Alix C Deymier
email: alix.c.deymier@gmail.com

Arun K Nair

Baptiste Depalle

Zhao Qin

Kashyap Arcot

Christophe Drouet

Claude H Yoder

Markus J Buehler

Stavros Thomopoulos

Guy M Genin

Jill D Pasteris
email: pasteris@levee.wustl.edu

Protein-free formation of bone-like apatite: New insights into the key role of carbonation

Keywords:

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

a b s t r a c t

The nanometer sized plate like morphology of bone mineral is necessary for proper bone mechanics and physiology. However, mechanisms regulating the morphology of these mineral nanocrystals remain unclear. The dominant hypothesis attributes the size and shape regulation to organic mineral in teractions. Here, we present data supporting the hypothesis that physicochemical effects of carbonate integration within the apatite lattice control the morphology, size, and mechanics of bioapatite mineral crystals. Carbonated apatites synthesized in the absence of organic molecules presented plate like morphologies and nanoscale crystallite dimensions. Experimentally determined crystallite size, lattice spacing, solubility and atomic order were modified by carbonate concentration. Molecular dynamics (MD) simulations and density functional theory (DFT) calculations predicted changes in surface energy and elastic moduli with carbonate concentration. Combining these results with a scaling law predicted the experimentally observed scaling of size and energetics with carbonate concentration. The experi ments and models describe a clear mechanism by which crystal dimensions are controlled by carbonate substitution. Furthermore, the results demonstrate that carbonate substitution is sufficient to drive the formation of bone like crystallites. This new understanding points to pathways for biomimetic synthesis of novel, nanostructured biomaterials.

Bone is a composite material composed of the protein type I collagen and plate like nanocrystals of mineral. The nanometer scale of the mineral crystallites has been shown to be essential to proper bone function and physiology. The nano size allows the crystallites to be flaw tolerant, thereby optimizing the fracture strength of the bone [START_REF] Gao | Materials become insensitive to flaws at nanoscale: lessons from nature[END_REF]. In addition, the large surface area of the nano plates increases the dissolution rate, which allows the body to quickly release necessary ions in response to pathologies such as acidosis [2e4]. Although the nanocrystal size and morphology are essential to bone's mechanical and physiological functions and must be considered in the creation of bone replacement bio materials, the mechanism by which these properties are controlled remains unclear.

The mineral phase in bone is often described as hydroxylapatite, a natural geologically occurring OH À containing calcium apatite. Geological hydroxylapatite forms large hexagonal prisms that derive from its hexagonal lattice structure [START_REF] Ropp | Encyclopedia of the Alkaline Earth Compounds[END_REF]. However, the min eral in bone resembles nanometer scale elongated plates or blades while still maintaining an internal hexagonal lattice structure [6e12]. More precisely, the nanocrystals of both bone and bio mimetic apatites exhibit a complex substructure composed of (1) an apatitic, often nonstoichiometric core containing ions in apatitic (i.e., lattice) chemical environments and (2) a surface layer con taining ions in non apatitic (i.e., non lattice) environments and surrounded by water molecules [START_REF] Rey | Surface properties of biomimetic nanocrystalline apatites; applications in biomaterials[END_REF]. In order to explain this discrepancy between the morphology of geological (inorganic) and biological apatites, many researchers have suggested that bone's crystallite size and shape are controlled via templating by or chemical interactions with surrounding proteins and organic moieties [10e12, [START_REF] Delgado-L Opez | Crystal size, morphology, and growth mechanism in bioinspired apatite nanocrystals[END_REF][START_REF] Blumenthal | Effect of carbonate and biological macromolecules on formation and properties of hydroxyapatite[END_REF]. It has been theorized that the fibrillar structure of collagen dictates the size of the crystallites via physical confinement within its nanometer sized gap zones [START_REF] Tomoaia | On the collagen mineralization[END_REF][START_REF] Landis | Structural relations between collagen and mineral in bone as determined by high voltage electron microscopic tomography[END_REF]. Others propose that proteins or other organic moieties bind to the mineral surface and control crystal growth [START_REF] Wang | A potential mechanism for amino acidcontrolled crystal growth of hydroxyapatite[END_REF][START_REF] Gu | Immobilization of a phosphonated analog of matrix phosphoproteins within cross-linked collagen as a templating mechanism for biomimetic mineralization[END_REF][START_REF] Delgado-L Opez | Crystal size, morphology, and growth mechanism in bioinspired apatite nanocrystals[END_REF][START_REF] Blumenthal | Effect of carbonate and biological macromolecules on formation and properties of hydroxyapatite[END_REF][START_REF] Chen | Refinement of collagen-mineral interaction: a possible role for osteocalcin in apatite crystal nucleation, growth and development[END_REF]. These results seem to suggest that organic interactions are necessary for the formation of plate like nanocrystals. However, what many of these studies fail to consider is the true composition of bioapatite.

The apatite structure is remarkably accommodating to sub stitutions [START_REF] Elliott | Calcium phosphate biominerals[END_REF]. The most common substitutions seen in bone mineral are the coupled substitution of CO 3 2-for PO 4 3À and Na þ for Ca 2þ , which lead to vacancies on the calcium and hydroxyl sites [18e20]. As a result, bone mineral is more accurately described as a carbonated, calcium and hydroxyl depleted apatite. The substitu tion of carbonate into the apatite structure can occur at one or both of two sites: in exchange for hydroxyl (OH À ), defined as A type, or for phosphate (PO 4

3À

), defined as B type [START_REF] Mcconnell | The problem of the carbonate-apatites III carbonate-apatite from magnet cove, Arkansas[END_REF]. In mammalian bone, carbonate levels span 2e8 wt%, depending on species and bone location, and are predominantly of B type substitution [18,22e24]. Empirical observations have qualitatively shown that carbonate content in the apatite lattice affects crystallite size [25e29], solu bility [START_REF] Okazaki | Solubility behavior of CO3 apatites in relation to crystallinity[END_REF][START_REF] Yoder | Synthesis, structure, and solubility of carbonated barium chlor-and hydroxylapatites[END_REF][START_REF] Grøn | The effect of carbonate on the solubility of hydroxylapatite[END_REF], and crystal mechanics [START_REF] Deymier | High energy xray diffraction measurement of load transfer between hydroxyapatite and collagen in bovine dentin[END_REF]. These results suggest that, in addition to possible organic interactions, carbonate con centration plays an essential role in regulating bone mineral morphology. Despite the importance of carbonate substitution to crystal properties, there is no proposed mechanism for this process. We hypothesized that, in the absence of proteins or other organic substances, carbonate substitution, through its control of crystal lattice spacing, atomic order, and solubility, is sufficient to control apatite crystals' nano scale size, shape, and mechanics via energetic effects [START_REF] Okazaki | Solubility behavior of CO3 apatites in relation to crystallinity[END_REF][START_REF] Markovic | Synthetical bone-like and biological hydroxyapatites: a comparative study of crystal structure and morphology[END_REF][START_REF] Yoder | Synthesis, structure, and solubility of carbonated barium chlor-and hydroxylapatites[END_REF][START_REF] Grøn | The effect of carbonate on the solubility of hydroxylapatite[END_REF]. This work proposes, for the first time, a direct mechanism for substitutional effects on crystallite morphology.

A protein free solution precipitation technique was used to prepare biomimetic apatites with carbonate concentrations ranging from 1.0 to 17.8 wt%. The composition, gross morphology, and lattice structure were measured experimentally, providing a comprehensive analysis of biomimetic carbonated apatites. Mo lecular dynamics (MD) and density functional theory (DFT) calcu lations, which were validated through comparison to experimental results, evaluated the energetics (i.e., conservation of energy and a trade off between surface energy and strain energy) of carbonate substitution. This work presents a situation in which nanoparticles are not created via templating to enhance the chemical properties, but rather in which the chemistry itself imposes a nanoscopic size. Organic templating is not necessary to form crystallites that exhibit physiological structures; rather, control of carbonate substitution is sufficient to precipitate crystallites with the correct nanometer scale size and morphology. Experimental and theoretical parame ters from these analyses were used in a continuum analysis to derive a proposed mechanism for how carbonate substitution controls crystallite size. Understanding this relationship provides new insight into the formation of biomimetic crystals for tissue engineering.

