Connectedness percolation in the random sequential adsorption packings of elongated particles
Nikolai I. Lebovka, Mykhailo O. Tatochenko, Nikolai V. Vygornitskii, Andrei V. Eserkepov, Renat K. Akhunzhanov, Yuri Yu. Tarasevich
CConnectedness percolation in the random sequential adsorption packings of elongatedparticles
Nikolai I. Lebovka,
1, 2, ∗ Mykhailo O. Tatochenko, Nikolai V.Vygornitskii, Yuri Yu. Tarasevich, † and Andrei V. Eserkepov Laboratory of Physical Chemistry of Disperse Minerals,F. D. Ovcharenko Institute of Biocolloidal Chemistry, NAS of Ukraine, Kyiv 03142, Ukraine Department of Physics, Taras Shevchenko Kiev National University, Kyiv 01033, Ukraine Laboratory of Mathematical Modeling, Astrakhan State University, Astrakhan 414056, Russia (Dated: December 8, 2020)Connectedness percolation phenomena in two-dimensional packings of elongated particles (dis-corectangles) were studied numerically. The packings were produced using random sequential ad-sorption (RSA) off-lattice model with preferential orientations of particles along a given direction.The partial ordering was characterized by order parameter S , with S = 0 for completely disor-dered films (random orientation of particles) and S = 1 for completely aligned particles along thehorizontal direction x . The aspect ratio (length-to-width ratio) for the particles was varied withinthe range ε ∈ [1; 100]. Analysis of connectivity was performed assuming a core-shell structure ofparticles. The value of S affected the structure of packings, formation of long-range connectivityand electrical conductivity behavior. The effects were explained accounting for the competitionbetween the particles’ orientational degrees of freedom and the excluded volume effects. For aligneddeposition, the anisotropy in electrical conductivity was observed and the values along alignmentdirection, σ x , were larger than the values in perpendicular direction, σ y . The anisotropy in local-ization of percolation threshold was also observed in finite sized packings, but it disappeared in thelimit of infinitely large systems. I. INTRODUCTION
Random packing of elongated particles onto the planeis a challenging problem that has continuously been thefocus of many researchers. The particle shape may af-fect the packing characteristics (e.g., packing density andcoordination numbers) [1–3], the aggregation [4], andgravity- and vibration-induced segregation [5]. A lotof interest to such systems is stimulated by practicalproblems related with preparation of advanced materi-als [6, 7] and composite films [8, 9], filled with elongatednanoparticles, e.g., carbon nanotubes [10] and silicateplatelets [11].For contraction of random packings, a random se-quential adsorption (RSA) model [12, 13] is frequentlyused. In this model, the particles are deposited ran-domly and sequentially onto a two-dimensional (2D) sub-strate without overlapping. In the so-called “jamminglimit” at saturated coverage concentration ϕ j no moreparticles can be adsorbed and the deposition processterminates. The saturated 2D RSA packings for dif-ferent particle shapes, including disks [14], ellipses [15],squares [16], rectangles [17, 18], discorectangles [19, 20],polygons [21], sphere dimers, sphere polymers, k -mersand extended objects [22–24], and other shapes [25–27]have been studied in details. Particularly, for elongatedparticles, the nonmonotonic dependencies of values ϕ j versus the aspect ratio, ε were observed. Similar depen- ∗ Corresponding author: [email protected] † Corresponding author: [email protected] dencies have also been observed for saturated RSA pack-ings of elongated particles in one-dimensional (1D) [28–30] and three-dimensional (3D) [31–33] systems. Theappearance of maximums of the jamming concentrationwas explained by a competition between the effects oforientational degrees of freedom and excluded volume ef-fects [31].The formation of long-range connectivity is the pri-mary issue to be solved for better understanding of per-colation phenomena of core-shell anisotropic particle inrandom packings. Core-shell composite particles con-sist of an inner layer of one material (core) and anouter layer of another material (shell). Core-shell par-ticles have already demonstrated promising applicationsin electrochemical, optical, wearable and gas adsorptivesensors [34], electrode materials [35], polymeric compos-ites [36] and drug delivery applications [37]. Practicalsignificance of the problem is also related description ofelectrical conductivity behavior of composites filled byelongated core-shell particles, e.g., carbon nanotubes andfibers, metallic nanorods and nanocables, and other core-shell particulates [36, 38–47]. In general case, the innermaterial can covered partially or fully by one or multipleouter layers. By regulation of shell properties, the mate-rials with enhanced optical, electrical, magnetic charac-teristics, and improved thermal stability or dispersibilitycan be obtained. For particles with core-shell structures,the electrical conductivity behavior can reflect the effectsof particle ordering, packing, connectivity rules and in-trinsic properties of the cores, the matrix, and the inter-face between particles and matrix (shells).In this paper, we’ll concentrate on the percolation ef-fects in 2D RSA packings of discorectangles. The hard a r X i v : . [ c ond - m a t . d i s - nn ] D ec a) y x S =1 S =0 S =0.5 S =1 S =0 S =0.5 b) ε =2 ε =5 FIG. 1. Examples of RSA packings in jamming state for discorectangles with aspect ratios ε = 2 (a) and ε = 10 (b), andat different values of the order parameters: S = 0 (random orientation), S = 0 . S = 1 (completealignment along the horizontal direction x ). core-soft shell structure of particles was assumed andanisotropic packings with preferential orientations of par-ticles along a given direction were considered. The effectsof particles aspect ratio, orientation ordering, and pack-ing fraction on electrical conductivity of packings andcritical thickness of shells required for a spanning paththrough the system were evaluated. The rest of the paperis organized as follows. In Sec. II, the technical details ofthe simulations are described and all necessary quantitiesare defined. In order to provide a better understandingon the precision of calculations some test results are alsogiven. Section III presents our principal findings and dis-cussions. Finally, Section IV concludes this work. II. COMPUTATIONAL MODEL
Discorectangle represents a rectangle with semicirclesat a pair of opposite sides. The discorectangles wererandomly and sequentially deposited until they reachedthe saturated coverage concentration ϕ j . An optimizedRSA algorithm, based on tracking of local regions, wasused [19, 20]. The aspect ratio (length-to-width ratio)was defined as ε = l/d , where l is the length of theparticle and d is its width. The discorectangles with ε ∈ [1; 100] were considered.The degree of orientation was characterized by the or-der parameter defined as S = (cid:104) cos 2 θ (cid:105) , (1)where (cid:104)·(cid:105) denotes the average, θ is the angle between thelong axis of the particle and the direction of the preferredorientation of the particles ( x directions).For generation of the aligned packings, the orientationsof the deposited particles were selected to be uniformly -2 ϕ S =1 j ϕ S =0 ε = ϕ tt S=
0 1
L/l
FIG. 2. Coverage concentration ϕ versus the deposition time, t , for the RSA packing of random ( S = 0) and perfectlyaligned ( S = 1) discorectangles with aspect ratio ε = 4 atdifferent values of L/l . Here, ϕ j is the jamming coverage. In-set shows the enlarged portion of ϕ ( t ) plot near the saturationconcentrations. distributed within some interval such that − θ m (cid:54) θ (cid:54) θ m , where θ m (cid:54) π/ S = sin 2 θ m θ m . (2)Figure 1 shows examples of the packing patterns injamming state for discorectangles with aspect ratios ε =2 (a) and ε = 10 (b). For random orientation of particles a) b) σ m σ c σ s po r e core ld shell δ d FIG. 3. To description of the connectivity analysis (a) and calculation of electrical conductivity (b) the RSA packing ofdiscorectangles on a 2D substrate. A core-shell structure of particles was assumed. Intersections of the particle cores wereforbidden. For the connectivity analysis, each particle was assumed to be covered by a soft (penetrable) shell with thickness δd . To calculate the electrical conductivity, σ , a discretization approach with supporting mesh was used. The mesh cells withcenters located at core, shell, or pore parts were assumed to have electrical conductivity of σ c , σ s and σ m , respectively. Thelarge contrasts in electrical conductivities were assumed, σ c (cid:29) σ s (cid:29) σ m . ( θ m = π/
