Bhushan B. Nanotribology and Nanomechanics: An Introduction

Подождите немного. Документ загружается.

14 Mechanical Properties of Nanostructures 787

51. E. Mazza, J. Dual: Mechanical behavior of a µm-sized single crystal silicon structure

with sharp notches, J. Mech. Phys. Solids 47, 1795–1821 (1999)

52. T. Yi, C.J. Kim: Measurement of mechanical properties for MEMS materials, Meas.

Sci. Technol. 10, 706–716 (1999)

53. H. Kahn, M.A. Huﬀ, A.H. Heuer: Heating eﬀects on the Young’s modulus of ﬁlms

sputtered onto micromachined resonators,Microelectromech. Struct. Mater.Res. Symp.

Proc. 518, 33–38 (1998)

54. S. Johansson, F. Ericson, J.A. Schweitz: Inﬂuence of surface-coatings on elasticity,

residual-stresses, and fracture properties of silicon microelements, J. Appl. Phys. 65,

122–128 (1989)

55. R. Ballarini, R.L. Mullen, Y. Yin, H. Kahn, S. Stemmer, A.H. Heuer: The fracture

toughness of polysilicon microdevices: Aﬁrst report, J.Mater. Res. 12, 915–922 (1997)

56. H. Kahn, R. Ballarini, R.L. Mullen, A.H. Heuer: Electrostatically actuated failure of

microfabricated polysilicon fracture mechanics specimens, Proc. R. Soc. London A

455, 3807–3823 (1999)

57. A.M. Fitzgerald, R.H. Dauskardt, T.W. Kenny: Fracture toughness and crack growth

phenomena of plasma-etched single crystal silicon, Sensor. Actuat. A 83, 194–199

(2000)

58. T. Tsuchiya, A. Inoue, J. Sakata: Tensile testing of insulating thin ﬁlms: Humidity eﬀect

on tensile strength of SiO

ﬁlms, Sens. Actuators A 82, 286–290 (2000)

59. J.A. Connally, S.B. Brown: Micromechanical fatigue testing, Exp. Mech. 33, 81–90

(1993)

60. K. Komai, K. Minoshima, S. Inoue: Fracture and fatigue behavior of single-crystal

silicon microelements and nanoscopic AFM damage evaluation, Microsyst. Technol. 5,

30–37 (1998)

61. T. Namazu, Y. Isono, T. Tanaka: Evaluation of size eﬀect on mechanical properties of

single-crystal silicon by nanoscale bending test using AFM, J. Microelectromech. Syst.

9, 450–459 (2000)

62. S. Sundararajan, B. Bhushan: Development of AFM-based techniques to measure me-

chanical properties of nanoscale structures, Sens. Actuators A 101, 338–351 (2002)

63. X. Li, B. Bhushan: Fatigue studies of nanoscale structures for MEMS/NEMS applica-

tions using nanoindentation techniques, Surf. Coat. Technol. 163-164, 521–526 (2003)

64. X. Li, B. Bhushan, K. Takashima, C.W. Baek, Y.K. Kim: Mechanical characteriza-

tion of micro/nanoscale structures for MEMS/NEMS applications using nanoindenta-

tion techniques, Ultramicroscopy 97, 481–494 (2003)

65. G. Wei, B. Bhushan, N. Ferrell, D. Hansford: Microfabrication and nanomechanical

characterization of polymer MEMS for biological applications, J. Vac. Sci. Technol. A

23, 811–819 (2005)

66. T. Hsu, N. Sun: Residual stresses/strains analysis of MEMS. In: Proc. Int. Conf. on

Modeling and Simulation of Microsystems, Semiconductors, Sensors and Actuators,ed.

by M. Laudon, B. Romanowicz (Computational Publications, Cambridge 1998) pp.82–

67. A. Kolpekwar, C. Kellen, R.D. (Shawn) Blanton: Fault model generation for MEMS.

In: Proc. Int. Conf. on Modeling and Simulation of Microsystems, Semiconductors, Sen-

sors and Actuators, ed. by M. Laudon, B. Romanowicz (Computational Publications,

Cambridge 1998) pp.111–116

68. H.A. Rueda, M.E. Law: Modeling of strain in boron-doped silicon cantilevers.In: Proc.

Int. Conf. on Modeling and Simulation of Microsystems, Semiconductors, Sensors and

