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Geometrical aspects of the glass-forming ability of dense binary hard-sphere mixtures
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Scripta Materialia 61 (2009) 261–264 www.elsevier.com/locate/scriptamat
Geometrical aspects of the glass-forming ability of dense binary hard-sphere mixtures V. Kokotin and H. Hermann* Leibniz Institute for Solid State and Materials Research Dresden, P.O. Box 270116, D-01171 Dresden, Germany Received 25 February 2009; revised 24 March 2009; accepted 29 March 2009 Available online 5 April 2009
Non-crystalline dense binary hard-sphere mixtures are generated where the size ratio of the spheres and the fraction of small spheres are varied from 1.0 to 2.0 in steps of 0.1 and from 0 to 100% in steps of 10%, respectively. A confined region within this parameter space is defined where non-crystalline structures are stabilized. Comparison of the present results with experimental data for binary bulk metallic glasses supports the validity of geometrical arguments regarding glass-forming ability of binary metallic melts. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Amorphous materials; Bulk amorphous materials modelling
The nature of the liquid amorphous state of metals and metallic alloys is of both fundamental interest and immense practical relevance. Improved microscopic understanding of phenomena like the glass-forming ability of metallic melts and the decomposition or crystallization of metallic glasses requires progress in the investigation of the basic structural properties of liquid and amorphous metallic alloys [1–3]. Two decades after the discovery of the existence of correlations of atomic distances in liquid mercury [4], Frank [5] suggested icosahedral clusters as the stabilizing structural feature of the liquid state. Bernal [6,7] proposed the model of random close packing of hard spheres as an approach to the structure of simple liquids. This model was transferred later to the structure of metallic glasses [8]. Unfortunately, experimental data like scattering curves do not provide unambiguous information about the structure of liquids and glasses [9,10]. Therefore, computational studies carry great weight for the investigation of related systems. While molecular dynamics studies using ab initio methods (see e.g. [11,12]) or optimized empirical interaction potentials (see e.g. [13,14]) are highly productive to simulate specific elements or alloys, computational approaches like Lennard–Jones liquids [15] and hard-sphere models [6–8,16–19] are useful to scrutinize more general aspects of the liquid and amorphous state. Additionally, the hard-sphere model * Corresponding author. E-mail: [email protected]
is of interest for colloidal suspensions [20], granular matter [21], random media [22–24], crystalline alloys [25], the Brazil nut phenomenon [26], etc. Problems of a mathematically rigorous definition [18,27], icosahedral clusters [28–32], poly-tetrahedral order [33,34], polydispersity [19,20,35–39] and glass-forming ability [40–49] are subjects of actual research in the field of random close packing of hard spheres and related metallic alloys. In this letter we present results of a systematic computational analysis of binary mixtures of hard spheres in the parameter space spanned by the size ratio, R2/ R1 = 1.0 to 2.0, and the relative composition, ci, of small/large spheres with c1 = csmall = 1 c2 = 0–100%, where csmall = 0%, csmall = 100% and R2/R1 = 1.0 correspond to the monodisperse case. Within this parameter range we find a confined region with significantly enhanced glass-forming ability. The models are generated using an improved version of the force-biased algorithm [39,50–52]. For each parameter set 20 models each consisting of 5000 spheres was prepared. Periodic boundary conditions were applied. The maximum packing fractions achieved correspond to previous results [36,53,54]. The models are characterized using the Laguerre tessellation or radical plane method [55,56], which attributes a uniquely defined polyhedral cell with volume vk to each sphere k. (In the special case of equal spheres, this method is identical with the Voronoi con-
1359-6462/$ - see front matter Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2009.03.058
262
V. Kokotin, H. Hermann / Scripta Materialia 61 (2009) 261–264
