apatite

Experimental method

Elastic constants

Bulk [GPa]

E [GPa]

Experimental

Bhimasenachar (1945)

Durango FAp single crystal

Yoon and Newham (1969) Katz and Ukraincik (1971)

Durango FAp single crystal 490% pure powdered natural OHAp 490% pure powdered natural OHAp 490% pure powdered natural FAp Single crystal FAp

Gilmore and Katz (1982) Gilmore and Katz (1982) Sha et al. (1994) Brunet et al. (1999) Brunet et al. (1999) Matsukage et al. (2004) Tofail et al. (2009) Simulation Mostafa and Brown (2007) Mostafa and Brown (2007) Snyders et al. (2007) Ching et al. (2009) MenéndezProupin et al. (2011) MenéndezProupin et al. (2011) Ren et al. (2013) Li et al. (2015)

Back-calculated elastic properties E11 [GPa]

E33 [GPa]

v

v

prism

basal

Yes

83.9

–

133.7

91.9

0.20

0.36

Yes

–

119.0

119.6

145.3

0.23

0.31

Yes

89.0

114.0

114.3

138.4

0.23

0.31

No

89.0

114.0

–

–

–

–

No

94.0

122.7

–

–

–

–

Yes

92.5

120.0

124.3

146.3

0.24

0.31

No

97.5

–

–

–

–

–

Synthetic OHAp

Transmission of longitudinal sound waves Ultrasonic pulse superposition High-pressure ultrasonic measurement High-pressure ultrasonic measurement High-pressure ultrasonic measurement Ultrasonic measurement High-pressure XRD

Synthetic FAp

High-pressure XRD

No

97.9

–

–

–

–

–

Single natural apatite crystal Synthetic (NIST) powder OHAp

High-pressure XRD

No

91.5

–

–

–

–

–

Ultrasonic measurement

Yes

80.3

–

105.4

91.3

0.29

0.29

Forced field modelling Forced field modelling Ab initio

Yes

90.7

–

–

–

–

–

Yes

91.8

–

–

–

–

–

Yes

82.0

132.1

102

189

0.17

0.39

Tensile test modelling Ab initio and force field modelling

Yes

84.5

120.6

116.2

137.5

0.23

0.32

Yes

77.0

–

90.8

105.7

0.22

0.45

FAp

Ab initio and force field modelling

Yes

81.0

–

100.9

118.1

0.23

0.39

carbonated apatite FAp

Ab initio modelling

No

E60.0

E80.0

–

–

–

–

Yes

84.4

113.5

107.1

122.2

0.24

0.36

OHAp FAp OHAp OHAp OHAp

Table 2 – Mechanically determined apatite elastic modulus reported in literature. Author

Type of material

Experimental method

E [GPa]

Teraoka et al. (1998) Angker et al. (2004) Viswanath et al. (2007) Snyders et al. (2007) Ang et al. (2010) Craig et al. (1961) Xu et al. (1998) Habelitz et al. (2001) Ang et al. (2010) Yilmaz et al. (2015)

Synthetic OHAp Biological cHAP Single crystalline OHAp OHAp magnetron coated OHAp Enamel (along rods) Enamel (along rods) Enamel (along rods) Enamel (along rods) Enamel (along rods)

Bending test Indentation and SEM-BSE signals Nanoindentation Nanoindentation Nanoindentation Indentation Indentation Nanoindentation N anoindentation Micro-pillar compression

54–79 131.9 126 (E11) 135 (E33) 147.0 120–129 83 94 87.5 115–120 E 70.0

174

journal of the mechanical behavior of biomedical materials 50 (2015) 171 –179

diffraction combined with in situ hydrostatic water-mediated pressurization to track changes in the crystal unit-cell dimensions. Water fully infiltrates dentine, interpenetrating between the organic and mineral components, and thus by pressurization, we apply equal stress directly onto all the particles in the dentine bulk. This exposes the nanometre-sized apatite mineral platelets to truly isotropic, uniform compression. We determine the crystal deformations based on the directly measured changes in the d-spacings of the (002), (004) and (310) atomic planes, which we further use to calculate the mineral bulk modulus. Additionally, we estimate the Young's moduli and Poisson's ratios of the cHAp particles in dentine.

2.

Materials and methods

2.1.

