# Study of Silver Ions Diffusion into Ti[O.sub.2]NT Using Finite Elements Method.

1. IntroductionThe application of titanium and its alloys is extensively studied in orthopedics and orthodontics in the use as bone and dental implants [1-5]. Among the most important characteristics of biomaterials, for this application, are the biocompatibility and resistance to corrosion [6], guaranteed by the formation of a chemically inert layer of titanium dioxide on the metal surface [7]. In this context, the use of implants with nanotubular structures, such as Ti[O.sub.2]NT, allows the improvement of adhesion of matrix proteins, which have structures in the nanometer scale, contributing to the osseointegration process [8].

The silver decoration of biomaterials aims to reduce, eliminate or prevent bacterial colonization and infection [9]. Materials decorated and doped with silver naturally release [Ag.sup.+], known by microbial growth inhibition [10]. Knowing the variables that influence the incorporation of ions into a nanotubular structure can bring great advances to the area of materials surface modification. Before starting an experimental methodology, a theoretical study can bring relevant information and reduce the experimental effort.

The diffusion of metallic cations into nanotubes can be studied by computer simulation using a versatile method, such as the finite element method (FEM). This method is based on the transformation of partial differential equations (PDEs) into algebraic equations [9,10] and can be applied to stationary and transient problems, in linear or nonlinear regions for a, one, two and three-dimensional domains [11]. That way, the use of the FEM, using a software, allows correlating the concentration distributions with the desired geometry.

The diffusion simulation using the finite element method has already been used for several purposes. Rahman et al. [13] used the method to study the migration of chlorides into concrete, [Cl.sup.-] ions are known by damage building structures, O'Malley et al. [14] studied the efficiency of different geometries for finite elements in the calculation of neutron diffusion and Krishnamurthy et al. [15] evaluated the diffusion of oxygen in cationic structures. Thus, in this work the diffusion of silver ions into nanotubular structures will be studied, utilizing Fick's second law, using the finite element method. To reduce the complexity of the problem the oxide film will consist only of a nanotube belonging to a two-dimensional model. The thickness of the oxide film, the nanotube diameter and the size of the ion supply phase will be investigated.

2. Results and Discussion

Calculation of final concentration of the system

The simulation results show the change in concentration, caused by the diffusion of Ag+ ions from the solution bulk into the Ti[O.sub.2] nanotube, is related to three parameters: solution area (equation (1)), nanotube area (equation (2)) and solution initial concentration. All the mathematical relations cited below, between these parameters and the concentration variation, are true for the conditions mentioned in section 3 (Materials and Methods).

[A.sub.sol] = [H.sub.s].[W.sub.s] (1)

where [A.sub.sol] is the solution area, [H.sub.s] is the height of solution bulk and [W.sub.s] is the width of solution bulk.

[A.sub.n] = n. [H.sub.n]. [W.sub.n] (2)

where [A.sub.n] is the area of nanotube , n is the number of nanotubes, [H.sub.n] is the height of a nanotube and [W.sub.n] is the width of the nanotube.

For different values of solution initial concentration, the ratio [DELTA]c/c is constant:

[DELTA]c [varies] c (3)

where [DELTA]c is the concentration variation of the system and c is the initial concentration of the solution.

In terms of diffusion, an increase in the initial concentration of the system raises the gradient intensity existent between the solution concentration in the bulk and the solution inside the nanotube, which has zero concentration. In this way, the silver ions will diffuse more strongly and in more quantity to the nanotube, leading to a greater concentration variation.

The concentration change, [DELTA]c, is inversely proportional to the solution area, [A.sub.sol]:

[DELTA]c [varies] (1/[A.sub.sol]) (4)

This correlation reflects that for large values of solution area, the concentration variation is smaller. This finding can be explained in terms of ratio R defined in equation (5). In a larger solution area, a smaller amount of diffused ions is required for the nanotube, about to total amount, so that the concentration of the system is homogeneous. That is, for smaller values of R and, consequently, higher [A.sub.sol] the concentration variation becomes larger.