Methods

Synthesis of powders

Apatite powders were made according to previously described methods [33e35] and summarized here. All water used in the synthesis was either deionized water that was further purified by passing through a calcium hydroxylapatite column or doubly deionized Milli Q (18 MU) water in order to minimize the amount of fluoride present. Nitrogen was then bubbled through the water for several hours in order to remove dissolved carbon dioxide. Approximately 250 mL of water was placed in a large beaker and heated to the desired reaction temperature (usually 60 C or 85 C). Mixtures were stirred magnetically. In accordance with the desired degree of carbonate substitution, an appropriate amount of NaHCO 3 or Na 2 CO 3 (Acros Organics, 99.5% or GFS Chemicals, Pri mary Standard) was dissolved directly in the 250 mL reactor bath, and the pH was elevated to 9.0 ± 0.5 and maintained with 0.1 M NaOH (Acros, 97%). To the bath, 25 mL of a 0.15 M Ca(NO 3 ) 2 $4H 2 O (Sigma Aldrich, 99%) and 25 mL of a 0.09 M NaH 2 PO 4 (Alrich, 98%) solution were added at a rate of 1 drop/2e4 s. After addition was complete, the mixture was digested at temperature for 2 h. After the digestion, heating and stirring were discontinued. The mixture was cooled to room temperature and then filtered in a medium porosity glass filter crucible. The product was washed three times with warm water, air dried in a desiccator overnight, and then dried in a 120 C vacuum pistol or an evacuated vacuum desiccator for a minimum of 24 h. 20 powders were made with CO 3 2-/PO 4 3À molar ratios ranging from 0.2 to 12 (Table 1).

Carbonate concentration determination 1.2.1. CHN analysis

In order to accurately measure the carbonate concentration of the apatite powders Carbon Hydrogen Nitrogen (CHN) analysis was performed by Galbraith Labs, Knoxville, TN. Analysis was performed by combusting the samples and measuring the levels of CO 2 , H 2 O and N 2 (or NO x ) by either an infrared cell or a thermal conductivity detector. Levels were reported as weight percents (wt %).

Raman analysis

Raman analysis was also used to determine the carbonate con centration as well as analyze the atomic order. The Raman micro probe apparatus (HoloLab Series 5000 fiber optically coupled Raman Microscope, Kaiser Optical Systems, Inc) has been previ ously described [START_REF] Wopenka | The tendon-tobone transition of the rotator cuff: a preliminary raman spectroscopic study documenting the gradual mineralization across the insertion in rat tissue samples[END_REF]. Spectra were acquired using a 532 nm laser focused to a ~1 mm spot size by an 80Â objective (N.A. 0.85) at a power of 10 mW on the sample surface. Scattered light was collected in 32 4 sec acquisitions through the objective lens and transferred to a 2048 channel CCD detector. Powders were manu ally compressed before analysis, and spectra were obtained at six independent locations within each powder.

Spectra for 20 of the powders were analyzed as described pre viously [36e39]. Briefly, the spectra were background corrected in the region of 700e1200 Dcm À1 and the peaks were deconvolved into their component bands by applying a mixed Gaussian Lorentzian peak fitting algorithm using the Grams32 software package (Galactic, Salem, NH). The relative carbonate concentration was determined from the ratio of the areas for the band associated with carbonate substituting for phosphate, i.e. 1070 Dcm À1 , and the n 1 P O stretch band of hydroxylapatite at 960 Dcm À1 (Fig. 1A).

Weight percent of carbonate was determined from a calibration curve developed in our laboratory. In addition, full width at half maximum (FWHM) of the 960 Dcm À1 peak was used to evaluate the relative atomic order of 12 of the samples. FWHM is inversely proportional to atomic order with a larger FWHM representing a less atomically ordered structure.

FTIR analysis

FTIR analysis was used to determine the carbonate concentra tion as well as the types of carbonate present in the samples. Transmission Fourier transform infrared spectroscopy (FITR) ana lyses were carried out at room temperature on a Thermo Nicolet 5700 spectrometer providing spectral acquisitions in the wave number range 400e4000 cm À1 . For each acquisition, 128 scans were recorded, with a resolution of 2 cm À1 . Of particular interest in this contribution were spectral analyses in the ranges 840 910 cm À1 and 1300 1600 cm À1 , where the vibrational modes n 2 (CO 3 ) and n 3 (CO 3 ) of carbonate ions in apatite compounds can be observed, respectively (Fig. 2). Prior to the comparison of spectral features (especially in terms of band integrated intensity), all spectra were baseline corrected and then normalized, taking as reference (fixed to an absorbance of 1) the maximum absorption between 1033 and 1048 cm À1 , depending on the samples, arising in the n 3 (PO 4 ) domain of phosphate ions.

The evaluation of carbonate concentrations via FTIR was un dertaken by following a recently reported methodology based on the peak area ratio between the n 3 (CO 3 ) and the n 1 n 3 (PO 4 ) contri butions [START_REF] Grunenwald | Revisiting carbonate quantification in apatite (bio)minerals: a validated FTIR methodology[END_REF]. This ratio was indeed previously demonstrated to be directly correlated to the carbonate concentrations of carbonated apatite compounds, whether of biological or synthetic origins, as validated by direct CO 2 coulometry titrations.

Solubility determination

An 80 mg sample of each of 6 carbonated apatite powders was added to 50 mL of phosphate buffer (prepared by dissolving 117.1 g NH 4 H 2 PO 4 in 1 L H 2 O) and equilibrated for 8 days at room tem perature. In each case, excess powder still remained in the beaker. Samples were then centrifuged for 10 min after which the super natant was removed and filtered through a 22 mm syringe filter. The pH was measured for each solution. The supernatant solutions were further filtered through 0.22 mm filters (PES membranes, Environmental Express), diluted, and acidified to 1% HNO 3 before analysis. Concentrations of elements were measured with induc tively coupled plasma mass spectrometry (ICP MS, Perkin Elmer). Laboratory blanks (1% HNO 3 ), calibration standards, and internal standards were included for quality control. Blanks were measured before, during, and after apatite samples. Standards for making calibration curves were prepared in a range of 0.05 mg/L to 20 mg/L for major elements (Ca, Mg, Na, and K). Internal standards (20 mg/L In, Li, and Bi) were added online to all samples and standards during ICP MS analysis. Weighted calibration curves were deter mined based on external standards and internal standards. The specific detection limit for each element was determined by ex amination of the blanks and the lowest concentration calibration standards. Phosphate concentrations were determined by the ascorbic acid method (wavelength at 880 nm for spectrophotom eter) [START_REF]Standard Methods for the Examination of Water and Wastewater[END_REF].

Assuming a simplified formula for carbonated apatite of Ca 10-

x Na 2x/3 (PO 4 ) 6-x (CO 3 ) x (OH) 2-x/3 , and considering 100% B type carbonation and the absence of protonated phosphate ions, x was calculated from the analyzed concentrations of Ca 2þ , Na þ , PO 4 3À , and CO 3 2-using Visual MINTEQ software. Formulas are listed in Table 1.

TEM

Small quantities of each of the 20 carbonated apatite powders were suspended in methanol and sonicated to minimize aggrega tion. Twenty microliters of each suspension was deposited on holey carbon TEM grids over filter paper. The grids were allowed to dry before imaging. The powders were imaged with a Hitachi H7500 (Hitachi High Technologies) equipped with a 5 megapixel AMT XR50 camera (Advanced Microscopy Techniques, Woburn, MA). All images were obtained at a beam energy of 80 kV.

XRD

X ray diffraction measurements were performed on 6 powder

Table 1

Values characterizing the composition of the apatite samples. Carbonate values in each sample were determined by CHN-analysis, Raman, and in some cases FTIR. The degree of carbonate substitution in the precipitated apatites was controlled by the CO 3 /PO 4 molar ratio in the precipitation solution. The crystallite dimension of the major axis was determined via visual inspection in the TEM. Stoichiometric formulas were calculated according to [START_REF] Montel | New concepts in the composition, crystallization and growth of the mineral component of calcified tissues[END_REF] based on the compositions acquired by ICP-MS and CHN-analysis.

