Modeling and simulation of AFM cantilever with two piezoelectric layers submerged in liquid over rough surfaces

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Modeling and simulation of AFM cantilever with two piezoelectric layers submerged in liquid over rough surfaces Moharam Habibnejad Korayem ∗ , Amir Nahavandi Department of Mechanical and Aerospace Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 24 September 2014 Received in revised form 19 April 2015 Accepted 22 May 2015 Available online xxx Keywords: Tapping-mode atomic force microscopy Liquid medium Submersion vibrations Rough surface Van der Waals force DLVO

a b s t r a c t In this paper, the vibration of an atomic force microscope (AFM) cantilever in tapping mode with two whole piezoelectric layers submerged in liquid medium is investigated. In the performed modeling, the sample surface has been considered as rough, and to show these surface asperities, two models of Rumpf and Rabinovich have been employed for analyzing the attractive Van der Waals force. This paper has been organized in two sections. The first section deals with the functioning of cantilever over rough surfaces, which accompanies the changes of the attractive Van der Waals force, and the second section involves the changes in the Van der Waals force which lead to a change in the liquid medium. The cantilever is totally submerged in the liquid. To show the effect of liquid on cantilever, first, only the cantilever tip is immersed in the liquid and it is dynamically analyzed. Then, the cantilever is totally submerged and then taken out of the liquid, so that the additional mass and damping of the cantilever could be calculated. In these two manners of cantilever immersion in liquid, the effects of the added mass and damping on the cantilever can be measured. When a cantilever vibrates totally in liquid, since the mass and damping of the liquid that is present on the cantilever cannot be determined, first, the cantilever’s natural frequency in liquid is estimated in the laboratory and then by using this frequency and the cantilever stiffness (which is not medium-dependent and is always considered as constant), the additional mass and damping of the cantilever are determined. © 2015 Elsevier Inc. All rights reserved.

1. Introduction The ability of the atomic force microscope in measuring then a Nano scale forces under physiological conditions has made this microscope an attractive tool in numerous biological research works involving medicine/protein, protein/protein, cell/cell or cell/protein interactions and in the investigation of many other phenomena such as the intermolecular forces. A molecular-level understanding of surface adsorption is necessary in detecting this phenomenon, and this knowledge can be potentially used in many applications including medication design, molecular electronics, fabrication of biomaterials and biosensor design. A high lateral resolution (less than one Angstrom) and the ability to measure very small forces, besides the ability to function in liquid and physiologic environments have made the AFM a suitable instrument for the study of biological systems. While AFM-based experiments have been conducted extensively for the mechanical touching of DNA, polysaccharides, proteins and other polymers, some researchers

∗ Corresponding author. Tel.: +98 612177240194. E-mail address: [email protected] (M.H. Korayem).

have investigated the mechanical properties of biopolymers on cellular surfaces. An AFM operating in tapping mode is capable of exploring surface topography and measuring the surface forces and mechanical properties of various surface at Nano scale. By applying the AFM in liquid medium, it would be possible to study the morphology and mechanical properties of biological samples in their original environments. Considering the extensive research works that are conducted on organic samples, liquid environments have become rather common mediums in atomic force microscopy works. The other advantages of liquid mediums include the elimination of adhesion forces, reduction of Van der Waals forces up to 10 times and more [1] and the reduction of contamination between tip and sample [2]. In the beginning, the imaging of biological samples in liquid mediums was carried out in the contact mode [3,4]. In the contact mode of AFM, the cantilever tip is always in contact with the surface of the sample. In such cases, the lateral force which is applied to the surface of a soft sample may damage the target particle. However, in the tapping-mode operation, the cantilever oscillates up and down, and when it approaches the surface, it only makes a brief contact with the surface and in this way, the side force diminishes considerably. Therefore, by imaging the biological samples in tapping mode, they will sustain