R = [q.sub.dif]/[q.sub.tot] (5)

where R is the ratio between diffused and total ions, [q.sub.dif] is the amount of diffused ions into the nanotube and [q.sub.tot] is the amount of total ions presents in system.

To determine the proportionality between the nanotube area and the concentration variation, it is necessary to change both the diameter and height of the nanotube and the number of nanotubes. In addition, it is pertinent to evaluate if the parameters mentioned above do not affect this relation. For this, thirteen sets of simulations are performed, described in Table 1. The results are expressed in the form of straight equations obtained by linear regression.

Each set represent ten simulations performed according to the parameters above and varying the size quoted. Therefore, the proportionality between the nanotube area ([A.sub.n]) and the concentration change ([DELTA]c) is:

[DELTA]c [varies] [A.sub.n] (6)

Observing the set of similar simulations, which were performed for one and two nanotubes, it is possible to conclude that it is the area of nanotube that influences the variation of the concentration of the system, regardless of the amount of nanotubes.

An increase in the nanotube area causes an increase in the ratio R, that is, more ions are diffused into the interior of the nanotube in relation to the total amount of ions. In this way, there is an increase in the concentration variation.

Considering all the proportionalities identified, it can be write the following mathematical relation:

[DELTA]c = a ([A.sub.n]c/[A.sub.sol]) (7)

Where the concentration variation is defined according to equation (8):

[DELTA]c = c - [c.sub.f] (8)

where [DELTA]c is the change in concentration, c is the initial concentration of the solution and [c.sub.f] is the final concentration of the system.

Substituting equation (8) into equation (7), and isolating [c.sub.f], the following mathematical function is obtained:

[c.sub.f] = -a ([A.sub.n]c/[A.sub.sol]) + c (9)

Performing linear regression again, but now using equation (9), where x = [A.sub.n].c/[A.sub.sol] and y = [c.sub.f], the following results are obtained (Table 2):

The value of the constant a is 1,000, obtained through the average of the angular coefficients of the equations of Table 2. Applying this value in equation (9) and showing the initial concentration, the following equation is obtained:

[c.sub.f] = c(1 - ([A.sub.n]/[A.sub.sol])) (10)

Considering a tridimensional model, where all Ti[O.sub.2] nanotubes have the same dimensions, the equation (10) can be generalized to a more realistic model, this way:

[c.sub.f] = c(1 - ([v.sub.n]c/[v.sub.sol])) (11)

where [c.sub.f] is the final concentration of the system, c is the initial concentration of the solution, [V.sub.n] is the nanotube volume, considering a cylindrical geometry (equation (12)), and [V.sub.sol] is the solution volume.

[V.sub.n] = (1/4) [pi].n.[H.sub.n].[W.sup.2.sub.n] (12)

where n is the number of nanotubes (equation (13)), [H.sub.n] is the height of a nanotube and [W.sub.n] is the diameter of a nanotube.

n = [mu]. A (13)

where [mu] is the amount of nanotube per area of Ti[O.sub.2], given in [m.sup.-2], and A is the surface area of the oxide, given in [m.sup.2].

The dilution equation

The simulation of the diffusion phenomenon of Ag+ cations gives concentration data as a function of time, which allows to relate how the solution area, the initial solution concentration and the nanotube area affect the final concentration of the system. However, this correlation can be analyzed through the dilution phenomenon, and its respective equation (equation (14)). In terms of the dilution, the solution containing [Ag.sup.+] in the bulk is diluted by the solution with zero concentration inside the nanotube.

[c.sub.i].[V.sub.i] = [c.sub.f].[V.sub.f] (14)

where [c.sub.i] is the initial concentration of solution, [V.sub.i] is the initial volume of the solution, [c.sub.f] is the final concentration and [V.sub.f] is the final volume.

