Ring of charge probed with Atomic Force Microscopy dielectric tip

Journal of Electrostatics 97 (2019) 95–100

Contents lists available at ScienceDirect

Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat

Ring of charge probed with Atomic Force Microscopy dielectric tip Yishai Eisenberg, Fredy R. Zypman

T

∗

Department of Physics, Yeshiva University, 2495 Amsterdam Avenue, Manhattan, NY, 10033, USA

ABSTRACT

We consider a dielectric sphere in the presence of a charged ring and derive expressions for their interaction force. New mathematical closed form expressions for electrostatic problems involving dielectric objects are always interesting, both for their intrinsic foundational value, and also for their practical use as test beds for numerical solutions. The chosen geometry is motivated by the importance to probe charged rings with Atomic Force Microscopy dielectric tips. The interaction forces obtained are further analyzed to identify notorious features of the force versus separation curves. This analysis provides practical guidelines to measure charge and size of rings from experiments.

1. Introduction We consider a charged ring probed by an Atomic Force Microscope (AFM) fitted with a microsphere dielectric tip. The goal of this paper is both practical and fundamental. From a practical standpoint, this article presents a mathematical method that allows measuring the charge and size of the ring from the AFM force-vs-separation curve. Second, from a fundamental electrostatics point of view, this paper presents analytical expressions for forces on AFM dielectric tips. In addition, the problem of rings of charge probed by AFM microsphere tips is of current interest at the nanometer/micron scale. On the one hand, AFM microsphere probes are routinely used to investigate electrostatics properties of materials [1,2]. For example, AFM microsphere probes have been used recently to measure charge of nanoparticles immersed in electrolytes [3]. Meanwhile, a number of systems of micrometer or nanometer size can be modeled as rings of charge. For instance, in molecular pumps [4–6], rings are driven, based on their charge content, to link amino acids into growing peptides. Knowledge of charge content and size is necessary to apply these pumps to growing other assemblies [7–10]. More generally, charge plays a key role in the structure attained by large molecules when they self-assemble [11]. There have also been proposals to use charged rings to physically store qubits for quantum computing [12]. These examples underscore the relevance of understanding electrostatics measurements on rings at the micro/nanoscale with microsphere tip fitted AFMs. In this paper, we first compute the force vs separation curve of electrostatic interaction between a charged ring and an AFM dielectric microsphere probe. Then, by focusing on the separation and value of the maximum force, which are practical features to identify in an experiment, two equations are obtained that permit the measurement of ∗

both the ring size and charge. From a fundamental electrostatic standpoint, we obtain for the first time the electrostatic field produced by the ring of charge in the presence of a dielectric sphere which represents the AFM microsphere. 2. Description of the system Our focus is on the interaction between the AFM dielectric microsphere and the charged ring. Fig. 1 shows a diagram displaying the attachment of the microsphere to the cantilever, and its position above the sample that supports the ring. The substrate's height is controlled from below by a piezoelectric actuator. The microsphere hovers above the substrate and moves under the influence of microsphere/substrate forces. This motion is routinely measured in AFM experiments and converted into forces. Here we are interested in deriving mathematical expressions of those forces for a charged ring on the substrate. Fig. 2 shows such a situation emphasizing the interacting microsphere and charged ring on the substrate. In addition, Fig. 2 illustrates the location of the image charges to be discussed shortly, the dimensions of the sphere and ring, and the distance between their centers. Thus, the problem that we solve here is that of deriving mathematical expressions for the sphere/ring interaction force as a function of their separation. In addition, we also show how to use those expressions to measure the charge of the ring. The main idea is that since the experimentally available information are force vs separation curves, by comparing them with the mathematical expressions derived here, the ring parameters can be attained.

Corresponding author. E-mail address: [email protected] (F.R. Zypman).

https://doi.org/10.1016/j.elstat.2019.01.002 Received 8 November 2018; Received in revised form 1 January 2019; Accepted 1 January 2019 0304-3886/ © 2019 Elsevier B.V. All rights reserved.

