nanomaterials
Article Surface Energy of Curved Surface Based on Lennard-Jones Potential
Dan Wang 1,*, Zhili Hu 1, Gang Peng 2 and Yajun Yin 2
1 Key Laboratory of Mechanics and Control of Mechanical Structures, Interdisciplinary Research Institute, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211100, China; [email protected] 2 Department of Engineering Mechanics, School of Aerospace, Tsinghua University, Beijing 100084, China; [email protected] (G.P.); [email protected] (Y.Y.) * Correspondence: [email protected]
Abstract: Although various phenomena have confirmed that surface geometry has an impact on surface energy at micro/nano scales, determining the surface energy on micro/nano curved surfaces remains a challenge. In this paper, based on Lennard-Jones (L-J) pair potential, we study the geomet- rical effect on surface energy with the homogenization hypothesis. The surface energy is expressed as a function of local principle curvatures. The accuracy of curvature-based surface energy is confirmed by comparing surface energy on flat surface with experimental results. Furthermore, the surface energy for spherical geometry is investigated and verified by the numerical experiment with errors within 5%. The results show that (i) the surface energy will decrease on a convex surface and increase on a concave surface with the increasing of scales, and tend to the value on flat surface; (ii) the effect of curvatures will be obvious and exceed 5% when spherical radius becomes smaller than 5 nm; (iii) the surface energy varies with curvatures on sinusoidal surfaces, and the normalized surface energy relates with the ratio of wave height to wavelength. The curvature-based surface energy offers new insights into the geometrical and scales effect at micro/nano scales, which provides a Citation: Wang, D.; Hu, Z.; Peng, G.; theoretical direction for designing NEMS/MEMS. Yin, Y. Surface Energy of Curved Surface Based on Lennard-Jones Keywords: surface energy; geometrical effect; curvatures; Lennard-Jones potential Potential. Nanomaterials 2021, 11, 686. https://doi.org/10.3390/nano11030686
Academic Editor: Frederik Tielens 1. Introduction Surface energy or surface tension plays an important role in determining various Received: 23 February 2021 natural and industrial phenomena, including surface bulking, phrase separation, welding Accepted: 5 March 2021 and solidification, diffusion and transportation, etc. [1–4]. At micro/nano scales, surfaces Published: 9 March 2021 are highly curved to achieve a lower energy, leading to a greatly increased ratio of surface area to volume. Surface effects become significant for mechanical properties at small Publisher’s Note: MDPI stays neutral scales, such as effective elastic moduli of nanocomposites, the bending, vibration, and with regard to jurisdictional claims in published maps and institutional affil- buckling responses of nanosized structural elements [5–7]. Hence, much effort has been iations. directed to determine the surface energy by means of experimental methods, dynamical simulations and theoretical modeling over the last few decades. Experiments determining surface tension mainly depend on the Laplace method, which relates the excess pressure at a curved surface with the principal radius of curvature at a point on the surface [8], for example maximum bubble pressure [9] and maximum pressure in a drop [10], sessile Copyright: © 2021 by the authors. drop [11], and pendant drop [12], etc. The tested surface energy values will be varied in Licensee MDPI, Basel, Switzerland. a range, which are affected by temperature, pressure as well as experimental errors [13]. This article is an open access article distributed under the terms and Atomic simulations can assist to calculate surface energy due to the development of conditions of the Creative Commons computing technology, such as classical thermodynamics calculations, molecular dynamics Attribution (CC BY) license (https:// simulations, ab initio calculations, and density functional theory [14]. Vollath et al. used creativecommons.org/licenses/by/ the atomistic approach to study surface energy of nanoparticles and discussed the influence 4.0/). of particle size and structure [15]. The surface energy density on rough metal surfaces is
Nanomaterials 2021, 11, 686. https://doi.org/10.3390/nano11030686 https://www.mdpi.com/journal/nanomaterials Nanomaterials 2021, 11, 686 2 of 14
investigated by atomistic calculations, and the simulation results suggest an analytical model to determine the surface energy density of rough surfaces by the surface geometric parameters and the surface energy density of planar surface [16]. Vega et al. estimated the surface tension of water with several models by using the test-area simulation method [17]. Besides experiments and simulations, some models have been proposed to study the surface energy and tension. For example, a unitary approach has been proposed for the calculation of surface energy and surface tension of nanoparticle being in equilibrium with its saturated vapor on both flat and curved surfaces at given temperature [18]. Through ex- tensive theoretical calculations of the surface tension of most of the liquid metals, Aqra et al. used the fraction of broken bonds in liquid metals to calculate surface tension and surface energy [19,20]. An analytical approach is developed to estimate the surface energy density of spherical surfaces through that of planar surfaces using a geometrical analysis and the scaling law [21]. Although many works have been done, determination of surface energy on curved surface remains unsolved, especially at small scales. For example, the testing of surface energy at nano scales will be very difficult considering environmental influence as well as non-negligible errors [22,23]. Experimental methods are hard to measure the surface energy on curved solid surface directly [24,25]. For the atomic simulations, Malfreyt et al. pointed out that the simulated system-sizes has an impact on the calculated surface ten- sion values [26]. Moreover, the uncertainty increases if one questions its dependence on particle size or more complicated geometry. Generally, considerations based on classical thermodynamics lead to the prediction of decreasing values of the surface energy with decreasing particle size [27]. However, the atomistic approach, based either on molecular dynamics simulations or ab initio calculations, generally leads to values with an opposite tendency. Vollath et al. points out that an insufficient definition of the particle size can lead to this result [15]. Besides size effect, calculating surface tension by atomic simulation takes a long time and a large amount of computation. Although some theoretical models have discussed the surface energy and tension on spherical particles using empirical parameters, there is a lack of models for more complicated curved surface. The present paper is aimed to investigate the effect of surface geometry on surface energy by theoretical modelling. The interaction pair potential between particles is as- sumed to be Lennard-Jones (L-J) potential, which can describe intermolecular interactions between electronically neutral atoms or molecules [28]. As the L-J potential is a short-range interaction, the surface particles will mainly interact with local particles in the neighbor- hood. The surface energy is expressed as a function of principle curvatures based on the homogenization hypothesis. The theoretical results for liquid metals are compared with experimental values in the literatures to confirm the reliability. The effect of curvatures are discussed based on spherical particles and compared with numerical experimental results. The geometrical effect on surface energy is investigated on more complicated structures such as sinusoidal surfaces based on curvature-based expression. This work provides a new perspective to understand the geometrical effect on surface energy at small scales, which will provide a direction on the design of MENS/NEMS.
