Citation: Jinhan Liu, Lin Zhang. Strain-induced packing transition of Ih Cun@Ag55-n(n = 0, 1, 13, 43) clusters from atomic simulations[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6390-6400. doi: 10.3934/mbe.2020336
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Experimental and theoretical models have been recently developed to examine fundamental aspects of collective motion exhibited by various biological systems. Following the seminal papers [18]-[17] and works cited therein we suggest that hydrodynamic interactions between the swimmers lead to collective motion when every bacterium interacts with other ones through the viscous environment. The implicit evidences of collective motion were confirmed by reduction in the effective viscosity [16]. The theoretical investigations of collective motion were based on the considering the motion in the framework of mechanical dynamical systems [13]-[20]. While the above presented theoretical and experimental results are related via the effective viscosity of suspension of swimmers, these relation lack simple and direct comparisons of viscosity for experimentally observed sets of bacteria and for simulated ensembles.
In the present paper, we propose a new quantitative criterion of collective behavior. The locations of bacteria are modeled by short segments having a small width randomly embedded in medium without overlapping. First, we theoretically simulate locations of particles called below by disordered sets of bacteria (DB sets for shortness) subjected to local viscous stresses and randomly reacted on hydrodynamic interactions. Second, we calculate the basic statistic moments of the constructed DB sets in terms of the generalized Eisenstein-Rayleigh sums (
Our criterion of the collective behavior is based on a new RVE (representative volume element) theory proposed in [6], on the invariance of the effective transport properties on the conformal mappings [4] and on the algebraic dependence of the viscous lattice sums on the
It is worth noting that the criterion can be applied to various two-phase dispersed media (biological systems, composites etc).
In the present section, we discuss theoretical simulations of elements embedded in a medium. Bacteria are modeled by short segments having non-overlapping thin
The complex numbers 1 and
Q(0,0)={z=t1+it2:−12<tk<12(k=1,2)}. |
The square lattice
Consider
The centers
Γk={z∈C:z=bk±l2eiαkt,0≤t≤1} |
generate a set of uniformly distributed non-overlapping segments. Theoretically, this distribution can be introduced as the distribution of the variable
The random variable
Introduce the density segments associated to the conformally invariant conductivity (capacity) [4]
ϱ(l,N)=N(l2)2, | (1) |
where
In order to study the distributions
e2=1N2N∑k=1N∑m=1E2(bk−bm),epp=(−1)pNp+1N∑m=1|N∑k=1Ep(bm−bk)|2,p=2,3,4. | (2) |
Here,
The theoretical probabilistic distributions corresponding to disordered locations of bacteria are modeled in the previous section. Now, we propose an effective computational tool to properly describe
Let
First, we determine the minimal
|1NN∑n=1Re[exp(iαn)]|≤0.15and|1NN∑n=1Im[exp(iαn)]|≤0.15, | (3) |
These conditions are satisfied for
In order to estimate
⟨e44⟩=1MM∑m=1(e44)m. | (4) |
The results are shown in Fig.3. One can observe that errors do not exceed
The values
The average
Table 1 contains the fundamental parameters of the uniform non-overlapping distribution
In the present section, we pay attention to experimental results partially presented in [18]. The images of Bacillus subtilis in
We use algorithms of image processing and analysis to determine number, centers, angles of inclinations and length of bacteria. The density of bacteria is calculated by formula (1) and oscillates around value
| |||||
In order to compare the distributions of DB sets with the distribution of bacteria we have made theoretical calculation of
| ||||
Comparing the results shown in the Tables 2 and 3 for the observed and theoretical distributions of bacteria, we can see that values of the corresponding
averaged | ||||
standard deviation of the | ||||
averaged | ||||
standard deviation of the |
In order to see that the theoretical and observed distributions are essentially different, we compare averaged basic sums (see Table 4) from Table 2 and Table 3. It is worth noting that changes of the parameters (
The above analysis of collective behavior can be considered as the first application of the RVE theory [6] which will be extended in the future. In particular, we plan to study dynamical parameters of the bacteria distributions in time using the data displayed in Fig.5.
Following [6], [7] we present constructive formulae for the Eisenstain-Rayleigh sums
The Eisenstein--Rayleigh lattice sums
The Eisenstein functions [21] are related to the Weierstrass function
E2(z)=℘(z)+S2,Em(z)=(−1)m(m−1)!dm−2℘(z)dzm−2,m=3,4,…. | (5) |
Every function (5) is doubly periodic and has a pole of order
We thank Alexei Sokolov who kindly presented the experimental results partially reported in [18] and Leonid Berlyand for stimulated discussion during his visit to Kraków and VM's visits to Penn State University.
