Bulk metallic glass Al2Ca from a biomedical application viewpoint: A molecular dynamics study

Molecular dynamics simulations were performed to investigate the glass transition temperature (Tg) and mechanical properties of Al–Ca-based bulk metallic glass (BMG), Al2Ca. The second-neighbour modified embedded atom method potential was used to simulate the system. We estimate the Tg between 600 to 660 K. To ensure the glassy state of the studied material, the total radial distribution function (RDF) has been calculated from 300 to 1500 K. Partial RDF for Al, Ca, and Al–Ca subsystems are also calculated. Elastic properties including bulk modulus, compressibility, Young’s modulus, shear modulus, tensile yield strength and Poisson ratio are calculated. From the obtained data of elastic constants, it is concluded that Al2Ca BMG can be the potential material to be used in the biomedical field.


Introduction
Zr-based bulk metallic glasses (BMGs) are routinely used for human bone implants due to their favorable elastic and non-toxic nature.Biomaterials require good fracture toughness and fatigue strength to find applications in the biomedical field.Currently, 316L SS, Ti, and Ti-6Al-4V are in-use metallic crystalline materials.They have relatively high Young's modulus and lower strength values, because of which often complications like implant failures occur [1][2][3][4].However, recently BMG with Ca and Al as the base elements are synthesized and analyzed as a potential substitute for these biomaterials [5].This paper reports various structural and elastic properties of Al2Ca BMG.We show that Al2Ca has elastic properties comparable to human bones and can find important applications in temporary implants in the human body and mimicking orthopedic replacements [2][3][4][5].These BMGs are amorphous solid materials, and exhibit properties such as biocompatibility, and biodegradability with low density.These materials show values for Young's and shear moduli comparable to human bone, and they are osteoconductive having high corrosion and wear resistance.The motivation to choose the binary alloy Al2Ca is two-fold: firstly, Al2Ca alloy is already synthesized experimentally in the crystalline form, which evidences the thermodynamic stability of the composition [6,7], and secondly, CaxAly BMG can be synthesized experimentally with various compositions such as (x,y) equal to (61,39), (8,3), (13,14).This corroborates that the Al-Ca-based binary glasses, having applications in the biomedical field, are already being synthesized in a passable size [5,8].
In this study, we examine glass forming ability, glass transition temperature (Tg), various elastic moduli such as bulk modulus (B), tensile yield strength (TYS), compressibility (K), shear modulus(G), Young's modulus (E), and Poisson ratio for Al2Ca BMG.As a routine method [9], we have employed classical molecular dynamics (MD) simulations using the modified embedded atom potential (MEAM) to account for the interaction.We show that Al2Ca BMG has mechanical properties that are comparable to the typical values for human bone, which justifies the use of studied BMG.The rest of the paper is organized as follows.The section-2 includes the details about the simulations followed in the study.Computed results are presented and discussed in section-3.The last section summarizes the work and draws important conclusions.We performed classical MD simulation using the LAMMPS code [10] employing the second-neighbor MEAM pair interaction [11].A cubical box containing 12288 atoms was used in the simulation with the periodic boundary to form the crystalline structure, the C15 crystal structure [6].The relatively large number of atoms ensures the minimum surface effect in the simulations.The time step was kept at 0.001 ps.To obtain the Tg, the step-increment and step-decrement procedures in the temperature were followed.For each annealing step, the simulation time was optimized and finally kept at 30 ps.Whereas for each heating and cooling step, the simulation time was kept at 150 ps.Heating, annealing, and cooling were performed within the NPT ensemble.From each annealing step, the average density and temperature were extracted to plot the graph for density as a function of temperature.To simulate the glass, the system was annealed at 300 K for 30 ps in the beginning.After that, the temperature was increased to and annealed at 1500 K, each for 30 ps.Subsequently, a quenching was performed from 1500 K to the desired temperature in steps of 100 ps.Finally, the system was again annealed at the given temperature for another 30 ps. Figure 1 summarizes the simulations flowchart.The resulting glass transition from crystalline alloy is depicted in Figure 2. To derive various properties like radial distribution function (g(r)), and coordination number (CN) initially, the Al2Ca glass was simulated using the NPT ensemble and then the properties were calculated with NVT ensemble.