Journal of Electrostatics 97 (2019) 95–100

Y. Eisenberg, F.R. Zypman

3.1. Image conic surface contribution We begin by analyzing the interaction between the real ring and the image cone. The image ring's charge is cylindrically symmetric. This allows to simply computing the force by dividing the cone into vertically infinitesimally thin rings, and integrating the differential force between each of these rings and the external ring. To do this, we consider the vertical component of the electric field produced by a ring of constant charge density λ and radius a [19] –see Fig. 3,

Ez =

cos f (µ ) 3 1 2) 2 a 0 (1 +

2

(1)

Where is the angle of the vector position of the observation point and 2 the z axis, µ = 1 + 2 sin , ξ the distance between the observation point and the center of the ring, in units of the ring's radius a , and 2µ 1+µ

2

f1 (µ ) =

(1

, with

µ) 1 + µ

the Elliptic integral of the first kind [20].

Taking advantage of cylindrical symmetry, and using the image linear charge density introduced in the previous section, the differential force between the real ring and an infinitesimal circular sliver of the cone, at location s from the center of the sphere in Fig. 2, is

Fig. 1. A flexible AFM cantilever beam fitted with a dielectric microsphere at the free end. The substrate is positioned to vary the microsphere-sample separation. The sample charged ring is not shown, but lies on top of the substrate.

3. Mathematical derivation

dFz =

Let us recall that for a point particle in the presence of a conducting sphere, the force of interaction is easily obtained via the well-known method of images. In this famous approach, Lord Kelvin was able to solve the electrostatic problem by mathematically replacing the sphere by an image charge [13]. Coulomb interaction between the real charge and the image charge thus provides the desired sphere-point interaction force. On the other hand, for extended charged objects in the presence of a conducting sphere the force can be calculated as a sum of Coulomb terms if the extended image charge distribution can be found. Successful applications of this approach have been done for rings of charges in air [14] and in electrolytes [15]. While there are important uses of metallic tips and microspheres in AFM, most AFM applications are done 4 ), silicon with tips and microspheres made of silicon nitride ( 4 ), diamond ( 4 ) [16,17]. For 6) or carbon ( 12 ), silica ( ( this reason, we consider here the case for arbitrary relative dielectric constant . In this way, we can model common dielectric tips and, by , also the conducting case is covered. letting To make progress, we use a generalized version of Kelvin's result [18], in which the sphere has an arbitrary dielectric constant . Although we will generalize the result for spatially extended charges below, we briefly outline the important results for a single point charge Q outside a dielectric sphere. The electrostatic problem outside the sphere is equivalent to that of three charge contributions. First, the real 1r Qa external point charge Q . Second, an image point charge Q = +1d

=

cos( ) 2

0a

2 0

f (µ ) 3 1 2) 2

(1 +

cos( )

f (µ ) 3 1 2) 2

(1 +

1 1 s ( + 1)2 r dk

=

1 Q ( + 1)2 r

( ) s dk

+1

+1

ds

ds

(2)

3.2. Image ring contribution Reference [14] derived the force of interaction between a uniformly charged ring and a conducting sphere. That situation has only an image ring charge located at dk from the center of the sphere. In addition, that image charge differs from the one considered here by a multiplicative constant, 1 . Since the electrostatic force is linear with respect to the +1 charge of one of the objects, we obtain the electrostatic force contribution from the image ring at the cone's base simply by multiplying the result in Ref. [14] by 1 . Thus we arrive at the force of interaction +1 between the ring of charge and the image ring in Fig. 2,

Fz =

distance dk = from the center of the sphere, on the line that joins Q with the center of the sphere; here r is the radius of the sphere, d is the distance from Q to the center of the sphere (see Fig. 2), and dk is defined as the Kelvin distance. Third, and image straight line charge that extends from the center of the sphere to the location of Q , with linear charge density

s dk

To be precise, the distance s along the surface of the sphere, on the line from the center of the sphere varies between 0 at the center of the sphere, and dk .