struction.) The density distribution, q, of a model is then defined by X 4p r3k qð~ rÞ ¼ pk dð~ r ~ rk Þ; pk ¼ 3 vk k where d is the Dirac’s delta function, vector ~ rk points to the center of sphere k, rk is the radius of sphere k and pk is the local packing fraction. Density–density correlations are discussed in terms of , X X CðrÞ ¼ hqqiX ¼ pk pl eðr rkl Þ eðr rkl Þ; k>l
rkl ¼ j~ rk ~ rl j;
k>l
rPr
1
where < . . . >X denotes the average over all spatial directions, eðxÞ ¼ 1 for x = 0 and eðxÞ ¼ 0 otherwise, and r1 is the next-nearest neighbor distance. Clearly, C(r) is the mean product of the values of the local density at two points with the distance r. It is determined for the partial cases of correlations of small–small (C11), small–big (C12) and big–big (C22), spheres. A given mosaic cell characterizes not only the local density but also the geometry of the cluster formed by the central sphere and its nearest neighbors. Moreover, each face is common to two adjacent clusters and describes how these clusters are interconnected. We use the number, ne, of vertices of a face connecting two neighboring mosaic cells as an approximate measure for the type of the local symmetry axis which is perpendicular to the considered face. If the number of vertices, ne, is 3, 4 or 6, then the local symmetry axis is approximately compatible with local translational order [57]. If ne = 5, 7, 8, . . . , then the local symmetry is incompatible with local translational order and, consequently, favours non-crystalline (liquid, amorphous, glassy) structures. The statistics of the number of vertices of the mosaic cell faces is discussed here in terms of the parameter , X X fnc ¼ ne ne e–3;4;6
Figure 1. Frequency distribution of parameter fnc describing the ratio of non-translational local symmetry axes to translational ones vs. fraction of small spheres, csmall (%), and size ratio of spheres for densepacked non-crystalline hard-sphere systems.
hard-sphere mixture to stabilize the non-crystalline state is particularly high. We test whether there are relations between the present results and experimental data. Figure 2 shows a contour plot obtained from Figure 1 by the condition fnc = 1 (closed line). The data points in Figure 2 marked by capitals A–K denote known binary bulk metallic (metal–metal) glasses. All these points are situated in the fnc > 1 region. (A comprehensive compilation of metallic glasses can be found, e.g. in Ref. [58].) To our knowledge, however, there are no corresponding experimental results for fnc > 1 and size ratio above 1.3. It would be interesting to search for new binary metallic glasses A1xBx in the light of the present geometrical arguments, for example at x 10% and RB/ RA > 1.3. Choosing, e.g. A = Fe, Co, Ni, . . . , with RA 0.125 nm, alkaline earth metals like Ca, Sr and Ba, or
e¼3;4;6
where the summation is taken over all cells of the considered model. The parameter fnc characterizes the affinity of a system to stabilize non-crystalline states and can also be considered as a quantitative measure for geometric frustration [33]. Strictly speaking, the incompatibility of ne = 5, 7, 8, . . . , with translational (crystalline) order applies to the symmetry of the elementary cell of a crystal, and it may happen that the Laguerre polyhedron of a single atom inside an elementary cell has faces with five edges. This means that the above condition – higher values of fnc mean higher affinity to stabilizing non-crystalline states – applies in the sense of spatial statistics. Figure 1 shows the results for parameter fnc and its dependence on size ratio and fraction of small spheres. The plot was generated by interpolation from the sampling points (R2/R1, csmall) = (1 + n/10, 10m); n = 0, 1, . . . , 10; m = 0, 1, . . . , 10. The presentation in Figure 1 resembles a ridge of mountains with a maximum height of fnc = 1.11 situated at (R2/R1, csmall) = (1.27, 79.8%). For fnc > 1, the tendency of the considered binary
Figure 2. Distribution of binary metal-metal glasses relative to atomic size ratio and fraction, csmall (%), of small atoms, and calculated region of enhanced glass-forming ability (closed loop); fnc > 1 inside the closed solid line and fnc < 1 outside. Data points: A, ZrNi [42]; B–E, ZrCu [42–44]; F, ZrCo [45]; G, ZrM, M=Fe,Co,Ni [46]; H, SnCu [47]; I, NbNi [48]; K, HfCu [49]; rectangles, data from molecular dynamics simulations [19]. T1 and T2 denote trajectories R2/R1 = 1.3 and csmall = 80%, respectively.
