- Email: [email protected]
Effects of surface tension of graphene sheet on impact and rebound behavior of colliding nanoparticle
Journal Pre-proof Effects of surface tension of graphene sheet on impact and rebound behavior of colliding nanoparticle Amin Sepahi-Boroujeni, Shahrokh Hosseini-Hashemi, Saeid Sepahi-Boroujeni PII:
S0749-6036(19)32048-8
DOI:
https://doi.org/10.1016/j.spmi.2020.106464
Reference:
YSPMI 106464
To appear in:
Superlattices and Microstructures
Received Date: 29 November 2019 Revised Date:
22 February 2020
Accepted Date: 23 February 2020
Please cite this article as: A. Sepahi-Boroujeni, S. Hosseini-Hashemi, S. Sepahi-Boroujeni, Effects of surface tension of graphene sheet on impact and rebound behavior of colliding nanoparticle, Superlattices and Microstructures (2020), doi: https://doi.org/10.1016/j.spmi.2020.106464. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.
Effects of surface tension of graphene sheet on impact and rebound behavior of colliding nanoparticle a,
a
Amin Sepahi-Boroujeni *, Shahrokh Hosseini-Hashemi , Saeid Sepahi-Boroujeni a
b
Impact Research Laboratory, Department of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran
b
Department of Mechanical Engineering, École Polytechnique de Montréal, Montréal, QC, Canada
Abstract In the present work, the classical theory of plate, the von Kármán nonlinear theory and the nonlocal theory of elasticity are adopted to analyze the impact between a nanoparticle and an orthotropic rectangular nanoplate with surface tensions. Employing an analytical contact law to explain the von der Waals forces, the equations of motion are solved for the indenter-plate system. As an application, this approach is taken for a nanoparticle colliding with a simply supported single-layered graphene sheet and the rebound behavior of the particle is discussed. Then, the influences of surface tension and the nonlocal parameter on the dynamics response of the particle are investigated. When enough surface tension is applied, the membrane theory gives an acceptable estimate for the restitution behavior of the indenter otherwise, the estimates are associated with a maximum error of 15%. The effects of disregarding the flexural stiffness of the graphene sheet and ignoring the attraction part of the von der Waals forces are also examined. Studying the dimensional effects on the system behavior revealed that optimum dimensions of the plate maximize its capacity for energy absorption when it is employed as an energy absorber against colliding particles. Finally, a contact law for large nanoparticles is proposed and then is applied to an example including an SLGS colliding with a multi-atom indenter.
Keywords: Graphene Sheet; Impact; Nonlocal plate theory; Surface tension.
Corresponding author: E-mail address: [email protected]
*
1
1. Introduction Mutual effects of nanoparticles and nanostructures have provided an increasing range of applications. Many nano-scaled devices such as mass sensors, pressure sensors, and oscillators work based on the interactions between foreign objects and nano-structures, e.g. nanobeams or nanoplates. Nano mass sensors have provided a vibration-based method to measure nano-scaled attached particles. Low-pressure gas sensors have been developed measuring response of a nanomembrane to colliding atoms. In addition, nanoplates have been utilized as implements to absorb, to store and then to deliver special foreign particles, such as ions, biocells, and nanodrugs. Depending on the applications and the investigation purposes of a nano-scaled impact phenomenon, studying both the nanoparticle (indenter) and the nanostructure (target) can be a matter of interest. Investigating the kinematic characteristics of a nano-indenter interacting with various structures enables some simple generalized rules to be derived, which reduce the massive calculations of many complex nano-scaled problems. Theoretical study of the response of a system including the collision between an indenter and a target structure, known as impact dynamics, concerns the dynamics responses of the system as well as the local displacements at the impact area. Despite a considerable number of studies on the impact dynamics of macro-scaled systems, few attempts have been made for modeling nanosized systems because of theoretical complexities and the necessity of employing nonlocal continuum theories in a transient media. The research works conducted on nano-scaled systems are limited to some recent studies. [1, 2] theoretically studied the response of a nanobeam to an external excitation and discussed the response of the target during and after the impact. [3] proposed a simple analytical approach in addition to a molecular dynamics (MD) simulation to study the impact response of a nanotube-nanoparticle system and measured the deflection of the nanotube during the impact. [4] developed a theoretical mass-spring model together with an MD simulation to address the problem of impact between a nanoparticle and a single-layered graphene sheet (SLGS), which was generalized later by [5] for two- and multi-layered graphene sheets. [6] used an MD simulation for modeling a suspended SLGS colliding with a nano-tube to estimate the energy absorption capacity of SLGSs. Employing a quasi-classical model for the problem of singleand few-layered graphene sheets colliding with a projectile, [7] investigated the influences of impact velocity and the number of layers on the coefficient of restitution and the critical rupture velocity. The full modeling of an impact phenomenon is known to include the equations of motion for both the target and the indenter, as well as the dynamics response of the system during and after 2
the impact period [8]. The only theoretical studies based on the full modeling are limited to [9-11], in which the dynamics responses of a nanoplate-nanoparticle system are derived based on an analytical contact law. These works estimate the interaction forces in terms of the indenter-plate distance. The contact laws utilized in these studies are based on estimating the interaction forces within a continuous force field. These contact laws that are described in [12] had been used in former studies like [13-15] to derive some conceptual models for the interactions between nanostructures and foreign nanoparticles. Utilizing a different method, [16] employed a densityfunctional ab-initio MD to identify the mechanisms driving precursor/surface reactions during epitaxy on the surface of SLGS. Focusing on the behavior of the indenter, the present research concerns the dynamics behavior of a nano-sized system including an indenter interacting with an SLGS with surface forces. The SLGS is assumed to be rectangular, orthotropic, and thin. The governing equation of motion is derived based on Kirchhoff’s classical plate theory (CPT) together with the nonlocal theory of elasticity. The utilized contact law is based on the Lennard-Jones (L-J) formula and gives an analytical approximation of the non-bonding van der Waals interaction forces. Moreover, the impact between a gold nanoparticle and an SLGS with normal surface tensions is investigated in detail. This approach can be employed to develop the general rules governing the behavior of nanoparticles after hitting nanostructures such as graphene sheets.
2. Formulation 2.1
Equations of motion Consider a thin rectangular nanoplate with a density of , a thickness of ℎ and dimensions
of and . The origin of the assigned , , Cartesian coordinate system is located at one corner of
the plate so that axes and are coincident with the edges of the undeformed mid-plane of the
nanoplate (i.e. 0 ≤ ≤ , 0 ≤ ≤ , −ℎ/2 ≤ ≤ ℎ/2). The plate is assumed to be orthotropic
and , , are taken as the moduli of elasticity in the corresponding directions. ν and ν are
the Poison's ratios respectively along directions and , and ! is the shear modulus of elasticity. According to the von Kármán nonlinear theory, the non-zero resultant strains in the CPT are [17]: 1 " = $%, − &, + &, ) 2
1 " = *%, − &, + &, ) 2
(1)
1 1 " = ($%, + *%, ) − &, + &, &, 2 2 3
in which $, *, & are the displacements in directions , , and $% , *% are the material displacements
at point (, , 0) along axes and , respectively. The strain field of Eq. (1) leads to the governing
differential equation of motion [17]: ..
-% & = . , + 2. , + . , + / + 0
(2)
where -% = ℎ is the mass momentum of inertia, . , . , . are the bending momentum
resultants, 0 is the transversal distributed force (measured per unit area), and / is a function of inplane tension resultants (1 , 1 , 1 ): /($, *, &) =
2 2 31 &, + 1 &, 4 + 31 & + 1 &, 4 2 2 ,
≈ 1 &, + 21 &, + 1 &,
(3)
in which the last expression is a linear approximation of /. Employing the strain-displacement and the stress-strain equations for nonlocal plates, Eq. (2) could be rewritten as: ..
-% ℒ& = −7 &%, − 237 + 7!! 4&%, − 7 &%, + ℒ/ + ℒ0
(4)
7, the flexural rigidities of the plate, are:
< ℎ: 87 , 7 , 7 , 7!! 9 = ; , , , = 12 1 − < < 1 − < < 1 − < < !