2. Models and Methods L-J potential models the soft repulsive and attractive interactions, which is usually expressed as, σ 12 σ 6 U = 4ε − , (1) LJ r r where r is the distance between two interacting particles, the depth of the potential well ε is referred to as dispersion energy, and σ is the distance at which the particle-particle potential energy ULJ is zero [28]. For simplification of calculation, a negative exponential potential form U(r) = C/rn is used to model L-J pair potential. For n = 12, C = 4εσ12 and n = 6, C = −4εσ6, L-J pair potential can be taken as a superposition of negative exponential form taking n = 6 and 12. Nanomaterials 2021, 11, x FOR PEER REVIEW 3 of 15
potential energy ULJ is zero [28]. For simplification of calculation, a negative exponential potential form U r C r n is used to model L-J pair potential. For n = 12, C 4 12 and n = 6, C 4 6 , L-J pair potential can be taken as a superposition of negative expo- Nanomaterials 2021, 11, 686 nential form taking n = 6 and 12. 3 of 14 As shown in Figure 1, particles are supposed to be hard spheres with the equilibrium radius . A homogenization hypothesis is used by assuming particles distributed uni- As shown in Figure1, particles are supposed to be hard spheres with the equilibrium formly in the body with the number density of particles v . Then, the interaction poten- radius τ. A homogenization hypothesis is used by assuming particles distributed uniformly intial the for body a particle with the numberP on the density surface of particles of the semiρv. Then,-infinite the interaction flat surface potential can be for expressed a as, particle P on the surface of the semi-infinite flat surface can be expressed as, 2vC Usurf n3 n 4 , (2) 2πρ C n 3 U (τ) = v n ≥ 4, (2) sur f (n − 3)τn−3
Figure 1. The schematic models: (a) The interaction for a particle P on the semi-infinite surface body; Figure 1. The schematic models: (a) The interaction for a particle P on the semi-infinite surface (b) The interaction for a particle P inside the body. body; (b) The interaction for a particle P inside the body. In the theory of solid physics, particles on the surface have higher energy than inside particles,In asthe the theory surface of particles solid physics, interact with particles less particles on the [29 surface]. The surface have higher energy andenergy than inside surfaceparticles, tension as havethe surface different particles physical meanings. interact with The additional less particles energy [29]. for the The surface surface energy and particle compared to inside particles is surface energy [30]. Surface tension is the external energysurface to formtension a new have unit different surface area physical [31]. For liquids,meanings. they areThe almost additional the same energy value, for the surface asparticle the atoms compared can move to freely inside from particles inside to theis surface surface. energy For solids, [30]. as theSurface surface tension layer is the external cannotenergy slide to freely,form thea new equilibrium unit surface for surface area tension [31]. For has liquids, to contain they the shearare almost stresses the same value, betweenas the theatoms interiors can of move the solid freely and the from surface inside layers, to which the surface. lead to quite For different solids, values as the surface layer ofcannot surface slide energy freely, and surface the equilibrium tension in solids. for surface Here, we tension use the definitionhas to contain of surface the shear stresses energy to calculate values, as the experimental calculated values from Young’s equation is surfacebetween energy. the interiors of the solid and the surface layers, which lead to quite different val- uesAs of shown surface in Figureenergy1b, and the interaction surface tension potential in for solids. a particle Here, inside we body use is twicethe definition the of surface valueenergy of a to surface calculate particle values, interacted as withthe semi-infiniteexperimental surface calculated body, values from Young’s equation is surface energy. 4πρ C U (τ) = v n ≥ 4, (3) As shown in Figureinside 1b, the interaction(n − 3)τn−3 potential for a particle inside body is twice the value of a surface particle interacted with semi-infinite surface body, According to the definition of surface energy density, it can be written as the additional energy for a surface particle compared to inside particle4 pervC area, i.e., Uinside n3 n 4 , (3) n 3 Usur f − Uinside 2πρ C 1 2ρ C γ = = v · = − v , (4) According to theA definition(n of− 3 surface)τn−3 πτ energy2 ( ndensity,− 3)τn−1 it can be written as the addi-