Actuators, ed. by M. Laudon, B. Romanowicz (Computational Publications, Cambridge

1998) pp.94–99

788 Bharat Bhushan

69. M. Heinzelmann, M. Petzold: FEM analysis of microbeam bending experiments using

ultra-micro indentation, Comput. Mater. Sci. 3, 169–176 (1994)

70. C.J. Wilson, P.A. Beck: Fracture testing of bulk silicon microcantilever beams sub-

jected to a side load, J. Microelectromech. Syst. 5, 142–150 (1996)

71. B. Bhushan, G.B. Agrawal: Stress analysis of nanostructures using a ﬁnite element

method, Nanotechnology 13, 515–523 (2002)

72. B. Bhushan, G.B. Agrawal: Finite element analysis of nanostructures with roughness

and scratches, Ultramicroscopy 97, 495–501 (2003)

73. K.E. Petersen: Silicon as a mechanical material, Proc. IEEE 70, 420–457 (1982)

74. B. Bhushan, S. Sundararajan, X. Li, C.A. Zorman, M. Mehregany: Mi-

cro/nanotribological studies of single-crystal silicon and polysilicon and SiC ﬁlms

for use in MEMS devices. In: Tribology Issues and Opportunities in MEMS, ed. by

B. Bhushan (Kluwer, Dordrecht 1998) pp.407–430

75. S. Sundararajan, B. Bhushan: Micro/nanotribological studies of polysilicon and SiC

ﬁlms for MEMS applications, Wear 217, 251–261 (1998)

76. X. Li, B. Bhushan: Micro/nanomechanical characterization of ceramic ﬁlms for mi-

crodevices, Thin Solid Films 340, 210–217 (1999)

77. H: Becker, C. Gaertner: Polymer Microfabrication Methods for Microﬂuidic Analytical

Applications, Electrophoresis 21, 12–26 (2000)

78. J.C. McDonald, D.C. Duﬀy, J.R. Anderson, D.T. Chiu, H. Wu, O.J.A. Schueller,

G.M Whitesides: Fabrication of microﬂuidic systems in poly(dimethylsiloxane), Elec-

trophoresis 21, 27–40 (2000)

79. B. Ellis: Polymers: A Property Database, Available on compact disk (CRC Press, Boca

Raton 2000) also see www.polymersdatabase.com

80. J. Brandrup, E.H. Immergut: Polymer Handbook (Wiley, New York 1989)

81. J.E. Mark: Polymers Data Handbook (Oxford Univ. Press, Oxford 1999)

82. B. Bhushan, X. Li: Nanomechanical characterization of solid surfaces and thin ﬁlms,

Int. Mater. Rev. 48, 125–164 (2003)

83. B.R.Lawn,A.G.Evans,D.B.Marshall:Elastic/plastic indentation damage in ceram-

ics: The median/radial system, J. Am. Ceram. Soc. 63, 574 (1980)

84. S. Sundararajan, B. Bhushan, T. Namazu, Y. Isono: Mechanical property measurements

of nanoscale structures using an atomic force microscope, Ultramicroscopy 91, 111–

118 (2002)

85. R.J. Roark: Formulas for Stress and Strain, 6th edn. (McGraw-Hill, New York 1989)

86. R.W. Hertzberg: Deformation and Fracture Mechanics of Engineering Materials,3rd

edn. (Wiley, New York 1989) pp.277–278

87. T.K. Ning (ed): Properties of Silicon, EMIS Datareviews Series No. 4 (INSPEC, Insti-

tution of Electrical Engineers, London 1988)

88. C.T.C. Nguyen, R.T. Howe: An integrated CMOS micromechanical resonator high-Q

oscillator, IEEE J. Solid-State Circ. 34, 440–455 (1999)

89. L.J. Hornbeck: A digital light processing

update – status and future applications,

Proc. Soc. Photo-Opt. Eng. 3634, 158–170 (1999)

90. M. Tanaka: Fracture toughness and crack morphology in indentation fracture of brittle

materials, J. Mater. Sci. 31, 749 (1996)

91. B. Bhushan, S. Venkatesan: Mechanical and tribological properties of silicon for mi-

cromechanical applications: A review, Adv. Info. Stor. Syst. 5, 211–239 (1993)