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the least amount of damage as a result of side forces. Simulating the vibrational motion of a cantilever in liquid medium is much more difficult than analyzing its vibrations in air. A large hydrodynamic damping and an additional mass on the cantilever due to the presence of the fluid cause the cantilever to display a vibrational behavior which is significantly different from its motion in vacuum or air. It’s been demonstrated that the vibrations of large cantilevers (with lengths larger than 1.0 m are not affected by fluid viscosity; while in microcantilevers, the frequency response is quite dependent on fluid viscosity [5]. Therefore, in light of the extensive use of AFM in liquid environments and the different behavior of cantilever in these mediums relative to the vacuum and air mediums, the vibrational motion of a cantilever with two piezoelectric layers in liquid medium is investigated in this paper. A self-exciting piezoelectric cantilever in the tapping mode was first used by [6] in a liquid medium. By employing a self-exciting and self-measuring PZT cantilever in Propanol medium, they could acquire clear images of a sample without the undesired external acoustic excitation effects. By using a piezoelectric cantilever operating in the tapping mode [7], obtained images of samples in different self-measuring (without the laser) and self-exciting (with the laser) modes in air and liquid environments; and they were able to achieve a bandwidth that was three times the bandwidth of tubular piezoelectric excitations. Certainly, the use of a piezoelectric cantilever in liquid mediums is only possible through proper coating; and Perylene, Silicon Oxide, Silicon Nitride and vacuum sealing wax are some of the coating which has been used for this purpose. In this paper, cantilever vibrations in three types of operations have been explored: (a) Cantilever tip inside the liquid, (b) Relative vibrations of cantilever in liquid and air mediums and (c) Full submersion of cantilever in liquid. The responses have been obtained by considering the fact that the additional mass and damping of the liquid medium influence the functioning of the cantilever with respect to amplitude, phase, tip movement velocity and natural frequency. Also, by changing the surface of the sample from smooth to rough, which alters the Van der Waals force, the three cases of cantilever placement in liquid have been examined for this type of surface by the Rabinovich and Rumpf models, and the results have been compared with those of the smooth surface. Regarding a liquid medium made up of homogeneous particles, this medium is modeled with the attractive Van der Waals force and the repulsive DMT contact force. In case the examined solution is a mix of undissolved particles in liquid (i.e., a colloidal solution), the Van der Waals force, which by nature arises by the presence of particles, leads to the use of DLVO model. The vibration of the AFM cantilever with two piezoelectric layers has also been investigated in this medium. The reason for analyzing the vibrations of such a cantilever in this type of environment is the refraction and scattering of light in this type of solution. Cantilevers that use laser rays for imaging run into difficulty in this type of medium. However, since one of the piezoelectric layers on the mentioned cantilever is used for measuring the instantaneous distance of tip from the surface, these kinds of cantilevers can perform more precise imaging operations in this environment.

three regions: the first region involves the beam length (0  X  L2 ), the second region designates the length of the piezoelectric layer (0  X  L1 ), and the third region outlines the cantilever tip zone (L2  X  L). In this model, a microcantilever is modeled by two piezoelectric layers placed on the top and bottom of the beam, and both of these layers are subjected to similar AC and DC voltages. Since the AC voltage can be positive or negative, and it also varies with time, the stable state component of this part of cantilever also changes with time; and the motion equation of cantilever includes a term which changes cantilever stiffness with time. The Hamilton principle is used to obtain the equation of motion. The potential and kinetic energies of the cantilever are expressed as Potential energy:

2. Dynamic modeling of the vibration of AFM cantilever with two piezoelectric layers submerged in liquid for smooth and rough surfaces

1 3 liq wl2 + wl a = 12 4

Usually, in common AFM cantilevers with two piezoelectric layers, the length and width of the piezoelectric layers are less than those of the microbeam. Therefore, a cantilever is divided into three sections, and the equations governing the vibrational behavior of cantilever are analyzed based on the boundary conditions and continuity principles. The cantilever in Fig. 1 has been divided into

ca = c∞ + cs

U (t) = Ubending-one part + Ubending-Piezoelectric Layer + Ubending-Tip + UVdc + UVac

(1)

The potential energy of the cantilever includes the potential energies of the cantilever section without the piezoelectric layer, the section with the piezoelectric layer, cantilever tip, and the potential energies of DC and AC voltages. Kinetic energy: T (t) = TBeam-one part + TPiezoelectric Layer + TBeam-Tip

(2)

Hamilton equation:

t2 ıH =





ıT (t) − ıU(t) + ıWnc dt = 0

(3)

t1

In the above equation, Wnc represents the work of nonconservative forces. Wnc = Fts (dts )(ı − L) + Fliq

(4)

Fig. 2 shows the manners of cantilever placement in liquid. In this paper, cases (a)–(c) refer to the illustrations of cantilever placement in liquid depicted in Fig. 2. In Eq. (4), denotes the cantilever tip-surface interaction force and (Fliq ) is the hydrodynamic force arising from the liquid around the cantilever. The magnitude of Fliq is determined for smooth and rough surfaces and for a colloidal solution. It is difficult to theoretically determine exactly how much water sits on the cantilever, and the estimations may contain error. For example, in the case of cantilever tip immersion in the liquid, it is uncertain whether just the tip or part of the cantilever end as well has entered the liquid; which in the latter case, more liquid mass and liquid damping will affect the microcantilever. Due to the difficulty of determining this force, the researchers tried to estimate it as follows [8]: 2

Fliq = −a

∂ v(x, t) ∂v(x, t) − ca 2 ∂t ∂t

(5)

In this equation, ca is the added hydrodynamic damping and a is the added density due to the liquid. The added mass can be calculated from the following equation [8].



