# Vacuum attraction, friction and heating of nanoparticles moving nearby a heated surface

**ABSTRACT** We review the fluctuation electromagnetic theory of attraction, friction and heating of neutral nonmagnetic nanoparticles moving with constant velocity in close vicinity to the solid surface. The theory is based on an exact solution of the relativistic problem of the fluctuation electromagnetic interaction in configuration sphere -plane in the dipole approximation.

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**ABSTRACT:**We calculate heating rate, attractive conservative and tangential dissipative fluctuation electromagnetic forces felt by a thick plate moving with nonrelativistic velocity parallel to a closely spaced another plate in rest using relativistic fluctuation electrodynamics. We argue that recently developed relativistic out of equilibrium theory of fluctuation electromagnetic interactions [A.I. Volokitin, B.N.J. Persson, Phys. Rev. B78 (2008) 155437; arXiv:/cond-mat.other/0807.1004v1, 2008] has serious drawbacks.Surface Science 04/2009; · 1.87 Impact Factor - SourceAvailable from: George Dedkov[Show abstract] [Hide abstract]

**ABSTRACT:**The heating rate and conservative—dissipative forces in a system of two parallel plates moving at a nonrelativistic velocity with respect to each other have been calculated in terms of the fluctuation electromagnetic electrodynamics for the first time. It has been demonstrated that the recently proposed relativistic theory of fluctuation electromagnetic interactions in the configuration under consideration offers fundamental disadvantages.Physics of the Solid State 02/2010; 52(2):409-418. · 0.77 Impact Factor -
##### Article: Friction and energy dissipation mechanisms in adsorbed molecules and molecularly thin films

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**ABSTRACT:**This review provides an overview of recent advances that have been achieved in understanding the basic physics of friction and energy dissipation in molecularly thin adsorbed films and the associated impact on friction at microscopic and macroscopic length scales. Topics covered include a historical overview of the fundamental understanding of macroscopic friction, theoretical treatments of phononic and electronic energy dissipation mechanisms in thin films, and current experimental methods capable of probing such phenomena. Measurements performed on adsorbates sliding in unconfined geometries with the quartz crystal microbalance technique receive particular attention. The final sections review the experimental literature of how measurements of sliding friction in thin films reveal energy dissipation mechanisms and how the results can be linked to film-spreading behavior, lubrication, film phase transitions, superconductivity-dependent friction, and microelectromechanical systems applications. Materials systems reported on include adsorbed films comprised of helium, neon, argon, krypton, xenon, water, oxygen, nitrogen, carbon monoxide, ethane, ethanol, trifluoroethanol, methanol, cyclohexane, ethylene, pentanol, toluene, tricresylphosphate, t-butylphenyl phosphate, benzene, and iodobenzene. Substrates reported on include silver, gold, aluminum, copper, nickel, lead, silicon, graphite, graphene, fullerenes, C60, diamond, carbon, diamond-like carbon, and YBa2Cu3O7, and self-assembled monolayers consisting of tethered polymeric molecules.Advances In Physics 01/2012; 61(3):155-323. · 34.29 Impact Factor

Page 1

Vacuum Attraction, Friction and Heating of Nanoparticles moving

nearby a heated Surface

G.V.Dedkov, A.A.Kyasov

Nanoscale Physics Group, Kabardino –Balkarian State University, Nalchik, 360004,

Russian Federation; e –mail: gv_dedkov@mail.ru

We review the fluctuation electromagnetic theory of attraction, friction and heating of neutral

non –magnetic nanoparticles moving with constant velocity in close vicinity to the solid surface.

The theory is based on exact solution of the relativistic problem of fluctuation electromagnetic

interaction in configuration sphere –plane in dipole approximation.

PACS: 41.20.q; 65.80.+n; 81.40.Pq

1. Introduction

Vacuum attraction, friction and heating of neutral non –magnetic particles moving nearby the

solid surface are the most accessible effects of electromagnetic fluctuations. Often, the

corresponding problems are considered separately each other despite their common physical

origin. Thus, vacuum attraction force between the bodies caused by fluctuation electromagnetic

interaction (FEI) is known as van –der –Waals (Casimir) force [1,2], vacuum friction force is a

dissipative component of this force [3,4], and heating (cooling) of a body in thermal field of

another body is a manifestation of radiation heat transfer mediated by evanescent

electromagnetic waves or by propagating waves [5,6]. In the latter case massive black bodies

emit thermal electromagnetic radiation according to the Stefan –Boltzman law. For particles with

dimensions

T

D

λ<

(

T λ is the wave length of the thermal radiation), or being situated nearby a

heated surface at a distance

T

d

λ<

, the radiation heat transfer via the evanescent waves plays a

dominating role.