92. B. Bhushan, B.K. Gupta: Handbook of Tribology: Materials, Coatings, and Surface

Treatments (McGraw-Hill, New York 1997)

14 Mechanical Properties of Nanostructures 789

93. T. Tsuchiya, O. Tabata, J. Sakata, Y. Taga: Specimen size eﬀect on tensile strength of

surface-micromachined polycrystalline silicon thin ﬁlms, J. Microelectromech. Syst. 7,

106–113 (1998)

94. T. Yi, L. Li, C.J. Kim: Microscale material testing of single crystalline silicon: Process

eﬀects on surface morphology and tensile strength, Sens. Actuators A 83, 172–178

(2000)

95. F.W.J. Billimeyer: Textbook of Polymer Science (Wiley, New York 1984)

96. Anonymous: Rohm and Haas General Information on PMMA

97. W. Wunderlich (ed): Physical Constants of Poly(methyl methacrylate), 2nd edn. (Wiley,

New York 1975)

98. E. Calvet, J.P. Bros, H. Pinelle: Compt. Rend. 260, 1164 (1965)

99. K. Eiermann: Kolloid-Z. 198, 5 (1965)

100. G. Hochstetter, A. Jimenez, J.P. Cano, E. Felder: An attempt to determine the true

stress-strain curves of amorphous polymer by nanoindentation, Tribol. Inter. 36, 973–

985 (2003)

101. S.P. Timoshenko, J.N. Goodier: Theory of Elasticity, 3rd edn. (McGraw-Hill, New

York 1970)

102. J.E. Shigley, L.D. Mitchell: Mechanical Engineering Design, 4th edn. (McGraw-Hill,

New York 1993)

Scale Eﬀect in Mechanical Properties and Tribology

Bharat Bhushan and Michael Nosonovsky

Summary. A model, which explains scale eﬀects in mechanical properties and tribology is

presented. Mechanical properties are scale dependent based on the strain gradient plastic-

ity and the eﬀect of dislocation-assisted sliding. Both single asperity and multiple asperity

contacts are considered. The relevant scaling length is the nominal contact length – contact

diameter for a single-asperity contact, and scan length for multiple-asperity contacts. For

multiple asperity contacts, based on an empirical power-rule for scale dependence of rough-

ness, contact parameters are calculated. The eﬀect of load on the contact parameters and

the coeﬃcient of friction is also considered. During sliding, adhesion and two- and three-

body deformation, as well as ratchet mechanism, contribute to the dry friction force. These

components of the friction force depend on the relevant real areas of contact (dependent on

roughness and mechanical properties), average asperity slope, number of trapped particles,

and shear strength during sliding. Scale dependence of the components of the coeﬃcient of

friction is studied. A scaledependent transition index, which is responsible fortransition from

predominantly elastic adhesion to plastic deformation has been proposed. Scale dependence

of the wet friction, wear, and interface temperature has been also analyzed. The proposed

model is used to explain the trends in the experimental data for various materials at nanoscale

and microscale, which indicate that nanoscale values of coeﬃcient of friction are lower than

the microscale values due to an increase of the three-body deformation and transition from

elastic adhesive contact to plastic deformation.

15.1 Nomenclature

a, a, a

, a

max

, a

max

: Contact radius, mean contact radius, macroscale value of mean

contact radius, maximum contact radius, mean value of maximum contact radius.

, A

re0

, A

rp0

, A

: Apparent area of contact, real area of con-

tact, real area of contact during adhesion, real area of elastic contact, macroscale

value of real area of elastic contact, real area of plastic contact, macroscale value of

real area of plastic contact, real area of contact during asperity summit deformation,

area of contact with particles

b: Burgers vector

c: Constant, speciﬁed by crystal structure

792 Bharat Bhushan and Michael Nosonovsky

: Constant required for normalization of p(d)

d, d

, d

, d, d

: Particle diameter, minimum for exponential distribution, mean

for normal distribution, exponential of mean of ln(d) for log-normal distribution,

mean trapped particles diameter, macroscale value of mean trapped particles diam-

eter

D: Interface zone thickness

, E

∗

: Elastic moduli of contacting bodies, eﬀective elastic modulus

F, F

, F

: Friction force, friction force due to ad-

hesion, friction force due to deformation, friction force during elastic adhesional

contact, plastic adhesional contact, summit deformation, particles deformation re-