2liq  ω

(6)

In Eq. (6), wb , ω, liq and ␩ are the cantilever width, natural frequency of cantilever, liquid density and liquid viscosity, respectively. The added damping coefficient is expressed as (7)

In Eq. (7), c∞ is the hydrodynamic damping due to the oscillation of cantilever in liquid, away from the surface. When we approach

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Fig. 1. (a) Geometry of the cantilever covered by two piezoelectric layers. (b) AFM cantilever with two piezoelectric layers oriented at angle ˛.

Fig. 2. Manners of cantilever placement in liquid medium. (a) Cantilever tip is immersed in liquid, (b) cantilever is completely submerged in and taken out of liquid, (c) cantilever is fully submerged in liquid.

Table 1 Properties and dimensions of the micro cantilever. Properties

Dimensions

Density of piezoelectric layer Density of micro cantilever Length of piezoelectric layer Length of micro cantilever Thickness of piezoelectric layer Thickness of micro cantilever Width of micro cantilever tip Width of piezoelectric layer Width of micro cantilever Elasticity modulus of piezoelectric layer Elasticity modulus of micro cantilever

6390 (kg/m3 ) 2330 (kg/m3 ) 375 ␮m 500 ␮m 4 ␮m 4 ␮m 55 ␮m 130 ␮m 250 ␮m 130 GPa 180 GPa

the surface, the damping due to the liquid film squeeze (cs ) emerges. This damping was expressed by [8] as follows: c∞ = 3 + cs =

3  b 2liq ω 4

b3

(8)

H0 (x, t)3

And [8] H0 (x, t) = D + l cos ˛ + v (x, t) cos ˛

(9)

In these equations, wl , ω, liq and  are the cantilever width, natural frequency of cantilever, liquid density and liquid viscosity, respectively. In Eq. (9), l is the length of cantilever tip for case (a),

in which the cantilever tip is inside the liquid, the damping due to the liquid film squeeze (cs ) is obtained from Eq. (9); and for case (b) of cantilever immersion in liquid (Fig. 2b), we have H0 (x, t) = D + L sin ˛ + l cos ˛ + v (x, t) cos ˛

(10)

L is the cantilever length. The force between tip and surface is determined from the following equation.



Fts =

Fvdw (dts ) cos ˛

dts > a0 dts ≤ a0

(Fvdw (a0 ) + FDMT ) cos ˛

(11)

In the above equation, Fvdw is the Van der Waals attractive force which tends to pull the cantilever tip toward the sample surface and FDMT is the repulsive contact force which tends to repel the tip from the surface of sample. This paper explores the influence of these two forces. The surface roughness of the sample causes a change in the Van der Waals force; and two models of this force are reviewed. In the Rumpf model, the Van der Waals force between a sphere and a rough surface with an asperity radius of Ra is expressed as [9,10] Fvdw (dts ) =

H 6dts



1+



R (1 + Ra /dts )

2

(12)

In this equation, H is the Hamaker constant, which can be determined between two mediums, and dts is the distance between

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Fig. 3. Vibrational mode shape of the AFM cantilever. (a) First mode. (b) Second mode. (c) Third mode.

probe tip and surface, which changes all the time. When this distance becomes less than the intermolecular distance a0 (about 0.2 nm), the Van der Waals force will be constant at this distance, and the repulsive contact force (which is determined by the DMT model) will repel the tip from the surface of sample. The repulsive contact force for this medium is expressed as [10] 4 √ 3/2 FDMT (dts ) = E ∗ R(a0 − dts ) 3



HR 6dts





1



Heff =

H1 + H2

(15)

(13)

E* denotes the effective modulus of elasticity between cantilever and sample surface. In the case of changing the liquid medium to a colloidal environment, the Van der Waals force will change, as will be explained later. The Rumpf model only considers the radius of surface asperities. If we want to consider the distance between two consecutive asperities in the equations, we have to use the Rabinovich model. This model expresses the Van der Waals force as follows [10–12]: Fvdw (dts ) =

In the above equation,  indicates the distance between two consecutive asperities. When the particle and the substrate are of different materials (material 1 and material 2), the Hamaker constant can be calculated from [13]:

+

1 + 58.144RRa /



1

1 + 1.817Ra /a0

3. Analyzing the motion equation of AFM cantilever with two piezoelectric layers submerged in liquid To obtain the vibrational motion equation of AFM cantilever with two whole piezoelectric layers of length L1 , we substitute Eqs. (1), (2) and (4) into Eq. (3), and after simplification, we get: q¨ (t) + ϕ1 q˙ (t) + ϕ2 q (t) =

