Received 2 September 2014Accepted 3 November 2014Published 8 January 2015
Application of single layered graphene sheets (SLGSs) as resonant sensors in detection of ultra-¯ne nanoparticles (NPs) is investigated via molecular dynamics (MD) and nonlocal elasticityapproaches. To take into consideration the e®ect of geometric nonlinearity, nonlocality andatomic interactions between SLGSs and NPs, a nonlinear nonlocal plate model carrying anattached mass-spring system is introduced and a combination of pseudo-spectral (PS) and in-tegral quadrature (IQ) methods is proposed to numerically determine the frequency shifts causedby the attached metal NPs. In MD simulations, interactions between carbon–carbon, metal–metal and metal–carbon atoms are described by adaptive intermolecular reactive empirical bondorder (AIREBO) potential, embedded atom method (EAM), and Lennard–Jones (L–J) potential,respectively. Nonlocal small-scale parameter is calibrated by matching frequency shifts obtainedby nonlocal and MD simulation approaches with same vibration amplitude. The in°uence ofnonlinearity, nonlocality and distribution of attached NPs on frequency shifts and sensitivity ofthe SLGS sensors are discussed in detail.
Keywords: Graphene; mass resonant sensors; nonlocal elasticity; molecular dynamics simulations.
Nowadays, nanotechnology has found a notable rolein our daily life due to its wide variety of potentialapplications in medicine, food industries, environ-ment and energy issues, manufacturing processes,etc., and it has been an attracting topic of investi-gation in all science and technology ¯elds in recentyears. The use of nanoparticles (NPs) as a group ofadvanced engineering materials is rapidly increasingin various industries like electronics, nanocomposites,fuel additives, food packaging, sensors and actuators,and drug delivery because of their unique propertiesin comparison with bulkmaterial.1–4 However, due tohigh surface to volume ratio, NPs can easily passthrough cell membranes in organisms, while theirin°uence on biological systems is considered to becritical even if not well understood. Recent researcheshave reported some evidences for detrimental e®ectsof NPs on human cells like lung toxicity.5–7 As therelease of NPs in environment may occur not only atworkplaces but also at any stage of the lifecycle ofproducts, investigation on possible solutions fordetecting these aerosolized NPs in environment canbe a signi¯cant issue in the ¯eld of nanosensors.
Having small sizes and superior mechanical andelectrical properties, nanostructural elements aremain candidates for nanosensing applications.8–12
Among various available chemical and physicalmethods for sensing nanosized objects, gigahertznanoresonant sensors which detect these objectsfrom vibration characteristics have received a no-table attraction owing to their high value of sensi-tivity.13 The detection criterion is established basedon measuring the resonant frequency shift of thesensor caused by the attached object due to changesin the total mass of the system. As decreasing thesize of resonant sensor is a main solution to improveits sensitivity,13 the ability of detecting ultra-¯neNPs is directly related to discover appropriatecandidates for resonant sensing applications even atthe nanosize. Graphene sheets, as the thinnest two-dimensional (2D) °at structures consisting of car-bon atoms settled in a hexagonal lattice, have aconsiderable potential in resonant sensing applica-tion due to their remarkable sensing privilege e.g.,large surface area, low mass per unit area and highbending sti®ness. The experimental e®orts on theresonant sensing application of graphene sheets arerare up to this time; however, many researchersinvestigate their capability as resonant sensors in
detection of nanosized objects using both continu-um and atomistic approaches.
Although continuum approaches have the promi-nent advantage of modeling nanostructures withoutany restriction for the number of consisting particles,it is essential to modify them for consideration of dis-continuities in atomic domains by introducing small-scale e®ects. In continuum approaches, a graphene-based nanoresonant sensor is usually consideredas a nanoplate with attached masses. Shen et al.14
studied the potential of SLGSs as nanomass sensorsby considering the graphene sheet as a rectangularnanoplate with concentrated attached masses basedon nonlocal Kirchho® plate theory and Galerkinmethod. The e®ects of the mass value and positionon the frequency shift were discussed. Also, usingKirchho® nonlocal plate theory and Galerkinmethod, Zhou et al.15 analyzed a circular graphenesheet carrying a nanoparticle as a nanoresonantmass sensor. Murmu and Adhikari16 proposed anonlocal mass sensor model based on vibratingmonolayer cantilever graphene sheets. Closed-formequations were derived for the frequency shift due tothe added mass. However, their work was limited tolined shape distribution of the masses and linearvibration analysis. Neglecting small-scale e®ects, ina similar work, Adhikari and Chowdhury17 investi-gated the possibility of implementing graphenesheets as nanoresonant sensors. Lee et al.18 analyzedmass detection using a graphene-based nanor-esonator in the framework of nonlocal elasticity.The graphene sheet was considered as a rectangularnanoplate with an attached mass and equations ofmotion are analytically solved for simply supportedboundary conditions. In°uence of the small-scalee®ect, size and aspect ratio of SLGS on sensitivityof sensor was discussed in detail. In order to inves-tigate the possibility of double-layered graphenesheets as resonant mass sensors, Natsuki et al.19
studied vibration of double-layered rectangulargraphene sheets resonators as nanoplates using thelocal continuum elasticity theory. In another re-sembling work,20 they explored the potential appli-cation of circular graphene sheets for nanoresonantmass sensing. Jalali et al.21 investigated the poten-tial application of planar nano structures with at-tached nano particles as nano resonant sensors byintroducing a nonlocal plate model and to take intoaccount an elastic connection between the nanoplate and the attached nanoparticle, they consid-ered the nano particle as a mass-spring system.
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Atomistic simulation of graphene-based resonantsensors has also been investigated by researchersusing both molecular dynamics (MD) and molecularstructural mechanics approaches. Sakhaee-Pouret al.22 implemented ¯nite element molecularstructural mechanics to model the vibrational be-havior of SLGSs and investigated the e®ect of pointmass on the fundamental frequencies for masssensing applications. Trivedi et al.23 investigatedsensor application for detection of acetone moleculespresent in human breath for cantilevered singlewalled-boron nitride nanotube and an atomistic 3Dspace frame model of ¯xed-free nanoresonator wasdeveloped. Arash et al.24 investigated the applica-tion of SLGSs as mass sensors in detection of noblegases by using MD simulations. The dependence ofnumber and location of gas atoms, size of graphenesheets and type of boundary of the sheets on thesensitivity were particularly studied.