The authors thank the anonymous reviewer for valuable comments and suggestions to improve the paper.
[1] |
M. Oezaslan, F. Haschxe, P. Strasser, In situ observation of bimetallic alloy nanoparticle formation and growth using high temperature XRD, Chem. Mater., 23 (2011), 2159-2165. doi: 10.1021/cm103661q
![]() |
[2] | F. Delogu, E. Arca, G. Mulas, Numerical investigation of the stability of Ag-Cu nanorods and nanowires, Phys. Rev. B, 78 (2008), 1-13. |
[3] |
R. Ferrando, J. Jellinek, R. L. Johnston, Nanoalloys: From theory to applications of alloy clusters and nanoparticles, Chem. Rev., 108 (2008), 845-910. doi: 10.1021/cr040090g
![]() |
[4] | J. Sopousek, J, Pinkas, P. Broz, Ag-Cu colloid synthesis: Bimetallic nanoparticle characterisation and thermal treatment, J. Nanomater., 36 (2014), 1-13. |
[5] | A. Aguado, J. M. López, Identifying structural and energetic trends in isovalent core-shell nanoalloys as a function of composition and size mismatch, J. Chem. Phys., 135 (2011), 1-11. |
[6] | B. M. Muñ ozflores, B. I. Kharisov, V. M. Jiménezpérez, Recent advances in the synthesis and main applications of metallic nanoalloys, Ind. Eng. Chem. Res., 50 (2011), 7705-7721. |
[7] | M. Faraday, Experimental Relations of Gold (and Other Metals) to Light, Royal Society, 1857. |
[8] | B. M. Muñ ozflores, B. I. Kharisov, V. M. Jiménezpérez, Recent advances in the synthesis and main applications of metallic nanoalloys, Ind. Eng. Chem. Res., 50 (2011), 7705-7721. |
[9] | H. J. Chen, Z. W. Li, Y. B. Zhao, Progress on the preparation of nanosized alloy materials, Prog. Chem., 16 (2004), 682-686. |
[10] |
R. Ferrando, J. Jellinek, R. L. Johnston, Nanoalloys: From theory to applications of alloy clusters and nanoparticles, Chem. Rev., 108 (2008), 845-910. doi: 10.1021/cr040090g
![]() |
[11] | K. Laasonen, E. Panizon, D. Bochicchio, Competition between icosahedral motifs in AgCu, AgNi, and AgCo nanoalloys: A combined atomistic-DFT study, J. Phys. Chem. C, 117 (2013), 26405-26413. |
[12] | F. Tournus, A. Tamion, N. Blanc, Chemical order in CoPt nanoclusters: Direct observation and magnetic signature, Phys. Rev. B, 77 (2008), 1-11. |
[13] | C. Langlois, Z. W. Wang, D. Pearmain, HAADF-STEM imaging of CuAg core-shell nanoparticles, J. Phys. Confer. Ser., 241 (2010), 1-4. |
[14] | S. Link, Z. L. Wang, M. A. El-Sayed, Alloy formation of gold-silver nanoparticles and the dependence on their absorption, J. Chem. Phys. B, 103 (1999), 3529-3533. |
[15] | M. Gaudry, J. Lerme, E. Cottancin, M. Ellarin, Optical properties of (AuxAg1-x)n clusters embedded in alumina: Evolution with size and stoichiometry, Phys. Rev., 64 (2001), 1-7. |
[16] | M. Tchaplyguine, T. Andersson, C. Zhang, Core-shell structure disclosed in self-assembled Cu-Ag nanoalloy particles, J. Chem. Phys., 138 (2013), 1-6. |
[17] | J. T. Jankowiak, M. A. Barteau, Ethylene epoxidation over silver and copper-silver bimetallic catalysts: I. Kinetics and selectivity, J. Catal., 236 (2005), 366-378. |
[18] | B. K. Hodnett, Heterogeneous Catalytic Oxidation, Wiley, 2000. |
[19] | E. Panizon, R. Ferrando, Strain-induced restructuring of the surface in core@shell nanoalloys, Nanoscale, 8 (2016), 15911-15919. |
[20] |