In a two-dimensional system, c, is the concentration of solution (c), [V.sub.i] can be replaced by the area of solution ([A.sub.sol]), [c.sub.f] is the final concentration of the system ([c.sub.f]) and [V.sub.f] is the sum of the solution area ([A.sub.sol]) with the area of solution inside the nanotube (An). Substituting these variables into (equation (14)) and isolating [c.sub.f], the following equation is obtained:

[c.sub.f] = c([A.sub.sol]/[A.sub.sol]+[A.sub.n]) (15)

Comparing the final concentration, [c.sub.f1], calculated by obtained equation (equation (10)) and the final concentration, [c.sub.f2], calculated by dilution equation (equation (15)), it is observed that the difference between the concentration calculated with the two equations is low. The results are represented in Table 3 for different values of nanotube area, and in Table 4 of different values of solution area.

The increase in the nanotube area, as well as, decrease in the solution area, leads to an increase in the difference between the final concentrations calculated with each equation. That is, the higher the [A.sub.n]/[A.sub.sol] ratio, the greater the difference of calculated values. This confirmation is in agreement with the observed experimentally, considering that the volume of solution is always higher, in order of magnitude, to the volume of nanotubes.

Influence of nanotube's height and diameter on diffusion

To discuss the influence of both the diameter and the height of the nanotube on the diffusion of silver cations, it is necessary to define a time parameter. The equilibrium time, [t.sub.eq], is defined as the time required for the net velocity to be zero, according to the derivative of the graphs c versus t plotted during the simulations (Figure 1).

The higher the nanotube, the longer the time required for the concentration of the system to be constant, and consequently the longer the time teq. This is because the distance to the base of the nanotube becomes bigger, increasing the time required for the [Ag.sup.+] to diffuse into the less concentrated region. Figure 2 shows the time [t.sub.eq] for different heights of nanotube.

The figure above shows that the equilibrium time is exponentially proportional to the height of the nanotube, indicating that the time for diffusion to occur completely is gradually larger for large values of height. This same relation is evidenced in the curves of the concentration at the bottom of the nanotube as a function of time for different height values also show the influence of height in the diffusion (Figure 3).

Differently from what is observed for different heights of nanotube, an increase in the diameter does not entail a significant change in the curve c versus t, that is, the value of the diameter does not interfere in the diffusion of silver ions into the interior of the nanotube. This can be seen in the graph of Figure 4.

3. Material and Methods

A nanotube, the Ti[O.sub.2] layer and the aqueous solution containing [Ag.sup.+] ions represent the two-dimensional domain (Figure 5). Nanotubes are formed during the anodization process, by the evolution of metastable pores into Ti[O.sub.2] oxide. The formation of nanotubes occurs by the attack of fluoride ions on the oxide barrier layer [16]. The barrier layer is formed by the anodization process and because it is compact [16], it does not interfere in the diffusion process of the ions that will be considered in this work. For the incorporation of silver ions, it is considered that a nanotubular film, after the anodizing process, is submerged in a solution containing a known concentration of silver ions. In this work, the ionic solution will be called bulk. For the purpose of implementing the physical model, a two-dimensional model was considered where the height and width of the solution are given by [H.sub.s] and [W.sub.s], respectively, and the height and diameter of nanotube are represented, respectively, by [H.sub.n] and [W.sub.n].

The commercial software Comsol Multiphysics[R] was used to perform the simulations using the finite element method, as well as for the construction of the two-dimensional model. The mesh is made up of finite triangular elements, and is refined in the region where the nanotube is located (Figure 6), because in this region there is an accelerated change of concentration [17].

For calculation purposes, the interior of the nanotube has zero concentration and the bulk solution concentration may assume any constant value other than zero. In addition, all boundaries presented, except at the top of the nanotube, do not allow flow of matter.

For the construction of the mathematical model of diffusion, the Einstein relation (equation (16)) was used to determine the diffusion coefficient of [Ag.sup.+] ions in aqueous solution. This constant is a necessary parameter for the Fick's second law (equation (17)).