spectively, meniscus force for wet contact, macroscale value of meniscus force

G: Elastic shear modulus

h: Indentation depth

: Liquid ﬁlm thickness

H, H

: Hardness, hardness in absence of strain gradient

k, k

: Wear coeﬃcient, macroscale value of wear coeﬃcient

, l

: Material-speciﬁc characteristic length parameters

L, L

lwl

, L

: Length of the nominal contact zone, long wavelength limit for

roughness parameters, long wavelength limit for contact parameters, length param-

eters related to l

and l

: Peclet number

m,n: Indices of exponents for scale-dependence of σ and β

∗

: Number of trapped particles divided by the total number of particles

, p

: Apparent pressure, critical apparent pressure

p(d), p

(d): Probability density function for particle size distribution, probability

density function for trapped particle size distribution

P(d): Cumulative probability distribution for particle size

R, R

, R

:Eﬀective radius of summit tips, radius of summit tip, mean radius of

summit tips, macroscale value of the mean radius of summit tips

R(τ): Autocorrelation function

s: Spacing between slip steps on the indentation surface

: Separation distance between reference planes of two surfaces in contact

N, N

: Total number of contacts, macroscale value of total number of contacts

T, T

: Maximum ﬂash temperature rise, macroscale value of temperature rise

x: Sliding distance

v: Volume of worn material

V: Sliding velocity

W: Normal load

z, z

min

, z

max

: Random variable, minimum and maximum value of z

α: Probability for a particle in the border zone to leave the contact region

∗

, β

∗

: Correlation length, macroscale value of correlation length

γ: Surface tension

Γ: Gamma function

ε:Strain

15 Scale Eﬀect in Mechanical Properties and Tribology 793

η: Density of particles per apparent area of contact

int

, η

: Density of dislocation lines per interfacearea, criticaldensity of dislocation

lines per interface area

κ:Curvature

: Thermal diﬀusivity

θ: Contact angle between the liquid and surface

: Indentation angle

: Roughness angle

μ, μ

, μ

ae0

, μ

ap0

, μ

ds0

, μ

dp0

, μ

re0

, μ

rp0

, μ

wet

:Co-

eﬃcient of friction, coeﬃcient of adhesional friction, coeﬃcient of adhesional elas-

tic friction, macroscale value of coeﬃcient of adhesional elastic friction, coeﬃcient

of adhesional plastic friction, macroscale value of coeﬃcient of adhesional plastic

friction, coeﬃcient of deformation friction, coeﬃcient of summits deformation fric-

tion, macroscale value of coeﬃcient of summits deformation friction, coeﬃcient of

particles deformation friction, macroscale value of coeﬃcient of particles deforma-

tion friction, ratchet component of the coeﬃcient of friction, macroscale value of

ratchet component of the coeﬃcient of friction, ratchet componentof the coeﬃcient

of elastic friction,macroscale value ofratchet componentof the coeﬃcient of elastic

friction, ratchet component of the coeﬃcient of plastic friction, macroscale value of

ratchet component of the coeﬃcient of plastic friction, and coeﬃcient of wet fric-

tion

, ν

: Poisson’s ratios of contacting bodies

ρc

: Volumetric speciﬁc heat

σ, σ

, σ

: Standard deviation of rough surface proﬁle height, macroscale

value of standard deviation of rough surface proﬁle height, standard deviation for

the exponential distribution, standard deviation for the normal distributions, stan-

dard deviation for ln(d) of the log normal distribution

ρ, ρ

, ρ

: Total densityof dislocationlines per volume,density of GND per volume,

density of SSD per volume

φ, φ

: Transition index, macroscale value of transition index

τ, τ

: Spatial parameter, value at which the autocorrelation function decays

, τ

ds0

, τ

dp0

, τ

: Adhesional shear strength during sliding,

macroscale value of adhesional shear strength, shear yield strength, shear yield

strength in absence of strain gradient, shear strength during summits deformation,

macroscale value of shear strength during summits deformation, shear strength dur-

ing particles deformation, macroscale valueof shear strength during particles defor-

mation, Peierls stress

15.2 Introduction

Microscale and nanoscale measurements of tribological properties, which became

possible due to the development of the Surface Force Apparatus (SFA), Atomic

Force Microscope(AFM), and Friction Force Microscope (FFM) demonstrate scale

794 Bharat Bhushan and Michael Nosonovsky

dependence of adhesion, friction, and wear as well as mechanical properties includ-

ing hardness [1–4]. Advances of micro/nanoelectromechanical systems (MEMS/

NEMS) technology in the past decade make understanding of scale eﬀects in

adhesion, friction, and wear especially important, since surface to volume ra-

tio grows with miniaturization and surface phenomena dominate. Dimensions of

MEMS/NEMS devices range from about 1 mm to few nm.