V. Kokotin, H. Hermann / Scripta Materialia 61 (2009) 261–264
rare earth elements like La, Ce, Pr, Nd, . . . , would be good candidates for component B from the geometrical point of view. Undoubtedly, it would be useful to include additional important aspects like the values of the mixing enthalpies in this search. Figure 3 shows the values of the mean partial local packing fractions for small and big spheres along the trajectories T1 and T2 sketched in Figure 2. The behavior is monotonous for all cases and there is no change of any characteristic of the curves when crossing the position of the maximum of fnc (intersection point of T1 and T2 in Fig. 2). Figure 4 shows density–density correlation functions at selected positions on the trajectories T1 and T2. The three partial correlation functions of the system with the maximum value of fnc show the same behaviour. The comparison of C11 of the two systems (R2/R1, csmall) = (1.3, 80) and (1.3, 20) shows differences in the subpeak at rred = 1 which changes from a peak to a shoulder of the main peak situated at rred = 0.7 when the fraction of small spheres is reduced by the factor of 4. Simultaneously, the peak at rred = 1.3 increases considerably due to the enhanced number of big spheres (R2/R1 = 1.3). There are also remarkable differences at rred > 1.5. However, the changes of the correlation functions along trajectory T1 are monotonous. The same is true for the changes of the correlation function along T2 illustrated by C11 of the two systems (R2/ R1, csmall) = (1.3, 80) and (1.7, 80) in Figure 4. It should be noted that the present results are valid for non-crystalline systems with maximum packing fractions. Different effects can be observed at lower packing fractions. For example, recent studies of binary mixtures of colloidal hard spheres showed dramatic changes of density fluctuations [59] and a marked change of the dominant wavelength in the pair-correlation function [20].
Figure 3. Mean values, ppartial, of partial local packing fraction of small (disks) and big (circles) spheres vs. position on the trajectories T1 and T2 shown in Figure 2.
263
Figure 4. Partial density–density correlation functions Cij(rred) for parameter values (R2/R1, csmall) on three different positions of trajectories T1 and T2 shown in Figure 2; rred = r (Ri + Rj). The Cij(rred) curves are shifted on the ordinate by
Scripta Materialia 61 (2009) 261–264 www.elsevier.com/locate/scriptamat
Geometrical aspects of the glass-forming ability of dense binary hard-sphere mixtures V. Kokotin and H. Hermann* Leibniz Institute for Solid State and Materials Research Dresden, P.O. Box 270116, D-01171 Dresden, Germany Received 25 February 2009; revised 24 March 2009; accepted 29 March 2009 Available online 5 April 2009
Non-crystalline dense binary hard-sphere mixtures are generated where the size ratio of the spheres and the fraction of small spheres are varied from 1.0 to 2.0 in steps of 0.1 and from 0 to 100% in steps of 10%, respectively. A confined region within this parameter space is defined where non-crystalline structures are stabilized. Comparison of the present results with experimental data for binary bulk metallic glasses supports the validity of geometrical arguments regarding glass-forming ability of binary metallic melts. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Amorphous materials; Bulk amorphous materials modelling
The nature of the liquid amorphous state of metals and metallic alloys is of both fundamental interest and immense practical relevance. Improved microscopic understanding of phenomena like the glass-forming ability of metallic melts and the decomposition or crystallization of metallic glasses requires progress in the investigation of the basic structural properties of liquid and amorphous metallic alloys [1–3]. Two decades after the discovery of the existence of correlations of atomic distances in liquid mercury [4], Frank [5] suggested icosahedral clusters as the stabilizing structural feature of the liquid state. Bernal [6,7] proposed the model of random close packing of hard spheres as an approach to the structure of simple liquids. This model was transferred later to the structure of metallic glasses [8]. Unfortunately, experimental