Fts (dts )∅(L) M

(16)

where

2



(14)

ϕ1 =

C , M



ϕ2 =

K M

(17)

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Fig. 4. Diagrams of cantilever vibration amplitude vs. separation distance of tip from surface in the three cases of cantilever placement in liquid [Case (a): cantilever tip inside the liquid, Case (b): full cantilever vibration in liquid and air, Case (c): total submersion of cantilever in liquid]. (a) Smooth sample surface, (b) rough sample surface in the Rumpf model, (c) rough sample surface in the Rabinovich model, (d) response obtained from the vibration of cantilever with piezoelectric layers in liquid [14], (e) Response obtained from the vibration of cantilever by magnetic excitation in liquid [15].

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Fig. 5. Diagrams of cantilever deflection vs. damping coefficient of the squeeze film [Case (a):Cantilever tip inside the liquid, Case (b): relative vibrations of cantilever in liquid and air, Case (c): full submergence of cantilever in liquid]; (a) smooth sample surface, (b) rough sample surface in the Rumpf model, (c) rough sample surface in the Rabinovich model.







In Eq. (17), K, M and C are the cantilever stiffness, mass and damping, respectively, and can be determined as follows:

where k1 (x) = Eb Ib1 (U0 − UL2 ) + 2Ep IP (U0 − UL1 ) + Eb Ib3 (UL2 − UL )



L



m (x) = (b wb tb ) (U0 − UL2 ) + (2p wp tp ) (U0 − UL1 )

m (x) ∅n (x)(r) ∅n (x)(r) dx

M=

+ (b wt tb a ) (UL2 − UL ) + madded

0 L



ˇ

K=

 k1 (x) ∅ n (x)(r) ∅n (x)(r) dx

0 L

ˇ

c (x) ∅n (x)(r) ∅n (x)(r) dx 0

k2 (x) = 2wp e¯ 31 [Vac cos (2ωt) + Vdc ] (U0 − UL1 ) k2 (x) ∅n (x)(r) ∅n (x)(r) dx

−

(18)

c(x) = c ((U0 − UL2 ) + (UL2 − UL )) + cadded

0



C=

L

(19)

The madded and cadded respectively denote the mass and damping added to the cantilever due to the presence of liquid around the cantilever; and they can be calculated for the three previously

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Fig. 6. Impacts of different surface asperity radiuses (0, 10, 20 and 30 nm) on the diagrams obtained in the Rumpf model. (a) Amplitude vs. tip-surface separation distance, (b) squeeze film damping vs. separation distance, (c) force vs. separation distance.

mentioned cases of cantilever immersion. In Eq. (20), Un is the Heaviside function, which is defined as follows:

Un = U (x − n) Ux =

0

x<0

1

x>0

(20)

In Eqs. (16) through (19), qn (t) is the time coordinates and (r) ∅n (x) is the nth mode shape. The deflection of cantilever has been obtained by employing the modal analysis method, as the following equation: ∞

(r)

to these sections, we have (Table 1)

∅n =

⎧ (1) (1) (1) (1) (1) (1) (1) ∅ (x) = An sin ˇn x + Bn cos ˇn x + Cn sin hˇn x ⎪ ⎪ ⎪ n ⎪ ⎪ (1) (1) ⎪ + Dn cos hˇn x, 0 < x < L1 /L ⎪ ⎪ ⎪ ⎪ ⎪ (2) (2) (2) (2) (2) (2) ⎨ ∅(2) n (x) = An sin ˇn x + Bn cos ˇn x + Cn sin hˇn x (2) (2) ⎪ + Dn cos hˇn x, L1 /L < x < L2 /L ⎪ ⎪ ⎪ ⎪ ⎪ (3) (3) (3) (3) (3) (3) (3) ⎪ ⎪ ∅n (x) = An sin ˇn x + Bn cos ˇn x + Cn sin hˇn x ⎪ ⎪ ⎪ ⎩ (3) (3)

+ Dn cos hˇn x,

L2 /L < x < L/L

(21)

(22)

Parameter (r) in the mode shape expression is due to the fact that the cantilever beam is not continuously uniform and that it comprises three sections. So, to obtain the mode shape with regards

where ˇn = ωn2 mr /EIr , An , Bn , Cn and Dn are constants which are determined by using the continuity of displacement, slope, bending moment and shear force boundary conditions and also by considering the orthogonally of mode shapes with respect to mass

v (x, t) =

∅n (x) qn (t)

n=1

(r)4

(r)

(r)

(r)

(r)

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Fig. 7. Impacts of different surface asperity radiuses (0, 10, 20 and 30 nm) on the diagrams obtained in the Rabinovich model. (a) Amplitude vs. tip-surface separation distance, (b) squeeze film damping vs. separation distance, (c) force vs. separation distance.