Sample preparation

Canine and incisor teeth, extracted from jaws of freshlyslaughtered adult sows were stored in a chloramine 0.5% solution at 4 1C in a refrigerator until further use. Although partially worn down due to the conditions under which these industrial reproduction adult animals are kept, the large sizes of the selected caries-free pig teeth (as compared with human teeth) facilitated harvesting uniform large segments of dentine. Enamel in the crown, and cementum in the root, easily identifiable microscopically by the distinct microstructure lacking tubules, were removed. Slices (see schematic, Fig. 2a) were cut from three different teeth using a slow speed water-cooled diamond saw (Isomet Buehler Ltd. Lake Bluff, Illinois, USA) followed by grinding and polishing using emery sheets (water used as a coolant) to reach uniform thicknesses of about 800 mm. Parallelepiped-shaped samples were then prepared by longitudinally sectioning each slice along the axis of the micron-sized dental tubules. In total, 16 samples sized 8  5  0.8 mm3 were used for the compression/ diffraction experiments: 8 samples from the crown and 8 samples from the root.

2.2. Hydrostatic pressurization and diffraction experiments Samples were mounted in a custom-made stainless steel water pressurization chamber, specifically designed for synchrotron diffraction experiments, with entry and exit diamond windows encasing the sample and minimizing stray scattering (Krywka et al., 2014, 2008). Water, although slightly compressible (K E3.4 GPa), is an excellent means to apply pressure to the particles suspended between the collagen and other organic matrix components in tooth dentine tissues, as it fully penetrates the structure through the elaborate network of tubules and collagen fibres, and it forms an essential component of the collagen fibril network (Masic et al., 2015). By pressurizing the water that fills the tubules and saturates dentine, a uniform isotropic pressure is applied to the particles in the intertubular matrix, and thus to the mineral crystals. During the experiments, an initial pressure of 5 MPa was used, needed to remove any gas bubbles entrapped within the system. This was followed by measurements at 100, 200, 300 and 400 MPa. This range of relatively low loads (o500 MPa), as compared with other apatite pressurization experiments reported in the literature (e.g. (Brunet et al., 1999)) is likely to be higher than the typical stresses from maximum biting loads, reported to reach 500 N for a single tooth (Pruim et al., 1980). This range of loads is thus assumed to span the full magnitude of realistic pressures that teeth are exposed to in the natural physiological environment in vivo. X-ray synchrotron measurements were performed with the compression apparatus mounted at the P07 High Energy Materials Science beam-line (HEMS) operated by Helmholtz Zentrum Geesthacht (HZG) PETRA III storage ring (DESY, Hamburg, Germany). Diffraction patterns were measured using a 1  0.8 mm2 beam size at 53 keV (λ¼ 0.23 Å) with a 20 s exposure time for each pattern. A Mar345 image plate detector (Mar USA, Evanston, USA), positioned 1 m away from the sample chamber was used to collect diffraction patterns. Each sample was measured three times at each pressure (5, 100, 200, 300, and 400 MPa) while diffraction patterns were recorded. Detector orientation, rotation and sample-detector

Fig. 2 – Schematic representation of sample preparation and diffraction experiments: (a) Slices from pig teeth, cut with a low speed water cooled diamond saw were used to produce bar-shaped samples with sizes of  8  5  0.8 mm3 from root and crown dentine. (b) Diffraction patterns were collected while maintaining the samples hydrated during in situ pressurization. Peak position of the (002) and (310) planes were used to track the local strain in the dentine biocomposite corresponding to the c- and a- lattice parameters of the cHAp particles, respectively.

journal of the mechanical behavior of biomedical materials 50 (2015) 171 –179

distance were calibrated using lanthanum hexaboride (LaB6) standards. An illustration of the experimental set-up is shown in supplementary Fig. S1.

2.3.

Data analysis

The diffraction patterns were analyzed using the XRDUA analysis kit (De Nolf et al., 2014). Diffraction patterns (see typical example in Fig. S2) contain well-filled Debye–Scherrer rings with no pronounced preferred orientations. This implies an almost isotropic distribution of the HAp crystallite orientations in our samples. The apatite c-axis lattice parameter was determined by tracking the (002) and (004) diffraction peaks, appearing at scattering vectors, q¼ 1.78–1.87 Å  1 and 3.58–3.70 Å  1 respectively. The a-axis parameter was determined by tracking the (310) diffraction peak appearing at q¼ 2.26–2.90 Å  1. The respective Debye–Scherrer rings were azimuthally integrated over 3601 around the beam center and the resulting 1D-data in the q-space were then fitted conventionally, using a pseudo-Voigt function and subtracting a 1st order orthogonal polynomial background (De Nolf et al., 2014). Each diffraction pattern was corrected to remove the effects of water scattering, by subtracting the water diffraction patterns collected under the corresponding pressure. Experimental results, obtained from multiple measurements at each given pressure, were averaged and used to determine the mean crystal lattice parameters, a and c, under each pressure.