(5)
and the nonlocal operator is ℒ = (1 − >? ) ) where ? ) = 2 ) /2 ) + 2 ) /2 ) is the 2-D Laplacian
operator and > = @%) ) is a nonlocal parameter introduced as a function of non-dimensional coefficient @%) . From this point forward, as in practical applications, boundary conditions are
assumed to be simply supported or clamped and surface shear force is disregarded (1 = 0), which allows the orthogonality statements to be derived. 2.2
Free vibrations In the case of free vibrations, external force 0 can be excluded from Eq. (4). Since the mode
shapes of vibration, represented here by ABC (, ) (D, E = 1,2, …), satisfy the equation of free vibrations, Eq. (4) can be restated as: ) −-% GBC ℒ ABC = −7 ABC, − 237 + 7!! 4ABC, − 7 ABC,
+ℒ(1 ABC, + 1 ABC, )
4
(6)
where GBC is the natural vibration frequency in mode (D, E). Multiplying Eq. (6) by an arbitrary mode shape AH! and then integrating over the area of plate (I) yield:
) GBC -% J 3AH! ABC + > AH!, ABC, + > AH!, ABC, 4 KI = L
J M7 ABC, AH!, + 237 + 7!! 4ABC, AH!, + 7 ABC, AH!, N KI − L
1 J 3AH!, ABC, + > AH!, ABC, + > AH!, ABC, 4 KI − L
(7)
1 J 3AH!, ABC, + > AH!, ABC, + > AH!, ABC, 4 KI L
in which the following relations, derived based on the partial integral properties, are employed
(O, P = , ),
L
L
L
L
L
L
J AH! ABC,RRSS KI = J AH!,RS ABC,RS KI
Q AH! ABC,RRRR KI = Q AH!,RR ABC,RR KI ;
(8)
J AH! ABC,RR KI = − J AH!,R ABC,R KI
The integrals in Eq. (7) are symmetric with respect to subscripts DE and TU. Therefore, a new equation could be formed by replacing GBC with GH! on the left side of Eq. (7). Eventually,
subtracting this equation from Eq. (7) results in the orthogonality properties of the mode shapes in a nonlocal classical plate:
L
L
-% J AH! ℒABC KI = J AH! M7 ABC, + 237 + 7!! 4ABC, + 7 ABC, N KI
+ J AH! ℒ(1 ABC, + 1 ABC, ) KI = 0 (T ≠ D WT U ≠ E)
(9)
L
In the case of simply supported boundaries, the mode shapes are harmonic functions of
and , which can be described as ABC = UDX
B Y Z
UDX
C Y [
, [18]. Substituting ABC in Eq. (6) and
employing orthogonality properties of Eq.(9) give the natural frequencies: \ ) GBC = ) ) ]
2.3
[^
Z^
D ] Z^ 7 + 2D ) E ) 37 + 7!! 4 + E ] [^ 7 -% _1 + \ ) @%) `D ) + E
^ )Z
ab [^
+ \)
D)
cdd Z^
Dynamics response Expanding displacement & as a series of mode shapes results: 5
+ E) -%
cee [^
(10)
&(, , f) = g
i B,Cjk
ABC (, )hBC (f)
(11)
in which hBC (f) (D, E = 1,2, …) are the time-dependent terms of the series. A mathematical process to derive the modal response of a nanoplate to an impact force is described in [9], which results in: hBC (f) = lWU GBC f hBC (0) +
UDX GBC f nBC q hm BC (0) − J o(p) UDX GBC (f − p) Kp GBC GBC %
(12)
where o(f) is the time-dependent part of the impact force and nBC represents the influence coefficient of the impact force for vibration mode (D, E), which can be formulated as: nBC =
3ℒABC 4(
r , r ) -% sL ABC ℒ ABC KI
(13)
As a result, it could be stated that the effect of surface forces on the dynamics response of the plate is limited to the shifts in the natural frequencies of the plate, and the equations of motion are the same for the plates with and without surface forces. The indenting particle exerts an opposing force, thus the equation of motion for the indenter with a displacement of t and a velocity of u is: t(f) = t% + u% f + 2.4
1 q J (f − p) o(p) Kp . %
(14)
Contact law Impact force may be represented in terms of the indenter-plate distance, known as “contact
law”. One of the most recently proposed contact laws to explain the non-bonding L-J interaction force between a concentrated nanoparticle and a nanoplate is [9]: z{ zk) o = 8 \ w x y } − kk ~ | |
(15)
in which material properties w and z are respectively the depth of the potential well and the distance where the inter-particle potential is zero. In this equation, x is the number of particles per
unit area of the plate and ∆ is the indenter-plate relative distance. Eq. (15) is derived by assuming that the particle is a concentrated point. A more general contact law would be needed for larger indenters that may not be treated as concentrated particles. A rigid spherical nanoparticle with a radius of is assumed to be standing above an infinite nanoplate, as shown in Fig. 1, and an T − − spherical coordinate system is assigned to the center of the particle, in which and
6
are respectively the azimuthal and the polar angles. Now, for a small arbitrary element of the particle, one can employ Eq. (15) to estimate the force imposed on the element by the plate:
Fig. 1. Schematic illustration of the system including a solid sphere model as a nanoparticle interacting with a continuum nanoplate. Ko = 8\ w x y
z{ zk) − ~ |′} |′kk
(16)
in which stands for the number of atoms allocated inside the assumed element and |′ is the vertical distance of the element from the plate, which can be rewritten in terms of T and : | = | + T lWU
(17)
where | is the distance between the particle center and the plate. Now, Eq. (16) can be integrated
over the volume of the nanoparticle, resulting in total force o imposed on the nanoparticle: { k) o = 8 \ w x J J J y − ~ T ) UDX KTK K } (| + T lWU )kk Hj% j% j% (| + T lWU )
Y
)Y
2 { | + 3 | − 3 k) | + 9 | − 9 ) = \ w x y − − − ~ 3 (| + ): (| − ): 45 (| + ) (| − )
(18)
For a single-atom particle, assuming that there is one atom inside a sphere with a radius of , one ]
can set = `: \ : a
k
and obtain Eq. (15) as approaches zero in Eq. (18). Since the relative
distance between the indenter and the plate depends on both the position of the indenter and the displacement of the plate at the contact point, the set of Eqs. (12) and (14) together with Eq. (15) or (18) should be solved simultaneously to derive the dynamics response of the system.
3. Results and discussion In the absence of an analytical solution for a set of nonlinear equations, employing numerical approaches can be helpful for simultaneously solving Eqs. (12), (14) and (15), which 7
gives the dynamics response of the plate-indenter system. In the following, the impact between an SLGS and a gold nanoparticle is studied. According to [19], the geometrical and the mechanical
features of the SLGS are assumed to be: × × ℎ = 9.496 × 4.877 × 0.145 nm: , = 5624 kg/m: , 8 , , ! 9 = {2.145, 2.097, 0.938} TPa, < = 0.223, < = < / , and the
nonlocal parameter is @% = 0.05. Without loss of generality, surface tensions are assumed to be equi-biaxial, i.e. 1 = 1 = 1. The indenting particle is taken as a gold atom with a mass of
. = 3.27 × 10)} Kg that hits the resting SLGS at point (% , % ) = (/2, /2). The L-J
parameters for the gold-carbon interactions are zL = 0.3187 nm, wL = 2.10@u =
3.364 × 10)) J [14], and x = 38.1770 nm) [9]. The response precision depends on the number of modes of vibration considered in the solution, whereas several modes of vibration may be
excited in the impact response of the plate. In the present study, × X = 30 × 30 = 900 modes are included to ensure that a sufficient number of higher modes of vibration are considered. Then, the following analyses are conducted on the above-mentioned problem: 3.1
Effects of surface tension As expected, increasing the surface tension stiffens the plate. The stiffer plate shows less
deflection against the impact load and, consequently, absorbs less energy from the indenter. Fig. 2 schematically illustrates the displacements calculated by Eqs. (12) and (14) respectively for the plate and the indenter with an initial velocity of 300 m/s. As this figure shows, in the cases
including the stiffer plates with higher values of surface tension, the indenter carries enough energy to recede into space after the collision. Eq. (13) describes the influence of the impact force on the vibration modes and shows how the surface tension affects the dynamics response of the plate. Because the mode shapes are independent of the surface tension, it can be concluded that influence coefficient of the impact force nBC is also independent of the surface tension. Thus, as can be
observed in Eq. (12), the only effect of tension on the dynamics response is the shift in the natural frequencies. In other words, in nonlocal plates, similar to local plates, the effects of surface tension are limited to the changes in natural frequencies, which determine whether the indenter sticks to the plate or escapes.