D. Nelli, R. Ferrando, Core-shell vs multi-shell formation in nanoalloy evolution from disordered configurations, Nanoscale, 11 (2019), 13040-13050. doi: 10.1039/C9NR02963J
![]() |
[21] | S. J. Kim, E. A. Stach, C. A. Handwerker, Fabrication of conductive interconnects by Ag migration in Cu-Ag core-shell nanoparticles, Appl. Phys. Lett., 96 (2010), 1-3. |
[22] | G. H. Wang, Stable structure and magic numbers of atomic clusters, Prog. Phys., 20 (2000), 53-93. |
[23] | H. R. Trebin, Quasicrystals: Structure and Physical Properties, Weinheim: Wiley-VCH, 2006. |
[24] | L. Zhang, C. B.Zhang, Y. Qi, Local structure changes of 54-, 55-, 56-atom copper clusters on heating, Phys. Lett. A, 372 (2008), 2874-2880. |
[25] | L. Zhang, S. N. Xu, C. B. Zhang, Modeling structural changes on cooling a molten Cu55 cluster by molecular dynamics, Acta. Metall. Sin., 44 (2008), 1161-1166. |
[26] |
W. Y. Li, F. Y. Chen, Effect of Cu-doped site and charge on the optical and magnetic properties of 55-atom Ag cluster: A density functional theory study, Comp. Mater. Sci., 81 (2014), 587-594. doi: 10.1016/j.commatsci.2013.09.022
![]() |
[27] |
P. L. Williams, Y. Mishin, J. C. Hamilton, An embedded-atom potential for the Cu-Ag system, Modelling. Simul. Mater. Sci. Eng., 14 (2006), 817-833. doi: 10.1088/0965-0393/14/5/002
![]() |
[28] | C. Mottet, G. Rossi, F. Baletto, Single impurity effect on the melting of nanoclusters, Phys. Rev. Lett., 95 (2005), 0355011-0355014. |
1. | Wojciech Nawalaniec, Efficient computation of basic sums for random polydispersed composites, 2018, 37, 0101-8205, 2237, 10.1007/s40314-017-0449-6 | |
2. | Wojciech Gwizdała, Roman Czapla, Wojciech Nawalaniec, 2020, 9780128188194, 269, 10.1016/B978-0-12-818819-4.00018-0 | |
3. | Piotr Drygaś, Simon Gluzman, Vladimir Mityushev, Wojciech Nawalaniec, 2020, 9780081026700, 1, 10.1016/B978-0-08-102670-0.00010-X | |
4. | Piotr Drygaś, Simon Gluzman, Vladimir Mityushev, Wojciech Nawalaniec, 2020, 9780081026700, 57, 10.1016/B978-0-08-102670-0.00011-1 | |
5. | 2020, 9780081026700, 333, 10.1016/B978-0-08-102670-0.00019-6 | |
6. | Vladimir Mityushev, Cluster method in composites and its convergence, 2018, 77, 08939659, 44, 10.1016/j.aml.2017.10.001 | |
7. | Roman Czapla, Random Sets of Stadiums in Square and Collective Behavior of Bacteria, 2018, 15, 1545-5963, 251, 10.1109/TCBB.2016.2611676 | |
8. | Wojciech Nawalaniec, Classifying and analysis of random composites using structural sums feature vector, 2019, 475, 1364-5021, 20180698, 10.1098/rspa.2018.0698 | |
9. | Wojciech Nawalaniec, Basicsums: A Python package for computing structural sums and the effective conductivity of random composites, 2019, 4, 2475-9066, 1327, 10.21105/joss.01327 | |
10. | Drygaś Piotr, 2022, Chapter 25, 978-3-030-87501-5, 253, 10.1007/978-3-030-87502-2_25 | |
11. | Wojciech Nawalaniec, 2022, Chapter 29, 978-3-030-87501-5, 289, 10.1007/978-3-030-87502-2_29 | |
12. | Natalia Rylko, Karolina Tytko, 2022, Chapter 30, 978-3-030-87501-5, 297, 10.1007/978-3-030-87502-2_30 | |
13. | Gael Yomgne Diebou, Well-posedness for chemotaxis–fluid models in arbitrary dimensions* , 2022, 35, 0951-7715, 6241, 10.1088/1361-6544/ac98ec | |
14. | Roman Czapla, 2022, Chapter 24, 978-3-030-87501-5, 241, 10.1007/978-3-030-87502-2_24 |
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