3.

Elastic constants determination

Hydroxyapatite crystals belong to the hexagonal symmetry system (Fig. 3) in which 5 independent elastic constants (C11, C12, C13, C33, C44) fully determine the tensor of elastic moduli (Nye, 1985; Zolotoyabko, 2011). Hydrostatic pressure (σ) applied to such crystals implies no shear deformation. As a result, both the strain (εik) and stress (σik) tensors have the following diagonal form in the standard Cartesian coordinate system shown in Fig. 3:

ε11 ¼ ε1

0

0

0

ε22 ¼ ε2

0

0

0

ε33 ¼ ε3

εik ¼

ð1Þ

σ 11 ¼ σ 1 ¼ σ

0

0

0

σ 22 ¼ σ 2 ¼ σ

0

0

0

σ 33 ¼ σ 3 ¼ σ

σ ik ¼

175

ð2Þ

Using Hooke's law and Eqns. (1) and (2), the relationship between strain and stress acting on each of the apatite nanocrystals in our experiment can be expressed as follows: σ ¼ C11 ε1 þ C12 ε2 þ C13 ε3 σ ¼ C12 ε1 þ C11 ε2 þ C13 ε3 σ ¼ C13 ε1 þ C13 ε2 þ C33 ε3

ð3Þ

Due to the hexagonal symmetry and by comparing the first and second equation in (3), we have ε1 ¼ε2 and the system of Eq. (3) reduces into σ ¼ ðC11 þ C12 Þε1 þ C13 ε3 σ ¼ 2C13 ε1 þ C33 ε3

ð4Þ

Solving system (4) following (Belen’ki| ̆ et al., 1988) yields the strain components ε1 and ε3 along the a- and c-axes of the hexagonal unit cell, respectively   C33  C13 Uσ ð5Þ ε1 ¼ ðC11 þ C12 ÞC33  2C13 2  ε3 ¼



C11 þ C12  2C13

ðC11 þ C12 ÞC33  2C13

2

Uσ

ð6Þ

Certainly, within linear theory of elasticity, the strains are proportional to the applied hydrostatic pressure σ. Additionally, the ratio η¼ ε1/ε3 is defined by the combination of the four elastic constants C11, C12, C13 and C33 η¼

ε1 C33 C13 ¼ ε3 C11 þ C12  2C13

ð7Þ

Thus, one fixed relationship between these four elastic constants in our system is given directly by the measurements of the strains, determined along the a- and c- apatite crystal axes (see typical experimental results for one sample, Fig. 4 inset). A second relationship between these constants under pressure is defined by the trace of the strain tensor, εik   ΔV C11 þ C12 þ 2C33  4C13 ¼ 2ε1 þ ε3 ¼ Uσ ð8Þ 2 V ½ðC11 þ C12 ÞC33 2C13 where ΔV/V depicts the unit cell volume change due to the pressure σ that we applied. The corresponding bulk modulus, K, is thus K¼

Fig. 3 – Orthogonal axes are defined using a standard Cartesian coordinate system with the X3 -axis oriented perpendicular to the basal facet, the X1 - axis being along one of the hexagon edges, and the X2 - axis, lying in the basal hexagon plane, perpendicularly to the X1 - axis.

½ðC11 þ C12 ÞC33 2C13 2 σ
¼ C11 þ C12 þ 2C33  4C13 ΔV=V

ð9Þ

Therefore, the bulk modulus K can also be directly determined from the measurements of lattice parameter changes measured under pressure. We note that the two measurable physical parameters obtained from our in situ hydrostatic X-ray experiment, namely η and K, do not allow us to unequivocally determine the four apatite elastic constants C11, C12, C13 and C33. However, our results allow us to provide reasonable estimates of these parameters in dentine cHAp, because much is known about the bounds on these constants from previously published

