The Ginstling-Brounstein (GB) model is applicable for describing diffusion-controlled processes, and its practical use has been investigated and reported for a wide range of applications [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18]. Known applications that mention the GB model for describing the kinetics include the hydration of the portland cement particles [1], solid-state synthesis of lanthanum manganite controlled by the three-dimensional solid-ionic diffusion [2], solid-state synthesis of the compound Zn_{2.5}VMoO_{8} [3], formation of magnesium ferrites [4], dehydration of the iron(III) phosphate dihydrate [5, 6], thermal degradation of carbohydrate polymers [7], oxidation and decomposition of three-dimensional braided carbon fibers [8], alkaline hydrolytic decomposition of the uncolored chromium leather wastes [9], thermal stability evaluation of silver sulfathiazole-epoxy resin networks [10], oxidation of boron powder [11], wollastonite fibre dissolution in acetic acid aqueous solution [12], xylan pyrolysis [13], thermal degradation of DGEBA epoxy crosslinked with natural hydroxy acids [14], and pyrolysis mechanisms of lignin-PVA blends [15]. The model has also been applied to some extent in the study of thermal modes of heterogeneous exothermic reactions [16] and the study of inclusion complexes in supramolecular host-guest architectures [17], as well as a methodological study on determination of kinetics from DTA and TG curves [18].

Wet precipitation (WP) is one of the most widely used hydroxyapatite (HAp) synthesis methods and is based on the reaction of calcium hydroxide with the orthophosphoric acid diffusing into the particle. Hydroxyapatite is the main mineral constituent of bones and teeth, and has found its application in various fields such as tissue engineering, orthopedics, prosthetics, drug transport and environmental remediation [19, 20, 21, 22, 23]. The orthophosphoric acid is added dropwise into the suspension. The reaction can be written as in the reaction Equation 1.

$$\begin{array}{}10\mathrm{C}\mathrm{a}(\mathrm{O}\mathrm{H}{)}_{2}+6{\mathrm{H}}_{3}\mathrm{P}{\mathrm{O}}_{4}\to \mathrm{C}{\mathrm{a}}_{10}(\mathrm{P}{\mathrm{O}}_{4}{)}_{6}(\mathrm{O}\mathrm{H}{)}_{2}+18{\mathrm{H}}_{2}\mathrm{O}\end{array}$$(1)

In this work, the diffusion model is analyzed in the context of diffusion of orthophosphoric acid into the calcium hydroxide particle in a water suspension. However, it has to be noted, that in the studied case, the chemical reaction is much faster than the diffusion, thus, the orthophosphoric acid is diffusing into the reaction product (hydroxyapatite) [24] in reality. The model is somewhat simplified, since it does not account for the presence of byproducts which are normally created in the case of hydroxyapatite synthesis [25].

Another important assumption is that particles can be considered as spherical, which is often only an approximation. The coefficient D is the effective diffusivity. It describes the diffusion in the porous material, and is macroscopic in its nature, e.g., describes not the individual pores, but all the available pores in total [26].

It is related to the molecular diffusivity D_{AB} as shown in Equation 2 [27].

$$\begin{array}{}{\displaystyle D={D}_{AB}\frac{\varphi \sigma}{\tau},}\end{array}$$(2)

where τ is tortuosity; ϕ is available porosity; σ is a contraction factor.

The available porosity is equal to the total porosity excluding pores, which are not accessible to the diffusant due to geometrical dimension restrictions, and excluding pores, which are not connected with the rest of the pore system and thus are closed. The contraction factor describes the decrease in the diffusion rate due to the increase of viscosity in the close proximity of pore walls (Renkin’s effect) [28, 29].

Since the diffusant reacts and diffuses into the particle, the resulting differential equation in the spherical coordinates in the steady state conditions can be written as in Equation 3 [27]. The diffusant’s concentration at the particle’s surface (r = 0) is C_{AS}.

$$\begin{array}{}{\displaystyle \frac{{d}^{2}{C}_{A}}{d{r}^{2}}+\frac{2}{r}\left(\frac{d{C}_{A}}{dr}\right)-\frac{{k}_{n}}{D}{C}_{A}^{n}=0,}\end{array}$$(3)

where n is the reaction order; k_{n} is the reaction rate constant.

Solving Equation 3 for the standard boundary conditions, Tile’s modulus always appears as shown in Equation 4 [27].

$$\begin{array}{}{\displaystyle \psi =\sqrt{\frac{{k}_{n}{R}^{2}{C}_{AS}^{n-1}}{D}}}\end{array}$$(4)

Based on the Equation 4, it can be deduced that the solution structure does not allow to separately obtain k_{n} and D, simultaneously, but rather only the ratio. Taking this into account the simplest model is chosen and is further described. In this work, GB diffusion model is chosen, which is based on earlier Jander’s work [30, 31].