L

A∅2n (x) dx = 1. Fig. 3 shows the validity of applying the boundary conditions to obtain the cantilever’s natural frequency and the coefficients of Eq. (22). Cantilever mass in air is M = 3.3342 × 10−9 kg, and in case (a), when the tip is inside the liquid (here assumed as water), its mass reaches M = 3.3497 × 10−9 kg. In case (b) in which the cantilever vibrates in two mediums of air and liquid, its mass becomes M = 7.7244 × 10−9 kg. Based on Eq. (8), the damping of the liquid film squeeze (cs ) is a variable and dependent on cantilever deflection, and it is directly entered into the equation of motion. It should be noted that because of the cantilever operating in Nano environment, all the values obtained in this article are in nanometer. When the cantilever is fully submerged in liquid (case (c)), the mass of the added liquid on cantilever cannot be obtained as easily as the first two cases. In this case, by considering the fact that cantilever stiffness is constant in liquid and in air, first, the natural frequency of cantilever in air, and then in liquid, is estimated in laboratory or via computer software, and then the liquid mass and damping added to the cantilever are determined by the following equations. 0

madded ω2 = na −1 2 m ωnf ς f ωna cadded −1 = c a ωnf

(23)

In the above equations, madded , ωnf , ς f ,  a , ωna and cadded are the added mass on cantilever, natural frequency of cantilever within liquid, damping coefficients in liquid and air mediums, natural frequency of cantilever in air, and the damping added to cantilever, respectively, and m and c are the cantilever mass and damping in air. The natural frequency of the cantilever with two piezoelectric layers in air in the first mode is 52.264 kHz and second mode is 287.513 kHz and third mode is 451.119 kHz. By placing the cantilever tip in liquid, the cantilever’s first mode frequency is 43.406, second mode reduces to184.882 kHz and third mode is 345.391 kHz. In the second case, in which the cantilever completely enters and then exits the liquid, its natural frequency in the first mode is 14.934 kHz, second mode reduces to 82.419 kHz and third mode is 127.174 kHz. In the third case, in which the cantilever becomes fully submerged in liquid, its frequency in the

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Fig. 8. Impacts of different  values (0, 10, 20 and 30 nm) on the diagrams obtained in the Rabinovich model. (a) Amplitude vs. tip-surface separation distance, (b) squeeze film damping vs. separation distance, (c) force vs. separation distance.

first mode reaches a value of about 2.281 kHz, second mode is 34.179 kHz and third mode is 83.013 kHz. In this paper, we have attempted to examine the influence of surface roughness on cantilever vibration in liquid environment. For this purpose, first, only the cantilever tip is placed in liquid and the above effects are investigated for smooth and also for rough surfaces without considering the distance between two consecutive asperities (Rumpf model) and considering the effect of two successive asperities (Rabinovich model). Then the cantilever is submerged relatively and fully into liquid and the obtained results are compared with those of the previous case in order to see the effect of the liquid medium on the vibrational behavior of cantilever. In some cases, images are needed from a sample in a liquid, and for this purpose, the cantilever has be immersed in that liquid. By adding a liquid film on cantilever, its natural frequency diminishes and consequently, the solution of the motion equation, which is directly related to system frequency, changes. An approximate analytical solution of the form shown below is frequently used to estimate the natural frequencies of the immersed beam. This is estimated based on the structure-only

natural frequencies, beam geometry, and the ratio of fluid-to-beam densities. The analytical expression approximately accounts for the added mass of the fluid that is displaced by the beam. It does not account for viscous effects. By knowing the cantilever’s natural frequency in liquid and vacuum mediums, its damping coefficient within liquid can be determined. ωnf = ωna



1 − 2f2

(24)

The submergence of cantilever in liquid can be analyzed by means of Eqs. (23) and (24). This paper investigates the effects of the existing parameters on the vibrational response of AFM cantilever in liquid medium. First, the vibrational responses of the cantilever in the three cases of cantilever placement in liquid are explored for smooth and rough surfaces, and the obtained responses are analyzed. The dynamics of cantilever vibration is associated with the parameters of effective damping of liquid surrounding the cantilever (based on the type of cantilever placement in liquid), cantilever stiffness (caused by two factors: the geometrical form

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Fig. 9. Impact of voltage on the diagrams obtained in the Rumpf model. (a) Amplitude vs. tip-surface separation distance. (b) squeeze film damping vs. separation distance.