2) we have S = 0 and for complete alignmentof particles along the horizontal direction x ( θ m = 0)we have S = 1. For intermediate values 0 < S < L along both the horizontal ( x ) and the vertical ( y ) di-rections, and periodic boundary conditions were appliedin both directions. The time was measured using dimen-sionless time units, t = n/L , where n is the numberof deposition attempts. Figure 2 shows examples of thecoverage concentration ϕ versus the deposition time, t ,for the RSA packing of random ( S = 0) and perfectlyaligned ( S = 1) discorectangles with aspect ratio ε = 4at different values of L/l . The similar dependencies wereobserved for other values of S and ε . The scaling testswith L/l = 16 , ,
64, and 128 evidenced the good con-vergence of the data at
L/l ≥
32. In the present work,the majority of calculations were done using L = 32 l andthe jamming coverage was assumed to be reached afterdeposition time of t = L × .The analysis of the connectivity was performed assum-ing a core-shell structure of particles. Each particle wascovered by an outer shell with a thickness δd (Figure 3a).Any two particles were assumed to be connected whenthe minimal distance between their hard cores does notexceed the value of δd . The connectivity analysis wascarried out using a list of near-neighbor particles [51].The minimum (critical) value of the relative outer shellthickness, δ c , (hereinafter, shell thickness) required forthe formation of spanning clusters along the x or y di-rection, was evaluated using the Hoshen—Kopelman al-gorithm [52].To calculate the electrical conductivity, σ , the 2D plane was covered by a supporting square mesh of size m × m (Figure 2b). The mesh cells with centers located atcore, shell, or pore parts were assumed to have electri-cal conductivity of σ c , σ s and σ m , respectively. Theneach cell was associated with a set of four resistors andthe system was transformed into a random resistor net-work (RRN). Location of the given mesh cell at the ob-ject boundaries (i.e. interfaces between core, shell, orpore parts) was identified using positions of 4 corners ofmesh square. The calculations were done for two situ-ations when all 4 corners beyond to the same part ofmedium (core, shell, or pore parts) and at least 1 cor-ner of mesh square beyond to the same part of medium.Then obtained data for these two situation were aver-aged. Preliminary studies evidenced that such approachallows obtaining better accuracy of calculations of elec-trical conductivity. Note that calculations at large valuesof m provided better accuracy, but required more signifi-cant computing resources. Therefore, the effects of values m ( m = 1024 , , σ werealso checked in some calculations.The large contrasts in electrical conductivities were as-sumed, σ c (cid:29) σ s (cid:29) σ m . We put σ c = 10 , σ s = 10 and σ m = 1 in arbitrary units. To calculate the electrical con-ductivity of RRN the Frank—Lobb algorithm based onthe Y-∆ transformation was applied [53]. More detailedinformation on electrical conductivity calculation can befound elsewhere [54, 55].For each given value of ε and S , the computer exper-iments were repeated using from 10 to 1000 independentruns. The error bars in the figures correspond to the stan-dard deviation of the mean. When not shown explicitly,they are of the order of the marker size. III. RESULTS AND DISCUSSIONA. Intrinsic conductivity
Concept of intrinsic conductivity is useful for descrip-tion of behavior of electrical conductivity in the lim-iting case of the infinitely diluted system. For ran-domly aligned and arbitrarily shaped particles with elec-trical conductivity σ p suspended in continuous mediumwith electrical conductivity σ m , the generalized Maxwellmodel gives the following virial expansion [56, 57] σσ m = 1 + [ σ ] ϕ + O( ϕ ) , (3)where [ σ ] = d ln ( σ/σ m )d ϕ (cid:12)(cid:12)(cid:12)(cid:12) ϕ → , (4)is called the intrinsic conductivity, and ϕ is the coverageconcentration. [ σ ] S ε [ σ ] [ σ ] x y ε = [ σ ] = [ σ ] ( ±κ S ) κ x → + , y → - κ I n t r i n s i c c ondu c ti v it y [ σ ] xy m = , σ c o r e = a)b) e lli p s e s FIG. 4. Examples of intrinsic conductivities [ σ ] versus theorder parameter, S . The data are presented along x and y directions for discorectangles with aspect ratios ε = 2 , , σ ] , κ (See Eq. 4) versus ε dependencies (b).. [ σ ] / [ σ ] ∞ x y ε /m xyS =1 m FIG. 5. Normalized intrinsic conductivity [ σ ] / [ σ ] ∞ along x and y direction versus the inverse mesh size 1 /m at differentaspect ratios ε = 4 ,