1 +1

d r

2

a 2 r

( ) +( ) a 2 r

0

1 a 2 d 2 + r r

1

()

d 2 r

()

a 2 r

r2 d

/( + 1)

1 2 a ( + 1) 2 r

2

a 2 r

+

+

a 2 d 2 + r r

d r

d r

3 2 2

f1 (µ)

a 2 d 2 + r r

(3) 3.3. Total ring-sphere force

, where s is the coordinate from

The total vertical force between the dielectric sphere and the charged ring is then the sum of the expressions in equations (2) and (3),

the center of the sphere. Thus, for our case of interest (Fig. 2), the ring of charge generates two image regions inside the dielectric sphere: first a ring of charge of uniform linear charge density, and second a conical region with varying surface charge density. The circular cone has a vertex at the origin of the sphere, and its base coincides with the image ring. Both charge densities will be written explicitly in what follows, to subsequently be able to compute the force between the ring and the AFM tip.

dk

2

0

0

Fz ( , ) =

1 +1

(s )cos( (s )) (1 +

3 2) 2

1 1 s ( + 1)2 r dk

2

+

2 2

+

(

2 2+ 2

+1

ds

1 2+ 2

1

2

2 0

f1 (µ (s ))

) ( 2

+

d r

3 2 2

2+ 2

)

f1 (µ (s ))

(4) Where we have introduced clarifying notation, namely that the force Fz d is a function of the sphere-ring dimensionless distance = r , and the 96

Journal of Electrostatics 97 (2019) 95–100

Y. Eisenberg, F.R. Zypman

Fig. 2. Perspective and zoomed view of Fig. 1 showing in inset (a) the charged ring on the substrate and the image charges; the separation d , between the center of the real ring and the sphere is also depicted. Inset (b) shows a magnification of the images, the opening angle of the cone 2 , the Kelvin distance dK from the vertex of the cone to the image ring, and the distance s from the apex of the cone to a generic point on the cone introduced in equation (2).

on the image cone between 0 and 1,

s dk

=

(6)

With which equation (4) becomes,

Fz ( , ) =

2

1

1 1 ( + 1) 2

0

( )cos( ( )) 3

( ) 2) 2

(1 +

0

2

2

+

+1

(

)

1

2

1 +1

f1 (µ ( ))

2

+

(

2 2+ 2

1 2+ 2

) ( 2

+

d

3 2 2

2+ 2

)

f1 (µ ( ))

(7) This is the charged-ring/dielectric-microsphere force that we sought. We will make a study of this force by evaluating it for different values of ring/sphere separation , and ring size . To continue, we need to evaluate the integral in equation (7). However, standard numerical approaches are slowly convergent due to 0 . To surmount this problem we the divergence of the integral as +1 , proceed as follows. First, note that the integrand diverges as with 0 < + 1 < 1 since the relative dielectric constant > 1. We then add and subtract the same term to the integrand in equation (7) so that the integral now is

Fig. 3. The ring of radius a , linear charge density , and axis of symmetry z , generates an electric field Ez at location r as shown in equation (1). The position vector r is characterized by its angle with the z axis, and its magnitude r =r=a .

dimensionless radius of the ring = r . To use equation (4) we need to find the functions (s ) and which can be obtained from the geometry of Figs. 2 and 3, a

(s ) = arctan

(s ) =

d

(s s in)[

s sin s cos ]2

+ (d

(s ) ,

1

(5a)

s c os)[

]2

a

(5b)

+ 0

(5c)

2 (s ) sin (s ) 1 + (s ) 2

f1 (µ ( ))

(1 + )

1

(1 + ) 2

3/2

+1

d

2

+1 2

+1

3/2

d

2

(8a)