Many researches in the literature have reportedthat geometric nonlinearity has a signi¯cant role inlarge amplitude vibration of graphene sheets andresonant frequency is related to the vibration am-plitude.25–30 On the other hand, the reported resultsusing nonlocal continuum elasticity with an unsub-stantiated value of small-scale parameter are notrealistic enough to predict the vibrational behaviorof graphene sheets as resonant sensors and it is vitalto calibrate small-scale parameter by comparingnonlocal and MD simulations results.13 However,with regard to authors' knowledge, simultaneousin°uence of geometric nonlinearity and nonlocalityon the frequency shift and sensitivity of nanor-esonant sensors for calibrating a proper small-scaleparameter has not been reported. In this study, ap-plication of SLGSs as resonant sensors in detection ofultra-¯ne metallic NPs is investigated. To take intoconsideration the e®ect of nonlinearity, nonlocalityand atomic interactions between SLGSs and NPs, anonlinear nonlocal plate model with an attachedmass-spring system is introduced and nonlocalsmall-scale parameter is calibrated by matchingfrequency shifts obtained by nonlocal and MD sim-ulation approaches with same vibration amplitude.
2. Nonlocal Plate Model for SLGS
Resonant Sensors
A nonlocal continuum model for nonlinear vibrationof an SLGS with attached NPs on its surface isproposed. According to its 2D geometry, a graphene
sheet may be represented by an elastic nanoplate oflength a, width b, the e®ective thickness h and massdensity �. To take into account the contact inter-actions in nonlocal formulation, the attachednanoparticle is considered as a mass-spring system(M0, K0) mounted on an arbitrary position (x0, y0)of the plate. The coordinate system (x, y, z) is lo-cated at the corner of plate such that the plate mid-plane coincides with xy-plane and z-axis is normal toit as it is illustrated in Fig. 1. Invoking von K�arm�anlarge de°ection assumption and the ¯rst-order sheardeformation plate theory (FSDT), nonlinear strain-displacement relations can be described as31:
The mid-plane strains and the curvatures aredescribed by
"x0 ¼ u0;x þ1
2ðw0 ;xÞ2; kx ¼ ’x;x; ð2aÞ
"y0 ¼ v0;y þ1
2ðw0 ;yÞ2; ky ¼ ’y;y; ð2bÞ
�xy0 ¼ u0;y þ v0;x þ w0;xw0;y; kxy ¼ ’x;y þ ’y;x;
ð2cÞ�xz0 ¼ ’x þ w0;x; ð2dÞ�zy0 ¼ ’y þ w0;y; ð2eÞwhere ð Þ;x and ð Þ;y indicate the di®erentiation withrespect to x and y, respectively. Also, u0, v0 and w0
are the mid-plane displacement components alongthe x-, y- and z-directions, while ’x and ’y de¯nerotation about the y- and x-axis, respectively.
Nonlocal continuum elasticity assumptions ap-pear in constitutive stress–strain relations. In con-ventional local elasticity, stress at a point dependsonly on the strain at that point. Then, the macro-scopic local stress component tij at a point is relatedto the strain tensor component "mn at that point bythe generalized Hooke's law as follows:
tij ¼ Cijmn"mn; ð3Þwhere Cijmn is the fourth-order elasticity tensorcomponent.31 On the other hand, based on nonlocalelasticity assumptions proposed ¯rst by Eringen,32
the stress at a point is related to the strain at everypoint of the elastic body through an integration on
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the whole elastic body domain. Then, Eringen32
explained that it is possible to represent the integralconstitutive relation in an equivalent simpler dif-ferential form as follows:
ð1� �r2Þ�ij ¼ tij ¼ Cijmn"mn; � ¼ ðe0a0Þ2: ð4ÞIn which � is nonlocal parameter to present thesmall-scale e®ects, e0 is a material constant, a0 is theinternal characteristic length (e.g., distance betweenconstitutive atoms), and r2 is the 2D Laplace op-erator and �ij is the nonlocal stress component.Setting the internal characteristic length equal tozero results in � ¼ 0, and the nonlocal elasticityreduces to the perfect continuous state of local elas-ticity. The nonlocal stress–strain relationship for theplane stress state of nanoplates can be written as:
�x
�y
�zy�xz�xy
8>>>><>>>>:
9>>>>=>>>>;
� �r2
�x�y�zy�xz�xy
8>>>><>>>>:
9>>>>=>>>>;
¼
Q �Q 0 0 0
vQ Q 0 0 0
0 0 G 0 0
0 0 0 G 0
0 0 0 0 G
2666664
3777775
"x"y�zy�xz�xy
8>>>><>>>>:
9>>>>=>>>>;;
Q ¼ E
ð1� � 2Þ; G ¼ E
2ð1þ �Þ:
ð5Þ
E, G and v are Young's modulus, shear modulusand Poisson's ratio of the nanoplates, respectively.One can calculate the nonlocal force and momentresultants through integrating stress components
across the plate thickness.
½Nxx;Nyy;Nxy� ¼Z h=2
�h=2
½�xx; �yy; �xy�dz; ð6aÞ
½Mxx;Myy;Mxy� ¼Z h=2
�h=2
½�xx; �yy; �xy�zdz; ð6bÞ
½Qx;Qy� ¼ Ks
Z h=2
�h=2
½�xz; �zy�dz: ð6cÞ
Ks is the shear correction coe±cient of FSDT whichis considered equal to 5/6.34 Combining Eqs. (5)and (6), these force and moment resultants can beexpressed in terms of displacement components:
Nxx
Nyy
Nxy
8<:
9=;� �r2
Nxx
Nyy
Nxy
8<:
9=;
¼A �A 0
�A A 0
0 0Að1� �Þ
2
2664
3775
"x0"y0�xy0
8<:
9=;; A ¼ Eh
ð1� � 2Þ:
ð7aÞMxx
Myy
Mxy
8<:
9=;� �r2
Mxx
Myy
Mxy
8<:
9=;
¼D �D 0
�D D 0
0 0Dð1� �Þ
2
2664
3775
kxkykxy
8<:
9=;;
D ¼ Eh3
12ð1� � 2Þ;
ð7bÞ
Fig. 1. Continuum and atomistic models of SLGS with an attached nanoparticle as a resonant sensor.
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Qx
Qy
� �� �r2
Qx
Qy
� �
¼ Ks
Að1� �Þ2
0
0Að1� �Þ
2
2664
3775 �xz
�zy
� �: ð7cÞ
The six governing equations of motion for freevibration of a shear deformable plate carrying amass-spring system can be obtained using theprinciple of minimum total potential energy as34:
(�) operator denotes di®erentiation with respect totime, t, and mass moments of inertia, I0 and I2 arecalculated by:
ðI0; I2Þ ¼Z h=2
�h=2
�ð1; z2Þdz: ð8gÞ
The nonlinear term Nðw0Þ in Eq. (8c), can bewritten as follows35:
Nðw0Þ ¼ Nxxw0;xx þ 2Nxyw0;xy þNyyw0;yy: ð8hÞAlso, the Dirac Delta function, given in Eq. (8c), isde¯ned as:
�ðx� x0Þ ¼ 0; x 6¼ x0;
�ðx� x0Þ ¼ 1; x ¼ x0;Z 1
0
fðxÞ�ðx� x0Þdx
¼Z a
0
fðxÞ�ðx� x0Þdx ¼ fðx0Þ; x0 < a: ð8iÞ
By the use of Eq. (7), one can rewrite Eqs. (8a)–(8e)in terms of the displacement components.
A ðu0;xx þ w0;xw0;xxÞ þ �ðv0;xy þ w0;yw0;xyÞ�
þ ð1� �Þ2
ðu0;yy þ v0;xy þ w0;yw0;xy þ w0;xw0;yyÞ�
¼ I0ð€u0 � �€u0;xx � �€u0;yyÞ; ð9aÞ
A �ðu0;xy þw0;xw0;xyÞ þ ðv0;yy þw0;yw0;yyÞ�
þ ð1� �Þ2
ðv0;xx þ u0;xy þw0;xw0;xy þw0;yw0;xxÞ�
¼ I0ðv::0 � �v::0;xx � �v
::0;yyÞ; ð9bÞ
A Ks
ð1� �Þ2
ðw0;xx þ w0;yy þ ’x;x þ ’y;yÞ�
þ w0;xx u0;x þ1
2w2
0;x
� �þ � v0;y þ
1
2w2
0;y
� �� �
þ w0;yy v0;y þ1
2w2
0;y
� �þ � u0;x þ
1
2w2
0;x
� �� �þð1� �Þw0;xyðu0;y þ v0;x þ w0;xw0;yÞ
�þ K0 z0 � w0ðx0; y0Þ½ ��ðx� x0Þ�ðy� y0Þ
¼ I0ð €w0 � � €w0;xx � � €w0;yyÞ; ð9cÞ
D ’x;xx þ �’y;xy þ1� �
2ð’x;yy þ ’y;xyÞ
� �
� KsAð1� �Þ
2ð’x þ w0;xÞ;
¼ I2ð’::x � �’::x;xx � �’
::x;yyÞ; ð9dÞ
D ’y;yy þ �’x;xy þ1� �
2ð’y;xx þ ’x;xyÞ
� �
� KsAð1� �Þ
2ð’y þ w0;yÞ
¼ I2ð’::y � �’::y;xx � �’
::y;yyÞ: ð9eÞ
K0½w0ðx0; y0Þ � z0� ¼ M0 z::0: ð9fÞ
Equation (9c) is singular at the point (x0, y0) wherethe mass-spring system is located. However, con-sidering Eq. (8i) one can integrate Eq. (9c) on theplate domain as follows:Z a
0
Z b
0
A Ks
ð1� �Þ2
ðw0;xx þ w0;yy þ ’x;x þ ’y;yÞ��
þ w0;xx u0;x þ1
2w2
0;x
� �þ � v0;y þ
1
2w2
0;y
� �� �
þ w0;yy v0;y þ1
2w2
0;y
� �þ � u0;x þ
1
2w2
0;x
� �� �
þ ð1� �Þw0;xyðu0;y þ v0;x þ w0;xw0;yÞ#
� I0ð €w0 � � €w0;xx � � €w0;yyÞ�dxdy
¼ K0½w0ðx0; y0Þ � z0�: ð9gÞ
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This integral form will be used in the Sec. 3 forthe solution procedure. Both clamped and simplysupported boundary conditions can be considered intransverse direction while in-plane boundary con-ditions are assumed to be fully immovable.
Clamped (CCCC):
At x ¼ 0; a and y ¼ 0; b : u0 ¼ 0;
v0 ¼ 0; w0 ¼ 0; ’x ¼ 0; ’y ¼ 0:ð10aÞ
Simply Supported (SSSS):
At x ¼ 0; a : u0 ¼ 0; v0 ¼ 0; w0 ¼ 0;
’y ¼ 0; ’x;x ¼ 0;
At y ¼ 0; b : u0 ¼ 0; v0 ¼ 0; w0 ¼ 0;
’x ¼ 0; ’y;y ¼ 0:
ð10bÞ
This set of partial di®erential equations should benumerically solved as a nonlinear eigenvalue prob-lem in order to determine the frequency response ofSLGSs carrying NPs.
3. Numerical Solution Method
3.1. PS solution procedure
Spectral methods are a class of numerical techniquesin applied mathematics which have been widelyapplied to scienti¯c problems.36 The collocationversion of spectral method called the pseudo-spectral (PS) method with use of Chebyshev poly-nomials can be pro¯table in the case of classicalelasticity problems like beams, plates and shellsanalysis.37,38 The main idea in this method is toapproximate the derivative of an unknown function,F, at a collocation point by an equivalent weightedlinear sum of the function values at all collocationpoints. In 1D domains it is explained as follows:
F ðnÞ;x ðxiÞ ¼
XNk¼0
dðnÞik F ðxkÞ or fF ðnÞ
;x gðNþ1Þ�1
¼ ½DðnÞ�ðNþ1Þ�ðNþ1ÞfFgðNþ1Þ�1; ð11Þ
where (N þ 1) is the number of collocation points,F
ðnÞ;x ðxiÞ indicates nth di®erentiation of function F
in ith collocation point and ½DðnÞ� is called the nthdi®erentiation matrix whose components based onChebyshev basic functions can be found in Ref. 39.The method could be extended to 2D domains byexplaining the nth partial derivative by use of
Kronecker products as follows:
@ ðnÞF@xðnÞ
� �ðNþ1Þ 2�1
¼ ½DðnÞ � I�ðNþ1Þ 2�ðNþ1Þ 2
� fFgðNþ1Þ 2�1; ð12aÞ@ ðmÞF@yðmÞ
� �ðNþ1Þ 2�1
¼ ½I �DðmÞ�ðNþ1Þ 2�ðNþ1Þ 2
� fFgðNþ1Þ 2�1; ð12bÞ@ ðnþmÞF@xðnÞ@yðmÞ
� �ðNþ1Þ 2�1
¼ ½DðnÞ � I�ðNþ1Þ 2�ðNþ1Þ 2
� ½I �DðmÞ�ðNþ1Þ2�ðNþ1Þ 2
� fFgðNþ1Þ 2�1: ð12cÞIf A and B are two matrices of dimensions p� q
and r� s, respectively, then the Kronecker product,A�B, is the matrix of dimension pr� qs with p� qblock form,where the i, jblock is aijB. Also, Idenotesthe ðN þ 1Þ � ðN þ 1Þ identity matrix.39 Chebyshevpolynomials are orthogonal in the range of ½�1; 1�.Therefore, the rectangular real domain of nanoplateneeds to be mapped to a 2� 2 square computationaldomain by the following transformations.
�x ¼ 2x
a� 1; �y ¼ 2y
b� 1; �x; �y 2 ½�1; 1�: ð13Þ
The grid points in both �y and �y directions are selectedbased on the Gauss–Lobatto interpolation points tooptimize the distribution.36 Also, the following di-mensionless parameters are introduced to make theproblem dimensionless.
where � and �! are the factual and dimensionlessnatural frequency of the system, respectively, andwmax is the maximum vibration amplitude. For thepurpose of frequency analysis, the dimensionlessdisplacement components are considered as:
Substituting Eqs. (13)–(15) into Eqs. (9)and (10), the dimensionless nonlinear eigenvalueequations of the present system (9a)–(9f) and thedimensionless form of the integral equation (9g) andboundary conditions (10a) and (10b) can beobtained. Then, one can obtain the discrete form ofequations based on the PS method by applyingEq. (12) to the dimensionless equations.
The standard matrix form of the eigenvalueproblem of Eqs. (9) and (10) could be presented asfollows:
ð½ ~K � þ �! 2½M �Þ
f�wgf�’xgf�’ygf�ugf�vg�z
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
¼ f0g; ð16Þ
where f�ug, f�vg, f�wg, f�’xg and f�’yg are the vectors
of dimension ðN þ 1Þ2 � 1, which indicate dimen-
sionless displacement components in the grid points.It should be noticed that the dimensionless discreteform of Eq. (9c) is valid in every grid points ofcomputational domain except kth grid point asso-ciated with the point (x0, y0) where the attachedmass-spring system is located, due to the singularityof the lateral governing equation in this point.Therefore, the lateral governing equation needs tobe replaced with the dimensionless discrete form ofintegral Eq. (9g) in the way which will be explainedin Sec. 3.2.
3.2. Integral quadrature procedure
In integral quadrature (IQ) method, the main idea isto evaluate the integration of an arbitrary function,H, on a domain by a weighted linear sum of thefunction values, Hi, at all grid points of the do-main.40 The IQ method for the present 2D compu-tational domain can be written as:Z þ1
�1
Z þ1
�1
Hð�x; �yÞd�xd�y ¼XðNþ1Þ 2
i¼1
liHi
¼ ½L�1�ðNþ1Þ 2fHgðNþ1Þ2�1:
ð17ÞFor applying the method, it is necessary to deter-mine the associated weighting coe±cients, li. Itcan be simply performed by introducing a set of
ðN þ 1Þ2 polynomial test functions as follows40:
Ht ¼ xmyn; m;n ¼ 0; . . . ;N : ð18ÞAs the values of these polynomials are known in thegrid points and the values of their integrals on thedomain can be easily computed, the weightingcoe±cients matrix, [L], will be evaluated through aninverse problem. Here, the IQ method is imple-mented to evaluate the discrete form of integralgoverning equation, Eq. (9g).
Now, the singular lateral governing equation inkth grid point in Eq. (16) can be replaced with IQdiscrete representation of integral equation usingthe following matrix form:
½I � ð½ ~K � þ �! 2½M �Þ
f�wgf�’xgf�’ygf�ugf�vg�z
8>>>>>>>>>>><>>>>>>>>>>>:
9>>>>>>>>>>>=>>>>>>>>>>>;
2666666666664
3777777777775¼ ½K �
f�wgf�’xgf�’ygf�ugf�vg�z
8>>>>>>>>>>><>>>>>>>>>>>:
9>>>>>>>>>>>=>>>>>>>>>>>;
;
ð19aÞwhere
I ij ¼1 i ¼ j; i 6¼ k
lj i ¼ k
0 otherwise
:
8<: ð19bÞ
Components of ½K � are equal to zero except thantwo components in kth row which contains termsfrom the right hand of Eq. (9g). Due to the simpleform of matrices ½I � and ½K �, Eq. (19a) can be re-written in the following form:
To establish the standard eigenvalue form of theproblem, the displacement vectors can be divided to
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the boundary and the domain parts as follows.
�wb
�’xb
�’yb
�ub�vb
8>>>><>>>>:
9>>>>=>>>>;
¼ fbg;
�wd
�’xd
�’yd
�ud
�vd�z
8>>>>>><>>>>>>:
9>>>>>>=>>>>>>;
¼ fdg; ð21Þ
where the subscripts b and d indicate boundary anddomain, respectively. Then, the resulting eigenvalueof equations can be written in the matrix form as:
where ½ �M � is the total mass matrix and ½ �K � is thetotal sti®ness matrix which contains the linear andnonlinear sti®ness terms and therefore is a functionof displacements.
The nonlinear frequency response of SLGSs withan attached mass-spring system will be iterativelydetermined with the following stepwise algorithm:At the ¯rst, the nonlinear terms in sti®ness matrix,½ �K � are neglected and the linear eigenvalue problemis solved to obtain the linear frequencies and asso-ciated mode shapes. Then, the mode shapes arescaled up to a given vibration amplitude, �wmax andare used to calculate nonlinear coe±cients. Afterthat, the updated eigenvalue problem is solved todetermine the nonlinear frequencies and modeshapes. The iteration continues until the nonlineareigenvalues converges with a desired accuracy.41
4. MD Simulation of SLGSs Resonant
Sensors
In nanoscale computations, theMDmethod is widelyused for simulating the interatomic interactions. The
method determines the trajectories of a group ofinteracting particles by solving the Newton'sequations of motion for the system, where interac-tions are de¯ned by potential functions. In thepresent study, MD simulation has been employedto investigate the vibration characteristics ofSLGSs carrying ultra-¯ne metallic NPs by using theopen source well-known software i.e., large-scaleatomic/molecular massively parallel simulator(LAMMPS) through a velocity-Verlet algorithmwith a time step of 0.5–1 fs to integrate the Newton'sequations of motion. Three types of interactionsexist in the system: carbon–carbon interactionsinside the SLGS, metal–metal interactions insideNPs and carbon–metal interactions which remainNPs on the surface of the SLGS. For the ¯rst one,carbon–carbon, the adaptive intermolecular reac-tive empirical bond order (AIREBO) potentialof Stuart et al.,42 which is addressed by manyresearchers for SLGSs, is applied. It contains thefollowing terms:
EAIREBO ¼ 1
2
Xi
Xj 6¼i
EREBOij þELJ
ij
"
þXk6¼i;j
Xl 6¼i;j;k
ETORSIONkijl
#; ð24Þ
EREBO is the famous hydrocarbon REBO po-tential,43 which represents short ranged C–C cova-lent interactions, while ELJ and ETORSION describethe standard Lennard–Jones (L–J) and four-bodytorsion potentials, respectively. To simulate metal–metal interactions, the embedded-atom method(EAM) potential of Daw and Baskes44 is applied.The metal–carbon interactions are assumed as anon-bonded van der Waals atomic force which isconsidered using the standard pairwise 6–12 L–Jpotential as follows:
ELJ ¼ 4�eff��eff
r
12� ��eff
r
6� �
; r < rc; ð25Þ
where �eff and ��eff , are respectively the e®ectivecoe±cients of well-depth energy and the equilibriumdistance, r is the interatomic distance and rc is thecut-o® radius. These e®ective coe±cients for car-bon–metal interactions can be approximated usingthe Lorentz–Berthelot mixing rules45:
�eff ¼ffiffiffiffiffiffiffiffiffiffi�m�c
p; ��eff ¼ ð��mþ��cÞ=2: ð26Þ
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Subscripts m and c indicate metal and carbon L–Jcoe±cients which are tabulated in Table 1 forvarious types of metal element.45,46
The MD simulations are performed based on thefollowing steps: At ¯rst, a minimization process isdone to locate the metallic NPs in an equilibriumdistance next to the surface of SLGS. Then, thesystem is allowed to be freely relaxed in a desiredtemperature for a proper time period by applyingthe Nose–Hoover thermostat.47 Boundary condi-tions are implemented by ¯xing one layer of carbonatoms on the edges of SLGS. It should be mentionedthat these boundaries determine the active area ofSLGS in vibrating motion and the length and thewidth of SLGS are speci¯ed based on these bound-aries. It means if in practical situation for strongersupporting, there are more ¯xed layers of carbonatoms in SLGS edges, they will not be considered asa part of vibrating SLGS. The vibrational motion ofthe SLGS sensor can be stimulated by applying aninitial displacement associated to its approximatefundamental mode shape. In order to avoid thee®ects of the thermostat during free vibration sim-ulation, a constant total energy ensemble (NVE) isapplied and the graphene sheet with attached me-tallic nanoparticle is allowed to vibrate freely. Asthe whole the system is vibrating with its funda-mental frequency, during the vibration motion allatoms of the system vibrate with an identical fre-quency and therefore the natural frequency can becaptured from the lateral position trajectory of anyindividual sample atom or a group of atoms byimplementing the fast Fourier transform (FFT)method. Here, we use the trajectory of the centralcarbon atom of the graphene sheet during a su±-cient time period.
5. Results and Discussion
In the following, results are presented to investigatethe potential application of SLGSs as resonant
sensors in detection of ultra-¯ne attached metallicNPs. At ¯rst, validation and convergence study ofnonlocal results is performed. Then, it is useful todraw attention to some characteristics of metallicNPs like average radius and e®ective spring con-stant. After these evaluations, the in°uence ofnonlinearity and nonlocality on frequency shift andsensitivity of the sensor is discussed in detail.Also, nonlocal continuum model and MD simulationresults are matched to calibrate the nonlocalparameter. In all forthcoming nonlocal continuumresults, the mechanical properties of SLGSs aretaken from Ref. 48 as: Young's modulus E ¼1:06TPa, Poisson's ratio � ¼ 0:16, density � ¼2250 kg/m3 and e®ective thickness h ¼ 0:34 nm.Observing MD simulations, carbon atoms next tothe boundaries of SLGSs can easily move in trans-verse direction and the slope of deformed SLGSsnext to the boundaries is considerable duringvibration. Accordingly, the associated boundaryconditions in the presented nonlocal results arechosen simply supported conditions, in spite ofpossibility of considering both clamped and simplysupported boundaries in the solution method. Also,since the nonlocal continuum model has beenestablished for the free vibration of a perfect °atnanoplate, in order to avoid thermal °uctuationsand for the possibility of comparison, all MDsimulations are performed at low temperature con-ditions, i.e., 1K.
5.1. Validation and convergence studyof nonlocal model
The validation and convergence study of nonlocalresults are done to be sure about the reliability andaccuracy of mathematical model and the numericalsolution procedure. Fundamental linear frequencyof a bare SLGS without any attached nanoparticlefor various values of small-scale parameter, lengthsize and aspect ratio is presented in Table 2.
Table 1. Properties of carbon and metallic elements.44,45
Atom type Atomic mass (g/mole) Lattice constant (Å) �"=kBðKÞa �� (Å)
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Excellent agreement with the exact results of non-local FSDT plate model in Ref. 49 con¯rms the ef-¯ciency of the present method. Also, an admissibleconvergence is obtained with N ¼ 8 grid points.Hence, this number is used for all the forthcomingresults.
In°uence of attached mass-spring system on fre-quency response of the plate is presented in Table 3and results are compared with exact results reported
in Ref. 50. For possibility of comparison, small-scaleparameter, ��, is considered equal to zero andthickness to length size ratio is set to � ¼ 0:001, asthe results in Ref. 50 are presented for a thin macroplate. The desirable match between present andexact results proves the accuracy and e±ciency ofcombined PS and IQ approach in prediction of vi-brational characteristic of systems with concen-trated objects.
Table 2. Fundamental frequencies (THz) of SLGSs without any attached NPs.
� a (nm) e0a0 (nm) N ¼ 5 N ¼ 8 N ¼ 10 Exact49 Error %
As the detection criterion of resonant sensors isestablished based on measuring mass of attachedNPs using frequency shift, presented conclusionswill be generally valid for any type of attachedparticle. However, in the present work, three typesof metallic NPs consist of copper (Cu) or silver (Ag)or gold (Au) elements are considered as the casestudy, due to their wide applications in nanotech-nology. The properties of these NPs are listed inTable 1. In order to ¯nd a relationship betweennumber of consisting atoms of a nanoparticle, Np,and the radius of nanoparticle, R, a sphere of theradius R is assumed randomly inside a face-centeredcubic (fcc) lattice of metal, as shown in Fig. 2(a). Itis obvious that the lattice constant, Lc, is the maine®ective parameter in this evaluation. Number ofconsisting atoms with respect to the dimensionlessparameter, ðR=LcÞ, is illustrated in Fig. 2(b) forvarious types of NPs. As the value of lattice con-stant for both Au and Ag NPs is almost equal, theyhave same number of consisting atoms for a certain
value of R. Applying a ¯tting procedure, the num-ber of consisting atoms and the radius of nanopar-ticle can be approximately related based on thefollowing cubic equation:
Np ffi 17R
Lc
� �3
: ð27Þ
Iron (Fe) nanoparticle with a constant lattice ofLc ¼ 2:866Å is also presented for more evaluation ofEq. (27). An acceptable match between predictedNp using Eq. (27) and the values obtained from MDsimulations is observed. As the mass of NPs can beexplained using atomic mass, ma, as M0 ¼ Npma, itis easy to relate the frequency shift of resonantsensor to the size of nanoparticle by use of Eq. (27).
In MD simulations, interactions between metallicNPs and SLGSs are considered using nonbondedL–J potentials. On the other hand, in continuumview, these interactions are represented by a linearspring which connects the nanoparticle to thesurface of the SLGS. Viewing snapshots of MDsimulations reveals that during vibrational motion,attached nanoparticle remains in an almost con-stant equilibrium distance next to the SLGS surface.It can be concluded that an equivalent spring con-stant near this equilibrium distance may beobtained to be used in nonlocal continuum model.To determine an approximation for this equivalentspring constant, an MD simulation procedure isperformed as following: The nanoparticle is locatednext to the surface of SLGS and a minimizationprocess is done to place it near the equilibrium dis-tance. Then, the nanoparticle and the SLGS areallowed to be relaxed during a long enough timeperiod. In this stage, a small displacement �x is ap-plied to the nanoparticle in both directions alongz-axis and the sum of associated interacting forces tothe nanoparticle in z-axis direction, Fzð�xÞ, isobtained [see Fig. 3(a)]. Then, the equivalent springconstant can be approximated as follows:
K0 ¼jFzð�xÞ � Fzð��xÞj
2�x: ð28Þ
Results reveal that as the small displacementincreases slightly, relationships between interactingforce and displacement is almost linear that con-¯rms the validity of Eq. (28) at the neighborhood ofequilibrium state. It is clear that the spring constantis related to the number of consisting atom of NPs,Np, asK0ðNpÞ. Since the atoms facing directly to the
R
(a)
0
100
200
300
400
500
Num
bero
fAto
ms,
Np
FeCuAg, Au17 (R/
R/Lc
u/Lc)^3
00 0.5 1 1.5 2 2.5 3
(b)
Fig. 2. Estimation of the nanoparticle radius using number ofconsisting atoms.
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surface of SLGS are more active during interactionin comparison with those on the top of nanoparticle,the spring constant may be proportional to thenumber of active atoms, N act
p . Two limiting casesare envisioned: all atoms within the particle volumeor only those on the projected area of the nano-particle active. This projected area is proportionalto R2, and R is related to the number of consistingatoms based on Eq. (27) with a power of 1/3.Therefore, it could be concluded that spring con-stant is proportional to the number of consisting
the dimensionless spring constant K0ðNpÞ=K0ð1Þwith respect to the number of consisting atoms, Np,for Au, Ag and Cu NPs. It is found by best ¯ttingthat the spring constant is proportional to Np raisedto the power of about 3/4, in agreement with thepredicted range of the scalling, thus:
K0ðNpÞ=K0ð1Þ ffi N 3=4p : ð29Þ
5.3. Frequency shift and sensitivityof SLGS sensors
The capability of an SLGS as a resonant sensor isrelated to its frequency shift, �f, due to changes in
the value of attached mass, M0. Frequency shift isde¯ned as the di®erence between fundamental fre-quency of an SLGS with attached NPs, f, and fun-damental frequency of a bare SLGS, f0, and relativedimensionless frequency shift is indicated asð�f=f0Þ ¼ 1� ðf=f0Þ. The resonant sensor mayhave the potential to operate in two general ways:The ¯rst one is to measure the mass value of onespeci¯ed object, here a metallic nanoparticle, whichis located at a certain position of SLGS surface. Inthe following, this situation is called \concentrated"case. The second way is to detect the molar con-centration of a group of distributed objects, heredistributed metallic NPs with various sizes, bymeasuring the total mass of them. In particular, theNPs are distributed on the surface of SLGS ran-domly. In the coming results this case is called\distributed" case.
In continuum model, distributed case may beconsidered by modifying density of SLGS as � ¼�þ�� where �� ¼ M0=ðabhÞ. This assumptionresults in perfect dispersing of attached mass on theSLGS surface. Since the resonant frequency of anSLGS is proportional to
ffiffiffiffiffiffiffiffi1=�
p, the dimensionless
frequency shift for distributed case can be calculatedas follows:
�f
f0¼ 1� 1ffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ �mp : ð30Þ
Figure 4 illustrates the variation of dimensionlessfrequency shift with respect to dimensionless mass,�m ¼ ��=�, for di®erent values of dimensionlessvibration amplitude and nonlocal parameter.The predicted shift by Eq. (30) is plotted aswell. Although both nonlinearity and nonlocality
Fig. 4. Dimensionless frequency shift versus dimensionlessmass for distributed case.
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signi¯cantly changes the resonant frequency, it isfound that the dimensionless frequency shift is in-dependent of these e®ects and matches well to theplot of Eq. (30) which proposes that this simpleequation can be used precisely for distributedcase with any combination of nonlinearity andnonlocality.
Concentrated case is indisputably more compli-cated than distributed one as the simple propor-tional ratio between resonant frequency and totalmass of the system cannot be used due to the pointload nature of the attached mass. The position ofconcentrated nanoparticle on the surface of SLGS isalso an e®ective issue. Figure 5 gives the in°uence ofthe nanoparticle position on linear dimensionlessfrequency shift of an SLGS with �� ¼ 0:01 carrying ananoparticle of the mass, �m ¼ 0:5. Although theabsolute maximum frequency shift occurs when thenanoparticle is exactly located at the center ofSLGS, for a central area equal to 25% of the totalsensing surface, the frequency shifts are at least 70%of the absolute maximum value; frequency shiftsdecrease dramatically by approaching the bound-aries. From a practical point of view, it is di±cult toplace the NPs exactly at the center; however, forconcentrated case the adsorbing mechanism shouldbe optimized to locate NPs as near as possible to thecenter. Both the proposed continuum and MDapproaches has the ability to take into account thee®ect of position on frequency shifts. However, herefor the concentrated case all the results are pre-sented for NPs located at the center to show themaximum performance of SLGS sensor as the upperlimit of operation.
Figure 6 illustrates the dimensionless frequencyshift versus dimensionless mass, attached at thecenter of SLGS, for various values of vibrationamplitude and nonlocal parameter. Results revealthat increasing �wmax as well as decreasing �� cause anincrease in dimensionless frequency shift. To o®er adeeper insight to the relationship between theseparameters, a general equation is proposed and is¯tted to numerical result during a best ¯tting pro-cess as follows:
The assumed general form of Eq. (31) is takenfrom the equation presented in Ref. 16 for lineardimensionless frequency shift of cantilever SLGSresonant sensors. Also, for adding nonlinear e®ects,an increasing factor is assumed since the resultsshow that nonlinearity increases dimensionlessfrequency shift proportional to the square of vi-bration amplitude that shows the well-knownhardening behavior. Figure 6(a) compares numer-ical results with plots of Eq. (31) for di®erentvalues of nonlocal parameter while nonlinear e®ectsare neglected and Fig. 6(b) presents also the non-linear e®ects. It is seen that the plots of Eq. (31) ¯twell with obtained results. It is noted that whenthe nonlocal e®ects is neglected, Eq. (31) becomesEq. (30) with an enhancing factor of 4.4 formass, that reveals that accumulating the attachedmass at the center of SLGS increases the e®ectivemass 4.4 times in comparison with the case where
Fig. 5. In°uence of nanoparticle position on dimensionless frequency shifts. Results are plotted on the dimensionless computationaldomain: �x; �y 2 ½�1; 1�.
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mass is uniformly distributed on the surface ofthe sensor.
Dimensionless sensitivity, S, is de¯ned as thepartial derivative of dimensionless frequency shiftwith respect to dimensionless mass, @ð�f=f0Þ=@ð �mÞ.In the other words, sensitivity is equal to the slope ofð�f=f0Þ � �m curves and therefore is a function ofdimensionless mass. It can be easily obtained fromEqs. (30) and (31) for distributed and concentratedcases, respectively:
Figure 7 demonstrates variation of dimensionlesssensitivity versus dimensionless mass for variouscombination of nonlinearity and nonlocality. Ingeneral, sensitivity increases when the dimensionlessmass decreases and the maximum sensitivity isachieved when the mass tends to zero and for thevalues of dimensionless mass greater than one,�m > 1, sensitivity reduces dramatically and thee®ects of nonlinearity, nonlocality and distributionis almost ignored. It means SLGSs are not muchsensitive to changes of the value of attached masswhen they are detecting NPs with the masses in theorder of the mass of SLGS. Also, it is shown thatdistributed case is less sensitive than concentrated
one and the maximum value of sensitivity for dis-tributed case is obtained equal to 0.5 when �m ! 0.For concentrated case, one can observe that in-creasing vibration amplitude as well as decreasingnonlocal parameter increases the sensitivity of theSLGS resonant sensor.
5.4. MD results and calibrationof nonlocal parameter
In what follows, frequency shifts and sensitivity ofSLGS resonant sensors will be presented and resultsfrom nonlocal continuum and MD simulations willbe compared to calibrate the nonlocal small-scaleparameter. An armchair SLGS of dimensions of6:96nm� 3:64 nm (a is the longer side for evalua-tion of ��) consisting 1044 carbon atoms with total
Fig. 7. Dimensionless sensitivity versus dimensionless mass fordistributed and concentrated cases with di®erent combinationsof nonlinearity and nonlocality.
Fig. 6. In°uence of (a) nonlocality and (b) nonlinearity on dimensionless frequency shift for concentrated case.
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area of 25.33 nm2 and aspect ratio of � ¼ 1:91 isselected for simulating a resonant sensor to detectultra-¯ne metallic NPs (see Fig. 8). Observing MDresults reveals that the frequency response is related
to vibration amplitude and hardening behavior alsooccurs in the molecular model. Hence, before mak-ing any comparisons, it is indispensable to choose aproper value of vibration amplitude. From MDsimulations point of view, the vibration amplitudeneeds to be large enough not to be disturbed by themovements of SLGS and attached NPs duringsimulations. In order to avoid these undesirablee®ects and to obtain the frequency response accu-rately, the value of vibration amplitude in all MDsimulations is selected equal to wmax ¼ 3Å which isassociated with nondimensional vibration ampli-tude �wmax ¼ 0:88 in nonlocal continuum model.Figure 9(a) demonstrates frequency spectrums plotsobtained from FFT for MD vibration analysis of theSLGS resonant sensor carrying various numbers ofCu atoms. The picks in frequency spectrum graphsshow the frequency of vibrating motion of the sensorwith connected NPs. As it is expected, increasingthe total mass of connected NPs decreases the fre-quency of the sensor. Also, Fig. 9(b) depicts thevibration motion of the central carbon atom of theSLGSs with respect to time. It is seen that the SLGScarrying 20Cu atoms vibrate slower than the baresensor. One can observe the variation of dimen-sionless frequency shift with respect to the dimen-sionless mass for concentrated case in Fig. 10. MDsimulation results are illustrated for the three typesof metallic NPs with various masses. The frequencyshifts obtained from nonlocal continuum model with�wmax ¼ 0:88 are also plotted to be compared withMD ones. In continuum results, the spring constantassociated with a copper nanoparticle with the same
Fig. 10. Comparison between MD simulation and continuumresults of concentrated case for calibrating nonlocal parameter.
3.5 Å
0 Å
Fig. 8. MD model of the SLGS resonant sensors.