tionalEquation energy (4) isfor the a surfacesurface energy particle density compared for a particle to oninside a flat particle surface body. per Thenarea, we i.e., consider the surface energy density on an arbitrary curved surface, as shown in Figure2 . UUsurf inside 2vCC1 2 v As mentioned above, L-J potential is short-range= and decreasesn3 fast 2 =with the increasing n 1 , of (4) distance, which means particle only interactsA withn particles3 in the nearest n zone. 3 A cut-off radius is used when calculating L-J potential in molecular simulations [32]. As shown in Figure2Equation, coordinate (4) system is theP − surfacexyz is built energy with original density point for at a particle particleP. Foron thea flat uniform surface body. Then ofwe expression, consider we the define surface that axisenergyz points density to the on concave an arbitrary side of outer curved surface surface,S of curved as shown in Figure surface2. As mentioned body. Coordinate above, surface L-J potential of x-y is coincide is short with-range tangent and surface decreases of S at fast particle with the increasing Pof, and distance, axes x and whichy are alongmeans the particle principle only tangent interacts direction with of P particleson S. In the in differential the nearest zone. A cut- off radius is used when calculating L-J potential in molecular simulations [32]. As shown in Figure 2, coordinate system P xyz is built with original point at particle P. For the uniform of expression, we define that axis z points to the concave side of outer surface S of curved surface body. Coordinate surface of x-y is coincide with tangent surface of S at
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Nanomaterials 2021, 11, 686 particle P, and axes x and y are along the principle tangent direction4 of of 14 P on S. In the differential geometry [33], the local shape of a curved surface S can be approximated by surface S0 depicted by two principle curvatures under principle coordinate system P-xyz geometry[34], [33], the local shape of a curved surface S can be approximated by surface S0 depicted by two principle curvatures under principle coordinate system P-xyz [34], 1 2 2 z1 f(,)() x y c1 x c 2 y , (5) z = f (x, y) ≈ (c x2 + c 2y2), (5) 2 1 2
Figure 2. The curved surface: (a) the convex surface S and approximated surface S ;(b) the Figure 2. The curved surface: (a) the convex surface S and approximated surface0 S0; (b) the con- concavecave surface. surface. In the previous work, we have expressed the potential of a particle on a curved surfaceIn as the [35 ],previous work, we have expressed the potential of a particle on a curved sur- _ 2πCρ n − 3 face as [35], U = v 1 + (ξτH) , (6) sur f (n − 3)τn−3 n − 4 2C n 3 Here, H is the mean curvature withUH the relation betweenv 1 two principle curvatures as, surf n3 , (6) n 3 n 4 1 H = (c + c ), (7) Here, H is the mean curvature with2 the1 relation2 between two principle curvatures as, Substituting Equation (6) into the definition of surface1 energy Equation (4) leads to the H= c c , (7) expression of surface energy density on the curved surface,2 1 2
_ Substituting EquationU (6)−U into the definition of surface energy Equation (4) leads to the γ = sur f inside = − 2πρvC 1 − n−3 (ξτH)· 1 expression of surface energyA density(n− 3on)τn− the3 curvedn−4 surface,πτ2 , (8) = − 2ρvC 1 − n−3 (ξτH) (n−3)τn−1 n−4 UUsurf inside 2 C n 3 1 = v 1 H n3 2 ξ is a sign operator with ξ = −1 Afor convex surfacen 3 and ξ = n1for4 concave surface. Equation (8) is the curvature-based surface energy based on L-J pair potential. When, the (8) 2 C n 3 principle curvatures of surface = tend tov be zero 1 (i.e., H = 0),H Equation (8) degenerates to n1 the expression of surface energy onn flat3 surface Equationn 4 (4). In addition, the value of τH mainly determines the magnitude of curvature effect. As τ is the equilibrium radius of particle is a andsignH operatoris the mean with curvature of1 curved for convex surface body,surfaceτH andwill be much1 for less concave than 1 as surface. Equa- the curved surface body is consisted of many particles, which means the term inside square tion (8) is the curvature-based surface energy based on L-J pair potential. When the prin- brackets of Equation (8) is positive. It is noted that the surface energy of Equation (8) is a localciple value curvatures which is relatedof surface with the tend mean to curvatures be zero on(i.e. each, H points, = 0), whichEquation means (8) it varies degenerates to the withexpression surface geometryof surface instead energy of a constanton flat surface value. Meanwhile, Equation as (4). it is In derived addition, from thethe value of H assumptionmainly determines of hard spheres the andmagnitude L-J potential, of curvature surface energy effect. is only As related is with the theequilibrium hard radius of sphereparticle equilibrium and H is radius the meanτ, the L-Jcurvature parameters, of thecurved number surface density body, of hard spheresH willρv andbe much less than the geometrical effect H. For the parameters such as temperature and pressure, their effect will1 as be the taken curved into account surface by body changing is consisted these parameters, of many which particles, will be discussedwhich means later. the term inside square brackets of Equation (8) is positive. It is noted that the surface energy of Equation 3.(8) Results is a local value which is related with the mean curvatures on each points, which means 3.1.it varies The Surface with Energy surface for Liquidgeometry Metals oninstead Flat Surface of a constant value. Meanwhile, as it is derived fromIn the the assumption last section, a modelof hard is proposedspheres and to calculate L-J potential, the surface surface energy energy based on is L-Jonly related with potential. The accuracy of the model (Equation (8)) is verified by comparing the theoretical the hard sphere equilibrium radius , the L-J parameters, the number density of hard
spheres v and the geometrical effect H. For the parameters such as temperature and pressure, their effect will be taken into account by changing these parameters, which will be discussed later.