8
Fig. 2. Schematic illustrations of the effects of surface tension on the deflection of the SLGS and the z-position of the indenter at various sequences. Fig. 3 shows how the bouncing behavior of the indenter varies by the surface tension. The
SLGS initially lies on the plane and interacts with the indenter based on the van der Waals forces expressed by Eq. (15). The long-range attraction part of this force absorbs the indenter, while the plate ejects the indenter by growing the short-range repulsion force as the relative distance decreases. This interaction transfers a portion of the indenter’s energy to the plate in the forms of kinetic and strain energies. The rest of the indenter’s energy determines whether it escapes or be trapped by the plate. On the other hand, the plate deflections have a major role in the force field of the system, so that the relative distance between the indenter and the corresponding point on the plate determines the amplitude of the interaction force. It can be observed in Fig. 3 that the indenter sticks to the SLGS with the tension loads less than a specific value, which occurs during an approach-rebound procedure and eventually results in a final integrated system including both the plate and the indenter. The smaller values of tension force cause shorter rebounds and faster absorption.
9
Fig. 3. Different rebound patterns for various tension loads applied to the target surface. As the indenter escapes, a kinematic coefficient of restitution (COR) can be defined as the
ratio of the post-impact velocity to the initial velocity of the indenter: COR = u1 /u0 . Fig. 4 presents
the variations of the COR vs. the surface tension for different initial impact velocities. As illustrated, the greater the u0 , the larger is the COR. This observation could be explained by the fact
that the impact procedure occurs faster for greater initial velocities of the indenter, then the plate has less time to absorb the indenter’s energy. Moreover, for every initial velocity, there is a specific
tensile load where the COR approaches zero and the indenter sticks to the plate. By rising 1, the
COR shows a stepwise increasing trend that is more obvious at lower initial velocities (u% <
200 m/s). The reason is that over the short relative distance between the indenter and the plate,
which comes along at the impact moment, the interaction force strongly varies within a wide range of amplitudes. Furthermore, the displacement of the plate depends on its excited mode of vibration and, consequently, the variation of frequency or the phase of modes alters the distance of the indenter from the plate. Although very small, this change can give rise to considerable momentary variations of the interaction force, resulting in a jump in the graph. In macro-scaled impacts, impact forces (e.g. the Hertz impact force) vary slightly over the impact period, then these fluctuations can be limited to the nano-scaled impacts.
10
Fig. 4. The COR vs. surface tension for various initial velocities of the indenter. 3.2
Effects of attraction forces In macro-scaled impact problems, the attraction effects of the force field are ignored in
contact laws. Ignoring the long-range attraction term in Eq. (15) makes it a rough estimation of the macro-scaled impact forces. Excluding the attraction forces, Fig. 5 presents the variations of the COR vs. the surface tension for different initial impact velocities. As illustrated, this system shows quite different behavior in comparison with the findings illustrated in Fig. 4. The COR reaches considerably higher levels for all initial velocities and the indenter does not stick to the plate, even for the small values of surface tension. In addition, unlike Fig. 4, increasing the initial velocity leads to a reduction in the COR due to changes in the behavior of SLGS in faster collisions. As observed in macro-scaled impact problems, it could be claimed that increasing the initial velocity of the indenter excites a larger number of vibration modes in the plate and, as a result, dissipates a greater amount of the indenter’s energy. These findings clearly show that the attraction part of the impact force has a pivotal role in the behavior of the system in a way that disregarding such influences of attraction forces gives an imprecise estimate for the dynamics behavior of the system.
11
Fig. 5. The COR vs. surface tension for various initial velocities of the indenter in the case of excluding the attraction forces from the contact law. 3.3
The validity range of the membrane theory in comparison with the classical plate theory Similar to a thin structure, a stretched SLGS can approximate to a membrane. This
simplification results in some errors due to neglecting the flexural rigidities. These errors can be investigated by comparing the dynamics response of an SLGS with and without flexural rigidities being involved, respectively given by Eqs. (4) and (10). Fig. 6 includes the variations of the COR vs. surface tension 1 measured based on both the CPT and the membrane theory (MT) at various
initial velocities. As expected, the agreement between these theories increases for higher tensile loads. Therefore, membrane theory gives an acceptable estimate when considerable surface tension is applied. On the other hand, for smaller values of tension, errors are less than 15%, which could be acceptable in the studies where a general view of the system behavior is expected, e.g. in studying the pressure sensors where several particles hitting an SLGS with various velocities. Based on the CPT and the MT, Fig. 7 includes the variations of the tension force threshold where the indenter sticks to the SLGS, vs. the initial velocity of the indenter. As this figure implies, there are acceptable agreements between the two theories in estimating the thresholds of sticking. As a general result, assuming a stretched SLGS to be a membrane can provide a reasonable perspective on its dynamics response to the impact of a foreign indenter.
Fig. 6. The COR vs. surface tension for various initial velocities of the indenter calculated based on the classical plate theory and the membrane theory (MT).
12
Fig. 7. Variations of the particle threshold of sticking vs. the initial velocity of the indenter estimated based on the CPT and the MT. 3.4
Effects of nonlocality Employing nonlocal parameter @% allows the results obtained from experiments or
simulations to be validated by the nonlocal theory. Notably different values are reported in the literature for the nonlocal parameter, which are due to various purposes of the investigations along with the errors resulting from the employed tools and methods. As an example, various values can be found for each mode of vibration of an SLGS. In any case, the question remains how different values of the nonlocal parameter affect the behavior of a system, such as its dynamics response to an impact load. First, it is necessary to discuss the theoretical effects of the nonlocal parameter. In the present example of the simply supported SLGS, the harmonic mode shapes lead to nonlocal operator ℒ = 1 + \ ) > (D ) ⁄) + E ) ⁄ ) ). Then, Eq. (13) can be simplified as: nBC =
4 D \ % E \ % UDX UDX -%
(19)
which is independent of @% for every mode and only depends on the weight of the plate (-% ) and
the non-dimensional location of the impact point (i.e. % / and % /). Therefore, similar to the surface tension, the effect of nonlocality is limited to certain changes in the natural frequencies of
the plate. Fig. 8-a displays the variations of the COR vs. nonlocal parameter @% for u% = 150 m/s and 1 = 10 N/m. As shown in this figure, two types of variation behavior can be attributed to the
COR. First, as stated in section 3.1, one can observe the consecutive jumps in the COR caused by slight variations in the frequencies or in the phases of modes. Another behavior includes the cyclic variations, which eventually converges to a constant value. Fig. 8-b compares the variations of energy for three most excited modes of vibration of the plate. As shown, in the present example, the 13
small variations of energy mostly belong to mode (1,1) and the majority of the changes are observable for modes (3,1) and (5,1). As illustrated in Fig. 8-b, increasing the nonlocal parameter reduces the energy absorbed by the most excited modes for the nonlocal parameter values less than 0.10. In general, the impact between a foreign particle and a stiff plate occurs over a shorter time compared with a more flexible plate. Eventually, a stiffer plate absorbs less energy from the indenter during the impact period. On the other hand, a light plate (with lower density) is likely to absorb more energy from the indenter rather than a more inert plate. Here, as illustrated in Fig. 8-b, for the values of the nonlocal parameter less than 0.10, energy absorption has been reduced by increasing the nonlocality. Two possible reasons could explain these observations: the increase in the plate stiffness, and/or the increase in the mass momentum of inertia. On the other hand, according to Eq. (10), the natural frequencies of the nonlocal plate decrease slightly as the nonlocal parameter increases. Therefore, the nonlocality is less likely to affect the rigidity and the increase in the mass of momentum inertia remains the main cause, which leads to the reduction in energy absorption and to the lower natural frequencies. In other words, the mass momentum of inertia of the SLGS is more likely to be affected by the nonlocal parameter rather than its stiffness, so that larger values of nonlocal parameter result in smaller deflections in the SLGS and, consequently, in less energy absorption from the indenter.