Sample

Measured samples on a Bruker d8 Advanced X ray Diffractometer with a Cu Ka tube source and a LynxEyeXE detector (Bruker, Madison, WI). Scans were performed from 20 60 with a step size of 0.02 and an acquisition rate of 0.5 s/step. To avoid issues with preferential orientation, the sample was rotated at rate of 15 rotations per minute. The system voltage and current were set to 40 kV and 40 mA, respectively. Peak fitting was performed on the x ray diffraction patterns using Diffract.Suite Topas software. Structural refinement was used to fit the (002) and (300) peaks to obtain crystal size and lattice spacing along the c and a axes, respectively (Fig. 3). Rietveld analysis was also performed on the entire diffractogram to obtain values of coherent domain size.

1.6. Modeling

Molecular simulations

An interatomic force field was used that has been empirically derived specifically for hydroxylapatite based on a variety of experimental parameters, such as the crystal structure and elastic constants, bond dissociation energies, and vibrational data on the phosphate group [START_REF] Mkhonto | A computer modelling study of the effect of water on the surface structure and morphology of fluorapatite: introducing a Ca10(PO4)6F2 potential model[END_REF][START_REF] De Leeuw | A computer modelling study of the uptake and segregation of fluoride ions at the hydrated hydroxyapatite (0001) surface: introducing a Ca10(PO4)6(OH)2 potential model[END_REF]. In this force field, the non bonded in teractions are modeled by a Buckingham potential, the covalent bond by a Morse potential, and the bond bending term by a simple harmonic function. This force field shows very good agreement with both density functional theory (DFT) calculations and exper iments regarding the crystal structure, the elastic properties, and surface energies [44e46]. This potential has been developed to be compatible with existing carbonate potential and has been used to study the structure of carbonated apatites [START_REF] Peroos | A computer modelling study of the uptake, structure and distribution of carbonate defects in hydroxy-apatite[END_REF]. In order to reduce the computational cost of the simulations, the electronic polariz ability of the oxygen atoms has not been taken into account.

All molecular dynamics simulations were performed using the LAMMPS code [START_REF] Plimpton | Fast parallel algorithms for short-range molecular dynamics[END_REF]. Buckingham and Coulomb interactions were computed with a switch function that ramps the energy and force smoothly to zero between an inner and outer cutoff of 8 Å and 10 Å respectively. The bulk and surface energy models were simulated using periodic boundary conditions and Ewald summation for long range interactions. The strain was applied by imposing a deformation on the sample. The energies and stresses were recor ded after energy minimization.

Atomistic models of apatite crystals

For computational simplification, an 88 atom orthorhombic unit cell was used instead of the classical 44 atom hexagonal unit cell, to model stoichiometric hydroxylapatite. Hydroxylapatite crystallizes in a hexagonal lattice. However, the modeling of me chanical properties can be simplified by mathematically trans forming it into an orthorhombic cell [START_REF] Ching | Ab initio elastic properties and tensile strength of crystalline hydroxyapatite[END_REF](Fig. 4) (Material Studio 4.4, Accelerys, Inc.). To maintain the periodicity of the lattice in the orthorhombic description, it is necessary to double the unit cell along the b axis. As a result, the orthorhombic unit cell is twice as large as the hexagonal unit cell and contains 88 atoms. The initial lattice parameters were chosen to match reported experimental measurements with the initial values a 9.424 Å, b 16.322 Å, c 6.879 Å, a b g 90 [START_REF] Hughes | Structural variations in natural F, OH, and Cl apatites[END_REF]. The orthorhombic unit cell fea tures characteristics almost identical to those in the hexagonal unit cell and can include substitution. The model samples were created by replication of the unit cell in three dimensions to form a rect angular prism with initial dimensions 2a  b  2c. All the surfaces of the samples have been structured in their charge neutral stoi chiometric form [START_REF] Astala | First-principles study of hydroxyapatite surfaces and water adsorption[END_REF].

In this study, only B type carbonate substitution, which is the dominant substitution mechanism in bone mineral and in our synthetic carbonated apatites, was investigated. Also, the proton ation of some phosphate ions, leading to HPO 4 2-ions instead of PO 4 3À , was not considered in the calculations: the presence of divalent anions was only accounted for here by the presence of carbonate species. In order to introduce substitutions in the models, phosphate ions were randomly selected in the original structure and replaced with carbonate ions (Visual Molecular Dy namics [START_REF] Humphrey | VMD: visual molecular dynamics[END_REF]). To make the model computationally more tractable, charge neutrality was simplified by removing one calcium atom for every two carbonate substitutions to compensate for the electron loss occurring during the substitution process [START_REF] Astala | First principles investigation of mineral component of Bone: CO3 substitutions in hydroxyapatite[END_REF]. Thus, OH vacancies were not considered. The extent of substitution was chosen to span from 0 to 17 wt%, beyond the physiological range of carbonate substitution measured experimentally in bone apatite [START_REF] Elliott | Calcium phosphate biominerals[END_REF][START_REF] Gibson | Novel synthesis and characterization of an AB-type carbonate-substituted hydroxyapatite[END_REF][START_REF] Legeros | Calcium phosphates in oral biology and medicine[END_REF]. Models were created for 0, 1.5, 3.1, 6.3, 9.8, 13.4, and 17.3 wt% of carbonate sub stitution. The position and orientation of the carbonate ions were selected randomly to explore the influence of carbonate location on the structure properties. The substitution sites were chosen to be at least 4.5 Å apart. For each amount of substitution modeled, the properties presented here are the average of 5 simulations with different substitution distributions.

Density functional theory calculations

Density functional theory (DFT) was used to compute the lattice constants and elastic constants, of carbonate substituted apatite and end member hydroxylapatite. The QUANTUM ESPRESSO package was used for DFT calculations with the exchange and correlation interaction defined by PBE functional and for the elec tronecore interaction, ultrasoft pseudopotentials were used [START_REF] Vanderbilt | Soft self-consistent pseudopotentials in a generalized eigenvalue formalism[END_REF]. The energy cutoff of 60 Ry is used for plane wave basis sets and 360 Ry for charge density. For bulk calculations a 2 Â 1 Â 2 k point grid has shown convergence of energy (energy difference between two consecutive steps is smaller than 10 À6 Ry), and forces are less than 30 meV/Å (10 À3 Ry/Bohr). DFT calculations have been per formed for 0 wt%, 6 wt% and 13 wt% carbonate concentration on a single unit cell to determine the lattice and elastic constants. In the DFT models, the positions of the substitutions have been selected based on previous studies to minimize the total energy of the system [START_REF] Astala | First principles investigation of mineral component of Bone: CO3 substitutions in hydroxyapatite[END_REF].

Elastic constants

The elastic constants were computed by fitting Hooke's law for both stress strain (Molecular Static) and energy strain (DFT) re lations for infinitesimal deformation [START_REF] Menendez-Proupin | Computer simulation of elastic constants of hydroxyapatite and fluorapatite[END_REF][START_REF] Fast | Elastic constants of hexagonal transition metals: theory[END_REF]. Six different strain modes ε ab , a; b 2=x; y; z= were applied with each ranging from 1.0% to 1.0% with a step of 0.2. The ground state energy versus lattice strains were plotted to obtain elastic constants cor responding to each direction x, y and z. The following equation was used for the computation of the elastic constants:

C ijkl 1 U v 2 E total vε ij vε kl (1)
where U E total U is the total energy per unit volume in the samples, ε ij is the strain tensor, U is the unit cell volume of the computational model, and E total is the ground state unit cell energy. Due to the symmetry of the hexagonal crystals, hydroxylapatite can be considered as a transversely isotropic material and its stiffness matrix can be expressed as:

C 0 B B B B B B B B B @ C 11 C 12 C 13
C 44 0 0 0 C 44 0 0 0 C 11 C 12 2 1 C C C C C C C C C A (2)
1.6.5. Surface energies The surface energy is the excess energy associated with the creation of a free surface for a crystallite with a particular compo sition. To compute the surface energy, free surfaces were created by adding a 40 Å thick vacuum on each side of the sample in the di rection orthogonal to the surface of interest (Fig. 4). The unit cell was duplicated 2 times in the direction orthogonal to the surfaces in order to avoid interactions between the two created surfaces. The dimensions of the sample and vacuum layer were chosen to provide a minimum distance of 10 Å between the free surfaces to avoid interaction between the formed interfaces. After relaxation of the structure, the surface energy is given by: g

E S E B A ( 3
)
Where g is the surface energy, E S is the energy of the sample with free surface, E B the energy of the equivalent bulk crystal, and A the free surface area. This approach was used to determine the (001) plane surface energy (not considering here the presence of a hy drated layer on the crystallites).