20, and S = 1. The value of the intrinsic conductivity [ σ ] can dependupon the electrical conductivity contrast ∆ = σ p /σ m , theparticle’s aspect ratio, ε , order parameter, S , and spatialdimension.Figure 4a demonstrates examples of intrinsic conduc-tivities [ σ ] versus the order parameter, S . The data arepresented along x and y directions for discorectangleswith different aspect ratios ε . These dependencies wereobtained using the mesh parameter of m = 4096 and 1000independent runs. Observed [ σ ] versus S were practicallylinear: [ σ ] = [ σ ] (1 ± κS ) , (5)where [ σ ] is the intrinsic conductivity for the isotropicsystem with S = 0, κ is the anisotropy coefficient, signs+ or − correspond to the x or y directions, respectively.Therefore, intrinsic conductivity [ σ ] x along alignmentdirection x exceeded value [ σ ] y in perpendicular direc-tion hence symmetric behavior with the same anisotropycoefficients κ was observed.Figure 4b presents values of [ σ ] and κ versus aspectratio ε . The intrinsic conductivity for the isotropic sys-tem [ σ ] increased with ε . Note, that the similar behaviorwas predicted by the theory for randomly aligned ellipses( S = 0) [56, 57] [ σ ] = (∆ − ε ) ε ∆)(∆ + ε ) . (6)For ∆ (cid:29)
1, this equation gives (see dashed line in Fig. 4b)[ σ ] = 1 + 12 (cid:18) ε + 1 ε (cid:19) . (7)The anisotropy coefficient κ also increased with ε reaching to the value of κ = 1 in the limit [ ε ] ∞ .Figure 5 illustrates the effect of mesh size m on preci-sion of [ σ ] determination at two values of aspect ratio ε .In the limit of the infinite large mesh m → ∞ . The dataevidenced that estimation errors of [ σ ] increased with in-creasing of ε and they can reach about 2% for ε = 20 and m = 1024. B. Connectivity
For discorectangle, the critical shell thicknesses δ c,x and δ c,y correspond to the formation of percolation clus-ters along the x and y direction, respectively. Forisotropic system with S = 0, the values of δ c,x and δ c,y coincide, i.e., δ c,x = δ c,y . For anisotropic systems with S (cid:54) = 0 these value may be different. At fixed value ofshell thicknesses, δ , the critical coverages ϕ c,x and ϕ c,y ,required for the formation of percolation clusters alongthe x and y direction, respectively, can be also defined. a)b) δ c /L ε =4 ϕ = ϕ = ϕ j ϕ c /L / ν ε =4 δ =0.3 δ = δ =0.8 ϕ =0.4 ϕ =0.557 Sx y Sx y FIG. 6. Scaling dependencies of the critical shell thickness δ c at different values of particles coverage, ϕ c , (a) and of thecritical particle coverage ϕ c at different fixed values of shellthickness, δ (b). The data are presented for aspect ratio of ε = 4 for completely disordered ( S = 0, dashed lines) andcompletely aligned ( S = 1, solid lines) packings. For S = 0the data along the x and y directions almost coincide. Here, L (= 16 , , , ν = 4 / Figure 6a shows the examples of the critical shell thick-ness δ c versus inverse systems size 1 /L at different ϕ .Here, L (= 16 l, l, l, l ) is the size of the system.The data are presented for aspect ratio of ε = 4 for com-pletely disordered ( S = 0, dashed lines) and completelyaligned ( S = 1, solid lines) packings. Increase in ϕ re-sulted in decrease of δ c and the minimum values of δ c were observed at the jamming coverage ( ϕ = ϕ j ≈ . ε = 4). For S = 0, the data along the x and y direc-tions almost coincide. However, for finite-sized alignedsystems ( S (cid:54) = 0), the value of δ c,y always exceeded thevalue of δ c,x , and both these values exceeded the value δ c for isotropic systems. Figure 6b shows the similarexamples of the critical coverage ϕ c versus the value of L − /ν at different fixed values of shell thickness, δ . Here, ν = 4 / ϕ c alsodemonstrated the presence of percolation anisotropy forthe finite-sized aligned systems ( S (cid:54) = 0). The similar per-colation anisotropy was observed in finite-sized discretesystems with aligned rods ( k -mers) and the finite size ef-fect were also more pronounced for systems with alignedrods [54, 59, 60]. Thus, it can be concluded that observedanisotropies in behavior the critical shell thickness, δ c ,and critical coverage ϕ c are the finite size scaling effectand they disappear in the limit of L → ∞ . Moreover,the scaling behaviors of value δ c for completely disor-dered ( S = 0) and of average value δ c = ( δ c,x + δ c,y ) / S (cid:54) = 0) packings were rather insignificant for L/l ≥