14
Fig. 8. Effects of the nonlocal parameter on a- the COR and b- the modal energies of the SLGS for u% = 150 m/s and 1 = 10 N/m.
3.5
Effects of plate dimensions Plate dimensions (, ) affect both the natural frequencies (Eq. (10)) and the influence
coefficient of impact force (Eq. (19)). Consequently, it is expected that plate dimensions affect its response to impact. To investigate this parameter, the collision between a square plate and a single
gold atom is studied. The results for various initial velocities of u% are plotted in Fig. 9. Comparing
different curves corresponding to various values of u% shows that the minimum values of COR
occur approximately at the same plate size (at = = 35 nm in this case). As a result, in the case
of employing a square SLGS as an energy absorber against colliding gold atoms, its sides should be equal to 35 nm in order to maximize energy absorption. The smaller plates are too stiff and the larger ones are too inert to significantly decrease the energy of indenter. 15
Fig. 9. The COR vs. plate dimensions for 1 = 10 N/m and various initial velocities of the indenter. 3.6
Effects of the indenter size In the case of studying the interactions between an indenter larger than a concentrated mass,
as discussed in section 2.4, Eq. (18) should be employed instead of Eq. (15) to obtain the dynamics response of the system. Since gold atoms have an FCC atomic structure, the gold atomic structure contains four atoms per unit cell. The unit cell is a cube with an edge length of √2 ¤LL . In Eq.
(18), the number of particles per unit area of the indenter is calculated as = 4/(√2¤LL ):
where the gold bond length is ¤LL = 0.288 nm [20]. Now, a spherical gold particle is assumed
to be colliding with a square SLGS. Illustrated in Fig. 10, are the effects of indenter radius and
the plate dimensions on the COR. In this figure, the particle size varies from 0.3 n, which is larger
than the size of a single atom, to 1.0 nm, which contains about 250 atoms. As this figure shows, by increasing the plate dimensions in span 10 nm ≤ = ≤ 50 nm, the COR decreases. At a
specific initial velocity, the larger indenters travel with more energy. Then, as Fig. 10 demonstrates, when colliding with the nanoplate of certain dimensions, the larger indenters bring about the higher values of COR.
16
Fig. 10. Effects of indenter radius and the plate dimensions on the COR for 1 = 10 N/m and u% = 200 m/s.
4. Conclusions Employing an analytical approach, the present research investigates the dynamics behavior of a nanoparticle hitting a nanoplate with surface tension. The plate was considered rectangular and orthotropic, to which Kirchhoff’s classical plate theory was applicable. Strain resultants were measured by adopting the von Kármán nonlinear theory, while the nonlocal theory of elasticity was utilized to derive the equations of motion for the plate. As an application, a particle-plate system, in which a gold particle hits an SLGS, was studied in detail. In this regard, a theoretical contact law, which is developed based on the non-bonding forces of Lennard-Jones, was used to solve numerically the simultaneous equations of motion. Then, the effects of surface tension and the nonlocal parameter on the rebound behavior of the particle were analyzed. In addition, concentrating on the dynamics response of the particle, the influences of ignoring the flexural stiffness of the SLGS, as well as disregarding the attraction part of the von der Waals forces were examined. Findings revealed that the effects of the surface tension, the nonlocal parameter and the flexural stiffness of the plate are limited to certain shifts in the natural frequencies of the plate. Furthermore, results showed that the attraction effects of the interaction forces play a pivotal role in the rebound behavior of the particle. Excluding the attraction effects from the contact law significantly changed the responses of the system. Moreover, a study on the effects of the system dimensions on its dynamics response showed that certain dimensions of the plate maximize its capacity for energy absorption, which are independent of the indenter initial velocity. Finally, the contact law for a large nanoparticle was derived through an example including an SLGS colliding with a multi-atom indenter. The proposed perspective on the nano-scaled impact phenomenon can 17
be utilized to describe the behavior of nanodevices, such as nanomass sensors, low-pressure gas sensors, and nanodelivery systems, in a simpler and more precise way.
References [1] S. Seifoori, G.H. Liaghat, Low velocity impact of a nanoparticle on nanobeams by using a nonlocal elasticity model and explicit finite element modeling, International Journal of Mechanical Sciences, 69 (2013) 85-93. [2] S.T. Talebian, M. Tahani, M.H. Abolbashari, S.M. Hosseini, Response of multiwall carbon nanotubes to impact loading, Applied Mathematical Modelling, 37 (2013) 5359-5370. [3] S. Seifoori, Molecular dynamics analysis on impact behavior of carbon nanotubes, Applied Surface Science, 326 (2015) 12-18. [4] S. Seifoori, H. Hajabdollahi, Impact behavior of single-layered graphene sheets based on analytical model and molecular dynamics simulation, Applied Surface Science, 351 (2015) 565-572. [5] S. Seifoori, M.J. Khoshgoftar, Impact and vibration response of multi-layered graphene sheets under different striker based on the analytical model and molecular dynamics, Superlattices and Microstructures, 135 (2019) 106249. [6] L. Yang, L. Tong, Suspended monolayer graphene traps high-speed single-walled carbon nanotube, Carbon, 107 (2016) 689-695. [7] S. Sadeghzadeh, L. Liu, Resistance and rupture analysis of single- and few-layer graphene nanosheets impacted by various projectiles, Superlattices and Microstructures, 97 (2016) 617-629. [8] S. Abrate, Modeling of impacts on composite structures, Composite Structures, 51 (2001) 129-138. [9] S. Hosseini-Hashemi, A. Sepahi-Boroujeni, Elastic impact response of a nonlocal rectangular plate, International Journal of Solids and Structures, 109 (2017) 93-100. [10] H. Liu, J. Liu, J.-L. Yang, X.-Q. Feng, Low velocity impact of a nanoparticle on a rectangular nanoplate: A theoretical study, International Journal of Mechanical Sciences, 123 (2017) 253-259. [11] S. Hosseini-Hashemi, A. Sepahi-Boroujeni, S. Sepahi-Boroujeni, Analytical and molecular dynamics studies on the impact loading of single-layered graphene sheet by fullerene, Applied Surface Science, 437 (2018) 366-374. [12] J.N. Israelachvili, Intermolecular and Surface Forces: Revised Third Edition, Elsevier Science, 2011. [13] T.A. Hilder, J.M. Hill, Theoretical comparison of nanotube materials for drug delivery, IET Micro & Nano Letters, 3 (2008) 18-24. [14] Y. Chan, J.M. Hill, Modelling interaction of atoms and ions with graphene, IET Micro & Nano Letters, 5 (2010) 247-250. [15] C. Yue, M.H. James, Hydrogen storage inside graphene-oxide frameworks, Nanotechnology, 22 (2011) 305403. [16] D.G. Sangiovanni, G.K. Gueorguiev, A. Kakanakova-Georgieva, Ab initio molecular dynamics of atomic-scale surface reactions: insights into metal organic chemical vapor deposition of AlN on graphene, Physical Chemistry Chemical Physics, 20 (2018) 17751-17761. [17] J.N. Reddy, Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates, International Journal of Engineering Science, 48 (2010) 1507-1518. [18] S.C. Pradhan, J.K. Phadikar, Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models, Physics Letters A, 373 (2009) 1062-1069. [19] L. Shen, H.-S. Shen, C.-L. Zhang, Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments, Computational Materials Science, 48 (2010) 680-685.
18
[20] W. Qi, B. Huang, W. Yang, Bond-Length and -Energy Variation of Small Gold Nanoparticles, Journal of Computational and Theoretical Nanoscience, 6 (2009) 635-639.