1.6.6. Scaling laws for crystallite size Scaling laws for crystallite size were generated from the first principles simulation data using the Griffith Irwin framework and Weibull statistics. The probability of fragmentation during crystal growth dominated by extension in a single direction was taken to follow the Weibull distribution [START_REF] Weibull | A statistical distribution function of wide applicability[END_REF]:

Fðs; L; CÞ 1 exp L L o ðCÞ s s o ðCÞ mðCÞ ; (4)
where F is the probability of fragmentation as a function of the substitution induced lattice stress (s) for an apatite crystallite of length L and carbonate concentration C; L o (C), s o (C), and m(C) are material parameters that depend upon C. From data for a range of ceramics, it is reasonable to approximate m(C), the Weibull modulus as a function of carbonate concentration, as invariant with respect to C: m(C) m o. [START_REF] Askeland | The Science and Engineering of Materials Cenage Learning[END_REF] In the carbonated apatite system there is not an external applied load, but rather an internal change in lattice spacing that modifies the atomic organization and crystal energetics. This change in lattice spacing can be defined as a lattice strain, which is simply the change in spacing with carbonate con centration divided by the original spacing at 0 wt% carbonate:

It was therefore assumed that the distribution of flaw sizes does not vary with C, which follows from the assumption that m is also invariant. The scaling procedure involves identifying the parame ters L o (C o ) from the experimental crystallite size data, and applying E(C o ) and ε(C o ) from the modeling results.

Because the Weibull parameter m can be reasonably approxi mated as invariant with respect to C, the distribution of critical flaws can also be approximated as such. The scaling arises from Griffith Irwin arguments by building from the work of Freudenthal, who noted that the density n c (C) of critical flaws (size a>a c (C)) in a specimen arises from the defect density g(a) [START_REF]Statistical Approach to Brittle Fracture[END_REF]:

n C ðCÞ Z ∞ aC ðCÞ gðaÞda (6) Z ∞ aC ðCÞ m o 2L o ðC o Þ a C ðC o Þ mo 2 a mo 2 þ1 ! da (7) 1 L o ðC o Þ a C ðC o Þ a C ðCÞ mo 2 (8)
Noting that, at an average fragmentation, n C ðCÞ 1=ðA o L o ðCÞÞ,

L o ðCÞ L o ðC o Þ a C ðCÞ a C ðC o Þ mo 2 (9) K Ic ðCÞs o ðC o Þ K Ic ðC o Þs o ðCÞ mo ( 10
)
where K Ic is the mode I fracture toughness. This relationship is equivalent to noting from equation ( 4) and the work of Dobbs et al. [START_REF] Gao | Calibration of Weibull stress parameters using fracture toughness data[END_REF] that the size of the crystallite will be limited by the following equivalences:

L o ðCÞ L o ðC o Þ s o ðCÞ s o ðC o Þ m K Ic ðCÞ K Ic ðC o Þ m (11)
Continuing from equation [START_REF] Wang | A potential mechanism for amino acidcontrolled crystal growth of hydroxyapatite[END_REF] and noting that K 2 Ic ðCÞfEðCÞgðCÞ,

where gðCÞ is the surface energy:

L o ðCÞ L o ðC o Þ K Ic ðCÞ K Ic ðC o Þ mo s o ðCÞ s o ðC o Þ m ( 12
)
EðCÞgðCÞ EðC o ÞgðC o Þ mo 2 aEðC o ÞεðC o Þ aEðCÞεðCÞ mo ( 13
)
EðC o Þ EðCÞ gðCÞ gðC o Þ εðC o Þ εðCÞ 2 !mo 2 (14)
All values in equation ( 14) can be determined from the modeling and XRD results except for m, the Weibull modulus.

Results

Synthesis of carbonated apatite powders with variable carbonate concentrations

Twenty samples of carbonated apatite were synthesized via aqueous chemical precipitation methods [START_REF] Yoder | Synthesis, structure, and solubility of carbonated barium chlor-and hydroxylapatites[END_REF]. Carbonate levels were controlled by changing relative proportions of carbonate and phosphate containing chemicals during the precipitation process. Precipitated powders of carbonated apatite were removed from solution and rinsed before analysis.

Carbonate concentrations in the apatites were measured using Carbon Hydrogen Nitrogen (CHN) combustion analysis, Raman spectroscopy, and Fourier Transform Infrared (FTIR) spectroscopy. CHN combustion analysis indicated that the carbonated apatite powders contained between 1.6 and 22.4 wt% carbonate, Table 1. Raman measurements showed similar values in carbonate con centrations. Differences between the Raman and the CHN analyses were more pronounced in samples with low carbonate levels.

X ray diffraction analysis of sample 20, which appeared to contain 22.4 wt% carbonate, indicated the presence of calcite in the powder sample, as did Raman analysis. Using the carbonate con centrations determined from the CHN analysis and the simplified formula for carbonated apatite as well as the level of calcite determined from Rietveld analysis of the XRD pattern, it was calculated that the carbonated apatite lattice contained 17.8 wt% carbonate.

Types of carbonate substitutions in the carbonated apatite powders

FTIR analyses were carried out on 7 of the carbonated apatite samples, with a focus on examining vibrational contributions of carbonate. Two spectral domains were targeted: the range of 840e910 cm À1 for the analysis of the carbonate vibration mode n 2 (CO 3 ), and the range of 1300e1600 cm À1 for n 3 (CO 3 ).

The evolution of the n 3 (CO 3 ) domain for the different samples analyzed showed a general trend of increasing area under the carbonate peaks in this interval, with increasing overall carbonate concentration in apatite (Fig. 1A). In carbonated apatite systems, this spectral region is difficult to decompose, as each type of car bonate species (A type, B type, or labile surface carbonates denoted LC) roughly gives rise to two main contributions that are over lapping [START_REF] Rey | Bioactive ceramics: physical chemistry[END_REF]. The general FTIR profiles in this region, however, suggest that more than one type/site of carbonate ion was present in all samples, although in variable amounts. In particular, B type carbonate appeared to be predominant; the proportion of A type carbonate was significantly lower than that of B type; LC type (labile non apatitic surface) carbonate occurred in much lower but non negligible concentration.

The analysis of the n 2 (CO 3 ) band (Fig. 2B), with the detection of several shoulders, confirmed the existence of several types of car bonate environments in these samples, as was concluded from the observation of the n 3 (CO 3 ) domain. However, examination of the n 2 (CO 3 ) contribution in apatite samples was also complicated by the superimposition of the P OH vibration band of HPO 4 2-ions, around 870 875 cm À1 , as a broad halo at the bottom of the band. B type carbonate, and to a lesser extent A type, were clearly identi fiable in the spectra, with n 2 (CO 3 ) bands around 872 and 878 cm À1 , respectively. Also, the band relevant to labile surface carbonate species can give rise to a large contribution; its position is difficult to assess exactly but is usually considered close to the value (around 866 cm À1 ) noted for amorphous carbonate containing calcium phosphate [START_REF] Rey | Bioactive ceramics: physical chemistry[END_REF]. As a general trend, the overall area un der the peaks in this spectral region increased with the total car bonate concentration of the apatite samples. The reported carbonate values in Table 1 for FTIR are a sum of all of the measured carbonate species.