176

journal of the mechanical behavior of biomedical materials 50 (2015) 171 –179

results, as summarized in Table 1. It is thus possible to calculate the strain ratio, η, and bulk modulus, K, using Eqns. (7) and (9) for published stiffness constants, and to compare literaturebased results with our own measurements. We found that the magnitudes of η, most closely related to our experimental data, are those published by Menéndez-Proupin et al. (2011) based on measurements of Tofail et al. (2009). We thus used previously reported elastic constants (Menéndez-Proupin et al., 2011 see Table 1), as starting points for further refining the elastic constants (C11, C12, C13 and C33) to match our measured values of K and η. This was practically achieved by refining the constants so that the equations are constrained to fit both η and K of each of our experimental measurements and by minimizing the residual sum of square differences (between the four original C11, C12, C13 and C33 values, and the refined coefficients that give the best fit with our experimental data). The solver function of Microsoft Excels (2010) was used for this process. We note that with this fitting method, it is not possible to provide any insight about the fifth independent elastic constant in the hexagonal system, C44.

Fig. 4 – Comparison of cHAp unit cell volume changes with increasing pressure, normalized by the initial unit-cell volume at 5 MPa. By extrapolation, it is possible to fit data reported by Brunet et al., Matsukage et al. and Mostafa et al. (see text) for comparison with the low (o500 MPa) pressures used in our experiment; this plot reveals an excellent agreement between our unit cell measurements and previously reported geological data. Inset: example of typical relative changes in the a- and c- lattice parameters observed with increasing pressure.

The four refined stiffness constants may be further used to derive the compliances, Sik (Freund and Suresh, 2008): S11 ¼

C33 C11  C13 2

½C33 ðC11 þC12 Þ  2C13 2 ðC11  C12 Þ C11 þ C12

S33 ¼

C33 ðC11 þ C12 Þ 2C13 2 C33 C12 þ C13 2 S12 ¼   C33 ðC11 þ C12 Þ 2C13 2 ðC11 C12 Þ C13 S13 ¼ C33 ðC11 þ C12 Þ 2C13 2

ð10Þ

Thus, Young's moduli of our biogenic dentine cHAp crystals can then be determined from (Li, 1976; Nye, 1985): E11 ¼ E22 ¼ 1=S11 E33 ¼ 1=S33

4.

ð11Þ

Results and discussion

Our results show that by applying a rather low pressure onto the mineral cHAp particles in tooth dentine, the compressibilities and a range of elastic constants can be derived. Fig. 4 plots the average, normalized volume change in all our samples, observed as a function of increasing pressure on the mineral particles. The results reported by Brunet et al. (1999), Matsukage et al., (2004) and Mostafa and Brown (2007) for geological apatite are also reproduced in the range from 0 to 6000 MPa. Bearing in mind the differences in the high pressure ranges used previously (4500 MPa), as compared with the current results measured under loads (o500 MPa) that are presumably encountered in vivo, we still found the slopes of the trendlines in the data of Brunet et al., Matsukage et al. and Mostafa et al., i.e., respectively, 1.38  10  5, 1.19.10  5 and  1.15  10  5 MPa  1, to well fit our result: 1.18(70.02)  10  5 MPa  1. Firstly, this implies that the mineral in dentine indeed deforms in situ despite our rather low hydrostatic pressure. Secondly, mineral particles in dentine, seemingly, are not shielded by the tooth organic matrix. Indeed, in the natural state in the oral cavity, water is a part of the dentine nanocomposite (comprising about 20% by volume (TenCate, 1998)). Therefore, water surrounds the mineral nanoparticles and transmits the externally applied pressure. Consequently water is an excellent proxy to deliver

Table 3 – Measured K [GPa], η [unitless] and corresponding standard-error estimates (ΔK, Δη) of each sample. Root