Experimental studies of scale dependence of tribological phenomena have been

conducted recently. AFM experiments provide data on nanoscale [5–10] whereas

microtriboapparatus[11,12] and SFA [13] provide data on microscale. Experimen-

tal data indicate that wear mechanisms and wear rates are diﬀerent at macro- and

micro/nanoscales [14,15]. During sliding, the eﬀect of operating conditions such

as load and velocity on friction and wear are frequently manifestations of the eﬀect

of temperature rise on the variable under study. The overall interface temperature

rise is a cumulativeresult of numerous ﬂash temperature rises at individual asperity

contacts. The temperature rise at each contact is expected to be scale dependent,

since it depends on contact size, which is scale dependent.

Friction is a complex phenomenon, which involves asperity interactions involv-

ing adhesion and deformation (plowing), Fig. 15.1. Adhesion and plastic deforma-

tion imply energy dissipation, which is responsible for friction. A contact between

two bodies takes place on high asperities, and the real area of contact (A

)isasmall

fraction of the apparent area of contact [16]. During contact of two asperities, a lat-

eral force may be required for asperities of a given slope to climb against each other.

This mechanism is known as ratchet mechanism, and it also contributes to the fric-

tion. Wear and contaminant particles present at the interface, referred as the “third

body”, also contributeto friction,Fig. 15.2a.In addition,during contact, even at low

humidity, a meniscus is formed. Generally, any liquid that wets or has a small con-

tact angle on surfaces will condense from vapor into cracks and pores on surfaces

as bulk liquid and in the form of annular-shaped capillary condensate in the contact

zone. Figure 15.2b shows a random rough surface in contact with a smooth surface

with a continuousliquid ﬁlm on the smooth surface. The presence of the liquid ﬁlm

of the condensate or pre-existing ﬁlm of the liquid can signiﬁcantly increase the

adhesion between the solid bodies [16]. The eﬀect of meniscus is scale-dependent.

A quantitative theory of scale eﬀects in friction should consider scale eﬀect on

physical properties relevant to these contributions. However, conventional theories

of contactand frictionlack characteristiclength parameters, whichwouldbe respon-

sible for scale eﬀects. The linear elasticity and conventional plasticity theories are

scale-invariant and do not include any material length scales. A strain gradient plas-

ticity theory has been developed, for microscale deformations, by Fleck et al. [17],

Nix and Gao [18] and Hutchinson[19]. Theirtheory predictsa dependenceof mech-

anical properties on the strain gradient, which is scale dependent: the smaller is the

size of the deformed region, the greater is the gradient of plastic strain, and, the

greater is the yield strength and hardness.

A comprehensive model of scale eﬀect in friction including adhesion, two- and

three-body deformations and the ratchet mechanism, has recently been proposed by

15 Scale Eﬀect in Mechanical Properties and Tribology 795

Friction mechanisms

Dry friction Wet friction

Adhesion

Deformation

(plowing)

Ratchet

Two-body Three-body

Elastic Plastic

Generation of dislocations

Statistically stored

dislocations

Geometrically

necessary dislocations

Propagation of dislocations

Gliding in bulk

and interface

Climbing in bulk

Rough surface

Single asperity

(Hertzian)

Multiple asperities

Statistical FractalNumerical

Fig. 15.1. (a) A block diagram

showing friction mechanisms

and generation and propaga-

tion of dislocations during

sliding, (b) a block diagram

of rough contact models

Bhushan and Nosonovsky [20–22] and Nosonovsky and Bhushan [23]. The model

for adhesional friction during single and multiple asperity contact was developed

by Bhushan and Nosonovsky [20] and is based on the strain gradient plasticity and

dislocation assisted sliding (gliding dislocations at the interface or microslip). The

model for the two-body and three-body deformation was proposed by Bhushan and

Nosonovsky [21] and for the ratchet mechanism by Nosonovsky and Bhushan [23].