D = uRT/zF (16)

where D is the diffusion coefficient, u is the ionic mobility, R is the gas constant, T is the temperature, z is the ion charge and F is the Faraday constant.

[partial derivative]c/[partial derivative]t = [nabla] x (D[nabla]c) (17)

The concentration at any point in the domain is calculated by this partial differential equation using the finite element method. The simulations were carried out using the parameters, constants and variables, described in Table 5.

In order to obtain the proportional mathematics presented in section 2 (Results and Discussion), the simulations are performed with all fixed parameters, except one, in which one is interested. This procedure was repeated for all the variable parameters mentioned in the Table 5 above.

4. Conclusions

Using a simple two-dimensional model, it was possible to determine which variables are directly involved in the change of the Ag+ concentration. The concentration change is caused by the diffusion of these ions from the bulk into the nanotube. The connection of the studies carried out leads to a mathematical relationship that calculates the final concentration as a function of the initial solution concentration, the solution volume, and the nanotube volume.

The analysis of the problem through two distinct phenomena, diffusion and dilution, shows that the mathematical data are close. This is evidenced when comparing the results calculated by the mathematical equation obtained and the dilution equation. The difference of the calculated final concentration between the two equations is minimal for An/Aso/ ratios, that is, for large values of solution area and small values of nanotube area. This result shows the spatial domain of the simulation performed should be carefully delimited to minimize such errors.

The mathematical relationship found may apply to real situations when extrapolated to three dimensions. The generality can also be covered for the study of diffusion of other ions in any material with nanotubular structure, because, the equation does not take into account the diffusion coefficient of the material.

The height of the nanotube, unlike the diameter, is directly related to the time required for the concentration of the system to be constant, tc, that is, the complete diffusion of the ions into the nanotube is slower for larger heights. This result shows that for the incorporation of chemical species into materials with nanotubular structure, the experimental conditions that favor the vertical growth of the nanotubes must be considered.

It is important to emphasize that this study considers only the concentration change caused by diffusion. The adsorption phenomenon of the silver ions in the nanotubular material was not considered, new studies must be carried out to implement this phenomenon.

Acknowledgments

The authors are grateful to CNPq (process no408250/2016-0) and to Prof. Dr. Ernesto Chaves Pereira de Souza by use of COMSOL Multiphysics.

References and Notes

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[11] Dhatt, G.; Touzot, G.; Lefrancois, E. Finite element method ISTE, 2012.

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Luis Felipe Pilonetto and Mariana de Souza Sikora *

Chemistry Department, Federal Technological University of Parana (UTFPR), Mail Box 571, Pato Branco, PR, 85503-390, Brazil.

Article history: Received: 19 September 2018; revised: 04 November 2019; accepted: 20 November 2019. Available online: 14 December 2019. DOI: http://dx.doi.org/10.17807/orbital.v11i6.1241

* Corresponding author. E-mail: marianasikora@utfpr.edu.br

Caption: Figure 1. Graph of the concentration as a function of time on the bottom of the nanotube, indicating the points ([t.sub.eq], [c.sub.f]) e ([t.sub.0], c), where [t.sub.eq] is the defined time parameter, [c.sub.f] is the final concentration of the system, [t.sub.0] is the initial time and c is the initial concentration ate the bottom of the nanotube.

Caption: Figure 2. Graph of the time [t.sub.eq] as a function of height of nanotube, where a, b and c are constants.

Caption: Figure 3. Graph of Ag+ concentration at the bottom of the nanotube as a function of time for different heights of nanotube.

Caption: Figure 4. Graph of [Ag.sup.+] concentration at the bottom of the nanotube as a function of time for different diameters of nanotube.

Caption: Figure 5. 2D model representing the solution of [Ag.sup.+], a Ti[O.sub.2] nanotube and the Ti[O.sub.2] barrier film, where (a) represents the entire domain, and (b) is an enlargement of the region where the nanotube is located.

Caption: Figure 6. Finite element mesh plotted on the geometric model.