Fig. 10. Impact of squeeze film damping on the diagram of amplitude vs. separation distance obtained in the Rumpf model for a cantilever submerged in liquid.

of the cantilever and also its constituent material), piezoelectric layer, mass, natural frequency and the force applied on cantilever probe tip. Fig. 4 shows the diagrams for the separation distance of tip from surface (smooth and rough) versus cantilever vibration amplitude in three types of cantilever immersion in liquid. By putting the cantilever into liquid, the slope of the amplitude curve decreases, the discrepancy between different tip-surface distances becomes less and the cantilever vibrates at amplitudes that are closer to each other. When the sample surface is smooth and the cantilever tip is into the liquid, the amplitude increases in a curvilinear form up to a distance of 30 nm and then assumes a constant slope. The blue diagram indicates the full submergence of cantilever in liquid, and its slope is much less than that of the other two cases (Fig. 4a). By changing the sample surface to rough (Fig. 4b), the changes of amplitude in case (a) and case (b) become more pronounced than those in the full submergence case.

At distances further away from the surface, amplitude is not dependent on the manner of cantilever placement in liquid and is almost constant. By solving Eqs. (16) and (22), the temporal and spatial variables of mode shapes have been respectively computed, and by inserting these two equations into Eq. (21), the deflection of cantilever has been determined. By substituting the cantilever deflection into Eq. (8), the changes of the damping coefficient of the squeeze liquid film have been obtained and sketched in Fig. 5. The Van der Waals force, which has an attractive nature and is dependent on surface radius, is the cause of difference between the smooth and rough surfaces. For the cantilever tip radius of R = 20 nm and the surface asperity radiuses shown in Figs. 6–8, the effects of asperity radius (Ra ) and  have been investigated. With the increase of surface asperity radius, the vibration amplitude approaches the amplitude for smooth surfaces; which this reduction of amplitude

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Fig. 11. (a) Non-dimensional amplitude frequency, (b) non-dimensional phase frequency (black, red, blue and green diagrams, respectively, indicate the cases of cantilever in air, cantilever tip in liquid, cantilever vibration in air as well as liquid mediums, and cantilever fully submerged in liquid). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

is due to the reduction of the Van der Waals force between the two surfaces. In view of Fig. 9, with the rise in the voltage of piezoelectric layer, the vibration amplitude increases due to the increase of cantilever stiffness, which leads to the reduction of the squeeze film damping (Figs. 10 and 11).

and substituting Eqs. (27) and (28) into Eq. (26) we get: J (A, B, C, E, ω) =

∞  1 2

(qq2 ) −



ϕ2 2 HR ϕ1 qq1 + q + cos ˛ dt 2 2 6M [D0 + q∅ (L) cos ˛]

ˇ

4. Finding the relationship between amplitude and time by using the calculus of variations principle The FTS numerical method that was used to solve the differential equations governing the vibrational behavior of AFM cantilevers is a time-consuming approach by nature. In this section, a relatively quicker numerical solution technique, namely, the calculus of variations, is proposed for solving the differential equations. For this purpose, using the motion equation, the functional of the equation is determined first, and then by employing an appropriate function that satisfies the cantilever’s initial conditions, the solution is obtained. The functional of the motion equation (Eq. (16)) is

 

J (q) =

ϕ2 2 1 ¨ qq − ϕ1 qq˙ + q −f 2 2

+

(25)

The above equation has been obtained from the Euler–Lagrange relation, where ∂f = Fts , and it is defined as follows:

2

(qq2 ) −

0

+

ϕ2 2 ϕ1 qq1 + q − 2 2

 H∅

(L) R q 6Ma0 2

  √ 8 RE ∗ 5/2 (a0 − D0 − q∅ (L) cos ˛) cos ˛ dt (28) 15M

where q = A cos (ωt) + B (cos (ωt) − cos (3ωt)) + C (cos (3ωt) − cos (5ωt)) + E (cos (5ωt) − cos (7ωt))



dt

ˇ  1

q1 =

dq = −Aω sin (ωt) − B (ω sin (ωt) − 3ω sin (3ωt)) dt

−C (3ω sin (3ωt)−5ω sin (5ωt))−E (5ω sin (5ωt)−7ω sin (7ωt))

∂q

⎧ HR ⎪ − cos ˛ ⎪ ⎪ ⎪ 6M [D0 + q∅ (L) cos ˛] ⎪ ⎪ ⎨ √ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

q (t) ≥ ˇ

H∅ (L) R 8 RE ∗ 5/2 (a0 − D0 − q∅ (L) cos ˛) q+ 2 15M 6Ma0

q2 =

 cos ˛

(26)



q (t) = A cos (ωt) + B (cos (ωt) − cos (3ωt)) q˙ (0) = 0



Parameters A, B, C, E and ω can be obtained by means of the Ritz method.