K ΔK η Δη

PS11R 83.16 1.05 1.212 0.032

PS12R 80.71 3.67 1.556 0.231

PS13R 78.15 2.51 1.573 0.106

PS21R 87.00 4.31 1.435 0.144

PS22R 82.11 3.57 1.630 0.138

PS32R 84.85 6.35 1.634 0.478

PS33R 85.17 4.92 1.348 0.116

PS34R 78.49 3.79 1.210 0.130

PS11C 80.93 1.95 1.118 0.072

PS12C 81.88 1.42 1.570 0.179

PS13C 83.23 1.16 1.292 0.034

PS21C 88.69 3.15 1.411 0.191

PS22C 84.92 2.28 1.515 0.139

PS32C 80.39 4.18 1.320 0.110

PS33C 79.31 3.18 1.585 0.169

PS34C 83.33 2.33 1.345 0.071

Crown K ΔK η Δη

journal of the mechanical behavior of biomedical materials 50 (2015) 171 –179

mechanical load to the mineral particles in the bone-tissue material in our experiment. The change in the apatite unit-cell volume takes place due to compression of the platelets containing these hexagonal crystals. The inset in Fig. 4 shows an example of the typical changes that we track in the a- and c-lattice parameters in a representative sample. We note the linearly increasing deformation along both axes with pressure. The ratio between the slopes of such curves, each representing the strain/stress relations for the different crystal axes during the experiment, yields the η-value as defined by Eq. (7). The bulk moduli of the individual samples, as well as the corresponding η values and their standard errors are presented in Table 3. Assuming that the bulk modulus is independent of the pressure for the relatively low and narrow load range used in our experiment, the resulting individual bulk moduli are calculated as the average of the bulk moduli of each load step derived from the relation K¼ V∂P/∂V. The average bulk modulus for our tooth samples is thus K¼82.772.9 GPa, which falls well within the range of previously reported measurements of cHAp bulk modulus (see Table 1). We found no statistically significant differences (t-test, p40.05) between bulk moduli determined for the root (K¼82.573.0 GPa) and crown (K¼ 82.872.8 GPa) regions. The values found for the η-range, 1.118–1.634, also fit the data that we back calculated from previous experiments, reported in the literature (Table 1). We found no statistically-significant difference between the measurement results obtained for root and crown dentine (t-test, p40.05). Due to slight variations in the slopes of the linear fits to both the a- and c-axis lattice parameters during each of the compression experiments, the η ratios contain an experimental uncertainty that allows us to determine Δη from standard errors in the fits. We used the range of Δη while refining the elastic constants, which have been published before by Menéndez-Proupin et al. (2011). We thus obtained upper and lower bounds of each refinement result, based on the error estimates of the η parameter. The stiffness matrix components, the derived Young's moduli and Poisson's ratios, as well as the residual sum of squares of the refinements, are presented in Supplementary Table S1. The calculated mineral moduli and Poisson's ratios are plotted in Figs. 5 and 6. Our elastic constants C11, C12, C13, C33 fall into the range of literature published values. When the elastic constants are converted into Young's moduli of the dentine mineral particles by using Eqns. (10) and (11), we observe that

Fig. 5 – Box plots of E11 and E33 Young's moduli in the root and crown samples.

177

Fig. 6 – Box plots of prismatic and basal Poisson's ratios in the root and crown samples.

the modulus, E33 ¼ 95.9 GPa, along the c-axis is about 5% higher than the modulus, E11 ¼ 91.1 GPa, along the a-axis. Indeed, according to the literature, geological apatite is slightly stiffer in the c-direction, as compared to the adirection (Brunet et al., 1999); the same is valid for our dental biogenic apatite. The prismatic (νprism) and basal (νbasal) Poisson's ratios for the hexagonal particles are derived from the compliance matrix (Tromans, 2011) using the following relationships: νprism ¼  ½ε2 þ ε3 =2ε1 ¼  ½ε1 þ ε3 =2ε1 ¼  ½S12 þ S13 =2S11

ð12Þ

νbasal ¼  ½ε2 þ ε1 =2ε3 ¼ S13 =S33

ð13Þ

While the basal Poisson's ratio (see Fig. 3) spans 0.33–0.36, averaging at νbasal ¼0.35, the prismatic Poisson's ratio appears to be smaller, νprism ¼0.30. This statistically significant difference (two sample t-test, equal variance, po0.05) reflects the small anisotropy in the elastic properties of the cHAp particles in the investigated dentine. We emphasize that the current work reports direct in situ measurements of strains along the c- and a-axes of cHAp which we used to determine the bulk modulus of dentine mineral. Our results are obtained with pristine cHAp particles still residing in the biocomposite and deformed within a relevant functional environment. Data were collected under low pressures, which span the range of mastication stresses typically encountered in vivo. While the extracted bulk modulus falls within the range previously reported for geological apatites, our derived Young's moduli (90–95 GPa) are lower than most of the previously published results (typically in the range of 110–120 GPa, see Table 1). We hypothesise that these numbers represent a true difference between the biogenic organically-embedded nanocrystals of cHAp and geological apatite. Biogenic cHAp particles in bones and teeth have typical dimensions of 100  25  3 nm3 (Weiner and Wagner, 1998), with the crystal c-axis aligned with the particle long axis. This suggests that in the thinnest direction corresponding to the a-axis of the crystals the platelets comprise about three unit cells only. As the mineral is tightly attached to the organic matrix within which it is embedded, it is likely that the mineral becomes effectively less stiff than its geological or synthetic counterparts due to changes in composition at the organic/mineral interfaces. Note that a possible reason for the somewhat reduced stiffness in biogenic cHAp may be