Table 1. Detail of the simulation sets. No Amount of c (mol.[L.sup.-1]) Variable [A.sub.sol] nanotubes dimension ([micro][m.sup.2]) 1 1 0.250 Hn 4000 2 1 0.250 Wn 4000 3 1 0.500 Hn 4000 4 1 0.500 Wn 4000 5 2 0.500 Hn 4000 6 2 0.250 Wn 4000 7 2 0.500 Hn 4000 8 2 0.500 Wn 4000 9 1 0.250 Hn 1000 10 1 0.250 Hn 3000 11 1 0.250 Hn 5000 12 1 0.250 Hn 10000 13 1 0.250 Hn 20000 No Equation of the line 1 [DELTA]c = 0.062x[10.sup.-3][A.sub.n] 2 [DELTA]c = 0.063x[10.sup.-3][A.sub.n] + 0.001x[10.sup.-3] 3 [DELTA]c = 0.125x[10.sup.-3][A.sub.n] + 0.001x[10.sup.-3] 4 [DELTA]c = 0.126x[10.sup.-3][A.sub.n] - 0.001x[10.sup.-3] 5 [DELTA]c = 0.062x[10.sup.-3][A.sub.n] 6 [DELTA]c = 0.062x[10.sup.-3][A.sub.n] 7 [DELTA]c = 0.125x[10.sup.-3][A.sub.n] - 0.001x[10.sup.-3] 8 [DELTA]c = 0.125x[10.sup.-3][A.sub.n] - 0.001x[10.sup.-3] 9 [DELTA]c = 0.248x[10.sup.-3][A.sub.n] + 0.003x[10.sup.-3] 10 [DELTA]c = 0.083x[10.sup.-3][A.sub.n] + 0.004x[10.sup.-3] 11 [DELTA]c = 0.050x[10.sup.-3][A.sub.n] 12 [DELTA]c = 0.025x[10.sup.-3][A.sub.n] 13 [DELTA]c = 0.012x[10.sup.-3][A.sub.n] Table 2. Equations parameters obtained through equation (9). No c (mol.[L.sup.-1]) [A.sub.sol] ([micro][m.sup.2]) 1 0.250 4000 2 0.250 4000 3 0.500 4000 4 0.500 4000 5 0.250 4000 6 0.250 4000 7 0.500 4000 8 0.500 4000 9 0.250 1000 10 0.250 3000 11 0.250 5000 12 0.250 10000 13 0.250 20000 No Equation 1 [C.sub.f] = - 0.997[A.sub.n]C/[A.sub.sol] + 0.250 2 [C.sub.f] = - 1.010[A.sub.n]C/[A.sub.sol] + 0.250 3 [C.sub.f] = - 0.998[A.sub.n]C/[A.sub.sol] + 0.500 4 [C.sub.f] = - 1.009[A.sub.n]C/[A.sub.sol] + 0.500 5 [C.sub.f] = - 0.998.[A.sub.n]C/[A.sub.sol] + 0.250 6 [C.sub.f] = - 1.000.[A.sub.n]C/[A.sub.sol] + 0.250 7 [C.sub.f] = - 1.001 .[A.sub.n]C/[A.sub.sol] + 0.500 8 [C.sub.f] = - 1.003. [A.sub.n]C/[A.sub.sol] + 0.500 9 [C.sub.f] = - 0.992. [A.sub.n]C/[A.sub.sol] + 0.250 10 [C.sub.f] = - 0.997. [A.sub.n]C/[A.sub.sol] + 0.250 11 [C.sub.f] = - 0.999.[A.sub.n]C/[A.sub.sol] + 0.250 12 [C.sub.f] = - 0.999..[A.sub.n]C/[A.sub.sol] + 0.250 13 [C.sub.f] = - 1.000.[A.sub.n]C/[A.sub.sol] + 0.250 Table 3. Comparison between the final concentrations calculated from the equation obtained and to the dilution equation for different values of [A.sub.n]. [A.sub.n] [A.sub.sol] [c.sub.f1] ([micro][m.sup.2]) ([micro][m.sup.2]) (mol.[L.sup.-1]) 0.7 4000 249.9563x[10.sup.-3] 1.4 4000 249.9125x[10.sup.-3] 2.1 4000 249.8688x[10.sup.-3] 2.8 4000 249.8250x[10.sup.-3] 3.5 4000 249.7813x[10.sup.