Simulations
The elastic constants (ECs) C11 and C12 were computed by giving a lateral strain of in the zdirection, whereas the C44 was computed by giving a shear strain of 3֯ in the xy-direction.Both lateral and shear strains were given for a period of 10 ps.By finding the slope from the linear part of the stressstrain curve, we determined these ECs.We also checked the stability criteria, viz; , , , and to confirm the mechanical stability of the studied BMG.These ECs are used to find other elastic and anisotropic mechanical quantities using the following equations.For cubic crystals, Voigt bounds ( , ) and Reuss bounds ( , ) can be calculated by equations ( 1) -( 3), from which, the bulk modulus (B), the shear modulus (G), the Young's modulus (E) and Poisson ratio are obtained using equation ( 4) -( 7): The glass transition temperature (Tg) was attained using the step-increment and step-decrement method as described in section-2.Computed results for density as a function of temperature are depicted in Figure 3.The crossing of a linear fit to heating and cooling density gives the Tg.The predicted Tg is ~660 K.For the application of BMG from the biomaterial viewpoint, we require a density compatible with human bones (1.81 gm/cm 3 ) [12].We have noticed that Al2Ca has a density of 2.35 gm/cm 3 in its alloy form but it decreases by 14% to 2.02 gm/cm 3 in the glassy state at 300 K.This indicates that Al2Ca has a suitable density to be used as an implant in the human body.The human body temperature, in normal to hyperpyrexia cases can range from 309 to 316 K (36-43 C).Thus, all the important properties were obtained at 300 K after the glass was formed.As a further verification, we have computed RDF (g(r)) at various temperatures, the results are shown in Figure 4. We found the following observations.(i) The slight distortion in the primary peak of g(r) is the characteristic of Al, which gradually disappears with temperatures.(ii) Inspection of Fig. 4 also indicates that at low temperatures the first dip in g(r) shows the humped structure, which corresponds to the principal peak of elemental Ca.The inset in Figure 5 shows the partial g(r) for Al, Ca, and Al-Ca subsystems at 1500 K, and confirms the peak position of elemental Ca.These observations suggest that Al2Ca behaves as a solidus solution comprising of Al-Ca network below Tg.Which upon heating converts into a uniform liquid with a smooth principal peak and first dip in g(r).(iii) The third observation is that at temperatures 600 K and below, the signature 'shoulder' feature of amorphous material in the second peak of the g(r) is visible [13,14].Figure 5 exemplifies this feature for a particular temperature and confirms the glassy state for Al2Ca composition.Figure 6 represents the temperature variation of coordination number.The computed results indicate that at T < Tg the CN ranges between 10.5-10.9 but for temperatures T > Tg it lies between 11-11.6.The range of computed CN for higher temperatures is a characteristic of a system melting from the closed-packed FCC structure.However, it is to be mentioned that the present 2NN-MEAM interaction potential is unable to predict the melting temperature correctly.

Results and Discussion
To verify the compatibility of Al2Ca BMG with the human bone, we have studied the elastic response and mechanical strength of Al2Ca.Figures 7 and 8 respectively show the stress-strain curve.The tensile yield strength (TYS) was obtained by giving the 20% tensile strain in the z-direction.The total strain was divided into 20 parts.Thus, the system would now experience 1% stain and then relax during each cycle, and the loop will run 20 times.Computed results for TYS were found to decrease linearly with temperature.It is respectively 12%, 10%, and 7.8% for temperatures 300, 600 (just below the Tg), and 900 K. Corresponding to human body temperature, the TYS of the Al2Ca BMG will be ~11.8%.This value is sufficiently large to find application in implants.Figure 7 represents the TYS at 600 K.The graph indicates that until 5% of the strain the TYS for Al2Ca BMG follows Hooks' law, between 5-10% the material undergoes the permanent deformation stage before breaking beyond 10% strain.Similarly, at lower temperatures, these values are 6-7%, 7-12% and the breaking point is >12%, respectively.The stress-strain curves were achieved by giving the lateral strain in the z-direction and the shear strain in the xy-direction.From the slope of the stress-strain curve shown in Figure 8, and using Eqs.
(1)-( 7) the ECs were calculated which are listed in Table 1.Table 1 shows the comparison of Al2Ca BMG and Al2Ca alloy along with the ECs of human bones as bracketed values.Table 1 clearly shows that the ECs of Al2Ca BMG are much lower than ECs in its alloy.This suggests that the glassy state is a soft supercooled liquid exhibiting the polymeric structure which is more useful for biomedical applications.It is also clear from Table 1 that Al2Ca BMG has ECs comparable to human bones and therefore suitable for various implants in the body.

Summary and Conclusion
In the present work, a classical MD has been performed to find elastic and structural properties and glass transition temperature for Al2Ca BMG.The results for density and elastic constants in the glassy state verify the material's capability to find application in the biomedical field.Particularly, the density of Al2Ca BMG is just 11% higher than the typical human bone, which is in better agreement than the commonly used Zr-based BMGs.Since the different compositions of Al and Ca (both stochiometric and a-stochiometric) are giving the stable glassy phase [5,8], in principle, by varying Al and Ca compositions, it is possible to tune both density and elastic properties in agreement with the human bone implant requirements.

Figure 3 .
Figure 3.The glass transition temperature of Al2Ca.

Figure 4 .
Figure 4.The radial distribution function of Al2Ca.Inset shows the partial RDF at 1500 K.

Figure 5 .
Figure 5.The radial distribution function of Al2Ca below glass transition temperature (660K)

Figure 6 .
Figure 6.Coordination number as a function of temperature for Al2Ca.

Figure 7 .
Figure 7. tensile yield Strength of Al2Ca BMG at 600 K below the glass transition temperature (660 K).

Table 1 .
Elastic moduli (in GPa) and Poisson ratio for Al2Ca BMG and Al2Ca alloy.Bracketed values are for femoral cortical bone in humans.