178

journal of the mechanical behavior of biomedical materials 50 (2015) 171 –179

potential intercalation of intra-crystalline organics in the course of biomineralization and slightly different stoichiometry, as compared with geological apatite. Whatever the reasons may be, the elastic properties of biogenic mineral in bone-like tissues, such as dentine, are significantly different from their geological counterparts..

5.

Conclusions

In situ high energy X-ray diffraction measurements under hydrostatic pressure (up to 400 MPa) allowed us to determine the bulk modulus and anisotropic compressibility characteristics of the mineral nanoparticles in the bone-like dentine tissue of teeth. This allowed us to refine four of the five elastic constants needed to describe the mechanical properties of biogenic carbonated hydroxyapatite (C11, C12, C13, C33). Consequently we summarize our main findings: a) The average bulk modulus K of cHAp mineral measured in pristine tooth dentine is 82.772.9 GPa. b) Based on the results in our investigated samples, we refined previously published elastic constants and found a narrow range of possible Young's moduli of biogenic cHAp. The mineral located in both root and crown dentine is less stiff than previously thought, showing that E11 ¼ 91.172.3 GPa and E33 ¼ 95.974.6 GPa. The prismatic and basal Poisson's ratios are, respectively, νprism ¼ 0.30 and νbasal ¼ 0.35. c) In an elastically isotropic body C11 ¼ C33, C12 ¼C13, C11  C12 ¼2C44 (Vitos, 2007; Zolotoyabko, 2011). Correspondingly, in that case E11 ¼ E33 (see Eqns. (10) and (11)). The small difference between E11 and E33 Young's moduli (about 5%) found in this work attests to the fact that the elastic properties of biogenic cHAp are nearly isotropic. Some remaining anisotropy is better revealed from the differences between the basal and prismatic Poisson ratios.

Acknowledgment This work was financially supported by the DFG through SPP1420. The Helmholtz-Zentrum Geesthacht (DESY) is gratefully acknowledged for beamtime allocation and we thank Norbert Schell and Rene Kirchhof for the excellent support at the P07 High Energy Materials Science beam-line (HEMS) PETRA III. The authors thank Peter Fratzl ( Max Planck Institute of Colloids and Interfaces, Potsdam, Germany) and John Currey (University of York, UK) for insightful feedback. E. Z. acknowledges the Technion Shore Fund in Advanced Composites for partial financial support.

Appendix A.

Supplementary Information

Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jmbbm.2015.06.005.