19
Highlights: • Effects of surface tension are mostly limited to shifts in natural frequencies of nanoplate. • Investigating nanoplate coefficient of restitution revealed that stretched graphene sheets can be reasonably approximated as membranes in impact problems. • A contact law is proposed for large nanoparticles interacting with nanoplates. • Depending on surface tension of nanoplate, there would be a critical initial velocity of nanoparticle under which indenter sticks to the target. • Motion study of rebounding nanoparticle proves the pivotal role of attraction effects in a reliable nano-scaled contact law. • There are optimum values for the nanoplate dimensions in order to maximize the energy absorbed from colliding particles
Declaration of interests ☑ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S0749-6036(19)32048-8
DOI:
https://doi.org/10.1016/j.spmi.2020.106464
Reference:
YSPMI 106464
To appear in:
Superlattices and Microstructures
Received Date: 29 November 2019 Revised Date:
22 February 2020
Accepted Date: 23 February 2020
Please cite this article as: A. Sepahi-Boroujeni, S. Hosseini-Hashemi, S. Sepahi-Boroujeni, Effects of surface tension of graphene sheet on impact and rebound behavior of colliding nanoparticle, Superlattices and Microstructures (2020), doi: https://doi.org/10.1016/j.spmi.2020.106464. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.
Effects of surface tension of graphene sheet on impact and rebound behavior of colliding nanoparticle a,
a
Amin Sepahi-Boroujeni *, Shahrokh Hosseini-Hashemi , Saeid Sepahi-Boroujeni a
b
Impact Research Laboratory, Department of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran
b
Department of Mechanical Engineering, École Polytechnique de Montréal, Montréal, QC, Canada
Abstract In the present work, the classical theory of plate, the von Kármán nonlinear theory and the nonlocal theory of elasticity are adopted to analyze the impact between a nanoparticle and an orthotropic rectangular nanoplate with surface tensions. Employing an analytical contact law to explain the von der Waals forces, the equations of motion are solved for the indenter-plate system. As an application, this approach is taken for a nanoparticle colliding with a simply supported single-layered graphene sheet and the rebound behavior of the particle is discussed. Then, the influences of surface tension and the nonlocal parameter on the dynamics response of the particle are investigated. When enough surface tension is applied, the membrane theory gives an acceptable estimate for the restitution behavior of the indenter otherwise, the estimates are associated with a maximum error of 15%. The effects of disregarding the flexural stiffness of the graphene sheet and ignoring the attraction part of the von der Waals forces are also examined. Studying the dimensional effects on the system behavior revealed that optimum dimensions of the plate maximize its capacity for energy absorption when it is employed as an energy absorber against colliding particles. Finally, a contact law for large nanoparticles is proposed and then is applied to an example including an SLGS colliding with a multi-atom indenter.
Keywords: Graphene Sheet; Impact; Nonlocal plate theory; Surface tension.
Corresponding author: E-mail address: [email protected]
*
1
1. Introduction Mutual effects of nanoparticles and nanostructures have provided an increasing range of applications. Many nano-scaled devices such as mass sensors, pressure sensors, and oscillators work based on the interactions between foreign objects and nano-structures, e.g. nanobeams or nanoplates. Nano mass sensors have provided a vibration-based method to measure nano-scaled attached particles. Low-pressure gas sensors have been developed measuring response of a nanomembrane to colliding atoms. In addition, nanoplates have been utilized as implements to absorb, to store and then to deliver special foreign particles, such as ions, biocells, and nanodrugs. Depending on the applications and the investigation purposes of a nano-scaled impact phenomenon, studying both the nanoparticle (indenter) and the nanostructure (target) can be a matter of interest. Investigating the kinematic characteristics of a nano-indenter interacting with various structures enables some simple generalized rules to be derived, which reduce the massive calculations of many complex nano-scaled problems. Theoretical study of the response of a system including the collision between an indenter and a target structure, known as impact dynamics, concerns the dynamics responses of the system as well as the local displacements at the impact area. Despite a considerable number of studies on the impact dynamics of macro-scaled systems, few attempts have been made for modeling nanosized systems because of theoretical complexities and the necessity of employing nonlocal continuum theories in a transient media. The research works conducted on nano-scaled systems are limited to some recent studies. [1, 2] theoretically studied the response of a nanobeam to an external excitation and discussed the response of the target during and after the impact. [3] proposed a simple analytical approach in addition to a molecular dynamics (MD) simulation to study the impact response of a nanotube-nanoparticle system and measured the deflection of the nanotube during the impact. [4] developed a theoretical mass-spring model together with an MD simulation to address the problem of impact between a nanoparticle and a single-layered graphene sheet (SLGS), which was generalized later by [5] for two- and multi-layered graphene sheets. [6] used an MD simulation for modeling a suspended SLGS colliding with a nano-tube to estimate the energy absorption capacity of SLGSs. Employing a quasi-classical model for the problem of singleand few-layered graphene sheets colliding with a projectile, [7] investigated the influences of impact velocity and the number of layers on the coefficient of restitution and the critical rupture velocity. The full modeling of an impact phenomenon is known to include the equations of motion for both the target and the indenter, as well as the dynamics response of the system during and after 2
the impact period [8]. The only theoretical studies based on the full modeling are limited to [9-11], in which the dynamics responses of a nanoplate-nanoparticle system are derived based on an analytical contact law. These works estimate the interaction forces in terms of the indenter-plate distance. The contact laws utilized in these studies are based on estimating the interaction forces within a continuous force field. These contact laws that are described in [12] had been used in former studies like [13-15] to derive some conceptual models for the interactions between nanostructures and foreign nanoparticles. Utilizing a different method, [16] employed a densityfunctional ab-initio MD to identify the mechanisms driving precursor/surface reactions during epitaxy on the surface of SLGS. Focusing on the behavior of the indenter, the present research concerns the dynamics behavior of a nano-sized system including an indenter interacting with an SLGS with surface forces. The SLGS is assumed to be rectangular, orthotropic, and thin. The governing equation of motion is derived based on Kirchhoff’s classical plate theory (CPT) together with the nonlocal theory of elasticity. The utilized contact law is based on the Lennard-Jones (L-J) formula and gives an analytical approximation of the non-bonding van der Waals interaction forces. Moreover, the impact between a gold nanoparticle and an SLGS with normal surface tensions is investigated in detail. This approach can be employed to develop the general rules governing the behavior of nanoparticles after hitting nanostructures such as graphene sheets.
2. Formulation 2.1
Equations of motion Consider a thin rectangular nanoplate with a density of , a thickness of ℎ and dimensions
of and . The origin of the assigned , , Cartesian coordinate system is located at one corner of
the plate so that axes and are coincident with the edges of the undeformed mid-plane of the
nanoplate (i.e. 0 ≤ ≤ , 0 ≤ ≤ , −ℎ/2 ≤ ≤ ℎ/2). The plate is assumed to be orthotropic
and , , are taken as the moduli of elasticity in the corresponding directions. ν and ν are
the Poison's ratios respectively along directions and , and ! is the shear modulus of elasticity. According to the von Kármán nonlinear theory, the non-zero resultant strains in the CPT are [17]: 1 " = $%, − &, + &, ) 2
1 " = *%, − &, + &, ) 2
(1)
1 1 " = ($%, + *%, ) − &, + &, &, 2 2 3
in which $, *, & are the displacements in directions , , and $% , *% are the material displacements
at point (, , 0) along axes and , respectively. The strain field of Eq. (1) leads to the governing
differential equation of motion [17]: ..
-% & = . , + 2. , + . , + / + 0
(2)
where -% = ℎ is the mass momentum of inertia, . , . , . are the bending momentum
resultants, 0 is the transversal distributed force (measured per unit area), and / is a function of inplane tension resultants (1 , 1 , 1 ): /($, *, &) =
2 2 31 &, + 1 &, 4 + 31 & + 1 &, 4 2 2 ,
≈ 1 &, + 21 &, + 1 &,
(3)
in which the last expression is a linear approximation of /. Employing the strain-displacement and the stress-strain equations for nonlocal plates, Eq. (2) could be rewritten as: ..
-% ℒ& = −7 &%, − 237 + 7!! 4&%, − 7 &%, + ℒ/ + ℒ0
(4)
7, the flexural rigidities of the plate, are:
< ℎ: 87 , 7 , 7 , 7!! 9 = ; , , , = 12 1 − < < 1 − < < 1 − < < !