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results on flat surface with experimental results in the literatures. Here, liquid metals are chosen as they are one element liquid with rich experimental values. As Equation (8) is derived based on the assumption that the body is made up with spherical particles homogeneously. The question is how to define equilibrium radius τ for liquid metals. We transfer the crystal structure to a homogeneous model using following assumption. As shown in Figures3 and4, taking body-centered crystal (bcc) as an example, there are two full particles in this crystal, the L-J potential between each atom in this crystal is supposed to be equal to the L-J potential between two spherical particles with equilibrium distance τ. Similarly, there are four equivalent atoms within a cubic close-packed (ccp) crystal lattice as shown in Figure3. The potential relation can be expressed as follows:
!12 !6 n n σ σ t t σ 12 σ 6 4εc c − = 4ε − , (9) ∑ ∑ i j r r ∑ ∑ τ τ i=1 j=1(j6=i) ij ij i=1 j=1(j6=i)
where n is the total number in a crystal lattice, i.e., n = 9 in bcc lattice and n = 14 in a ccp lattice. ci and cj is the equivalent ratio for atoms i and j in a lattice, for example, c = 1/8,1/2,1 for atoms on point angle, surface of lattice and inside crystal respectively. t is the equivalent atoms number in a crystal lattice, i.e., t = 2 for bcc and t = 4 for ccp. The 3 number density in Equation (8) is calculated by ρv = t/a , where a is the side length of lattice shown in Figures3 and4. For other lattice types, Equation (9) is also adopted to estimate the equilibrium radius τ. It is noted that Equation (9) is an estimation method to define equilibrium radius τ, which considers the L-J potential in the nearest neighbor contributions as crystal are periodic structures. It will be more precious to consider a larger system to define equilibrium radius τ, but here the numerical results have shown an acceptable approximation by considering one lattice. We have listed the crystal type, L-J parameters, number density ρv, the equilibrium radius τ for several metals in Table1. The abbreviation of crystal type is cubic close-packed (ccp), body-centred cubic (bcc), rhombohedral (rhomb), monoclinic (mo), trigonal (tri), and hexagonal close-packed (hcp). The surface energy is calculated using Equation (8) and compared with the experi- mental data in the literature for flat surface in Table2. The error is defined as,
|Ther − Exp| Error = × 100%, (10) Exp
From Table2, the curvature based surface energy model is confirmed to be feasible, as Nanomaterials 2021, 11, x FOR PEER REVIEWthe error is within 5% for most cases. As the surface energy varies in a range according to 6 of 15 different experimental methods, it is acceptable when the theoretically estimated surface energy is around experimental values.
FigureFigure 3. 3.The The equivalent equivalent atoms atoms in a body-centered in a body-centered crystal (bcc) crystal lattice (bcc) lattice
Figure 4. The equivalent atoms in a cubic close-packed (ccp) crystal lattice
Table 1. The parameters in Equation (8).
3 Metals Crystal (J) v (/m ) (nm) (nm) Ref for L-J Ag ccp 2.2438 × 10−19 5.8464 × 1028 0.2648 2.2812 [36] Au ccp 2.6426 × 10−19 5.8895 × 1028 0.2646 0.2841 [37] Al ccp 2.3552 × 10−19 6.0214 × 1028 0.2548 0.2933 [38] Ni ccp 4.2299 × 10−19 9.1417 × 1028 0.2239 0.1915 [39] Cu ccp 3.3300 × 10−19 8.4678 × 1028 0.2297 0.2869 [40] Pb ccp 1.2224 × 10−19 3.2980 × 1028 0.3189 0.2795 [41] Pd ccp 2.9760 × 10−19 6.7953 × 1028 0.2451 0.2133 [42] Pt ccp 4.3684 × 10−19 6.6192 × 1028 0.254 0.2189 [43] Fe bcc 4.5210 × 10−19 8.4922 × 1028 0.2267 0.2804 [40] Mo bcc 7.1165 × 10−19 6.4183 × 1028 0.2489 0.2132 [40] Cr bcc 4.3086 × 10−19 8.3377 × 1028 0.2281 0.2831 [40] W bcc 9.2891 × 10−19 6.3071 × 1028 0.2503 0.2127 [40] V bcc 5.5791 × 10−19 7.2389 × 1028 0.2391 0.2042 [40] B rhomb 2.6376 × 10−19 2.5641 × 1028 0.3543 2.4731 [44] Bi mo 1.9520 × 10−19 2.8170 × 1028 0.059 0.0829 [45] Sb tri 3.6896 × 10−19 3.1745 × 1028 0.354 0.8336 [46] Zn hcp 1.0074 × 10−19 7.5140 × 1028 0.244 0.2217 [47]
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Nanomaterials 2021, 11, 686 6 of 14 Figure 3. The equivalent atoms in a body-centered crystal (bcc) lattice
FigureFigure 4. 4.The The equivalent equivalent atoms atoms in a cubic in close-packeda cubic close (ccp)-packed crystal (ccp) lattice crystal lattice
TableTable 1. The1. The parameters parameters in Equation in Equation (8). (8).