Determination of crystallite size

The size of the crystallites within the carbonated apatite pow ders was measured manually from transmission electron micro scopy (TEM) images of the dispersed powders. In all cases, the crystallites that composed the apatite powders appeared to be flat plates (Fig. 5). Large crystallites looked blade like, with clearly defined edges and faceting; however, as the crystals became more carbonated and smaller, the crystallite shape became less well defined and facets were harder to identify. The long axis of the plates decreased with increasing carbonate substitution, from 291.8 ± 222 nm to 22.1 ± 5.7 nm for an increase in carbonate concentration of 10 wt% (Table 1). The variability in the crystallites' long dimension also decreased with increasing carbonate substitution.

XRD was performed on 6 representative samples to obtain in formation about how lattice parameters of the carbonated apatite crystals were modified with increased carbonate content. XRD patterns indicated that all the samples were solely carbonated apatite, except for the sample containing 22.4 wt% total carbonate, which also exhibited small calcite peaks in XRD patterns and Raman spectra. As explained above, this calcite was accounted for and a true lattice incorporated carbonate concentration of 17.8 wt% was calculated for this sample. D spacing values calculated from the peak center for the (002) and (300) peaks indicated a 0.5% in crease in lattice spacing along the c axis (6.887e6.919 Å) and a decrease of 1.5% along the a axis (9.417e9.274 Å) as the carbonate value increased from 1 to 17.8 wt% (Fig. 3). With that same increase in carbonate, the coherent domain size, defined as the distance between grain boundaries or significant defects within the crys tallites, along the c axis decreased from 49.8 to 16.3 nm, repre senting a 67% decrease in domain size. The aspect ratio c:a of the coherent domains also decreased from 3.4 to 0.63 with the same change in carbonate.

The FWHM of the 960 Dcm À1 Raman band, which is inversely proportional to the degree of atomic order of the sample, increased with increasing carbonate concentration. The increase was not linear with respect to wt% CO 3 , but rather exhibited a decrease in slope with increasing carbonate level (Fig. 1).

Modeling of carbonated apatite

Molecular statics and molecular dynamics (MD) methods as well as density functional theory (DFT) were used to predict the lattice, elastic and surface energy values of hydroxylapatite and carbonate substituted apatite. These methods were applied to a geometrically simplified orthorhombic model of apatite, which actually is crystallographically hexagonal (Fig. 4). Since DFT calcu lations are computationally expensive, DFT was used to validate the MD predictions of elastic and lattice constants for the 0 wt%, 6 wt% and 13 wt% cases. Below, comparisons of both simulation and experimental results are discussed.

Modeled lattice constants and strain

The evolution of the modeled lattice parameters, as a function of carbonate concentration, is presented in Fig. 3BeC. Both DFT and MD results were in good agreement with experimental analyses. Introducing carbonate ions into hydroxylapatite led to a significant decrease in both a and b lattice constants, indicating a decrease in a in the true hexagonal unit cell. The transverse lattice parameters a and b decreased linearly with the amount of substitution, and the lattice strain rose to 1.5% for 13 wt% substitution according to the simulation results (Fig. 3C). In contrast, the lattice parameter along the crystal c axis increased linearly with carbonate substitution. The lattice strain in the c direction was more moderate than in the a and b directions, reaching 0.5% for 13 wt% substitution (Fig. 3B). The modeling results showed that the lattice strain along the c axis was more sensitive than along the a axis to the location of the substitution sites in the crystal structure.

Elastic moduli

Simulations indicated that carbonate substitution changes the mechanical properties of bone at the nanoscale, i.e., at the crys tallite level. The elastic constants of end member (non carbonated) hydroxylapatite computed using both DFT and MD (Table 2 and in Fig. 6) were in close agreement with previous simulations [START_REF] Menendez-Proupin | Computer simulation of elastic constants of hydroxyapatite and fluorapatite[END_REF][START_REF] Hughes | Structural variations in natural F, OH, and Cl apatites[END_REF] and experiments [START_REF] Gilmore | Elastic properties of apatites[END_REF]. The simulations predicted a decrease in moduli with increasing carbonate substitution, with significant differences for the constants C 33 , C 13 , and C 44 . These values decreased almost linearly as a function of carbonate concentration.

In both the MD and DFT calculations, the transversely isotropic behavior of the crystal was well conserved, as would be expected in the true hexagonal apatite crystal (Table 2).

Surface energy considerations

The computed surface energy value for the (001) surface using MD is given in Table 2. Introducing CO 3 2-into the crystal lattice led to a decrease in the surface energy for the (001) surface. For example, the computed surface energy value for the (001) surface decreased from g 1.04 J/m 2 for pure hydroxylapatite to g 0.53 J/m 2 for a sample containing 13 wt% of substituted carbonate (Table 2). This surface energy evaluation corresponds to an idealized estimate, not considering the presence of a non apatitic layer on the surface of the nanocrystals. Although this estimate is not fully representative of the nanocrystals' behavior, it provides insights into how the presence of carbonate ions in the crystalline core may influence the surface energies of a crystallite.

Discussion

The mineral in bone exhibits a nanometer scale size and plate like morphology. The nanometer size plays an essential role in bone mechanics allowing the mineral crystallites to be flaw insensitive, thereby increasing bone strength [START_REF] Gao | Materials become insensitive to flaws at nanoscale: lessons from nature[END_REF]. In addition, the high surface area of the nanocrystals allows for effective dissolution to regulate body pH [START_REF] Bushinsky | Decreased bone carbonate content in response to metabolic, but not respiratory, acidosis[END_REF][START_REF] Hankermeyer | Dissolution rates of carbonated hydroxyapatite in hydrochloric acid[END_REF]. Although the importance of the mineral nano morphology has been established, the mechanisms describing nano crystal formation, size control, morphology, and mechanics remain unclear despite extensive study [START_REF] Gibson | Novel synthesis and characterization of an AB-type carbonate-substituted hydroxyapatite[END_REF][START_REF] Xiao | Hydrothermal preparation of nanocarbonated hydroxyapatite crystallites[END_REF][START_REF] Markovic | Synthetical bone-like and biological hydroxyapatites: a comparative study of crystal structure and morphology[END_REF][START_REF] Yoder | Synthesis, structure, and solubility of carbonated barium chlor-and hydroxylapatites[END_REF][START_REF] Legeros | Apatite crystallites: effects of carbonate on morphology[END_REF][START_REF] Drouet | Apatite formation: why it may not work as planned, and how to conclusively identify apatite compounds[END_REF]. Here, wide ranging carbonate concentrations were examined experi mentally to establish relationships between carbonate concentra tion and crystallite size, crystallite shape, and lattice spacing. In addition, MD and DFT modeling approaches successfully inter preted the experimental results. The comprehensive approach of combining experiment with modeling allowed us to define the relationships between carbonate substitution and bioapatite crys tal properties in the absence of protein control. Understanding the precipitation of mineral nanocrystals in an in vitro system will help to elucidate physicochemically controlled mineralization mecha nisms in bone tissues and provide unique information for tailoring mineral properties in tissue engineering systems.

Successful creation of nano crystalline bone mineral analogues was accomplished by aqueous precipitation methods at well below 100 C. CNH combustion analysis, Raman spectroscopy, and FTIR spectroscopy showed that the suite of apatite samples under analysis contained carbonate levels from 1.6 to 17.8 wt%, which includes and extends beyond the physiological values of 2e8 wt% CO 3 2-in mammalian bone mineral [START_REF] Elliott | Calcium phosphate biominerals[END_REF][START_REF] Gibson | Novel synthesis and characterization of an AB-type carbonate-substituted hydroxyapatite[END_REF][START_REF] Legeros | Calcium phosphates in oral biology and medicine[END_REF]. Both biological and synthetic precipitation techniques result in some amount of car bonate deposition on the surface of the crystallites, i.e., not inte grated into the lattice structure of the nanocrystals, known as labile carbonate [START_REF] Rey | The carbonate environment in bone mineral: a resolution-enhanced fourier transform infrared spectroscopy study[END_REF]. Within the lattice, CO 3 2À can substitute in place of OH À (A type substitution), PO 4 3À (B type substitution), or both [START_REF] Mcconnell | The problem of the carbonate-apatites III carbonate-apatite from magnet cove, Arkansas[END_REF].