32. Therefore, in the present work, the averagedvalues δ c and ϕ c in both directions were always used andall connectivity analysis tests were done using L/l = 32.Figure 7 and 8 demonstrate the examples of the criticalshell thickness δ c (Fig. 7) and critical coverage ϕ c (Fig. 8)versus aspect ratio ε for completely disordered, S = 0,(a) and completely aligned, S = 1, (b) packings. For forcompletely disordered systems ( S = 0) the maximums on δ c ( ε ) (Fig. 7a) and ϕ c ( ε ) (Fig. 8a) curves, at some valuesof ε max , were observed. Positions of these maximumswere affected values of ϕ (Fig. 7a) and δ (Fig. 8a).The observed maximums in percolation characteristics δ c and ϕ c can reflect internal structure of RSA packingsof elongated particles. Particularly, the the maximums injamming coverage ϕ j versus ε dependencies were also ob-served for disordered packings, and they were explainedby the competitions of the effects of orientation degreesof freedoms and excluded volume effects. The jamminglimit decreased with ε [20], and for elongated particles inthe vicinity of percolation packings the terminations ofcurves δ c ( ε ) (Fig. 7a) and ϕ c ( ε ) (Fig. 8a) at some criticalvalues of ε were observed.These maximums became less pronounced for partiallyaligned systems, and they completely disappeared forand completely aligned, S = 1, packings (Fig. 7b andFig. 8b). For the case of S = 1, the values of δ c (Fig. 7b)and ϕ c (Fig. 8b) grown with ε , and for relatively smallshell thickness, δ , the termination of ϕ c ( ε ) was observed a)b) δ c ε S =00.2 ε max ϕ =0.5 δ c ε S =1 ϕ =0.50.40.30.2 FIG. 7. Critical shell thickness δ c versus aspect ratio ε atdifferent coverages, ϕ for completely disordered, S = 0, (a)and completely aligned, S = 1, (b) packings. when the values ϕ c exceed the jamming coverage, ϕ j . C. Electrical conductivity
For each independent run the electrical conductivity σ displayed the jump at some percolation concentration ϕ σ . Figure 9 presents σ , versus the difference, d ϕ = | ϕ − ϕ σ | , for RSA packings of disks ( ε = 1) at differentshell thickness, δ = 0 . δ = 0 . s and t were estimated from the scaling relationsfor the electrical conductivities just below, σ ∝ (d ϕ ) − s ,and above, σ ∝ (d ϕ ) t , the percolation threshold [58].The classical values for 2D percolation are s = t ≈ / δ = 0 . δ = 0 . a)b) ϕ c ε ε max ϕ j ( ε ) δ =0.80.50.30.20.150.1 δ =0.80.50.30.20.150.1 ε ϕ c S =1 S =0 ϕ j ( ε ) FIG. 8. Critical coverage ϕ c versus aspect ratio ε at differentshell thickness, δ , for completely disordered, S = 0, (a) andcompletely aligned, S = 1, (b) packings. fects of shell thickness on value of ϕ σ . Above the percola-tion threshold, such effects were insignificant. Figure 10compares σ , versus the difference, d ϕ = | ϕ − ϕ σ | , depen-dencies, for RSA packings of discorectangles ( ε = 4) atfixed value of δ = 0 . S = 0,(a) and completely aligned, S = 1, (b) packings. Foraligned packings, the significant anisotropy in electricalconductivity was observed and the values along align-ment direction, σ x , significantly exceeded the values inperpendicular direction, σ y . Importantly, the obtaineddata for the mesh sizes of m = 1024 and m = 2048 wereapproximately the same within data errors.Finally, figure 11 compares electrical conductivity σ ,versus the difference, d ϕ = | ϕ − ϕ σ | rather long dis-corectangles ( ε = 10). The data are presented at fixedvalue of δ = 0 . S = 1) RSA pack-ings at two values of m . Observed behavior for ε = 10 wassimilar to that observed for ε = 4 (Fig. 10b). Above per-colation threshold ( ϕ > ϕ σ ) the effect of m was insignif-icant. However, below percolation threshold ( ϕ < ϕ σ ) -3 -2 -1 σ σ c o r e = , σ s h e ll = ε = ( D i s k s ) m = d ϕ = |ϕ - ϕ σ | t =4/3 s =-4/3 δ FIG. 9. Electrical conductivity, σ , versus the difference, d ϕ = | ϕ − ϕ σ | , for RSA packings of disks ( ε = 1) at different shellthickness δ . Here, the value of ϕ σ was identified from theconcentration of percolation jump for each independent run,and calculation were done using the mesh size of m = 1024.Dashed lines corresponds to the classical exponents s = t ≈ / the electrical conductivities estimated at m = 1024 weresystematically smaller as compared to that estimated at m = 2048. IV. CONCLUSION