In this way, the singularity has been removed from the first integral, while the second integral can be evaluated analytically to render

Finally,

µ (s ) =

3

( )2) 2

1

With the half-angle opening of the cone being

a = arctan d

( )cos( ( )) (1 +

0

2 ( + 1) 2 2 3/2 ( + )

(5d)

To make the ensuing analysis of the force more transparent, we consider the following change of variables that measures the position

Then we arrive at this practical expression for the force 97

(8b)

Journal of Electrostatics 97 (2019) 95–100

Y. Eisenberg, F.R. Zypman

Fig. 4. Sphere/ring force versus separation for rings of various sizes . These were obtained for

Fig. 5. Locus of the ( 0 ,β) points that make Fz ( ,

0)

= 2.

vanish using equation (9).

4. Results Fz ( , ) =

2 0

+

1

1 1 ( + 1) 2 1 +1 ( 1 +1

( )cos( ( )) 3

( ) 2) 2

(1 +

0

f1 (µ ( ))

1

(1 + )

2 3/2

+1

d

The expression for the force in equation (9) is the central mathematical result of this paper. In this section we study its properties by and sphere/ring seevaluating it for different values of ring sizes parations . Fig. 4 is an example of such computation where force-separation curves are shown for various ring sizes. Some universal features of this function are already apparent in the figure: for each ring size, the force-separation curve exhibits a maximum force and a minimum force. Moreover, the locations of the maxima (minima) move monotonically to the right with increasing ring size. On looking at equation (9), one notices that the charge of the ring appears in the force expression as an overall multiplicative constant. Thus, the shape of the force curve is independent of the charge up to an overall vertical stretch. Consequently, the location where the force vanishes 0 , the location of the maximum force max , and the location of

2

2 2

2 )3/2

+

1

2 2

+

2 2

+

(

2 2+ 2

(

1 2+ 2

) +(

) 3 2 2

2

2+ 2

)

f1 (µ ( ))

(9)

0<

The integral in equation (9) is now regular in the whole range < 1 and presents no numerical difficulties in its evaluation. 98

Journal of Electrostatics 97 (2019) 95–100

Y. Eisenberg, F.R. Zypman

Fig. 6. Ring radius

for which the distance (

min

0)

is given –in practice the distance that is measured experimentally.

Fig. 7. Dimensionless force value of the minimum force

the minimum force min , are all independent of the charge. This observation provides a practical scheme to obtain experimentally the size of the ring as follows. Before we give the full correct answer which emphasizes measurement constraints, we argue thus. Fig. 5 shows the locus of the ( , ) points that make Fz ( , ) zero in equation (9). Hence by experimentally measuring the location 0 , where to force is zero one could find from the graph, or equation (9) directly, the corresponding value of the ring radius . However, although the argument is mathematically correct, it is not generally suitable since experimentally the values of are only known up to an arbitrary additive constant. This is because no unambiguous zero tip/sample separation can be defined in AFM. One exception to this restriction exists when the ring is large, and the spherical tip can touch the substrate allowing for a = 0 absolute reference. To overcome this difficulty in the general case, we consider instead the positive quantity min , the 0 . Since this is a difference of

min

=

0 2

Fmin as a function of the radius of the ring.

arbitrary additive constant mentioned above cancels, that is min 0 is experimentally accessible. Then, in practice one would pull out the AFM tip starting from contact and record the separation 0 where the force is zero. Afterwards the tip continues to withdraw until the minimum force is reached, and min is logged (see Fig. 6). With the value of the ring radius known, the charge of the ring can be obtained by looking for a prominent feature in the experimental force-separation curve. Fig. 4 suggests the use of the location of either the maximum or the minimum force. If we selected to locate and measure the minimum, we can use the result in Fig. 7 to obtain the charge: Fig. 7 shows a graph of min = 02 Fmin . By equating Fmin to the experimental measured value Fmin , one can get the charge from known quantities,