Diffusion in the spherical particles is a limiting stage in various chemical processes, including the heterogeneous reaction of orthophosphoric acid with the calcium hydroxide. Describing such processes in a general sense, a compound A reacts with a compound B, resulting in a compound AB, whose layer thickness continuously grows during the process. Furthermore, the compound A diffuses into the compound’s B surface through the layer of compound AB with such rate that is much lesser than the chemical reaction rate of the A with the B [30].

Since the outer resistance to diffusion is significantly less than the inner resistance of AB, compound’s A concentration in the plane dividing A and AB can be considered constant. In the plane dividing AB and B, the concentration of compound A is always equal to zero due to much higher chemical reaction rate between A and B than that of diffusion [30].

As the starting point in GB model a classical diffusion equation for spherical particles is used, represented by Equation 5.

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}C(r)}{\mathrm{\partial}\tau}=D\left(\frac{{\mathrm{\partial}}^{2}C(r)}{\mathrm{\partial}{r}^{2}}+\frac{2}{r}\phantom{\rule{negativethinmathspace}{0ex}}\frac{\mathrm{\partial}C(r)}{\mathrm{\partial}r}\right)}\end{array}$$(5)

The initial and boundary conditions are shown in Equations 6 - 9.

$$\begin{array}{}\begin{array}{c}C(r)\\ r\\ \tau \end{array}|\begin{array}{c}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}={C}_{0}\\ =R\\ >0\end{array}\end{array}$$(6)
$$\begin{array}{}\begin{array}{c}C(r)\\ r\end{array}|\begin{array}{c}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=0\\ =R-x\end{array}\end{array}$$(7)
$$\begin{array}{}{\displaystyle \frac{dx}{d\tau}=\frac{D}{\epsilon}(\frac{\mathrm{\partial}C(r)}{\mathrm{\partial}r})r=R-x}\end{array}$$(8)
$$\begin{array}{}\begin{array}{c}x\\ \tau \end{array}|\begin{array}{c}=0\\ \phantom{\rule{thinmathspace}{0ex}}=0\end{array}\end{array}$$(9)

where the $\begin{array}{}\epsilon =\frac{\rho n}{\mu}\end{array}$ is the proportionality coefficient when concentration is in the molar units; n is the stoichiometric reaction coefficient which represents compound’s A quantity in moles reacting with a single mole of compound’s B; μ is compound’s A molecular mass.

As one of the solutions intermediate forms to this model the Equation 10 is obtained.

$$\begin{array}{}{\displaystyle {x}^{2}(1-\frac{2}{3}\phantom{\rule{negativethinmathspace}{0ex}}\frac{x}{R})=K\tau}\end{array}$$(10)

Equation 10 describes the relationship between the thickness of the reacted layer x and the elapsed time τ. It can be deduced that the rate of the change in layer thickness of compound’s AB continuously decreases in time at
$\begin{array}{}\frac{x}{R}\end{array}$ values from 0 up to 0.5, but then symmetrically increases at $\begin{array}{}\frac{x}{R}\end{array}$ values from 0.5 up to 1.

Since it is extremely challenging or even close to impossible to experimentally determine the thickness of the reacted layer in time, Brounstein and Ginstling proposed to use the conversion G in place of x, where G varies from 0 to 1.

In the G – τ coordinates, model can be represented by the Equation 11.

$$\begin{array}{}{\displaystyle 1-\frac{2}{3}G-(1-G{)}^{\frac{2}{3}}=\frac{K\tau}{{R}^{2}}}\end{array}$$(11)

where K is the nominal kinetic constant, and is related to the effective diffusivity as represented by Equation 12.

$$\begin{array}{}{\displaystyle \frac{\sqrt{\pi \epsilon K}}{2{C}_{0}\sqrt{D}}=\frac{exp\left(-\frac{K}{4D}\right)}{\mathit{\Phi}\left(\frac{K}{\sqrt{4D}}\right)}}\end{array}$$(12)

where Φ(y) is the error integral defined as in Equation 13.

$$\begin{array}{}{\displaystyle \mathit{\Phi}(y)=\frac{2}{\sqrt{\pi}}{\int}_{0}^{y}{e}^{-{n}^{2}}du}\end{array}$$(13)

Equation 12 can be solved numerically and the value of D
can be obtained. However, in this work, from the practical standpoint, it is more efficient to look into the ways of experimentally evaluating the nominal kinetic constant, since effective diffusivity includes both the diffusivity and reaction kinetic terms, separate evaluation of which is an unnecessary complication in this case and thus to be avoided.

The final GB model is given by Equation 14.

$$\begin{array}{}{\displaystyle 1-\frac{2}{3}G-(1-G{)}^{\frac{2}{3}}=\frac{K\tau}{{R}^{2}}}\end{array}$$(14)

where G is conversion (-); K is nominal kinetic constant (m^{2}/s); τ is time (s); R is particle radius (m).

The nominal kinetic constant K is a parameter in GB model which describes diffusion and is related to the effective molecular diffusivity. When the value of K is known, it becomes possible to predict the required time to achieve the desired conversion and design the synthesis accordingly.

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