Numerical study of two-dimensional RSA deposition ofaligned discorectangles on a plane was done. The partialordering was characterized by order parameter S , with S = 0 for random orientation of particles and S = 1 forcompletely aligned particles along the horizontal direc-tion x . Analysis of connectivity was performed assum-ing a core-shell structure of particles. The values of theaspect ratio, ε , and order parameter, S , significantly af-fected the structure of packings, formation of long-rangeconnectivity and electrical conductivity behavior. Theobserved effects can reflected the competition betweenthe particles’ orientational degrees of freedom and theexcluded volume effects [32]. For aligned systems, dif-ferent anisotropies in intrinsic conductivity, long rangedconnectivity, and behavior of electrical conductivity wereobserved. For example, the significant anisotropy in elec-trical conductivity was observed and the values alongalignment direction, σ x , were large than values in per-pendicular direction, σ y . For aligned finite-size systems,the percolation thresholds along the x and y directionswere different. However, these differences disappeared inthe limit of infinitely large systems. -3 -2 -1 -3 -2 -1 σ d ϕ t =4/3 s =-4/3 a)b) d ϕ x,yx,yy xy σ c o r e = , σ s h e ll = ε = m = ε = , δ = S =0 xx y ε = , δ = S =1 t =4/3 s =-4/3 FIG. 10. Electrical conductivity, σ , versus the difference, | ϕ − ϕ σ | , for RSA packings of discorectangles at different val-ues of aspect ratio, ε , for fixed shell thickness of δ = 0 . S = 0, (a) and completely aligned, S = 1, (b) packings. Here, the value of ϕ σ was identifiedfrom the concentration of percolation jump for each indepen-dent run, and calculation were done using the mesh size of m = 1024. Dashed lines corresponds to the classical expo-nents s = t ≈ / ACKNOWLEDGMENTS
We are thankful to R.K. Akhunzhanov andI.V.Vodolazskaya for stimulating discussions andA.G.Gorkun for technical assistance. We acknowl-edge funding from the National research founda-tion of Ukraine, Grant No. 2020.02/0138 (M.O.T.,N.V.V.), National Academy of Sciences of Ukraine,Project Nos. 7/9/3-f-4-1230-2020, 0120U100226and 0120U102372/20-N (N.I.L.), and funding fromthe Foundation for the Advancement of Theoreti-cal Physics and Mathematics “BASIS”, Grant No.20-1-1-8-1. (Y.Y.T. and A.V.E.).. -3 -2 -1 σ d ϕ t =4/3 s =-4/3 x y m= ε = , S =1, δ =0.3 σ c o r e = , σ s h e ll = ε = ( D i s k s ) xyxy FIG. 11. Electrical conductivity, σ , versus the difference, | ϕ − ϕ σ | . The data are presented for discorectangles withaspect ratio ε = 10, shell thickness δ = 0 . S = 1, RSA packings. Here, the value of ϕ σ wasidentified from the concentration of percolation jump for eachindependent run, and calculation were done using the meshsize of m = 1024 and of m = 2048. Dashed lines correspondsto the classical exponents s = t ≈ / , 71 (1996).[2] R. Guises, J. Xiang, J.-P. Latham, and A. Munjiza, Gran-ular packing: numerical simulation and the characterisa-tion of the effect of particle shape, Granul. Matter ,281 (2009).[3] A. V. Kyrylyuk and A. P. Philipse, Effect of parti-cle shape on the random packing density of amorphoussolids, Phys. Status Solidi A , 2299 (2011).[4] A. K. H. Kwan and C. F. Mora, Effects of variousshape parameters on packing of aggregate particles, Mag.Concr. Res. , 91 (2001).[5] C. R. A. Abreu, F. W. Tavares, and M. Castier, Influenceof particle shape on the packing and on the segregationof spherocylinders via Monte Carlo simulations, PowderTechnol. , 167 (2003).[6] L. Bokobza, Natural rubber nanocomposites: A review,Nanomaterials , 12 (2019).[7] L. Yang, Z. Zhou, J. Song, and X. Chen, Anisotropicnanomaterials for shape-dependent physicochemical andbiomedical applications, Chem. Soc. Rev. , 5140(2019).[8] J. Hirotani and Y. Ohno, Carbon nanotube thin filmsfor high-performance flexible electronics applications, in Single-Walled Carbon Nanotubes (Springer, 2019) pp.257–270.[9] I. Tiginyanu, P. Topala, and V. Ursaki, eds.,
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