To date, the field of FEI has a lot of puzzles, especially in the friction problem (see, for

instance, discussions in [7-9]). And even in the case of conservative van –der –Waals and

Casimir forces, some new issues have attracted enormous efforts (see, for example, [10,11] and

references there in). Quite recently, some aspects of the aforementioned problems have been

reviewed in Refs. [9,12], but the authors did not give necessary attention to several important

results obtained by us in [7,8,13-16]. Thus, the criticism raised in [9,17] (see also references

Page 2

2

there in), referring to our earlier publications [18,19] is now of only historical value. Therefore,

the basic aim of this paper is to picturing our present day understanding of the subject.

We treat all the problems of attraction, friction and heating of a small particle using a closed

relativistic theory of FEI developed in [13,14]. This allows to reproduce not only all existing

results being addressed to the involved nonrelativistic statement of the problem [4,7-9, 12, 20-

23], but also, to get a lot of new results, greatly improving general understanding of the subject.

Our approach is based on relativistic and fluctuating electrodynamics, using local approximation

to the dielectric and magnetic permeability functions of contacting materials. A moving particle

has both electric and magnetic fluctuating dipole moments in its rest frame. A fluctuating

magnetic moment appreciably influences FEI and appears even on a resting nonmagnetic object

due to stochastic Foucault currents being induced by random external magnetic fields penetrating

inside a volume of an object having non –zero magnetic polarizability [24]. An advantage of the

relativistic framework is due to an automatic incorporation of retardation, magnetic and thermal

effects which manifest strong and nontrivial interplay.

Due to the lack of space, we leave aside another approach to the problems of attraction, friction

and heating, which is related with the geometry of two parallel plates in rest or in relative

motion. The necessary information can be found in Refs. [9,10,12, 30].

The paper is organized as follows :

In sect.2 we summarize the problem statement, main definitions and assumptions relevant to

further theoretical consideration.

In sect.3 we obtain an equivalent form for the Lorentz electromagnetic force and heating rate

of a particle embedded into the fluctuating electromagnetic field.

In sect.4 we derive relativistic formulas for the conservative and dissipative forces on a

moving particle and its heating rate.

In sect.5 we get the expression for an attractive nonrelativistic force (Casimir force)

In sect.6 we obtain the expression for the retarded vacuum friction force up to the terms linear

in velocity .

In sect.7 we obtain the expression for the heat flux between a resting particle and the surface.

In sect. 8 we consider application of the results to the problems of damping motion of an

AFM tip, friction and heating of nano- and micro –sized particles in the nonretarded and

retarded regimes of interaction with metallic surfaces.

Finally, in sect.9 we present concluding remarks.

2. Problem statement

Page 3

3

Let us consider a small –sized neutral spherical particle of radius R that moves in vacuum

parallel to the boundary of a semi –infinite medium at a distance

0z from the boundary (Fig.1).

The particle velocity V is directed along the

−

x

axis and is arbitrary. The half –space

0

<

z

is

filled by homogeneous and isotropic medium characterized by complex dielectric permittivity ε

and permeability µ depending on the frequency ω . It is assumed that the particle has the

temperature

1 T and the medium and surrounding electromagnetic vacuum background -

2 T . The

global system is out of thermal equilibrium, but in a stationary regime. Also, it is assumed that

the particle is non –magnetic in its rest frame and has the electric and magnetic polarizabilities

)(ωαe

and )(ωαm

. The laboratory system K is related to the resting surface and the coordinate

system K′ is related to the moving particle. In dipole approximation

Rz >>

0

, the particle can

be considered as a point fluctuating dipole with random electric and magnetic moments

( ),

t

( )

t

dm

. Our aim is to determine the force of FEI applied to a moving particle and its heating

rate in the process of radiation heat transfer.