(5)
and the nonlocal operator is ℒ = (1 − >? ) ) where ? ) = 2 ) /2 ) + 2 ) /2 ) is the 2-D Laplacian
operator and > = @%) ) is a nonlocal parameter introduced as a function of non-dimensional coefficient @%) . From this point forward, as in practical applications, boundary conditions are
assumed to be simply supported or clamped and surface shear force is disregarded (1 = 0), which allows the orthogonality statements to be derived. 2.2
Free vibrations In the case of free vibrations, external force 0 can be excluded from Eq. (4). Since the mode
shapes of vibration, represented here by ABC (, ) (D, E = 1,2, …), satisfy the equation of free vibrations, Eq. (4) can be restated as: ) −-% GBC ℒ ABC = −7 ABC, − 237 + 7!! 4ABC, − 7 ABC,
+ℒ(1 ABC, + 1 ABC, )
4
(6)
where GBC is the natural vibration frequency in mode (D, E). Multiplying Eq. (6) by an arbitrary mode shape AH! and then integrating over the area of plate (I) yield:
) GBC -% J 3AH! ABC + > AH!, ABC, + > AH!, ABC, 4 KI = L
J M7 ABC, AH!, + 237 + 7!! 4ABC, AH!, + 7 ABC, AH!, N KI − L
1 J 3AH!, ABC, + > AH!, ABC, + > AH!, ABC, 4 KI − L
(7)
1 J 3AH!, ABC, + > AH!, ABC, + > AH!, ABC, 4 KI L
in which the following relations, derived based on the partial integral properties, are employed
(O, P = , ),
L
L
L
L
L
L
J AH! ABC,RRSS KI = J AH!,RS ABC,RS KI
Q AH! ABC,RRRR KI = Q AH!,RR ABC,RR KI ;
(8)
J AH! ABC,RR KI = − J AH!,R ABC,R KI
The integrals in Eq. (7) are symmetric with respect to subscripts DE and TU. Therefore, a new equation could be formed by replacing GBC with GH! on the left side of Eq. (7). Eventually,
subtracting this equation from Eq. (7) results in the orthogonality properties of the mode shapes in a nonlocal classical plate:
L
L
-% J AH! ℒABC KI = J AH! M7 ABC, + 237 + 7!! 4ABC, + 7 ABC, N KI
+ J AH! ℒ(1 ABC, + 1 ABC, ) KI = 0 (T ≠ D WT U ≠ E)
(9)
L
In the case of simply supported boundaries, the mode shapes are harmonic functions of
and , which can be described as ABC = UDX
B Y Z
UDX
C Y [
, [18]. Substituting ABC in Eq. (6) and
employing orthogonality properties of Eq.(9) give the natural frequencies: \ ) GBC = ) ) ]
2.3
[^
Z^
D ] Z^ 7 + 2D ) E ) 37 + 7!! 4 + E ] [^ 7 -% _1 + \ ) @%) `D ) + E
^ )Z
ab [^
+ \)
D)
cdd Z^
Dynamics response Expanding displacement & as a series of mode shapes results: 5
+ E) -%
cee [^
(10)
&(, , f) = g
i B,Cjk
ABC (, )hBC (f)
(11)
in which hBC (f) (D, E = 1,2, …) are the time-dependent terms of the series. A mathematical process to derive the modal response of a nanoplate to an impact force is described in [9], which results in: hBC (f) = lWU GBC f hBC (0) +
UDX GBC f nBC q hm BC (0) − J o(p) UDX GBC (f − p) Kp GBC GBC %
(12)
where o(f) is the time-dependent part of the impact force and nBC represents the influence coefficient of the impact force for vibration mode (D, E), which can be formulated as: nBC =
3ℒABC 4(
r , r ) -% sL ABC ℒ ABC KI
(13)
As a result, it could be stated that the effect of surface forces on the dynamics response of the plate is limited to the shifts in the natural frequencies of the plate, and the equations of motion are the same for the plates with and without surface forces. The indenting particle exerts an opposing force, thus the equation of motion for the indenter with a displacement of t and a velocity of u is: t(f) = t% + u% f + 2.4
1 q J (f − p) o(p) Kp . %
(14)
Contact law Impact force may be represented in terms of the indenter-plate distance, known as “contact
law”. One of the most recently proposed contact laws to explain the non-bonding L-J interaction force between a concentrated nanoparticle and a nanoplate is [9]: z{ zk) o = 8 \ w x y } − kk ~ | |
(15)
in which material properties w and z are respectively the depth of the potential well and the distance where the inter-particle potential is zero. In this equation, x is the number of particles per
unit area of the plate and ∆ is the indenter-plate relative distance. Eq. (15) is derived by assuming that the particle is a concentrated point. A more general contact law would be needed for larger indenters that may not be treated as concentrated particles. A rigid spherical nanoparticle with a radius of is assumed to be standing above an infinite nanoplate, as shown in Fig. 1, and an T − − spherical coordinate system is assigned to the center of the particle, in which and
6
are respectively the azimuthal and the polar angles. Now, for a small arbitrary element of the particle, one can employ Eq. (15) to estimate the force imposed on the element by the plate:
Fig. 1. Schematic illustration of the system including a solid sphere model as a nanoparticle interacting with a continuum nanoplate. Ko = 8\ w x y
z{ zk) − ~ |′} |′kk
(16)
in which stands for the number of atoms allocated inside the assumed element and |′ is the vertical distance of the element from the plate, which can be rewritten in terms of T and : | = | + T lWU
(17)
where | is the distance between the particle center and the plate. Now, Eq. (16) can be integrated
over the volume of the nanoparticle, resulting in total force o imposed on the nanoparticle: { k) o = 8 \ w x J J J y − ~ T ) UDX KTK K } (| + T lWU )kk Hj% j% j% (| + T lWU )
Y
)Y
2 { | + 3 | − 3 k) | + 9 | − 9 ) = \ w x y − − − ~ 3 (| + ): (| − ): 45 (| + ) (| − )
(18)
For a single-atom particle, assuming that there is one atom inside a sphere with a radius of , one ]
can set = `: \ : a
k
and obtain Eq. (15) as approaches zero in Eq. (18). Since the relative
distance between the indenter and the plate depends on both the position of the indenter and the displacement of the plate at the contact point, the set of Eqs. (12) and (14) together with Eq. (15) or (18) should be solved simultaneously to derive the dynamics response of the system.
3. Results and discussion In the absence of an analytical solution for a set of nonlinear equations, employing numerical approaches can be helpful for simultaneously solving Eqs. (12), (14) and (15), which 7
gives the dynamics response of the plate-indenter system. In the following, the impact between an SLGS and a gold nanoparticle is studied. According to [19], the geometrical and the mechanical
features of the SLGS are assumed to be: × × ℎ = 9.496 × 4.877 × 0.145 nm: , = 5624 kg/m: , 8 , , ! 9 = {2.145, 2.097, 0.938} TPa, < = 0.223, < = < / , and the
nonlocal parameter is @% = 0.05. Without loss of generality, surface tensions are assumed to be equi-biaxial, i.e. 1 = 1 = 1. The indenting particle is taken as a gold atom with a mass of
. = 3.27 × 10)} Kg that hits the resting SLGS at point (% , % ) = (/2, /2). The L-J
parameters for the gold-carbon interactions are zL = 0.3187 nm, wL = 2.10@u =
3.364 × 10)) J [14], and x = 38.1770 nm) [9]. The response precision depends on the number of modes of vibration considered in the solution, whereas several modes of vibration may be
excited in the impact response of the plate. In the present study, × X = 30 × 30 = 900 modes are included to ensure that a sufficient number of higher modes of vibration are considered. Then, the following analyses are conducted on the above-mentioned problem: 3.1
Effects of surface tension As expected, increasing the surface tension stiffens the plate. The stiffer plate shows less
deflection against the impact load and, consequently, absorbs less energy from the indenter. Fig. 2 schematically illustrates the displacements calculated by Eqs. (12) and (14) respectively for the plate and the indenter with an initial velocity of 300 m/s. As this figure shows, in the cases
including the stiffer plates with higher values of surface tension, the indenter carries enough energy to recede into space after the collision. Eq. (13) describes the influence of the impact force on the vibration modes and shows how the surface tension affects the dynamics response of the plate. Because the mode shapes are independent of the surface tension, it can be concluded that influence coefficient of the impact force nBC is also independent of the surface tension. Thus, as can be
observed in Eq. (12), the only effect of tension on the dynamics response is the shift in the natural frequencies. In other words, in nonlocal plates, similar to local plates, the effects of surface tension are limited to the changes in natural frequencies, which determine whether the indenter sticks to the plate or escapes.