By considering q(t) as + C (cos (3ωt) − cos (5ωt)) + E (cos (5ωt) − cos (7ωt)) + · · ·



(29)

a0 − D0 ∅(L) cos ˛

q (0) = dts (0);



− C 9ω2 cos (3ωt) − 25ω2 cos (5ωt)

− E 25ω2 cos (5ωt) − 49ω2 cos (7ωt)

0 ≤ q (t) < ˇ

B=−

  d2 q = −Aω2 cos (ωt) − B ω2 cos (ωt) − 9ω2 cos (3ωt) dt 2

(27)

∂J ∂J ∂J ∂J ∂J =0 ; =0 ; =0 ; =0 ; =0 ∂ω ∂A ∂B ∂C ∂E

(30)

Fig. 12 shows the solution obtained through the above mentioned method for the case of total cantilever submergence in

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Fig. 12. (a) Results obtained from the calculus of variations and the Runge–Kutta method for the case of cantilever submergence in liquid, (b) Displacement of cantilever tip vs. time [16,17].

liquid. In the Runge–Kutta approach, the cantilever is at a distance of 90 nm from the sample surface, and in the calculus of variations, it is 20 nm away from the sample surface. 5. Cantilever vibration in colloidal liquid medium by changing the Van der Waals force (DLVO theory) This theory expresses the accumulation and dispersion of liquid particles and studies the forces acting on a charged surface in a liquid medium. In principle, this theory is where technology meets the sciences of physics, chemistry and biology. This theory mostly expresses the existence and lack of existence of equilibrium in colloidal systems. A colloid is a state between solution and mixture, in which the dissolvable particles are larger than solution particles and scatter the light, while at the same time, they are smaller than mixture particles and do not precipitate. Colloidal mixes are also

called gluey solutions. The dispersion of their particles is not as ionic and molecular dispersion, but in the form of molecular accumulations known as Miscels, which are easily distinguishable from the solvent; like chalk particles or olive oil droplets in water. False solutions, i.e. colloidal solutions, are not uniform compounds. A colloid is a solution mixture with particles larger than the particles in the solution. The particles present in a colloid are scattered and suspended. Colloids are made of at least two phases (a dispersed phase and a dispersing phase); like starch in water and milk. The word “colloid” was coined by Thomas Graham in 1861; and today, colloids have gained a scientific and technological reputation. Paints, ceramics, cosmetics, cleaning solutions, foodstuff and other material on which our daily lives depend, all remind us of the importance of colloids. Colloidal components have a size range of 1–100 nm. Since light is refracted in a colloidal medium, cantilevers which are monitored by laser light cannot be used. Therefore, piezoelectric

Fig. 13. Differences between the diagrams of (a) amplitude vs. tip-surface separation distance and (b) cantilever deflection vs. squeeze film damping coefficient obtained in water and colloidal liquid mediums [Case (a): cantilever tip inside the liquid, Case (b): relative vibrations of cantilever in liquid and air, Case (c): full submersion of cantilever in liquid].

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cantilevers are employed to study the effects of colloidal solutions and to investigate the performance of AFM in these environments. The classical models of DLVO and XDLVO have modeled the contact between a sphere and a smooth flat surface. According to the DLVO model, the force between cantilever tip and surface, obtained by summing the Van der Waals force and the electrostatic repulsive force between two layers, has been presented by Deryagin & Landau and by Verwey & Overbeek [18]. DLVO LW EL ESP = ESP + ESP

(31)

LW is the Van der Waals energy and E EL In the above equation, ESP SP is the electrostatic energy of interaction between medium (1) and liquid medium (2) in which it is situated. Both of these energies are functions of distance dts , and they have been expressed by [19], as follows: LW (d ) = − ESP ts

HR 6dts



EL (d ) = ε ε R 2ϕ ϕ ln ESP ts 0 r 1 2





 1 + exp

(−kdts ) 1 − exp (−kdts )



+ ϕ12 + ϕ22 ln[1 − exp (−2kdts )]

 (32)

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H is the Hamaker coefficient, ϕ1 and ϕ2 are the potentials between the two surfaces, ε0 = 8.854 × 10−12 cv−1 m−1 and εr = 79 are the dielectric permeability values in vacuum and liquid mediums, respectively, and k = 3.28 × 109 I1/2 m−1 is the inverse of the Debye length (I is the ionic power of the electrolyte in molarity conditions). Fig. 13 illustrates the vibrational response of cantilever in colloidal environment, which has been modeled using the DLVO theory. The point to consider in this case is the higher amplitude and phase of cantilever relative to the liquid medium that consists of identical particles. The cantilever has a lower vibrational amplitude and higher velocity when just its tip enters the colloidal medium than when it is fully submerged in liquid (Fig. 14). In Fig. 15, the diagrams of the force exerted on cantilever tip have been plotted for smooth and rough sample surfaces and for the DLVO model. The similar behavior of the force exerted on cantilever tip for the smooth sample surface and water medium and for the colloidal medium (Fig. 15c) has produced similar diagrams for these two cases. The force exerted on cantilever tip inside liquid drops for rough surfaces relative to smooth surfaces. The lesser value of this effect relative to cantilever stiffness and damping has caused the cantilever to achieve a constant amplitude at a distance further away from the sample surface. The force diagram in the Rumpf