=

0

experimental Fmin min

99

(10)

Journal of Electrostatics 97 (2019) 95–100

Y. Eisenberg, F.R. Zypman

5. Conclusions

[4] N. Sperelakis (Ed.), Cell Physiology, Academic Press, San Diego, California, 1998. [5] Z. Siwy, A. Fulinski, Phys. Rev. Lett. 89 (2002) 198103. [6] D. Lichtenberg, A.E. Filippov, M. Urbakh, Directed Molecular Transport in an Oscillating Symmetric Channel, arXiv:cond-mat/0307158 [cond-mat.soft]. [7] M. Peplow, March of the machines, Nature 525 (2015) 18–21. [8] K. Rykaczewski, A. Marshall, W.B. White, A.G. Fedorov, Dynamic growth of carbon nanopillars and microrings in electron beam induced dissociation of residual hydrocarbons, Ultramicroscopy 108 (2008) 989–992. [9] Lin Wang Zhong, Nanostructures of zinc oxide, Mater. Today 7 (6) (June 2004) 26–33. [10] K.M. Liew, Jian-Wei Yan, Lu-Wen Zhang, Mechanical Behaviors of Carbon Nanotubes: Theoretical and Numerical Approaches, Elsevier, 978-0-323-43137-8, 2017. [11] Z. Zhang, S. Witham, E. Alexov, On the role of electrostatics on protein-protein interactions, Phys. Biol. 8 (2011) 035001https://doi.org/10.1088/1478-3975/8/3/ 035001. [12] Y. Wang, N. Yang, J.-L. Zhu, Aharonov-Bohm phase operations on a double barrier nanoring charge qubit, Phys. Rev. B 74 (2006) 035432. [13] J.D. Jackson, Classical Electrodynamics, third ed., John Wiley and Sons, USA, 1998. [14] D. Lazarev, F.R. Zypman, Determination of charge and size of rings by atomic force microscopy, J. Electrost. 83 (October 2016) 69–72. [15] D. Lazarev, F.R. Zypman, Charge and size of a ring in an electrolyte with atomic force microscopy, J. Electrost. 87 (June 2017) 243–255. [16] David R. Lide, Handbook of Chemistry and Physics, seventy fifth ed., CRC Press, Boca Raton, 1994, p. 45 section 12. [17] S. Bhagavantam, D.A.A.S. Narayana Rao, Dielectric constant of diamond, Nature 161 (1948) 729. [18] I.V. Lindell, Image theory for electrostatic and magnetostatic problems involving a material sphere, Am. J. Phys. 61 (1993) 39. [19] F.R. Zypman, Off-axis electric field of a ring of charge, Am. J. Phys. 74 (4) (2006). [20] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1964 Chapter 17.

In this paper we have considered a ring of charge probed by a dielectric microsphere fitted at the end of an Atomic Force Microscopy cantilever. We have developed the mathematical theory that gives rise to the interaction force between ring and microsphere. Moreover, this force function is found explicitly as depending on the dielectric constant of the microsphere, the ring/microsphere separation and the ring's charge. By studying the properties of this function, we were able to propose rules whereby from experimental data of forces and their locations, the charge and size of the ring can be measured. Acknowledgments This project is funded by the National Science Foundation Grant No. CHE-1508085. References [1] T. Jiang, Y. Zhu, Measuring graphene adhesion using atomic force microscopy with a microsphere tip, Nanoscale 7 (2015) 10760 https://doi.org/10.1039/ C5NR02480C. [2] Novasacan microsphere probes. http://www.novascan.com/products/afm_bead_ sphere_colloid_probes.php. [3] F. Zypman, Nanoparticle charge in fluid from atomic force microscopy forces within the nonlinear Poisson-Boltzmann equation, J. Appl. Math. Phys. 6 (2018) 1315–1323, https://doi.org/10.4236/jamp.2018.66110.

100