8
Fig. 2. Schematic illustrations of the effects of surface tension on the deflection of the SLGS and the z-position of the indenter at various sequences. Fig. 3 shows how the bouncing behavior of the indenter varies by the surface tension. The
SLGS initially lies on the plane and interacts with the indenter based on the van der Waals forces expressed by Eq. (15). The long-range attraction part of this force absorbs the indenter, while the plate ejects the indenter by growing the short-range repulsion force as the relative distance decreases. This interaction transfers a portion of the indenter’s energy to the plate in the forms of kinetic and strain energies. The rest of the indenter’s energy determines whether it escapes or be trapped by the plate. On the other hand, the plate deflections have a major role in the force field of the system, so that the relative distance between the indenter and the corresponding point on the plate determines the amplitude of the interaction force. It can be observed in Fig. 3 that the indenter sticks to the SLGS with the tension loads less than a specific value, which occurs during an approach-rebound procedure and eventually results in a final integrated system including both the plate and the indenter. The smaller values of tension force cause shorter rebounds and faster absorption.
9
Fig. 3. Different rebound patterns for various tension loads applied to the target surface. As the indenter escapes, a kinematic coefficient of restitution (COR) can be defined as the
ratio of the post-impact velocity to the initial velocity of the indenter: COR = u1 /u0 . Fig. 4 presents
the variations of the COR vs. the surface tension for different initial impact velocities. As illustrated, the greater the u0 , the larger is the COR. This observation could be explained by the fact
that the impact procedure occurs faster for greater initial velocities of the indenter, then the plate has less time to absorb the indenter’s energy. Moreover, for every initial velocity, there is a specific
tensile load where the COR approaches zero and the indenter sticks to the plate. By rising 1, the
COR shows a stepwise increasing trend that is more obvious at lower initial velocities (u% <
200 m/s). The reason is that over the short relative distance between the indenter and the plate,
which comes along at the impact moment, the interaction force strongly varies within a wide range of amplitudes. Furthermore, the displacement of the plate depends on its excited mode of vibration and, consequently, the variation of frequency or the phase of modes alters the distance of the indenter from the plate. Although very small, this change can give rise to considerable momentary variations of the interaction force, resulting in a jump in the graph. In macro-scaled impacts, impact forces (e.g. the Hertz impact force) vary slightly over the impact period, then these fluctuations can be limited to the nano-scaled impacts.
10
Fig. 4. The COR vs. surface tension for various initial velocities of the indenter. 3.2
Effects of attraction forces In macro-scaled impact problems, the attraction effects of the force field are ignored in
contact laws. Ignoring the long-range attraction term in Eq. (15) makes it a rough estimation of the macro-scaled impact forces. Excluding the attraction forces, Fig. 5 presents the variations of the COR vs. the surface tension for different initial impact velocities. As illustrated, this system shows quite different behavior in comparison with the findings illustrated in Fig. 4. The COR reaches considerably higher levels for all initial velocities and the indenter does not stick to the plate, even for the small values of surface tension. In addition, unlike Fig. 4, increasing the initial velocity leads to a reduction in the COR due to changes in the behavior of SLGS in faster collisions. As observed in macro-scaled impact problems, it could be claimed that increasing the initial velocity of the indenter excites a larger number of vibration modes in the plate and, as a result, dissipates a greater amount of the indenter’s energy. These findings clearly show that the attraction part of the impact force has a pivotal role in the behavior of the system in a way that disregarding such influences of attraction forces gives an imprecise estimate for the dynamics behavior of the system.
11
Fig. 5. The COR vs. surface tension for various initial velocities of the indenter in the case of excluding the attraction forces from the contact law. 3.3
The validity range of the membrane theory in comparison with the classical plate theory Similar to a thin structure, a stretched SLGS can approximate to a membrane. This
simplification results in some errors due to neglecting the flexural rigidities. These errors can be investigated by comparing the dynamics response of an SLGS with and without flexural rigidities being involved, respectively given by Eqs. (4) and (10). Fig. 6 includes the variations of the COR vs. surface tension 1 measured based on both the CPT and the membrane theory (MT) at various
initial velocities. As expected, the agreement between these theories increases for higher tensile loads. Therefore, membrane theory gives an acceptable estimate when considerable surface tension is applied. On the other hand, for smaller values of tension, errors are less than 15%, which could be acceptable in the studies where a general view of the system behavior is expected, e.g. in studying the pressure sensors where several particles hitting an SLGS with various velocities. Based on the CPT and the MT, Fig. 7 includes the variations of the tension force threshold where the indenter sticks to the SLGS, vs. the initial velocity of the indenter. As this figure implies, there are acceptable agreements between the two theories in estimating the thresholds of sticking. As a general result, assuming a stretched SLGS to be a membrane can provide a reasonable perspective on its dynamics response to the impact of a foreign indenter.
Fig. 6. The COR vs. surface tension for various initial velocities of the indenter calculated based on the classical plate theory and the membrane theory (MT).
12
Fig. 7. Variations of the particle threshold of sticking vs. the initial velocity of the indenter estimated based on the CPT and the MT. 3.4
Effects of nonlocality Employing nonlocal parameter @% allows the results obtained from experiments or
simulations to be validated by the nonlocal theory. Notably different values are reported in the literature for the nonlocal parameter, which are due to various purposes of the investigations along with the errors resulting from the employed tools and methods. As an example, various values can be found for each mode of vibration of an SLGS. In any case, the question remains how different values of the nonlocal parameter affect the behavior of a system, such as its dynamics response to an impact load. First, it is necessary to discuss the theoretical effects of the nonlocal parameter. In the present example of the simply supported SLGS, the harmonic mode shapes lead to nonlocal operator ℒ = 1 + \ ) > (D ) ⁄) + E ) ⁄ ) ). Then, Eq. (13) can be simplified as: nBC =
4 D \ % E \ % UDX UDX -%
(19)
which is independent of @% for every mode and only depends on the weight of the plate (-% ) and
the non-dimensional location of the impact point (i.e. % / and % /). Therefore, similar to the surface tension, the effect of nonlocality is limited to certain changes in the natural frequencies of
the plate. Fig. 8-a displays the variations of the COR vs. nonlocal parameter @% for u% = 150 m/s and 1 = 10 N/m. As shown in this figure, two types of variation behavior can be attributed to the
COR. First, as stated in section 3.1, one can observe the consecutive jumps in the COR caused by slight variations in the frequencies or in the phases of modes. Another behavior includes the cyclic variations, which eventually converges to a constant value. Fig. 8-b compares the variations of energy for three most excited modes of vibration of the plate. As shown, in the present example, the 13
small variations of energy mostly belong to mode (1,1) and the majority of the changes are observable for modes (3,1) and (5,1). As illustrated in Fig. 8-b, increasing the nonlocal parameter reduces the energy absorbed by the most excited modes for the nonlocal parameter values less than 0.10. In general, the impact between a foreign particle and a stiff plate occurs over a shorter time compared with a more flexible plate. Eventually, a stiffer plate absorbs less energy from the indenter during the impact period. On the other hand, a light plate (with lower density) is likely to absorb more energy from the indenter rather than a more inert plate. Here, as illustrated in Fig. 8-b, for the values of the nonlocal parameter less than 0.10, energy absorption has been reduced by increasing the nonlocality. Two possible reasons could explain these observations: the increase in the plate stiffness, and/or the increase in the mass momentum of inertia. On the other hand, according to Eq. (10), the natural frequencies of the nonlocal plate decrease slightly as the nonlocal parameter increases. Therefore, the nonlocality is less likely to affect the rigidity and the increase in the mass of momentum inertia remains the main cause, which leads to the reduction in energy absorption and to the lower natural frequencies. In other words, the mass momentum of inertia of the SLGS is more likely to be affected by the nonlocal parameter rather than its stiffness, so that larger values of nonlocal parameter result in smaller deflections in the SLGS and, consequently, in less energy absorption from the indenter.