3 Metals Crystal ε (J) ρv (/m ) σ (nm) τ (nm) Ref for L-J 3 Metals Crystal (J) v (/m ) (nm) (nm) Ref for L-J Ag ccp 2.2438 × 10−19 5.8464 × 1028 0.2648 2.2812 [36] × −19 × 28 AuAg ccpccp 2.6426 ×2.243810−19 105.8895 ×5.84641028 100.2646 0.28410.2648 [372.2812] [36] Au ccp 2.6426 × 10−19 5.8895 × 1028 0.2646 0.2841 [37] Al ccp 2.3552 × 10−19 6.0214 × 1028 0.2548 0.2933 [38] Al ccp 2.3552 × 10−19 6.0214 × 1028 0.2548 0.2933 [38] Ni ccp 4.2299 × 10−19 9.1417 × 1028 0.2239 0.1915 [39] Ni ccp 4.2299 × 10−19 9.1417 × 1028 0.2239 0.1915 [39] Cu ccp 3.3300 × 10−19 8.4678 × 1028 0.2297 0.2869 [40] Cu ccp 3.3300 × 10−19 8.4678 × 1028 0.2297 0.2869 [40] Pb ccp 1.2224 × 10−19 3.2980 × 1028 0.3189 0.2795 [41] Pb ccp 1.2224 × 10−19 3.2980 × 1028 0.3189 0.2795 [41] × −19 × 28 PdPd ccpccp 2.9760 2.976010 × 106.7953−19 6.795310 × 100.245128 0.21330.2451 [420.2133] [42] −19 28 PtPt ccpccp 4.3684 ×4.368410 × 106.6192−19 ×6.619210 × 100.25428 0.21890.254 [430.2189] [43] −19 28 FeFe bccbcc 4.5210 ×4.521010 × 108.4922−19 ×8.492210 × 100.226728 0.28040.2267 [400.2804] [40] MoMo bccbcc 7.1165 ×7.116510−19 × 106.4183−19 ×6.4181028 3 × 100.248928 0.21320.2489 [400.2132] [40] CrCr bccbcc 4.3086 ×4.308610−19 × 108.3377−19 ×8.33771028 × 100.228128 0.28310.2281 [400.2831] [40] WW bccbcc 9.2891 ×9.289110−19 × 106.3071−19 ×6.30711028 × 100.250328 0.21270.2503 [400.2127] [40] VV bccbcc 5.5791 ×5.579110−19 × 107.2389−19 ×7.23891028 × 100.239128 0.20420.2391 [400.2042] [40] −19 28 BB rhombrhomb 2.6376 ×2.637610−19 × 102.5641 ×2.56411028 × 100.3543 2.47310.3543 [442.4731] [44] −19 28 BiBi momo 1.9520 ×1.952010−19 × 102.8170 ×2.81701028 × 100.059 0.08290.059 [450.082] 9 [45] × −19 × 28 SbSb tritri 3.6896 ×3.689610−19 103.1745 ×3.17451028 100.354 0.83360.354 [460].8336 [46] Zn hcp 1.0074 × 10−19 7.5140 × 1028 0.244 0.2217 [47] Zn hcp 1.0074 × 10−19 7.5140 × 1028 0.244 0.2217 [47]
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Table 2. Comparison of surface energy by Equation (8) and experimental data.