FTIR analysis of the samples indicated that the carbonate was indeed in the lattice and predominantly incorporated via B type substitution (with labile species in a non negligible amount), as shown in Fig. 2. In all tested samples, there were also peak shoul ders indicative of lesser amounts of A type substitution. Incorporation of the carbonate into the crystal lattice resulted in specific modifications of the apatite lattice structure. When car bonate substitutes for hydroxyl, the a axis lattice spacing increases [START_REF] Suetsugu | Structure analysis of a-type carbonate apatite by a single-crystal x-ray diffraction method[END_REF]. Conversely, B type substitution for PO 4 3À causes the a axis spacing to decrease and the c axis spacing to increase [18,67e69]. Lattice spacing values measured via XRD for the carbonated apatite powders showed that, for all of the apatites, the c axis spacing increased while the a axis spacing decreased with increasing car bonate concentrations (Fig. 3), indicative of predominant B type substitution, the substitution type most commonly seen in bone [START_REF] Bigi | Chemical and structural characterization of the mineral phase from cortical and trabecular bone[END_REF]. Analysis of the composition and lattice structure of the apatite crystals confirms that the aqueous precipitation technique pre sented here created some carbonated calcium apatites similar to those found in bone.

Examination of the TEM images indicated that the apatites exhibited decreased crystallite size with an increase in carbonate (Fig. 5). A carbonate related decrease in crystallite size has been reported previously in the literature [8,25,27e29,70]. At carbonate concentrations below approximately 10 wt% the maximum crys tallite dimension decreased rapidly from values nearing 1 mm to tens of nanometers. Above 10 wt% CO 3 À the crystallite size plateaued at around 30 nm (Table 1 and Fig. 7). Although there is no distinct shift in lattice spacing at this composition, a drastic change in the OH À resonance upon approaching 8 wt% carbonate substitution has been previously observed via NMR [START_REF] Wang | Impact of collagen confinement vs. ionic substitutions on the local disorder in bone and biomimetic apatites[END_REF]. Coherent domain size, the average distance between grain boundaries (or large defects in a crystalline structure), was measured via examination of the width of the XRD peaks and application of Scherrer's equation [START_REF] Noyan | Residual Stress: Measurement by Diffraction and Interpretation[END_REF]. Coherent domain size was smaller than the TEM visualized crys tallite size. This apparent inconsistency indicates that the visual ized crystallites were polycrystalline [START_REF] Ziv | Bone crystal sizes: a comparison of transmission electron microscopic and x-ray diffraction line width broadening techniques, connect[END_REF]. The aspect ratio of both the crystallites and the coherent domains within them also decreased according to both the TEM images and XRD patterns, as seen previously [START_REF] Zahedi | Synthesis and characterization of sodium-carbon apatite nano-crystals by chemical sedimentation method[END_REF]. This suggests that the relative growth rates along the dominant axes were in part controlled by the addition of carbonate. Carbonate concentration seems to be a major controller of crystallite size, maintaining the bone like mineral in a nano meter size range that most benefits the bone mechanics and physiology.

The morphology of the apatites was also distinctly affected by carbonate concentration (Fig. 5). The shape of biological and bio mimetic apatites has been described as plate like [7e9], needle like [START_REF] Liao | Morphological effects of variant carbonates in biomimetic hydroxyapatite[END_REF][START_REF] Markovic | Synthetical bone-like and biological hydroxyapatites: a comparative study of crystal structure and morphology[END_REF][START_REF] Barralet | Carbonate substitution in precipitated hydroxyapatite: an investigation into the effects of reaction temperature and bicarbonate ion concentration[END_REF], elongated drop like [START_REF] Markovic | Synthetical bone-like and biological hydroxyapatites: a comparative study of crystal structure and morphology[END_REF], rod like [START_REF] Okazaki | Solubility behavior of CO3 apatites in relation to crystallinity[END_REF][START_REF] Xiao | Hydrothermal preparation of nanocarbonated hydroxyapatite crystallites[END_REF], and sphe roidal [START_REF] Okazaki | Solubility behavior of CO3 apatites in relation to crystallinity[END_REF][START_REF] Barralet | Carbonate substitution in precipitated hydroxyapatite: an investigation into the effects of reaction temperature and bicarbonate ion concentration[END_REF]. Variations in the size and shape of crystallites are Table 2 Elastic moduli and lattice parameters for pure hydroxylapatite and for 6 and 13.4 wt% carbonated hydroxylapatite computed using both MD and DFT. The elastic constants (C ij ), the bulk modulus (B), and the elastic modulus (E) are expressed in GPa; the lattice parameters (a, b, c) are in Å, and the surface energies (g) are in J/m 2 . Experimental data from Refs. [START_REF] Hughes | Structural variations in natural F, OH, and Cl apatites[END_REF][START_REF] Gilmore | Elastic properties of apatites[END_REF][START_REF] Katz | On the anisotropic elastic properties of hydroxyapatite[END_REF] recognized to be a function of their composition and formation temperature [8,9,25e27,63]. The powders in this study exhibited plate like morphologies, in which two dimensions appeared much larger than the third, at all carbonate levels. At low levels of car bonate, the crystallites were larger and blade like, with visibly straight edges. As carbonate levels increased, the edges became less linear and more uneven. It has been suggested that the loss of obvious faceting on the crystallites may be caused by a decrease in atomic order associated with carbonate incorporation [START_REF] Okazaki | Solubility behavior of CO3 apatites in relation to crystallinity[END_REF]. This proposal is supported by our experimental results showing a decrease in both faceting and atomic order (Figs. 1 and5) with an increase in carbonate concentration. In the range of physiological carbonate concentrations for mammalian bone [START_REF] Elliott | Calcium phosphate biominerals[END_REF][START_REF] Gibson | Novel synthesis and characterization of an AB-type carbonate-substituted hydroxyapatite[END_REF][START_REF] Legeros | Calcium phosphates in oral biology and medicine[END_REF], the crystallite length (~50e100 nm) and plate like morphology are similar to the mineral sizes and shapes generally seen in bone [START_REF] Ziv | Bone crystal sizes: a comparison of transmission electron microscopic and x-ray diffraction line width broadening techniques, connect[END_REF].

Although the mechanisms for mineralization of the collagen matrix in bone are poorly understood, both collagenous and non collagenous proteins have been intimated as responsible for the plate like nano scale morphology of the apatite crystallites [10e12,14]. Collagen molecules self assemble into well organized fibrils that exhibit periodic gap and overlap zones every 67 nm [START_REF] Orgel | The in situ supermolecular structure of type I collagen[END_REF][START_REF] Miller | 1] the Collagens: an Overview and Update, Methods in Enzymology[END_REF]. It has been suggested that collagen residues or acidic non collagenous proteins promote mineralization in the gap zones by either physically or chemically constraining crystal growth into nanometer scale plates [10,75e78]. However, our results indicate that there is no need for collagen templating or interference from other organic moieties to produce crystallites that exhibit physio logical morphologies and nanometer sizes; instead, controlling carbonate concentrations is sufficient to precipitate such crystallites.