14
Fig. 8. Effects of the nonlocal parameter on a- the COR and b- the modal energies of the SLGS for u% = 150 m/s and 1 = 10 N/m.
3.5
Effects of plate dimensions Plate dimensions (, ) affect both the natural frequencies (Eq. (10)) and the influence
coefficient of impact force (Eq. (19)). Consequently, it is expected that plate dimensions affect its response to impact. To investigate this parameter, the collision between a square plate and a single
gold atom is studied. The results for various initial velocities of u% are plotted in Fig. 9. Comparing
different curves corresponding to various values of u% shows that the minimum values of COR
occur approximately at the same plate size (at = = 35 nm in this case). As a result, in the case
of employing a square SLGS as an energy absorber against colliding gold atoms, its sides should be equal to 35 nm in order to maximize energy absorption. The smaller plates are too stiff and the larger ones are too inert to significantly decrease the energy of indenter. 15
Fig. 9. The COR vs. plate dimensions for 1 = 10 N/m and various initial velocities of the indenter. 3.6
Effects of the indenter size In the case of studying the interactions between an indenter larger than a concentrated mass,
as discussed in section 2.4, Eq. (18) should be employed instead of Eq. (15) to obtain the dynamics response of the system. Since gold atoms have an FCC atomic structure, the gold atomic structure contains four atoms per unit cell. The unit cell is a cube with an edge length of √2 ¤LL . In Eq.
(18), the number of particles per unit area of the indenter is calculated as = 4/(√2¤LL ):
where the gold bond length is ¤LL = 0.288 nm [20]. Now, a spherical gold particle is assumed
to be colliding with a square SLGS. Illustrated in Fig. 10, are the effects of indenter radius and
the plate dimensions on the COR. In this figure, the particle size varies from 0.3 n, which is larger
than the size of a single atom, to 1.0 nm, which contains about 250 atoms. As this figure shows, by increasing the plate dimensions in span 10 nm ≤ = ≤ 50 nm, the COR decreases. At a
specific initial velocity, the larger indenters travel with more energy. Then, as Fig. 10 demonstrates, when colliding with the nanoplate of certain dimensions, the larger indenters bring about the higher values of COR.
16
Fig. 10. Effects of indenter radius and the plate dimensions on the COR for 1 = 10 N/m and u% = 200 m/s.
4. Conclusions Employing an analytical approach, the present research investigates the dynamics behavior of a nanoparticle hitting a nanoplate with surface tension. The plate was considered rectangular and orthotropic, to which Kirchhoff’s classical plate theory was applicable. Strain resultants were measured by adopting the von Kármán nonlinear theory, while the nonlocal theory of elasticity was utilized to derive the equations of motion for the plate. As an application, a particle-plate system, in which a gold particle hits an SLGS, was studied in detail. In this regard, a theoretical contact law, which is developed based on the non-bonding forces of Lennard-Jones, was used to solve numerically the simultaneous equations of motion. Then, the effects of surface tension and the nonlocal parameter on the rebound behavior of the particle were analyzed. In addition, concentrating on the dynamics response of the particle, the influences of ignoring the flexural stiffness of the SLGS, as well as disregarding the attraction part of the von der Waals forces were examined. Findings revealed that the effects of the surface tension, the nonlocal parameter and the flexural stiffness of the plate are limited to certain shifts in the natural frequencies of the plate. Furthermore, results showed that the attraction effects of the interaction forces play a pivotal role in the rebound behavior of the particle. Excluding the attraction effects from the contact law significantly changed the responses of the system. Moreover, a study on the effects of the system dimensions on its dynamics response showed that certain dimensions of the plate maximize its capacity for energy absorption, which are independent of the indenter initial velocity. Finally, the contact law for a large nanoparticle was derived through an example including an SLGS colliding with a multi-atom indenter. The proposed perspective on the nano-scaled impact phenomenon can 17
be utilized to describe the behavior of nanodevices, such as nanomass sensors, low-pressure gas sensors, and nanodelivery systems, in a simpler and more precise way.
References [1] S. Seifoori, G.H. Liaghat, Low velocity impact of a nanoparticle on nanobeams by using a nonlocal elasticity model and explicit finite element modeling, International Journal of Mechanical Sciences, 69 (2013) 85-93. [2] S.T. Talebian, M. Tahani, M.H. Abolbashari, S.M. Hosseini, Response of multiwall carbon nanotubes to impact loading, Applied Mathematical Modelling, 37 (2013) 5359-5370. [3] S. Seifoori, Molecular dynamics analysis on impact behavior of carbon nanotubes, Applied Surface Science, 326 (2015) 12-18. [4] S. Seifoori, H. Hajabdollahi, Impact behavior of single-layered graphene sheets based on analytical model and molecular dynamics simulation, Applied Surface Science, 351 (2015) 565-572. [5] S. Seifoori, M.J. Khoshgoftar, Impact and vibration response of multi-layered graphene sheets under different striker based on the analytical model and molecular dynamics, Superlattices and Microstructures, 135 (2019) 106249. [6] L. Yang, L. Tong, Suspended monolayer graphene traps high-speed single-walled carbon nanotube, Carbon, 107 (2016) 689-695. [7] S. Sadeghzadeh, L. Liu, Resistance and rupture analysis of single- and few-layer graphene nanosheets impacted by various projectiles, Superlattices and Microstructures, 97 (2016) 617-629. [8] S. Abrate, Modeling of impacts on composite structures, Composite Structures, 51 (2001) 129-138. [9] S. Hosseini-Hashemi, A. Sepahi-Boroujeni, Elastic impact response of a nonlocal rectangular plate, International Journal of Solids and Structures, 109 (2017) 93-100. [10] H. Liu, J. Liu, J.-L. Yang, X.-Q. Feng, Low velocity impact of a nanoparticle on a rectangular nanoplate: A theoretical study, International Journal of Mechanical Sciences, 123 (2017) 253-259. [11] S. Hosseini-Hashemi, A. Sepahi-Boroujeni, S. Sepahi-Boroujeni, Analytical and molecular dynamics studies on the impact loading of single-layered graphene sheet by fullerene, Applied Surface Science, 437 (2018) 366-374. [12] J.N. Israelachvili, Intermolecular and Surface Forces: Revised Third Edition, Elsevier Science, 2011. [13] T.A. Hilder, J.M. Hill, Theoretical comparison of nanotube materials for drug delivery, IET Micro & Nano Letters, 3 (2008) 18-24. [14] Y. Chan, J.M. Hill, Modelling interaction of atoms and ions with graphene, IET Micro & Nano Letters, 5 (2010) 247-250. [15] C. Yue, M.H. James, Hydrogen storage inside graphene-oxide frameworks, Nanotechnology, 22 (2011) 305403. [16] D.G. Sangiovanni, G.K. Gueorguiev, A. Kakanakova-Georgieva, Ab initio molecular dynamics of atomic-scale surface reactions: insights into metal organic chemical vapor deposition of AlN on graphene, Physical Chemistry Chemical Physics, 20 (2018) 17751-17761. [17] J.N. Reddy, Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates, International Journal of Engineering Science, 48 (2010) 1507-1518. [18] S.C. Pradhan, J.K. Phadikar, Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models, Physics Letters A, 373 (2009) 1062-1069. [19] L. Shen, H.-S. Shen, C.-L. Zhang, Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments, Computational Materials Science, 48 (2010) 680-685.
18
[20] W. Qi, B. Huang, W. Yang, Bond-Length and -Energy Variation of Small Gold Nanoparticles, Journal of Computational and Theoretical Nanoscience, 6 (2009) 635-639.
19
Highlights: • Effects of surface tension are mostly limited to shifts in natural frequencies of nanoplate. • Investigating nanoplate coefficient of restitution revealed that stretched graphene sheets can be reasonably approximated as membranes in impact problems. • A contact law is proposed for large nanoparticles interacting with nanoplates. • Depending on surface tension of nanoplate, there would be a critical initial velocity of nanoparticle under which indenter sticks to the target. • Motion study of rebounding nanoparticle proves the pivotal role of attraction effects in a reliable nano-scaled contact law. • There are optimum values for the nanoplate dimensions in order to maximize the energy absorbed from colliding particles
Declaration of interests ☑ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: