2.1. Cantilever The Hooke's law

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Clement Lester
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1 .1. Cantiever.1.1 The Hooke's aw The cantiever is the most common sensor of the force interaction in atomic force microscopy. The atomic force microscope acquires any information about a surface because of the cantiever beam mechanica defections which are detected by an optica system. In noncontact microscopy, resonators of the tuning fork-type are frequenty used. Such sensors require tracking of the resonance frequency shift upon the probe-surface interaction onset. Normay, cantiever is a beam in the form of a rectanguar paraepiped (Fig. 1a) having ength, thickness t (t ) and width w ( w ) or in the form of two beams connected at an ange (Fig. 1b) having a with ength at its free end. Let us examine beow the rectanguar cantiever in detai. Its characteristic dimensions are shown in Fig. 1a. The probe's interacts with the surface. Assume that the point force acting from the sampe is appied to the 's apex. The force acting on the probe has sometimes not ony vertica but aso horionta components. Therefore, the cantiever can defect not ony aong the O -axis but in two other directions: Ox and Oy (see Fig. 1a). Let's ca the force vertica component F the norma force and ongitudina F and transverse F components the atera forces. x y Because in AFM the -sampe interaction infuences the cantiever deformation, to determine the force one shoud know the cantiever deformation stiffness in various directions. Consider that the defection vector (having components x, y, ) is ineary dependent on the appied force F in accordance with the Hooke's aw [1]: 1 = C F. (1) The «constant» of proportionaity is the second rank tensor C which we ca the inverse stiffness tensor. It contains a the information about eastic properties of the cantiever. To find the components of the tensor C it is necessary to sove the probem of the cantiever static deformation under the infuence of forces directed aong different axes. For the sake of carity, we write the formua (1) as a matrix expression: x cxx cxy cx Fx y = c yx c yy c y Fy. () cx cy c F Notice that the optica system detects not defection but incination of the cantiever top surface near its free end. Two anges are measured: defection of the norma from vertica in the Oy pane (ange α ) and in the orthogona direction in the pane Ox (ange β ). Instead of (), we can write for the mathematica convenience the matrix expression reating anges α and β directy with force F components.. b b b F x α αx αy α = Fy β bβx bβy b β F () 1

The introduced matrix, however, does not contain fu information about the cantiever eastic properties in contrast to tensor C. Fig. 1a. Rectanguar cantiever with a. Fig. 1b.

2 The introduced matrix, however, does not contain fu information about the cantiever eastic properties in contrast to tensor C. Fig. 1a. Rectanguar cantiever with a. Fig. 1b. V-shaped cantiever with a Summary: 1. The information about a sampe in AFM can be obtained ony from the cantiever deformation. The optica system aows to measure two anges defining the cantiever top pane incination.. To determine the force acting on the cantiever one shoud know its eastic properties which are described by the second rank tensor of the cantiever inverse stiffness.. The deformation-force reation is modeed by the inear Hooke's aw written as the tensor expression.

3 .1. Defections under the vertica (norma) force component ( F ) Let us determine the magnitude and direction of the deformation arising from the vertica force F. Soution to this probem wi aow to find components of the third coumn of tensor C (). x = c F, (4) x y y = c F, (5) = cf. (6) Deformation that we ca here the vertica bending is shown in Fig.. Fig.. Vertica defection of the -type Next, et's examine a section of the beam. We wi cut the beam and consider the deformation of the beam eement having ength L between the two cross sections (Fig. ). Since this eement is bent, the materia at the outer edge is stretched in tension whie at the inner edge it is compressed. Hence, there is a neutra pane of ero stress between the two surfaces. For cacuations simpification we assume that the beam cross-sections remain panar and norma to their centroida axis (pure bending of the uniform cross-section beam). This assumption is vaid if t 8 [] which is true in our case. Fig. a. Section of the bent beam Fig. b. Beam cross-section

4 At the pure bending the neutra pane passes through the centroid of the beam cross-sectiona area [], i.e. in our case the paraepiped ong axis beongs to the neutra pane. The materia ongitudina extension L is proportiona to the distance from the neutra pane: LL= R(see Fig. ). According to the Hooke's aw, the force acting on a unit area in a sma strip near with square ds is equa to df = EdS R where E Young's moduus, R beam curvature radius. If any cross-section is considered, the forces are acting in one direction over the neutra surface and in the other direction beow it. This makes a coupe of forces producing the bending moment M which is a moment of forces with respect to the neutra axis: E. (7) M = df = J R S The quantity J is caed the axia moment of inertia of the beam section about the axis that passes through its centroid. For the beam with rectanguar cross-section it is given by: ( ) J wt. (8) 1 = ds = S By u y we denote the defection of the beam point at the distance y from the fixed end in the -direction. The curvature of the u( y ) curve at sma bends ( du dy 1) is given by 1 R ( y) = d u d y. Then, taking into account expression (8), the bending moment M can be expressed as On the other hand, d u M ( y) = EJ. (9) dy M is a moment of forces with respect to point y due to the action of mg mg force F ( M F = F ( y) ) and the beam own weight ( M ( mg = pdp y = ( y ) ). Thus, du F mg = y y ). (10) ( ) dy EJ EJ du Integration of (10) having for boundary conditions u y=0 = 0 and y=0 = 0, gives: dy F mg uy ( ) = ( yy ) ( 6 y y 6EJ 4EJ ). (11) The beam end defection is (Fig. ): F 5mg = u y= =. (1) EJ 4EJ The second summand is the defection under own weight. For a typica cantiever, it is of the fraction of the angstrom and can be negected because in AFM experiments the first term is hundreds of time more. Reation (1) is nothing but expression (6) in which we shoud suppose: 4

5 c = = c. (1) EJ The beam defection ange cacuated without the second term in (1) is as foows: du F α tg α = y= = = = cf. (14) dy EJ The coefficient of inverse stiffness c is the argest among the tensor C components. In (1) this parameter is speciay denoted as " c " without indexes. In particuar, magnitude of 1 c characteries the cantiever stiffness and is one of its major parameters. Beow, for the purpose of obviousness, we wi take c outside as a common muier of a the matrix () (section.1.1) components. For a cantiever with rectanguar cross-section, (1) can be rewritten as 4 c =. (15) Ewt From formua (14) and diagram for the beam vertica bending of -type (Fig. ) it is easy to derive the defection y induced by the force F appication: y = α =. (16) From (16) and (4)-(6) it is cear that c y = c. (17) Taking into consideration that x = 0, we get c x = 0. (18) Finay, we cacuate the components of the matrix () third coumn. From expressions (1)- (15) it foows that bα = c. (19) Because under the infuence of the force F the top cantiever surface does not bend in the Ox direction, then Summary: b β = 0. (0) 1. The -type defection is a resut of the vertica bending force action.. To find the components of the inverse stiffness tensor corresponding to the -type defection, one shoud sove the probem of the beam static defection which is reduced to the ordinary differentia equation of the second order.. The vertica force resuts in the defection in vertica and ongitudina directions and in the defection ange α = F c EJ = F appearance. 4. Besides the supporting force from the sampe, the cantiever is infuenced in vertica direction by its own gravity. Under this oad the cantiever free end is defected but such a deformation is sma compared to minima detected dispacement. 5

6 .1. Defections under the ongitudina force ( F y ) In this section we determine the magnitude and direction of the deformation produced by the axia force F y. Soution to this probem wi give the midde coumn () of tensor C. x = c F, (1) The force Fy xy yy y y = c F, () y y = c F. () acting in the cantiever axis direction produces moment in deformation caed here the vertica bending of y-type (Fig. 4). y M = F that resuts y Fig. 4. Vertica defection of the y-type In spite of the forma resembance to vertica bending of -type (see section.1.), the deformation profie in this case is quite different. The equation describing the y-type bending reads du F y =. (4) dy EJ du Boundary conditions remain the same: u y=0 = 0 and y=0 = 0. For the soution we find: dy F y uy ( ) = y. (5) EJ Thus, the vertica defection due to this type of deformation is as foows: F y = u() = cfy EJ =. (6) Comparing (6) and () and taking into account the expression for the common muier c (1), we get: cy = = c. (7) EJ The ange of the beam end defection α is given by the foowing formua: 6

7 du F y α = y= = = = cf y. (8) dy EJ From formua (8) and diagram for the beam vertica bending of y-type (Fig. 4) it is easy to derive the defection y induced by the force appication: F y y = α =. (9) From (), (7) and (9) it is easy to obtain: Taking into account that x = c 0, we get: = cy = c. (0) yy c = 0. (1) xy Finay, we cacuate the components of the matrix () third coumn. From expressions (5)- (7) it foows that Because under the infuence of the force direction, then Summary: b αy Fy = c. () the top cantiever surface does not bend in the Ox b βy = 0. () 1. The y-type defection is a resut of the axia bending force action.. To find the components of the inverse stiffness tensor corresponding to the y-type defection, one shoud sove the probem of the beam static defection which is reduced to the ordinary differentia equation of the second order.. The axia force resuts in the defection not ony in the ongitudina but aso in vertica direction = cf y and in the defection ange α = Fy = cf y appearance. EJ 7

.1.4 Defections under the transverse force ( F x ) In this section we determine the magnitude and direction of the deformation produced by the transverse force.

(6) As a resut of the transverse force action, the compicated deformation is induced which is a superposition of simpe bending and twisting (Fig. 5a and 5b)

8 .1.4 Defections under the transverse force ( F x ) In this section we determine the magnitude and direction of the deformation produced by the transverse force. Soution to this probem wi give the first coumn of tensor C () components. F y x = c F, (4) xx yx x y = c F, (5) = cxfx. (6) As a resut of the transverse force action, the compicated deformation is induced which is a superposition of simpe bending and twisting (Fig. 5a and 5b). x Fig. 5a. Simpe bending Fig. 5b. Torsion It is easy to obtain the inverse stiffness of simpe bending (Fig.5a). This deformation is anaogous to the vertica bending of -type (Fig. in section.1.) with the ony difference that in fina expression for the inverse stiffness (15) we must interchange the beam width with its thickness ( w t): 4 t = = c. (7) c bend Ew t w The soution to the probem of the rectanguar beam torsion is much more compicated. Therefore, we give the formua reating the torsion ange β and appied to the beam end force moment M [4] without derivation: M β = Gwt, (8) where G shear moduus. If the atera force F acts on the having ength then torque is given by M = F. The x atera defection is, in turn, reated with the torsion ange as stiffness coefficient reads: c x x = β. Hence, the inverse tors x tors =. (9) F Gwt tors = x 8

9 E Knowing that G = 1+ ν ( ), the Poisson's ratio is ν 1/ (for the majority of materias) and taking into account the expression for c (15), the coefficient is cacuated as: 8 c tors Ewt = c. (40) To find the resuting defection at superposition of simpe bending and torsion it is just enough to sum the corresponding defections (assuming that deformations are sma): x = xbend + xtors = cbend Fx + ctorsfx = cxxf x. (41) Thus, the resuting inverse stiffness is a sum of the simpe bending and torsion inverse stiffness, too: t c = xx + c. (4) w Note that for the most of cantievers the simpe bending inverse stiffness c (7) exceeds much the torsion inverse stiffness c (40) so normay the simpe bending can be negected. For tors the standard AFM cantiever CSC1 having the foowing parameters: = 90мкм, bend = 10мкм, w = 5мкм, t = 1мкм, stiffness 1 с = 0.5Нм, the atera stiffness constants are: 1 м 1 м ctors c 0.05, cbend c (4) 40 Н 10 Н Notice that besides the x defection, both the simpe bending and torsion induce the deformations y and, respectivey. However, the magnitude of these dispacements is of the next order of smaness as compared with x and their reation with appied force is noninear (quadratic), i.e. "non-hookean". We can prove this, e.g. for torsion. Fig. 6. On cacuating c yx. whie: Referring to Fig. 6, we can write: β = ( 1 cosβ), (44) x = β. (45) 9

10 Since β 1, then x. Simiary, at the simpe bending y x. Hence, we can suppose: c = c = 0. (46) yx x Finay, we cacuate the components of the first matrix () coumn. The norma to the top surface of the cantiever, subjected to the transverse force Fx, defects in the Ox pane, so from (44)-(46) we can derive: βx b = c. (47) Accordingy, there is no defection in the Oy direction, so: b αx = 0. (48) Note that nonero ange β arises ony from the torsiona deformation. At the simpe bending the cantiever surface remains horionta so this bending can not be detected and its magnitude can ony be cacuated. However, to determine the transverse force experimentay, it is enough to detect the torsiona deformation. Summary: 1. The transverse force resuts in a compicated deformation which is a superpositon of the simpe bending and torsion of the cantiever beam. The simpe bending is anaogous to the - type defection. Soution to the more difficut probem of the rectanguar beam torsion is given in iterature [4].. Ony transverse defection of the cantiever (in the first order of smaness according to the Hooke's aw) occurs as a resut of the transverse force action.. The optica detection system registers ony torsiona deformation β = cf x. The simpe bending can not be measured directy. 10

11 .1.5 Cantiever inverse stiffness tensor Let us write the obtained components of inverse stiffness tensor C () into the representative matrix for mathematica convenience: cxx cxy cx C = cyx cyy cy. cx cy c The (x,x) coefficient of inverse stiffness - c is the argest of other tensor C components. In formua (1) this parameter is speciay denoted as c without indexes. It is namey the quantity 1 c that characteries a cantiever stiffness which is one of its major parameters. Beow, for the sake of cearness, we wi take c outside as a common muier of a the matrix eements [1]: t w C = c 0. (49) 0 1 Tensor C is symmetric. That is true. The eft side of expression (1) in section.1.1,, is a poar vector, therefore, the right side must have the same transformationa properties. The force F is a poar vector; therefore, the tensor must be symmetric in order to propery transform the expression (1) at co-ordinates refections. The presence of non-diagona eements eads to the difference in directions of the appied force and of the defection vector and is an evidence of imited appicabiity of the eastic cantiever simpified mode based on three perpendicuar springs. To appy this mode, one shoud determine not ony stiffness but true directions of three springs that do not coincide with coordinate axes. This probem is reduced to the tensor (49) diagonaiation in order to obtain its eigenvaues as we as to determine the directions of transformed co-ordinates aong which the mode springs shoud be oriented. It is seen that the cantiever eastic properties are competey defined by five parameters. We can find a tensoria components knowing geometrica characteristics of the cantiever and its stiffness constant. To faciitate computation at experiments, the obtained components are introduced into the matrix in formua () of section.1.1: 0 Fx α = c y β F. (50) 0 0 F Summary: 1. The tensor component c corresponding to the vertica dispacement at -type defection is the argest. Its inverse vaue is the beam stiffness constant that characteries a cantiever.. Tensoria nature of a cantiever eastic properties eads to the imited appicabiity of simpified modes based on springs.. To propery mode a cantiever eastic properties by three springs, one shoud determine their parameters correcty by diagonaiation of the inverse stiffness tensor. 11

12 .1.6 Effective mass and eigenfrequency of the cantiever In AFM, there exist techniques that are based not ony on static beam defection detection but aso on cantiever vibration. To use them one shoud know the cantiever resonant frequency. Let us cacuate the resonant frequency of isotropic cantiever with mass m in the form of paraepiped having ength, thickness h (h ) and width w (w ) to the free end of which the vertica point force F is appied (see Fig. 7). Fig. 7. Rectanguar cantiever with a Determine kinetic Eкин and potentia Eпот energy of the cantiever. Consider the beam eement having ength dy at distance y from the fixed end. Kinetic energy of such an eement is given by: ( ( τ, )) u y mdy deкин =, (51) where u(τ, y) dispacements of the beam axia points at distance y from the fixed end at time τ. Using formuas (11), (1) of section.1., u ( τ, y) can be obtained as a function of the beam free ( ) end defection u τ, : ( τ, ) u y y u( τ, y) =. (5) Substituting the expression for u (τ, y into (51) and integrating over the beam ength, we get: E пот E кин ) L ( u( τ, y) ) mdy m ( ( τ, )) = = u. (5) Potentia energy cacuation is easier. Because the point force F acts ony on the free end, is evidenty equa to the work done to move the beam end the distance u τ, : ( ) ( ) u τ, u τ, 0 0 ( τ, ) 1 u Eпот = Fdu = udu = c c ( ), (54) 1

13 where 1 c coefficient of norma stiffness defined by formua (1). If system vibrations are considered to occur without tota energy W dissipation, i.e. W = Eкин + Eпот = const, then, differentiating W with respect to time, we get equation of the cantiever free end move:.. ( ) mu τ, 1 + u( τ, ) = 0. (55) 140 c Therefore, the cantiever effective mass is: m эфф = m. (56) 140 Thus, cacuating m эфф and knowing the coefficient of stiffness 1 c defined by formua (15) in section.1. the eigenfrequency of the cantiever osciation can be expressed as a function of its parameters in the foowing way: ω = 1 = 1.09t E ρ, (57) 0 cmэфф where ρ cantiever density, E Young's moduus. As can be seen from (6), ω 0 is inversey as the square of the beam ength. This fact shoud be taken into consideration when choosing a cantiever. The cantiever eigenfrequency must be as high as possibe, otherwise its natura osciations wi be readiy excited due to the probe trace-retrace move during scanning or due to externa vibrations infuence. Summary: 1. To empoy AFM techniques based on the probe vibration, one shoud know the cantiever eigenfrequency and effective mass.. The effective mass is given by: mэфф = m t E. The eigenfrequency is: ω0 =. ρ 1

14 References. 1. Handbook of Micro/Nanotriboogy / Ed. by Bhushan Bharat. - d ed. - Boca Raton etc.: CRC press, p.. Feynman R., Leighton R., Sands M. The Feynman Lectures on Physics, voume 7. MIR, p. (in Russian).. Gorshkov A.G., Troshin V.N., Shaashiin V.I. Strength of Materias. FIZMATLIT, p. (in Russian). 4. Feodosev V.I. Strength of Materias, MSTU pubishing, p. (in Russian). 14

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