r e f e r e n c e s

Almer, J.D., Stock, S.R., 2007. Micromechanical response of mineral and collagen phases in bone. J. Struct. Biol. 157, 365–370. Almer, J.D., Stock, S.R., 2005. Internal strains and stresses measured in cortical bone via high-energy X-ray diffraction. J. Struct. Biol. 152, 14–27. Angker, L., Nockolds, C., Swain, M.V., Kilpatrick, N., 2004. Correlating the mechanical properties to the mineral content of carious dentine—a comparative study using an ultra-micro indentation system (UMIS) and SEM-BSE signals. Arch. Oral Biol. 49, 369–378. Ang, S.F., Bortel, E.L., Swain, M.V., Klocke, A., Schneider, G.A., 2010. Size-dependent elastic/inelastic behavior of enamel over millimeter and nanometer length scales. Biomaterials 31, 1955–1963. Bar-On, B., Wagner, H.D., 2012. Enamel and dentin as multi-scale bio-composites. J. Mech. Behav. Biomed. Mater. 12, 174–183. Belen’kiı˘, G.L., Salaev, E´.Y., Suleı˘manov, R.A., 1988. Deformation effects in layer crystals. Sov. Phys. Uspekhi 31, 434. Bhimasenachar, J., 1945. Elastic constants of apatite. Proc. Indian Acad. Sci. A 22, 209–214. Brunet, F., Allan, D.R., Redfern, S.A.T., Angel, R.J., Miletich, R., Reichmann, H.J., Sergent, J., Hanfland, M., 1999. Compressibility and thermal expansivity of synthetic apatites, Ca5(PO 4)3X with X¼OH, F and Cl. Eur. J. Mineral 11, 1023–1035. Ching, W.Y., Rulis, P., Misra, A., 2009. Ab initio elastic properties and tensile strength of crystalline hydroxyapatite. Acta Biomater. 5, 3067–3075. Craig, R.G., Peyton, F.A., Johnson, D.W., 1961. Compressive properties of enamel, dental cements, and gold. J. Dent. Res. 40, 936–945. De Nolf, W., Vanmeert, F., Janssens, K., 2014. XRDUA : crystalline phase distribution maps by two-dimensional scanning and tomographic (micro) X-ray powder diffraction. J. Appl. Crystallogr. 47, 1107–1117. Deymier-Black, A.C., Almer, J.D., Stock, S.R., Dunand, D.C., 2012. Variability in the elastic properties of bovine dentin at multiple length scales. J. Mech. Behav. Biomed. Mater. 5, 71–81. Deymier-Black, A.C., Almer, J.D., Stock, S.R., Haeffner, D.R., Dunand, D.C., 2010. Synchrotron X-ray diffraction study of load partitioning during elastic deformation of bovine dentin. Acta Biomater. 6, 2172–2180. Freund, L.B., Suresh, S., 2008. Thin Film Materials: Stress, Defect Formation and Surface Evolution, 1st ed. Cambridge University Press, Cambridge, UK. Gao, H.J., Ji, B.H., Jager, I.L., Arzt, E., Fratzl, P., 2003. Materials become insensitive to flaws at nanoscale: lessons from nature. Proc. Natl. Acad. Sci. USA 100, 5597–5600. Ge, J., Cui, F.Z., Wang, X.M., Feng, H.L., 2005. Property variations in the prism and the organic sheath within enamel by nanoindentation. Biomaterials 26, 3333–3339. Gilmore, R.S., Katz, J.L., 1982. Elastic properties of apatites. J. Mater. Sci. 17, 1131–1141. Habelitz, S., Marshall, S.J., Marshall Jr, G.W., Balooch, M., 2001. Mechanical properties of human dental enamel on the nanometre scale. Arch. Oral Biol. 46, 173–183. Hellmich, C., Barthe´le´my, J.-F., Dormieux, L., 2004. Mineral– collagen interactions in elasticity of bone ultrastructure – a continuum micromechanics approach. Eur. J. Mech. - A/Solids 23, 783–810. Ja¨ger, I., Fratzl, P., 2000. Mineralized collagen fibrils: a mechanical model with a staggered arrangement of mineral particles. Biophys. J. 79, 1737–1746.

journal of the mechanical behavior of biomedical materials 50 (2015) 171 –179

Jantou-Morris, V., Horton, M.A., McComb, D.W., 2010. The nanomorphological relationships between apatite crystals and collagen fibrils in ivory dentine. Biomaterials 31, 5275–5286. Johansen, E., Parks, H.F., 1960. Electron microscopic observations on the three-dimensional morphology of apatite crystallites of human dentine and bone. J. Biophys. Biochem. Cytol. 7, 743–746. Katz, J.L., Ukraincik, K., 1971. On the anisotropic elastic properties of hydroxyapatite. J. Biomech. 4, 221–227. Krywka, C., Krasnov, I., Figuli, R., Burghammer, M., Mu¨ller, M., 2014. Determination of silkworm silk fibroin compressibility using high hydrostatic pressure with in situ X-ray microdiffraction. Macromolecules 47, 7187–7193. Krywka, C., Sternemann, C., Paulus, M., Tolan, M., Royer, C., Winter, R., 2008. Effect of osmolytes on pressure-induced unfolding of proteins: a high-pressure SAXS study. ChemPhysChem 9, 2809–2815. Li, C.-X., Duan, Y.-H., Hu, W.-C., 2015. Electronic structure, elastic anisotropy, thermal conductivity and optical properties of calcium apatite Ca5(PO4)3X (X ¼F, Cl or Br). J. Alloy. Compd. 619, 66–77. Li, Y., 1976. The anisotropic behavior of Poisson’s ratio, Young’s modulus, and shear modulus in hexagonal materials. Phys Status Solidi A 38, 171–175. Masic, A., Bertinetti, L., Schuetz, R., Chang, S.-W., Metzger, T.H., Buehler, M.J., Fratzl, P., 2015. Osmotic pressure induced tensile forces in tendon collagen. Nat. Commun. 6, 5942. Matsukage, K.N., Ono, S., Kawamoto, T., Kikegawa, T., 2004. The compressibility of a natural apatite. Phys. Chem. Miner. 31, 580–584. McNally, E.A., Schwarcz, H.P., Botton, G.A., Arsenault, A.L., 2012. A model for the ultrastructure of bone based on electron microscopy ofion-milled sections. PLoS ONE 7, e29258. Mene´ndez-Proupin, E., Cervantes-Rodrı´guez, S., Osorio-Pulgar, R., Franco-Cisterna, M., Camacho-Montes, H., Fuentes, M.E., 2011. Computer simulation of elastic constants of hydroxyapatite and fluorapatite. J. Mech. Behav. Biomed. Mater. 4, 1011–1020. Mostafa, N.Y., Brown, P.W., 2007. Computer simulation of stoichiometric hydroxyapatite: structure and substitutions. J. Phys. Chem. Solids 68, 431–437. Nye, J.F., 1985. Physical Properties of Crystals: Their Representation by Tensors and Matrices. Oxford University Press, Oxford, UK. Pruim, G.J., de Jongh, H.J., ten Bosch, J.J., 1980. Forces acting on the mandible during bilateral static bite at different bite force levels. J. Biomech. 13, 755–763. Ren, F., Lu, X., Leng, Y., 2013. Ab initio simulation on the crystal structure and elastic properties of carbonated apatite. J. Mech. Behav. Biomed. Mater. 26, 59–67.