γ in Equation (8) γ in Experiment Ref for γ in Table 2. ComparisonMetals of surface energy by Equation (8) and experimental data. Error (%) (J/m2) (J/m2) Experiment in Equation (8) in Experiment Ref for in Ex- Metals Ag 0.9006 0.954Error [48] (%) 5.6 (J/m2) 2 periment Au 1.1134(J/m ) 1.1154 [49] 0.18 Ag Al0.9006 1.02100.954 1.1050[48] [505.6] 7.60 Au 1.1134 1.1154 [49] 0.18 Ni 1.8464 1.854 [49] 0.41 Al 1.0210 1.1050 [50] 7.60 Ni Cu1.8464 1.29661.854 1.285[49] [510.41] 0.90 Cu Pb1.2966 0.43901.285 0.458[51] [520.9]0 4.15 Pb Pd0.4390 1.53910.458 1.500[52] [534.15] 2.61 Pd 1.5391 1.500 [53] 2.61 Pt 1.918 1.865 [54] 2.84 Pt 1.918 1.865 [54] 2.84 Fe 1.8194 1.820 [55] 0.03 Fe 1.8194 1.820 [55] 0.03 Mo Mo2.5601 2.56012.510 2.510[56] [561.99] 1.99 Cr Cr1.6865 1.68651.700 1.700[55] [550.79] 0.79 W W2.4828 2.48282.500 2.500[57] [570.69] 0.69 V 1.9852 1.950 [57] 1.81 V 1.9852 1.950 [57] 1.81 B 0.9701 1.060 [58] 8.48 Bi B0.3785 0.97010.379 1.060[55] [580.13] 8.48 Sb Bi0.3787 0.37850.380 0.379[59] [550.34] 0.13 Zn Sb0.8104 0.37870.816 0.380[60] [590.69] 0.34 Zn 0.8104 0.816 [60] 0.69 3.2. The Surface Energy for Spherical Surface To test the3.2. reliability The Surface of Energycurvature for- Sphericalbased surface Surface energy (Equation (8)) further, the surface energy on spherical particles is calculated by numerically integration method. As To test the reliability of curvature-based surface energy (Equation (8)) further, shown in Figure 5, the sphere structures for metals are created by assuming 2 2the 2 surface energy2 on spherical particles is calculated by numerically integration x i y i method. z i As a shown2 , where in Figure a is the5, thelength sphere of lattice, structures and the for metalsradius of are sphere created is by assuming set to a 2 ,x where2(i) + y2( iis) +anz 2integer.(i) ≤ (η The∗ a/2 surface)2, where particlesa is the are length chosen of lattice,by satisfying and the the radius of sphere is set to η ∗2a/2, where 2η 2is an integer.2 The surface particles are chosen by satisfying condition 1a 2 x i y i z i a 2 , supposing that the total number the condition [(η − 1)a/2] ≤ x2(i) + y2(i) + z2(i) ≤ (η ∗ a/2)2, supposing that the total of surface particles is n. Based on the L-J pair potential, the surface energy is calculated by number of surface particles is n. Based on the L-J pair potential, the surface energy is adding up the external energy for all surface particles compared to inside particles and calculated by adding up the external energy for all surface particles compared to inside 2 then divided byparticles the total and surface thendivided area n by. the Here, total Fe surface and Al area atomsnπτ are2. Here,chosen Fe to and compare Al atoms are chosen the theoreticalto surface compare energy the with theoretical numerical surface integration energy withresults, numerical which are integration shown in results,Fig- which are ures 6 and 7, respectively.shown in Figures The specific6 and7, calculatingrespectively. errors The are specific put as calculating figures in errorsAppendix are put as figures A. in AppendixA.
(a) (b)
Figure 5. TheFigure schematic 5. The model schematic of Al model and Fe of spherical Al and Fe particles: spherical (a) particles: The Al particles (a) The withAl particles surface with and insidesurface atoms; (b) The Fe particle withand surfaceinside atoms; and inside (b) The atoms. Fe particle with surface and inside atoms.
NanomaterialsNanomaterials 20212021,,,11 11,,, 686 xx FORFOR PEERPEER REVIEWREVIEW 88 of 1415
FigureFigure 6.6. TheThe comparisoncomparison between theoretical surface energy and numerical integration results for AlAl sphericalspherical particles.particles..
FigureFigure 7.7. TheThe comparisoncomparison between theoretical surface energy and numericalnumerical integration results results for for FeFe sphericalspherical particles.particles..
FromFrom FiguresFigures6 6 and and7 ,7, it it is is confirmed confirmed that that the the curvature-based curvature-based surface surface energy energy based based onon L-JL-J potentialpotential is inin accordanceaccordance withwith numericalnumerical integrationintegration resultsresults withwith anan errorerror aroundaround 5%.5%. ThisThis errorerror isis acceptableacceptable atat nanonano scalesscales asas aa resultresult ofof homogenizationhomogenization hypothesis.hypothesis. TheThe curvescurves inin FiguresFigures6 6 and and7 7also also indicate indicate that that the the values values of of surface surface energy energy decrease decrease with with the the increasingincreasingincreasing ofofof sphericalsphericalspherical radius.radius.radius. Here,Here, spheresspheres withwith largerlarger radiusradius isis notnot consideredconsidered forfor twotwo reasons,reasons, oneone is is that that the the curvature curvature effect effect will will be be negligible negligible on sphereson spheres with with larger larger radius, radius, and anotherand another reason reason is that is thethat estimated the estimated error error of curvature of curvature based based surface surface energy energy will decrease will de- withcrease the with increasing the increasing of radius. of radius. In addition, In addition, the calculation the calculation task will task also will be also huge be and huge take and a longtake a time, long thus time, the thus testing the testing error for error sphere for sphere with larger with radiuslarger isradius omitted is omitted here. here. ThenThen wewe considerconsider several several metals metals with with two two different different surface surface geometries, geometries, i.e., i.e. the,, the convex con- sphericalvex spherical surface surface and and concave concave spherical spherical surface, surface the,, thethe surface surfacesurface energy energyenergy is shown isis sshown in inFigure Figure8 . It8. isIt confirmedis confirmed that that the the surface surface energy energy will will increase increase with with the the decreasing decreasing ofof curvaturescurvatures onon concaveconcave surface.surface. WhileWhile forfor convexconvex surface,surface, thethe surfacesurface energyenergy willwill decreasedecrease withwith thethe decreasingdecreasing ofof curvatures.curvatures. ComparedCompared toto surfacesurface energyenergy onon aa flatflat surface,surface, thethe surfacesurface energyenergy willwill increaseincrease onon aa convexconvex surfacesurface andand decreasedecrease onon aa concaveconcave surfacesurface withwith thethe decreasingdecreasing of scales. This is consistent with the facts that the surface energy comes from the lack of of scales. This is consistent with the facts that the surface energy comes from the lack of bonds for surface particles compared to the inside particles. For particles on convex surface, the lacking bonds will increase, leading to surface energy larger than flat surface, and
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Nanomaterialsbonds2021, 11 for, 686 surface particles compared to the inside particles. For particles on convex sur- 9 of 14 face, the lacking bonds will increase, leading to surface energy larger than flat surface, and this effect becomes significant on small scales. However, particles on concave surface will have smaller lackingthis bonds effect becomes compare significantd to particles on small on scales. flat surface, However, which particles leads on concave to smaller surface will surface energy. Fromhave Figure smaller 8 lacking, it is also bonds noted compared that the to particleseffect of on curvatures flat surface, on which surface leads en- to smaller ergy will exceed 5%surface compared energy. to From one Figure for flat8, itsurface is also noted when that the the radius effect of curvaturesspherical onparticles surface energy becomes smaller thwillan exceed5nm. Hence, 5% compared the curvature to one for effect flat surface will be when non the-negligible radius of sphericalwhen the particles becomes smaller than 5nm. Hence, the curvature effect will be non-negligible when the surface curvatures become larger than 1/5 nm−1, which is in consistent with the conclusion surface curvatures become larger than 1/5 nm−1, which is in consistent with the conclusion mentioned in Ref. mentioned[21]. in Ref. [21].