Fig. 14. Effect of liquid medium’s potential in the case of full cantilever submergence on the diagrams of (a) amplitude vs. tip-surface separation distance, (b) cantilever deflection vs. squeeze film damping coefficient and (c) force vs. separation distance.

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(a)

(b)

(c) Fig. 15. Effects of surface asperity radius (Ra ) and distance between two consecutive asperities () on the force exerted on cantilever tip in the (a) Rabinovich, (b) Rumpf and (c) DLVO models.

model has a similar slope, but a lower value, to the case in which the tip vibrates over a smooth surface; thus, the diagrams of cantilever vibration amplitude vs. separation distance in the Rumpf model are identical for rough and smooth surfaces, but they are different in size. The diagram of the Rabinovich model is totally different from the other three cases; however, due to the small effect of this force, not much difference is observed in the responses of Rumpf and Rabinovich models. As the cantilever submerges in liquid, its added mass and damping increase; and this causes the responses to lose their dependency on the applied forces. In the Rabinovich model, at a distance of less than 10 nm, the applied force has a descending to ascending trend, which causes a breaking point in the amplitude diagram in this region (Fig. 4c).

6. Conclusion The extensive research works on biological samples has made the liquid medium one of the most common environments in AFM

research. By solving the motion equation of an AFM cantilever (Eq. (16)), its vibrations in liquid medium were analyzed. The aim of this paper was to investigate the effects of additional mass and damping added to cantilever and also the influence of surface roughness. First, the cantilever was made to vibrate so that only its tip entered the liquid. In this case, a noticeable drop was observed in the natural frequency of cantilever; and as more of the cantilever entered the liquid and became fully submerged, its natural frequency decreased considerably. The effects of cantilever submergence on the diagrams of amplitude, cantilever tip deflection and also the coefficient of damping due to the squeeze liquid film have been illustrated. The difference between the highest and lowest amplitude values with respect to the separation distance of tip from surface is less for a cantilever immersed in liquid than for the case in which the tip is inside the liquid. Regarding a smooth surface, as the cantilever moves away from the surface, its vibration amplitude increases quickly and at a distance of more than 30 nm, it becomes constant and independent of cantilever position relative to surface. When the sample surface is rough, the amplitude increases

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nonlinearly; and as the cantilever enters more into the liquid, the slope of the amplitude curve diminishes and becomes flat at a distance far from the surface. The discrepancy between the amplitude values for rough and smooth surfaces is due to the interaction force between cantilever tip and surface. The vibration amplitude behaves the same in the Rumpf and Rabinovich models and the only difference is the lesser value of amplitude in the Rumpf model. In the Van der Waals force models used for rough surfaces, with the increase in the radius of surface asperities, the diagrams approach the smooth surface diagram; while in the Rabinovich model, the variation of parameter  has no effect on the obtained diagrams. The damping coefficient of the squeeze film (shown in Eq. (8)) is dependent on cantilever deflection and the existing parameters in the motion equation. When the cantilever is immersed in liquid, the diagram of the damping coefficient of squeeze film during the immersion of tip has a smaller slope than that in the other two cases. The reason for this discrepancy is that the vibrations of an immersed cantilever produce smaller amplitudes and displacements than when just the tip is vibrating in the liquid. This difference arises from the change of tip position relative to surface. Because of the sensitivity of soft samples, and in order to avoid damaging these samples when using a cantilever in liquid medium, the cantilever can be immersed in liquid. The use of the DLVO theory, which is necessitated by the liquid medium becoming colloidal, produces a substantial reduction in the amplitude of vibration near the surface; and by moving the tip away from the surface, the diagram shows a constant trend. This applies to the case in which the cantilever tip is inside the liquid. With the submersion of cantilever in liquid, the amplitude in the DLVO model falls in the operating range of cantilever over smooth surfaces and exhibits limited variations. By increasing the potential between two surfaces in the DLVO theory, the diagrams change; and the same way the change of surface asperity radius in the Rumpf and Rabinovich models alters the relevant diagrams, the liquid medium also affects the performance of the cantilever. It was also demonstrated that by employing the calculus of variations and choosing a function that applies to the governing boundary conditions of the cantilever, a quicker solution can be achieved in the range of solutions obtained through other methods.

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