179

Sha, M.C., Li, Z., Bradt, R.C., 1994. Single-crystal elastic constants of fluorapatite, Ca5F(PO4)3. J. Appl. Phys. 75, 7784–7787. Snyders, R., Music, D., Sigumonrong, D., Schelnberger, B., Jensen, J., Schneider, J.M., 2007. Experimental and ab initio study of the mechanical properties of hydroxyapatite. Appl. Phys. Lett. 90, 193902. TenCate, A.R., 1998. Oral histology: development, structure, and function. Mosby-Elsevier, Philadelphia, PA, USA. Teraoka, K., Ito, A., Maekawa, K., Onuma, K., Tateishi, T., Tsutsumi, S., 1998. Mechanical properties of hydroxyapatite and OH-carbonated hydroxyapatite single crystals. J. Dent. Res. 77, 1560–1568. Tofail, S.A.M., Haverty, D., Cox, F., Erhart, J., Ha´na, P., Ryzhenko, V., 2009. Direct and ultrasonic measurements of macroscopic piezoelectricity in sintered hydroxyapatite. J. Appl. Phys. 105, 064103. Tromans, D., 2011. Elastic anisotropy of HCP metal crystals and polycrystals. Int. J. Res. Rev. Appl. Sci, 6; 462–483. Vercher-Martı´nez, A., Giner, E., Arango, C., Javier Fuenmayor, F., 2015. Influence of the mineral staggering on the elastic properties of the mineralized collagen fibril in lamellar bone. J. Mech. Behav. Biomed. Mater. 42, 243–256. Viswanath, B., Raghavan, R., Ramamurty, U., Ravishankar, N., 2007. Mechanical properties and anisotropy in hydroxyapatite single crystals. Scr. Mater. 57, 361–364. Vitos, L., 2007. Computational quantum mechanics for materials engineers: the EMTO method and applications. Springer Science & Business Media, Springer-Verlag, London, UK. Weiner, S., Wagner, H.D., 1998. The material bone: structuremechanical function relations. Annu. Rev. Mater. Sci. 28, 271–298. Xu, H.H.K., Smith, D.T., Jahanmir, S., Romberg, E., Kelly, J.R., Thompson, V.P., Rekow, E.D., 1998. Indentation damage and mechanical properties of human enamel and dentin. J. Dent. Res. 77, 472–480. Yilmaz, E.D., Jelitto, H., Schneider, G.A., 2015. Uniaxial compressive behavior of micro-pillars of dental enamel characterized in multiple directions. Acta Biomater. 16, 187–195. Yoon, H.S., Newham, R.E., 1969. Elastic properties of fluorapatite. Am. Mineral. 54, 1193–1197. Zolotoyabko, E., 2011. Basic Concepts of Crystallography. WileyVCH, Weinheim, Germany. Zuo, S., Wei, Y., 2007. Effective elastic modulus of bone-like hierarchical materials. Acta Mech. Solida Sin. 20, 198–205.