(a) (b)
(c) (d)
(e) (f)
FigureFigure 8. The 8. curvatureThe curvature effect oneffect surface on surface energy for energy spherical for convexspherical and convex concave and surface concave of different surface crystals: of dif- (a) Al particles;ferent (b )crystals: Ag particles; (a) Al (c) particles; Au particles; (b ()d Ag) Cu particles; particles; ((ec)) FeAu particles; particles; (f) Pt(d particles.) Cu particles; (e) Fe particles; (f) Pt particles.
3.3. The Curvature Effect on Curved Body with Sine Surface In this section, a more complex surface geometry with a sine surface will be consid- ered. As shown in Figure 9, the equation of the sinusoidal surface is set to,
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3.3. The Curvature Effect on Curved Body with Sine Surface 2 x z hsin , (11) In this section, a more complex surface geometry with a sine surface will be considered. As shown in Figure9, the equation of the sinusoidal surface is set to, where h and are the amplitude and wavelength of the sinusoidal surface respectively. 2πx z = h sin , (11) The principle curvatures of the sinusoidalλ surface are, where h and λ are the amplitude and wavelength42h of the sinusoidal 2 x surface respectively. 1 The principle curvatures of thec1 sinusoidal0, c 2 surface 2 are, sin 3 2 , 42h 2 2 x (12) 2 2 4π h 2πx 1 1 2 cos c1 = 0, c2 = − sin , (12) 2 λ h 2 2 i3/2 λ + 4π h 2 2πx 1 λ2 cos λ
Figure 9. The schematic model of sinusoidal surface. Figure 9. The schematic model of sinusoidal surface.
The mean curvature is defined as H = (c1 + c2)/2. Taking Cu as an example, its crystal type is fcc with lattice constant a = 3.615A. The ratio of surface energy on the sinusoidalThe surface mean to onecurvature on flat surface is defined is drawn in as Figure H 10a for c1h = c 21 nm2and. Takingλ = 5 nm Cu, as an example, its crys- whichtal type confirms is fcc the with variation lattice of the constant surface energy a = 3 at.615A. different The positions ratio on of sinusoidal surface energy on the sinusoidal surface. The distribution of surface energy shown in Figure 10a is very similar to the one calculatedsurface byto DFT one method on inflat Ref. surface [16]. The relation is drawn between in surface Figure energy 10a and wavelengthfor h1 nm and 5nm , which λ as well as amplitude h can also be obtained, which are shown in Figure 10b,c respectively. Theconf valuesirms of the surface variation energy will of increase the withsurface the increasing energy of at amplitude differenth and positions decrease on sinusoidal surface. withThe the distribution increasing of wavelength of surfaceλ. If energy we define shown the normalized in Figure surface 10a energy is onvery convex similar to the one calculated part of sinusoidal period as, by DFT method in Ref. [16].2 TheZ λ/2 relation between surface energy and wavelength as γe = γ(x)dx, (13) well as amplitude h can alsoλ be0 obtained, which are shown in Figure 10b,c respectively. TheThe values variation of surface of normalized energy surface will energy increase compared with to flat the surface increasing energy with of amplitude h and decrease different ratio of wavelength and amplitude is drawn in Figure 10d. The results show the normalizedwith the surfaceincreasing energy isof larger wavelength with higher ratio . ofIf wavelengthwe define and the amplitude, normalized and surface energy on con- tendsvex part to the surfaceof sinusoidal energy on flatperiod surface as, when wavelength becomes more than 20 times of lattice constant a. 2 2 x dx , (13) 0 The variation of normalized surface energy compared to flat surface energy with dif- ferent ratio of wavelength and amplitude is drawn in Figure 10d. The results show the normalized surface energy is larger with higher ratio of wavelength and amplitude, and tends to the surface energy on flat surface when wavelength becomes more than 20 times of lattice constant a.