Comparison of the experimental XRD values with the DFT and MD models demonstrated that the models, which assumed 100% B type substitution, were successful in predicting changes in lattice spacing with increased carbonate concentration (Fig. 3BeC). The theoretically derived values, especially in the MD simulations, were overestimates of the experimentally measured changes in lattice spacing. This discrepancy may be a result of mixed A and B type substitutions in the powders or the release of lattice strain by additional substitutions, such as the incorporation of Na as a sub stitute for Ca, which was not accounted for in the models presented here. ICP MS analyses indicated that there was significant sodium substitution in the structure, which increased with carbonate incorporation, as seen in bone mineral [18e20]. Despite the Fig. 6. Elastic constants computed with both DFT and Molecular Statics for A) C11 and C33 and B) C12, C13, and C44. The results computed using the two methods show good agreement. Except for C12, all elastic modulus constants decrease significantly with increasing carbonate content in the apatite structure. Fig. 7. A) Normalized crystallite length vs. carbonate content as calculated from the modified Weibull analysis. Crystallite length exhibits a rapid decrease at low carbonate levels, which plateaus at higher carbonate levels, where the slope and endpoint of the drop are controlled by the Weibull modulus (m). B) Assuming a starting crystallite length of 400 nm at a CO 3 concentration of 0 wt%, the calculated variation in crystallite size as a function of carbonate content matches well with the experimental data when m 1. models' limitations, their success in predicting the lattice param eters of the carbonated apatites makes them credible for further investigation of the crystal properties.

The models were also used to calculate the mechanical prop erties and theoretical surface energies of the crystals, which are difficult to obtain experimentally. The model results showed that the elastic modulus of the crystals, i.e., their resistance to elastic deformation, decreased with increasing carbonate substitution. These results are in agreement with previous work that has shown that biological bone apatites, which contain carbonate, have a lower modulus compared to carbonate poor geological apatites [START_REF] Deymier-Black | Synchrotron X-ray diffraction study of load partitioning during elastic deformation of bovine dentin[END_REF][START_REF] Forien | In situ compressibility of carbonated hydroxyapatite in tooth dentine measured under hydrostatic pressure by high energy X-ray diffraction[END_REF]. If bone is considered to be a composite of mineral and collagen, changes to the mineral stiffness due to composition and size effects, as seen here, will have significant consequences for the overall bone mechanics.

The idealized estimated surface energy of the crystals (espe cially on the (001) face), not considering the presence of a non apatitic surface layer on the nanocrystals, also decreased signifi cantly as carbonate incorporation increased. A decrease in surface energy will promote the formation of nanocrystals with high sur face areas and low volumes. The trends in lattice strain, modulus, and surface/interfacial energy all indicate that the substitution of carbonate into the apatite structure results in important modifi cations to the crystal energetics.

Beyond simply correlating the changes in crystal energetics and size, a model was developed to elucidate the mechanistic rela tionship between these two factors. Estimation of crystallite size from first principles analysis is a well known challenge due to the number of atoms that must be considered [START_REF] Mutaftschiev | The Atomistic Nature of Crystal Growth[END_REF]. A scaling law was developed using the Griffith Irwin framework and Weibull statis tics [START_REF] Abernethy | The New Weibull Handbook[END_REF] to relate crystal energetics, including surface energy and the elastic parts of the lattice strain, to the experimentally measured crystallite size. As described in the Methods section, the Weibull type scaling law relating the probability F that a growing crystal with carbonate concentration C will fragment before reaching length L or elastic stress s:

where L o (C) and s o (C) are normalization parameters, and m is the Weibull modulus, a material parameter that is related to the en ergetics of surface formation and the distribution of flaws in a material; for a nano ceramic, a Weibull modulus on the order of 1 is expected. As appropriate for brittle materials that fracture at critical defects, this scaling law predicts an increased probability of failure with increasing crystallite volume, which is associated with an increased probability that the crystallite houses a critical defect. The statistical distribution of defect sizes was held invariant with respect to C; therefore, as described in the Methods section, m was also held invariant. This enabled us to test the hypothesis that the change in the crystal energetics due to carbonate substitution controls the crystallite size. Adapting the Weibull relationship for elastic strain of the crystal lattice due to atomic substitution and applying Griffith Irwin en ergetic arguments, the scaling relationship between carbonate content and crystallite size as related to crystal energetics is:

L o ðCÞ L o ðC o Þ EðC o Þ EðCÞ gðCÞ gðC o Þ εðC o Þ εðCÞ 2 !mo 2 ( 14
)
where E is the Young's modulus, g is the idealized surface energy on the growing crystal face, and ε is the lattice strain. All values on the right side of this equation can be determined from the modeling and XRD results except for m, the Weibull modulus. For all values of m, crystallite length decreased nonlinearly with carbonate con centration, with a rapid drop off at small carbonate concentrations followed by a plateau (Fig. 7A). The smaller the m value the steeper the drop in size towards an asymptote. As expected for a ceramic of this character, m z 1 provided a good fit to the experimentally determined trends for crystallite size vs. carbonate concentration (Fig. 7B); supporting our hypothesis that carbonate content con trols the nanometer scale crystallite size.

A necessary consideration was the possible effect of differences in the saturation state of the solutions from which apatites with increasing carbonate concentration were precipitated. The effect of saturation on the differences in crystallite size was investigated by measuring the solubility of individual samples. The reasoning was that differences in solubility at a fixed temperature could lead to variations in the degree of supersaturation given the constancy of calcium and phosphate in the synthesis solutions. Differences in the degree of supersaturation could in turn cause changes in nucleation rate from sample to sample. Solubility products, K sp , were measured from the ICP MS results. K sp values increased from 6 Â 10 À115 to 5 Â 10 À103 for apatite containing 3.6 to 17.8 wt% carbonate, respectively (Fig. S1). Although this is a large increase in solubility, it is unclear what effects this difference would have on the crystal nucleation and growth rate, since many factors, including the initial crystal shape and kinetic coefficient, are un known. In addition, it is possible that the increase in solubility with increasing carbonate is not related to the crystallite size, but rather to the change in crystal energetics. It has been shown that the solubility of hydroxylapatite powders is more closely related to the degree of atomic order of the crystallite than to the crystallite size [START_REF] Baig | Relationships among carbonated apatite solubility, crystallite size, and microstrain parameters[END_REF]. Furthermore, the presence of a non apatitic surface layer is expected to modulate solubility behaviors [START_REF] Rey | Bioactive ceramics: physical chemistry[END_REF]. In the samples examined here, the degree of atomic order of the samples was shown to decrease with increasing carbonate substitution. The carbonate induced change in lattice structure (XRD) and atomic ordering (Raman), and therefore the crystal energetics, was likely the most important contributor to the change in solubility [START_REF] Baig | Relationships among carbonated apatite solubility, crystallite size, and microstrain parameters[END_REF].

Conclusions

Through aqueous precipitation methods, biomimetic apatite nanocrystals were synthesized with carbonate concentrations from 1 to 17.8 wt%. The carbonate was present predominantly in B type substitution for phosphate, as found in natural bone mineral. The morphology of the crystallites was plate like and exhibited decreasing size, aspect ratio, and faceting with increasing carbon ate concentration, Fig. 8. These modifications to the morphology appear to be caused by a change in crystal energetics, including surface energies, as seen by the change in atomic order and lattice spacing with carbonate incorporation. In the range of physiological carbonate levels, 2e8 wt% CO 3 , the crystallites exhibited mor phologies similar to those seen in natural bone mineral. Organic (e.g., proteins, citrates) templating was not necessary for the for mation of bone like apatite nanocrystals. Instead, carbonate sub stitution within the physiological range was sufficient for the production of crystallites with the nano scale size, shape, and composition under the conditions presented here. MD and DFT models of B type substituted apatite were able to accurately predict relative changes in the lattice spacing of the crystals as well as in the compliance and theoretical surface energy of the apatites caused by carbonate incorporation. For the first time, a mechanism for how substitution controls morphology was proposed via a modified Weibull analysis. The resulting normalized size vs. car bonate concentration predictions successfully replicated the experimentally measured values. These results strongly suggest carbonation of apatites plays a critical role in controlling crystal energetics and thus the nanometer size of crystallites, which are essential to maintain proper bone mechanics and physiology. With the current focus on tissue engineering and development of bio mimetic materials, this study provides a better comprehension of the processes that underlie compositional control of size and shape of crystalline particles. Such understanding can enable the tailoring of crystallite size at the nanometer scale by modification of composition.

Fig. 1 .