data like scattering curves do not provide unambiguous information about the structure of liquids and glasses [9,10]. Therefore, computational studies carry great weight for the investigation of related systems. While molecular dynamics studies using ab initio methods (see e.g. [11,12]) or optimized empirical interaction potentials (see e.g. [13,14]) are highly productive to simulate specific elements or alloys, computational approaches like Lennard–Jones liquids [15] and hard-sphere models [6–8,16–19] are useful to scrutinize more general aspects of the liquid and amorphous state. Additionally, the hard-sphere model * Corresponding author. E-mail: [email protected]
is of interest for colloidal suspensions [20], granular matter [21], random media [22–24], crystalline alloys [25], the Brazil nut phenomenon [26], etc. Problems of a mathematically rigorous definition [18,27], icosahedral clusters [28–32], poly-tetrahedral order [33,34], polydispersity [19,20,35–39] and glass-forming ability [40–49] are subjects of actual research in the field of random close packing of hard spheres and related metallic alloys. In this letter we present results of a systematic computational analysis of binary mixtures of hard spheres in the parameter space spanned by the size ratio, R2/ R1 = 1.0 to 2.0, and the relative composition, ci, of small/large spheres with c1 = csmall = 1 c2 = 0–100%, where csmall = 0%, csmall = 100% and R2/R1 = 1.0 correspond to the monodisperse case. Within this parameter range we find a confined region with significantly enhanced glass-forming ability. The models are generated using an improved version of the force-biased algorithm [39,50–52]. For each parameter set 20 models each consisting of 5000 spheres was prepared. Periodic boundary conditions were applied. The maximum packing fractions achieved correspond to previous results [36,53,54]. The models are characterized using the Laguerre tessellation or radical plane method [55,56], which attributes a uniquely defined polyhedral cell with volume vk to each sphere k. (In the special case of equal spheres, this method is identical with the Voronoi con-
1359-6462/$ - see front matter Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2009.03.058
262
V. Kokotin, H. Hermann / Scripta Materialia 61 (2009) 261–264
struction.) The density distribution, q, of a model is then defined by X 4p r3k qð~ rÞ ¼ pk dð~ r ~ rk Þ; pk ¼ 3 vk k where d is the Dirac’s delta function, vector ~ rk points to the center of sphere k, rk is the radius of sphere k and pk is the local packing fraction. Density–density correlations are discussed in terms of , X X CðrÞ ¼ hqqiX ¼ pk pl eðr rkl Þ eðr rkl Þ; k>l
rkl ¼ j~ rk ~ rl j;
k>l
rPr
1
where < . . . >X denotes the average over all spatial directions, eðxÞ ¼ 1 for x = 0 and eðxÞ ¼ 0 otherwise, and r1 is the next-nearest neighbor distance. Clearly, C(r) is the mean product of the values of the local density at two points with the distance r. It is determined for the partial cases of correlations of small–small (C11), small–big (C12) and big–big (C22), spheres. A given mosaic cell characterizes not only the local density but also the geometry of the cluster formed by the central sphere and its nearest neighbors. Moreover, each face is common to two adjacent clusters and describes how these clusters are interconnected. We use the number, ne, of vertices of a face connecting two neighboring mosaic cells as an approximate measure for the type of the local symmetry axis which is perpendicular to the considered face. If the number of vertices, ne, is 3, 4 or 6, then the local symmetry axis is approximately compatible with local translational order [57]. If ne = 5, 7, 8, . . . , then the local symmetry is incompatible with local translational order and, consequently, favours non-crystalline (liquid, amorphous, glassy) structures. The statistics of the number of vertices of the mosaic cell faces is discussed here in terms of the parameter , X X fnc ¼ ne ne e–3;4;6
Figure 1. Frequency distribution of parameter fnc describing the ratio of non-translational local symmetry axes to translational ones vs. fraction of small spheres, csmall (%), and size ratio of spheres for densepacked non-crystalline hard-sphere systems.