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(a) (b)
(c) (d)
Figure 10. FigureThe surface 10. The energy surface on energy sinusoidal on sinusoidal surface: ( asurface:) The ratio (a) The of surface ratio of energy surface on energy sinusoidal on sinusoidal surface to one on flat surface whensurfaceh = to1 nm oneand on flatλ = surface5 nm.( whenb) The h surface 1 nm and energy 5nm distribution. (b) The surface on sinusoidal energy distribution surface with on different h when = = λ 5 nm;(scinusoidal) The surface surface energy with distribution different h when on sinusoidal 5nm ; surface(c) The withsurface different energyλ distributionwhen h 1 onnm sinusoidal;(d) The normalized surface energy on convex part as a half of sinusoidal periodic surface at different wavelength λ with several ratio h/λ. surface with different when h 1 nm ; (d) The normalized surface energy on convex part as a half of sinusoidal4. periodic Discussion surface at different wavelength with several ratio h . In this section, some discussions towards curvature-based surface energy are pre- 4. Discussion sented. Firstly, we want to compare the curvature based surface energy with the Tol- In this section,man equationsome discussions which builds towards the curvature relation between-based surface surface energy energy are and pre- spherical radius sented. Firstly, forwe nanoparticleswant to compare [61]. the curvature based surface energy with the Tolman equation which builds the relation between surfaceγ energy1 and spherical2δ radius for nano- = ≈ 1 − , (14) particles [61]. γ 2δ r 1 + r 1 2 where γ represents the surface energy 1 of, spherical particle, γ is the surface energy for flat 2 (14) surface. δ is the length parameter1 whichr is related with the particle’s property, but the physical meaning is unknown.r where representsAccording the surface to energ curvaturey of spherical based surface particle, energy, is thethe ratiosurface for energy spherical for surface energy flat surface. andis the flat length energy parameter is, which is related with the particle’s property, but γ n − 1 τ the physical meaning is unknown. = 1 − n ≥ 4, (15) According to curvature based surface energy,γ0 the ration forr spherical surface energy and flat energyIf is, we compare Equation (15) with Equation (14), it is interesting that two become the same = n−1 if assuming δ 2n τ, whichn may1 give a physical definition for δ. As mentioned above,1 it is noted that n 4parameters, of temperature and(15) pressure have not n r appeared in curvature-based0 surface energy (Equation (8)). However, they can affect the If we compare Equationequilibrium (15) radius with Equation indirectly. (14), Generally, it is interesting higher that temperature two become and the lower same pressure lead to largern 1 equilibrium radius, causing a decrease in surface energy. However, this effect is not if assuming , which may give a physical definition for . that2n obvious for solids, as the distance between atoms is not easy changed by temperature As mentionedand pressure.above, it is noted that parameters of temperature and pressure have not appeared in curvatureAt last,-based we want surface to discuss energy the(Equation application (8)). However, conditions they for curvaturecan affect based surface the equilibriumenergy. radius indirectly. Firstly, itis Generally, derived basedhigher on temperature the L-J pair and potential lower pressure between lead particles, which is
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a short-range interaction potential. It has not taken the long range interaction such as electrostatic interaction into consideration. However, the geometrical effect of long range interaction on surface energy is very small compared to short range interaction. Secondly, the surface energy is derived based on the model of hard sphere model, which means the determination of sphere equilibrium radius is important to the value of surface energy. It is not easy to determine the equilibrium radius for system with various particles. The equilibrium radius maybe decided based on the known surface energy for a flat surface, and then the geometrical effect for surface energy on a curved surface can be estimated.
5. Conclusions The curvature based surface energy is discussed based on the L-J pair potential. The surface energy for flat surface and spherical particles are compared with results by experimental and numerical integration method to confirm the reliability of the curvature- based surface energy. By considering the surface energy on a spherical convex and concave surface, it is confirmed that the surface energy will increase on the concave surface and decreased on the convex surface with the increasing of scales, which will degenerate to the value on the flat surface. The curvature effect will be obvious for the spherical surface with the curvatures larger than 1/5 nm-1 with an increment more than 5%. The curvature- based surface energy can estimate the surface energy on more complex surface such as sine surface, and provide a physical meaning of the Tolmann length to understand the curvature effect of surface energy for spherical particles.
Author Contributions: Conceptualization, D.W.; investigation, G.P.; writing—original draft prepara- tion, D.W.; writing—review and editing, Z.H.; supervision, Y.Y.; funding acquisition, Y.Y. and D.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Natural Science Foundation of China, grant number 11902151, 11802121, 12050001, Natural Science Foundation of Jiangsu Province, grant number BK20180411, BK20180416. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Conflicts of Interest: The authors declare no conflict of interest.
Appendix A The parameters for the calculation of surface energy on metal particles are listed in Figure A1, with the errors calculated by, Nanomaterials 2021, 11, x FOR PEER REVIEW 13 of 15 |Ther − Exp| Error = × 100% (A1) Exp
(a) (b)
Figure A1. FigureThe comparison A1. The comparison errors of theerrors theoretical of the theoretical values and values numerical and numerical results in results Figures in6 Figures and7:( 6 aand) The errors for spherical Fe7: particles; (a) The errors (b) The for errors spherical for spherical Fe particles; Al particles.(b) The errors for spherical Al particles.
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