 1 Fig. 1. A) Raman spectra for samples ranging from 1.6 to 17.8 wt% carbonate. B) Plot of the width of the 960 Dcm 1 Raman peak as a function of wt% carbonate content asdetermined by CHN analysis. The peak width is inversely proportional to the atomic order of the sample; Raman-derived atomic order of the samples decreases with increasing carbonate substitution.

Fig. 2 .

 2 Fig. 2. FTIR spectra in the ranges of (A) 1300 1600 cm 1 for the analysis of the carbonate vibration mode n 3 (CO 3 ), and the range of (B) 840 910 cm 1 for n 2 (CO 3 ). Peaks at 1462, 1412, and 872 Dcm 1 indicate B-type carbonate substitution for phosphate. Their intensity increases with increased carbonate content. Shoulders and small peaks at 1542, 1465, and 878 Dcm 1 indicate lesser levels of A-type substitution in most samples. In addition all samples exhibit the presence of surface labile carbonate (LC) by peaks at 1480, 1417, and 866 Dcm 1 .

Fig. 3 .

 3 Fig. 3. A) XRD pattern for samples containing 0, 5.2, and 17.8 wt% carbonate. The (002) and (300) peaks shift in opposite directions with increased carbonate content. These shifts represent changes in the lattice spacing. B) Lattice spacing along the c-axis as a function of carbonate content measured experimentally from the XRD pattern and theoretically via DFT and MD methods. c-axis lattice spacing increases with the addition of carbonate. C) Lattice spacing along the a-axis as a function of carbonate content measured experimentally from the XRD and theoretically via DFT and MD methods. a-axis lattice spacing decreases with the addition of carbonate.

C 12 C 11 C 13 C 13 C 13

 1211131313

Fig. 4 .

 4 Fig. 4. Schematic showing B-type carbonate (shown as wt%) substitution (atoms explicitly shown) for phosphate groups (only bonds shown) in the model. Apatite lattice models shown here are used for the MD and DFT calculations.

  ε(C) (D D o )/D o . As for substitutional solid solution strengthening within the Hume Rothery framework, a lattice strain of ε(C) along the length of the crystallite results in both elastic and inelastic ef fects on the lattice, of which we take the elastic part to be pro portional to ε(C): s aEðCÞεðCÞ, in which a is a constant of proportionality.L o (C) is chosen such that it represents the length expected to contain a single critical flaw when s s o (C). This choice is equiv alent to arbitrarily choosing a value F 1 1/e as the probability of fracture that defines the mean crystallite size at a certain value of C.

Fig. 5 .

 5 Fig. 5. TEM images of three carbonated apatites containing 1.6%, 6.6%, and 17.8% carbonate by weight. Right hand images are 4Â enlargements of indicated areas on the left. As carbonate content increases, the particles tend be smaller and have a greater propensity for aggregation. Scale bars are 500 nm for the left column, 100 nm for the right column.

Fðs

Fig. 8 .Figure S1 :

 8S1 Fig. 8. Schematic showing the changes in crystal morphology and energetics as a function of carbonate concentration. At low carbonate concentrations the bioapatite crystals are long faceted blades with lattice spacing close to that of geological hydroxylapatite. However, carbonate incorporation into the lattice changes the lattice parameters and decreases the atomic order, coherent domain size, and surface energy. This results in a decrease in crystal size, faceting, stiffness and aspect ratio.

  CO 3 wt% CO 3 /PO 4 in soln.

						Size (TEM)	Size Std. Dev.	Stoichiometric Formula
		CNH-Analysis	Raman	FTIR		nm	nm
	OHAp	0			0.0	71.0	52.1	Ca 10 Na 0 (PO 4 ) 6 (OH) 2
	1	1.625	2.5	1.2	0.0	133.6	101.3
	2	3.63	5.17		1.6	291.8	222.0	Ca 9.4 Na 0.4 (PO 4 ) 5.4 (CO 3 ) 0.6 (OH) 1.8
	3	4	4.1		0.2	171.2	92.2
	4	4	5.17	3.3	0.3	121.0	138.0
	5	5.05	5.05		0.9	109.3	50.5
	6	5.1	7.39	5.3	0.2	32.0	18.7
	7	6.25	6.95	5.5	0.4	40.6	17.6
	8	6.35	6.35		1.1	62.0	39.7
	9	6.55	6.55		0.5	58.2	25.4
	10	8.2	8.79	7.8	1.3	69.5	68.5
	11	9.15	10.2		2.5	85.2	78.0	Ca 8.6 Na 0.9 (PO 4 ) 4.6 (CO 3 ) 1.4 (OH) 1.5
	12	11.3	12.5		5	35.2	12.7	Ca 8.3 Na 1.1 (PO 4 ) 4.3 (CO 3 ) 1.7 (OH) 1.4
	13	11.6	11.6		3.3	26.5	12.8
	14	12	11.92	10.9	2.9	54.4	28.5
	15	12.8	12.35	12.7	4.3	68.1	28.0
	16	13.3	13.3		7.0	35.5	12.5
	17	13.6	13.5		4	22.1	5.7	Ca 8.0 Na 1.3 (PO 4 ) 4 (CO 3 ) 2 (OH) 1.3
	18	14	14		4.0	105.5	47.3
	19	15.5	15.5		12.0	33.7	10.4
	20	17.8 (22.4)	(23.7)		3.0	36.0	28.4	Ca 7.

3 Na 1.8 (PO 4 ) 3.3 (CO 3 ) 2.7 (OH) 1.1

  .

		0 wt% CO 3			6 wt% CO 3		13.4 wt% CO 3
		MD	DFT	Experimental	MD	DFT	MD	DFT
	Elastic Constants (GPa)
	C11	137.4 ± 0.3	118.58	137	126.6 ± 1.5	117.63	114.0 ± 2.5	118.21
	C22	138.2 ± 0.9			124.5 ± 2.6		117.0 ± 1.5
	C33	185.4 ± 0.3	166.38	172	140.0 ± 3.4	148.18	106.1 ± 1.7	128.02
	C12	49.4 ± 0.3	32.27	43	52.2 ± 1.2	49.28	48.0 ± 1.7	29.95
	C13	70.5 ± 0.1	65.73	55	53.6 ± 1.7	47.39	45.0 ± 1.1	45.35
	C23	70.7 ± 0.3			52.8 ± 2.2		43.6 ± 1.0
	C44	52.1 ± 0.1	40.4		37.1 ± 1.9	28.49	30.6 ± 0.9	24.26
	C55	51.4 ± 0.1			37.5 ± 1.8		30.0 ± 0.9
	C66	44.3 ± 0.1			35.4 ± 1.4		33.0 ± 0.7
	Moduli (GPa)
	B	91.7 ± 0.2	77.9	89	78.5 ± 1.4	74.3	67.5 ± 2.6	67
	E	120.9 ± 0.1	103.2	114	96.8 ± 2.1	87.7	82.8 ± 3.8	86.2
	Lattice Parameters (Å)
	a	9.36 ± 0.00	9.56	9.42	9.29 ± 0.08	9.5	9.24 ± 0.11	9.48
	b	16.19 ± 0.00	16.56	16.31	16.09 ± 0.01	16.43	15.92 ± 0.11	16.33
	c	6.87 ± 0.00	6.89	6.88	6.9 ± 0.04	6.93	6.93 ± 0.11	6.91
	Surface Energy (J/m 2 )
	g 001	1.04 ± 0.00			0.75 ± 0.04		0.54 ± 0.03

Acknowledgments

The authors would like to acknowledge Prof. Daniel Giammar at Washington University who performed all of the ICP MS experi ments. This project was funded by the National Institute of Health (NIH) R01 EB016422 and R01 AR057836. Dr. Deymier was funded in part by the National Space Biomedical Research Institute NSBRI RFA 13 01 Postdoctoral First Fellow Award. Arun Nair would like to acknowledge support from Department of Mechanical Engineering, University Arkansas. This work was supported in part by the Na tional Science Foundation under the grants ARI#0963249, MRI#0959124 and EPS#0918970, and a grant from Arkansas Sci ence and Technology Authority, managed by Arkansas High Per formance Computing Center.

Appendix A. Supplementary data

Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.biomaterials.2017.02.029.