-3] 4.2 4000 249.7375x[10.sup.-3] 4.9 4000 249.6938x[10.sup.-3] 5.6 4000 249.6500x[10.sup.-3] 6.3 4000 249.6063x[10.sup.-3] 7.0 4000 249.5625x[10.sup.-3] [A.sub.n] ([micro][m.sup.2]) [c.sub.f1] (mol.[L.sup.-1]) difference 0.7 249.9563x[10.sup.-3] 0.0 1.4 249.9125x[10.sup.-3] 0.0 2.1 249.8688x[10.sup.-3] 0.0 2.8 249.8251x[10.sup.-3] 1x[10.sup.-07] 3.5 249.7814x[10.sup.-3] 1x[10.sup.-07] 4.2 249.7378x[10.sup.-3] 3x[10.sup.-07] 4.9 249.6941x[10.sup.-3] 3x[10.sup.-07] 5.6 249.6505x[10.sup.-3] 5x[10.sup.-07] 6.3 249.6069x[10.sup.-3] 6x[10.sup.-07] 7.0 249.5633x[10.sup.-3] 8x[10.sup.-04] Table 4. Comparison between the final concentrations calculated from the equation obtained and to the dilution equation for different values of [A.sub.n]. [A.sub.n] [A.sub.sol] [c.sub.f1] ([micro][m.sup.2]) ([micro][m.sup.2]) (mol.[L.sup.-1]) 3.5 1000 249.1250x[10.sup.-3] 3.5 3000 249.7083x[10.sup.-3] 3.5 5000 249.8250x[10.sup.-3] 3.5 10000 249.9125x[10.sup.-3] 3.5 20000 249.9563x[10.sup.-3] [A.sub.n] ([micro][m.sup.2]) [c.sub.f1] (mol.[L.sup.-1]) difference 3.5 249.1280x[10.sup.-3] 3x[10.sup.-6] 3.5 249.7086x[10.sup.-3] 3x[10.sup.-7] 3.5 249.8251x[10.sup.-3] 2x[10.sup.-7] 3.5 249.9125x[10.sup.-3] 0.0 3.5 249.9563x[10.sup.-3] 0.0 Table 5. Description of the parameters used during the simulations. Name (symbol) Classification Height of solution bulk Variable ([H.sub.s]) Height of nanotube ([H.sub.n]) Variable Diffusion coefficient of Constant [Ag.sup.+.sub.(aq)] (D) Initial concentration of Variable solution (c) Diameter of nanotube ([W.sub.n]) Variable Width of solution bulk Variable ([W.sub.s]) Temperature (T) Constant Simulation time (t) Constant Name (symbol) Minimum value; interval; maximum value [unit] Height of solution bulk 10; 10; 100 e 100; 50; ([H.sub.s]) 600 [[micro]m] Height of nanotube ([H.sub.n]) 10; 10; 150 [[micro]m] Diffusion coefficient of 1.603x[10.sup.-9] [[m.sup.2]/s] [Ag.sup.+.sub.(aq)] (D) Initial concentration of 0.05; 0.05; 0.50 [mol.[L.sup.-1]] solution (c) Diameter of nanotube ([W.sub.n]) 0.07; 0.07; 0.7 [[micro]m] Width of solution bulk 50; 50; 150 [[micro]m] ([W.sub.s]) Temperature (T) 298.15 [K] Simulation time (t) 20 [s]

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Author: | Pilonetto, Luis Felipe; de Souza Sikora, Mariana |

Publication: | Orbital: The Electronic Journal of Chemistry |

Date: | Nov 15, 2019 |

Words: | 4435 |

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