The model has been extended for wet contacts, wear and interface temperature by

Bhushan and Nosonovsky [22]. The detailed model is presented in this chapter.

The chapter is organized as follows. In the next section of this chapter, the scale

eﬀect in mechanical properties is considered, including yield strength and hard-

ness based on the strain gradient plasticity and shear strength at the interface based

on the dislocation assisted sliding (microslip). In the fourth section, scale eﬀect in

surface roughness and contact parameters is considered, including the real area of

contact, number of contacts, and mean size of contact. Load dependence of contact

parameters is also studied in this section. In the ﬁfth section, scale eﬀect in friction

is considered, including adhesion, two- and three-body deformation, ratchet mech-

796 Bharat Bhushan and Michael Nosonovsky

Solid-solid contact

Two body contact Three body contact

Plowing during sliding

Reference plane

of rough surface

W Loaded

surface

Rough

surface

Meniscus around

noncontacting asperity

Smooth surface

Meniscus around

contacting asperity

Liquid thin film

Contact area

Meniscus area

Meniscus

)

Fig. 15.2. Schematics of

(a) two-bodies and three-

bodies during dry contact of

rough surfaces, (b) formation

of menisci during wet contact

anism, meniscus analysis, total value of the coeﬃcient of friction and comparison

with the experimental data. In the sixth and seventh sections, scale eﬀects in wear

and interface temperature are analyzed, respectively.

15.3 Scale Eﬀect in Mechanical Properties

In this section, scale dependence of hardness and shear strength at the interface is

considered. A strain gradient plasticity theory has been developed, for microscale

15 Scale Eﬀect in Mechanical Properties and Tribology 797

deformations, by Fleck et al. [17], Nix and Gao [18], Hutchinson [19], and others,

which is based on statistically stored and geometrically necessary dislocations (to be

described later). Their theory predicts a dependence of mechanical properties on the

strain gradient, which is scale dependent: the smaller is the size of the deformed re-

gion, the greater is the gradient of plastic strain, and, the greater is the yield strength

and hardness. Gao et al. [24] and Huang et al. [25] proposed a mechanism-based

strain gradient (MSG) plasticity theory, which is based on a multiscale framework,

linking the microscale (10–100nm) notion of statistically stored and geometrically

necessary dislocationsto the mesoscale (1–10µm) notion of plastic strain and strain

gradient. Bazant [26] analyzed scale eﬀect based on the MSG plasticity theory in

the limit of small scale, and found that corresponding nominal stresses in geomet-

rically similar structures of diﬀerent sizes depend on the size according to a power

exponent law.

It was recently suggested also, that relative motion of two contacting bodies dur-

ing sliding takes place due to dislocation-assisted sliding (microslip), which results

in scale-dependent shear strength at the interface [20]. Scale eﬀects in mechanical

properties (yield strength, hardness, and shear strength at the interface) based on the

strain gradient plasticity and dislocation-assisted sliding models are considered in

this section.

15.3.1 Yield Strength and Hardness

Plastic deformation occurs during asperity contacts because a small real area of

contact results in high contact stresses, which are often beyond the limits of the

elasticity. As stated earlier, during loading, generation and propagation of dislo-

cations is responsible for plastic deformation. Because dislocation motion is irre-

versible, plastic deformation provides a mechanism for energy dissipation during

friction. The strain gradient plasticity theories [17–19] consider two types of dis-

locations: randomly created Statistically Stored Dislocations (SSD) and Geomet-

rically Necessary Dislocations (GND). The GND are required for strain compati-

bility reasons. Randomly created SSD during shear and GND during bending are

presented in Fig. 15.3a. The density of the GND (total length of dislocation lines

per volume) during bending is proportional to the curvature κ andtothestraingra-

dient

∂ε

∂z

∝∇ε, (15.1)

where ε is strain, b is the Burgers vector, and ∇ε is the strain gradient.

The GND during indentation, Fig. 15.3b, are located in a certain sub-surface

volume. The large strain gradients in small indentations require GND to account for

the large slope at the indented surface. SSD, not shown here, also would be created

and would contribute to deformation resistance, and are function of strain rather

than strain gradient. According to Nix and Gao [18], we assume that indentation is

accommodated by circular loops of GND with Burgers vector normal to the plane