0.1 0.30.15 0.2 0.25 0.35 0.4
Frequency (THz)
Np=0Np=5Np=15Np=20
(a)
-4
-3
-2
-1
0
1
2
3
4
0 5 10 15
Ampl
itude
(Å)
Time (ps)
Np=0 Np=20
(b)
Fig. 9. (a) Frequency spectrum plots and (b) vibration motionversus time for the resonant SLGS sensor carrying Cu atoms asthe attached NPs.
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mass is used. It is seen that classical continuumelasticity with �� ¼ 0 overestimates the frequencyshift, while the nonlocal model with calibratednonlocal parameter equals �� ¼ 0:0075ðe0a0 ¼ 0:6nmÞ can ¯t MD results accurately. In order to clarifysome misunderstandings in calibration of small-scaleparameter, linear nonlocal results with �wmax ¼ 0 arealso demonstrated in Fig. 10. Because of neglectingthe hardening e®ect, the frequency shifts predictedbased on linear vibration analysis are signi¯cantlylower than nonlinear ones with same nonlocal pa-rameter. At the ¯rst glance, one may conclude thatclassical linear model with �� ¼ 0 and �wmax ¼ 0makes a good match to MD frequency shifts; how-ever, comparing the linear analysis, which neglectsconsiderable slops in strains, with MD simulationresults of large amplitude vibration is fundamen-tally wrong. In fact, in linear classical (local) con-tinuum analysis, the underestimation of frequencyshift because of neglecting the hardening e®ect iscountervailed by overestimating frequency shift dueto neglecting the nonlocal e®ects and the resultantpredicted frequency shift gets close to MD simula-tion results by chance. Therefore, neglecting non-linear e®ects may make the wrong conclusion thatnonlocal e®ects are not signi¯cant in prediction offrequency shift. Also, the frequency shifts calculatedby Eq. (31) with �wmax ¼ 0:88 and �� ¼ 0:0075 arealso plotted as ¯tted with MD results accurately.
Finally comparison between MD and nonlocalresults are done for distributed case. Figure 11(a)illustrates the molecular model for a random dis-tribution of metallic NPs with various sizes and atotal number of consisting atoms, Np, adsorbed onthe surface of SLGS sensor. Figure 11(b) demon-strates MD dimensionless frequency shift for thethree types metallic NPs with total dimensionlessmass in the range, 0–2. The dimensionless frequencyfor distributed case obtained from both nonlocalmodel with �wmax ¼ 0:88 and �� ¼ 0:0075 andEq. (31) are also presented. It is observed thatMD results are close to continuum results especiallywhen the dimensionless mass increases. The reasonis that since a maximum size is assumed for ultra-¯ne attached nanoparticle, increasing the total massresults in more uniform distribution of NPs. Itcan be concluded that continuum model canpredict frequency shifts with an acceptable accura-cy. For distributed case, it is shown in the previoussection that, although resonant frequency is signif-icantly a®ected by nonlinearity and nonlocality,
dimensionless frequency and sensitivity are almostindependent from these e®ects and can be estimatedsimply by the use of Eqs. (30) and (32a).
6. Conclusion
In the present study, the potential application ofSLGSs as nanoresonant mass sensors in detection ofultra-¯ne metallic NPs is investigated by using bothMD and nonlocal elasticity approaches. A combina-tion of PS and IQ methods is implemented to nu-merically determine frequency shift and sensitivity ofthe sensor. The conclusions are listed as follows.
. Vibration amplitude has a signi¯cant e®ect onpredicted frequency shift of SLGS sensors in bothMD simulation and nonlocal continuum approa-ches and neglecting this e®ect may result in wrongcalibrated nonlocal small-scale parameter.
. Increasing the small-scale parameter causes adecrease in both frequency and frequency shift. Itmeans neglecting nonlocal small-scale parameterresults in overestimating the frequency, frequency
Fig. 11. Comparison between MD simulation and continuumresults for the distributed case.
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shift and sensitivity of SLGS resonant sensor. Aproper value of small-scale parameter can befound that matches the nonlinear nonlocal fre-quency shifts with those obtained from MDsimulations.
. Distribution of metallic NPs on the surface ofSLGS resonant sensors has a signi¯cant e®ect asthe relative frequency shift and sensitivity de-crease when NPs are dispersed more uniformly onthe surface of SLGS rather than they are accu-mulated near the center of SLGS. However, thein°uence of nonlinearity and nonlocality on rela-tive frequency shift decrease when NPs are moredispersed on the surface.
Acknowledgments
NMP is supported by the European ResearchCouncil (ERC StG Ideas 2011 BIHSNAM on \Bio-Inspired hierarchical super-nanomaterials", ERCPoC 2013-1 REPLICA2 on \Large-area replicationof biological anti-adhesive nanosurfaces", ERCPoC2013-2 KNOTOUGH on \Super-tough knotted¯bres") by the European Commission under theGraphene Flagship and by the Provincia Autonomadi Trento (Graphene Nanocomposites).
References
1. L. Ding, A. M. Bond, J. Zhai and J. Zhang, Anal.Chim. Acta 797, 1 (2013).
2. J. P. M. Almeida, E. R. Figueroa and R. A. Drezek,Nanomed-Nanotechnol. 10, 503 (2014).
3. F. Iskandar, Adv. Powder Technol. 20, 283 (2009).4. L.-T. Yan and X.-M. Xie, Prog. Polym. Sci. 38, 369
(2013).5. D. B. Warheit, C. M. Sayes, K. L. Reed and K. A.
Swain, Pharmacol. Therapeut. 120, 35 (2008).6. H.-R. Paur et al., J. Aerosol Sci. 42, 668 (2011).7. D. Cohen et al., Toxicol. in Vitro 27, 292 (2013).8. M. D. Angione et al., Mater. Today 14, 424 (2014).9. N. Pugno, Proc. Inst. Mech. Eng., Part N,
J. Nanoeng. Nanosyst. 219, 29 (2005).10. C. H. Ke, N. Pugno, B. Peng and H. D. Espinosa,
J. Mech. Phys. Solids 53, 1314 (2005).11. H. F. Zhan and Y. T. Gu, J. Appl. Phys. 111, 124303
(2012).12. H. F. Zhan, Y. T. Gu and H. S. Park, Nanoscale 4,
6779 (2012).13. Q. Wang and B. Arash, Comput. Mater. Sci. 82, 350
(2014).
14. Z.-B. Shen, H.-L. Tang, D.-K. Li and G.-J. Tang,Comput. Mater. Sci. 61, 200 (2012).