hard-sphere mixture to stabilize the non-crystalline state is particularly high. We test whether there are relations between the present results and experimental data. Figure 2 shows a contour plot obtained from Figure 1 by the condition fnc = 1 (closed line). The data points in Figure 2 marked by capitals A–K denote known binary bulk metallic (metal–metal) glasses. All these points are situated in the fnc > 1 region. (A comprehensive compilation of metallic glasses can be found, e.g. in Ref. [58].) To our knowledge, however, there are no corresponding experimental results for fnc > 1 and size ratio above 1.3. It would be interesting to search for new binary metallic glasses A1xBx in the light of the present geometrical arguments, for example at x 10% and RB/ RA > 1.3. Choosing, e.g. A = Fe, Co, Ni, . . . , with RA 0.125 nm, alkaline earth metals like Ca, Sr and Ba, or
e¼3;4;6
where the summation is taken over all cells of the considered model. The parameter fnc characterizes the affinity of a system to stabilize non-crystalline states and can also be considered as a quantitative measure for geometric frustration [33]. Strictly speaking, the incompatibility of ne = 5, 7, 8, . . . , with translational (crystalline) order applies to the symmetry of the elementary cell of a crystal, and it may happen that the Laguerre polyhedron of a single atom inside an elementary cell has faces with five edges. This means that the above condition – higher values of fnc mean higher affinity to stabilizing non-crystalline states – applies in the sense of spatial statistics. Figure 1 shows the results for parameter fnc and its dependence on size ratio and fraction of small spheres. The plot was generated by interpolation from the sampling points (R2/R1, csmall) = (1 + n/10, 10m); n = 0, 1, . . . , 10; m = 0, 1, . . . , 10. The presentation in Figure 1 resembles a ridge of mountains with a maximum height of fnc = 1.11 situated at (R2/R1, csmall) = (1.27, 79.8%). For fnc > 1, the tendency of the considered binary
Figure 2. Distribution of binary metal-metal glasses relative to atomic size ratio and fraction, csmall (%), of small atoms, and calculated region of enhanced glass-forming ability (closed loop); fnc > 1 inside the closed solid line and fnc < 1 outside. Data points: A, ZrNi [42]; B–E, ZrCu [42–44]; F, ZrCo [45]; G, ZrM, M=Fe,Co,Ni [46]; H, SnCu [47]; I, NbNi [48]; K, HfCu [49]; rectangles, data from molecular dynamics simulations [19]. T1 and T2 denote trajectories R2/R1 = 1.3 and csmall = 80%, respectively.
V. Kokotin, H. Hermann / Scripta Materialia 61 (2009) 261–264
rare earth elements like La, Ce, Pr, Nd, . . . , would be good candidates for component B from the geometrical point of view. Undoubtedly, it would be useful to include additional important aspects like the values of the mixing enthalpies in this search. Figure 3 shows the values of the mean partial local packing fractions for small and big spheres along the trajectories T1 and T2 sketched in Figure 2. The behavior is monotonous for all cases and there is no change of any characteristic of the curves when crossing the position of the maximum of fnc (intersection point of T1 and T2 in Fig. 2). Figure 4 shows density–density correlation functions at selected positions on the trajectories T1 and T2. The three partial correlation functions of the system with the maximum value of fnc show the same behaviour. The comparison of C11 of the two systems (R2/R1, csmall) = (1.3, 80) and (1.3, 20) shows differences in the subpeak at rred = 1 which changes from a peak to a shoulder of the main peak situated at rred = 0.7 when the fraction of small spheres is reduced by the factor of 4. Simultaneously, the peak at rred = 1.3 increases considerably due to the enhanced number of big spheres (R2/R1 = 1.3). There are also remarkable differences at rred > 1.5. However, the changes of the correlation functions along trajectory T1 are monotonous. The same is true for the changes of the correlation function along T2 illustrated by C11 of the two systems (R2/ R1, csmall) = (1.3, 80) and (1.7, 80) in Figure 4. It should be noted that the present results are valid for non-crystalline systems with maximum packing fractions. Different effects can be observed at lower packing fractions. For example, recent studies of binary mixtures of colloidal hard spheres showed dramatic changes of density fluctuations [59] and a marked change of the dominant wavelength in the pair-correlation function [20].
Figure 3. Mean values, ppartial, of partial local packing fraction of small (disks) and big (circles) spheres vs. position on the trajectories T1 and T2 shown in Figure 2.
263
Figure 4. Partial density–density correlation functions Cij(rred) for parameter values (R2/R1, csmall) on three different positions of trajectories T1 and T2 shown in Figure 2; rred = r (Ri + Rj). The Cij(rred) curves are shifted on the ordinate by
2 plus a multiple of 0.01.
The results obtained in Ref. [19] by means of molecular dynamics of binary hard-sphere systems with size ratio of R2/R1 = 1.11 in the composition range csmall = 0.05–1.00 correspond to the present results though the packing fraction used in [19] was 0.58 (0.643–0.644 in the present study). While the simulations for csmall 6 0.3 and csmall P 0.825 readily underwent a large degree of crystallization, for 0.5 6 csmall 6 0.7 the crystallization was almost totally suppressed on the time scale of the simulations [19]. The data points corresponding to suppressed crystallization are situated in the fnc > 1 region in Figure 2 (marked by rectangles). The results presented here suggest that for binary dense random hard-sphere systems there exists a confined region in the size-ratio/concentration parameter range accentuated by an enhanced glass-forming ability. This specific region is determined by the condition that the probability of finding 5-, 7-, . . . , -fold local symmetry is higher than that for the 3-, 4- and 6-fold one. Both experimental and computer simulated data points for, respectively, binary metal–metal glasses and non-crystallizing hard-sphere systems are situated in the specified region. We conclude from this that geometric aspects are important for the glass-forming ability of binary systems, especially of binary metallic alloys. Empirical rules for good glass-forming ability of bulk metallic glasses have been established in recent years by Inoue [2] and Egami [3]. These rules include (i) presence of polydispersity (increase the number of elements [3]), (ii) specific atomic interaction (negative enthalpy of mixing of the three main elements [2], attractive interaction of small and large atoms [3]), and (iii) high atomic size ratio (difference among the three main components should be above 12% [2]). Item (iii) of the empirical rules for good glass-forming ability of multi-component metallic alloys is related to the present results: the demand for the minimum size ratio of 12% corresponds quantitatively to the lower bound of the fnc > 1 region
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in Figure 2 for the concentration range of 30 < csmall < 90. This result substantiates the supposition that, irrespective of specific electronic properties, geometrical effects contribute essentially to the glass-forming ability of metallic alloys. The results shown in Figures 1 and 2 also demonstrate that, from the geometrical point of view, there is also a maximum size ratio for good glass-forming ability of binary hard-sphere mixtures. This ratio depends distinctly from the concentration of small/big spheres as shown in Figures 1 and 2. To our knowledge, the existence of a maximum size ratio for good glass-forming ability has not been discussed until now. A final remark concerns the type of structure parameters which could be relevant for the degree of glassforming ability of a binary system. It seems that, at least regarding dense binary hard-sphere systems, first- and second-order parameters such as density and correlation functions are not helpful to discriminate between good and minor glass-forming ability. Instead, higher order parameters like local symmetry of interlinking neighboring clusters are shown to be essential for this. Part of this work was supported by the Deutsche Forschungsgemeinschaft. We thank Joachim Ohser for fruitful discussions on spatial statistics and Ulrike Nitzsche for technical support. [1] [2] [3] [4] [5] [6] [7] [8] [9]
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