3. Fluctuation electromagnetic force and heating rate: general relations

First of all, we obtain several relationships corresponding to the fluctuation electromagnetic

force and rate of radiation heating of a particle embedded into electromagnetic field. In the

−

K

system (Fig.1), vectors of electric and magnetic polarization of a moving particle laboratory

are

( , )

P r

( ) (

t

δ

)

tt

=−

drV (1)

( , )( ) (

t

δ

)

tt

=−

M rmrV (2)

Using (1), (2), the Maxwell equations

1

c

rot , div0

t

∂

∂

= −=

B

EB

and general relations for the

charge and current densities,

div , rot

c

t

ρ

∂

∂

= −=+ ⋅

P

P jM, the expression for an averaged Lorentz

force can be represented in the form [16]

()()( )()()()

33

1

c

1

c

1

c

1

c dt

d r d r

d

t

ρ=+×=

∂

∂

= ∇++×+∇× = ∇++×

∫∫

FE j B

dE mB d BV d B dE mB d B

(3)

where the angular brackets ... imply total quantum and statistical averaging and the integrals

are taken over the particle volume. If the electromagnetic field is non fluctuating, all the terms in

(3) must be taken without averaging. In the case of stationary fluctuations of electromagnetic

field, the last term of (3) equals zero and we get

Page 4

4

()

= ∇+

F dE mB (4)

The heating rate of a particle in the K′ system is, obviously, given by the dissipation integral

′

′′′

=⋅

′ ∫j E

3

dQ

dt

d r

(5)

Furthermore, making use of the relativistic transformations for the current density, electric field

and volume in the integrand (5), we obtain [14]

(

∫∫

)

323

d r d r

γ

′′′

⋅=⋅−⋅

j Ej E F V

(6)

where

2 / 1

−

2) 1 (

−=

βγ

is the Lorentz –factor,

cV /

=

β

and F is given by (4). According to the

Planck’s formulation of relativistic thermodynamics,

dt dQt d

/

Qd

/

2

γ

=

′′

[26], and Eq.(6) yields

3

dQ

dt

d r

=⋅−⋅

∫j E F V (7)

Formula (7) has clear physical meaning: the energy dissipated is transformed into heat and work

of the fluctuation electromagnetic force performed over the particle. The dissipation integral in

(7) can be rewritten by analogy with the Lorentz force (the points above vectors d and m denote

the time differentiation):

(

(

)

)

()()

()

333

rot

d rd rc d r

t

d

dt

dd

dt c dt

∂

∂

⋅=⋅+⋅=

=⋅+⋅+ ⋅∇⋅+⋅−⋅=

=⋅+⋅+⋅−⋅−×

∫∫∫

P

j EE M E

d E m B

&

V d E m B m B

V

d E m B

&

V Fm B d B

&

&

(8)

Again, assuming the case of stationary fluctuations, the last two terms in (8) are equal to zero

after averaging, and from (7) and (8) we finally get

/

dQ dt =⋅+⋅

d E m B

&

&

(9)

Moreover, it is not difficult to show that in the K′ system the form of Eq.(9) does not change:

/

dQ dt

′′′′′′

=⋅+⋅

d E

&

m B

&

. Eqs. (4) and (9) are the primary ones in further calculations.

4. General relativistic expressions for conservative and dissipative

components of the interaction force and heating rate

Following our method [7,8,13,14], the quantities in the right hand sides of (4) and (9) have to be

conveniently written as the products of spontaneous and induced terms, i.e.

()

sp ind ind spsp ind indsp

= ∇⋅+⋅+⋅+⋅

FdEdEmBmB

(10)

spind indspspind ind sp

/

dQ dt =⋅+⋅+⋅+⋅

d

&

Ed

&

Em

&

Bm

&

B

(11)

Page 5

5

The projections of the force

z F and

x

F onto directions

xz, of the system K correspond to the

attractive conservative (Casimir) and dissipative (frictional) force. Important technical details of

these calculations can be found in [13,14], thus we only briefly touch upon the main ideas. So,

all the quantities in the right hand side of (10), (11) have to be Fourier transformed over the time

and space variables

yxt

,,

. The induced field components

ind ind

,

EB

must satisfy the necessary

boundary conditions at

0

=

z

and Maxwell equations, which contain the extraneous fluctuating

currents induced by the fluctuating moments

sp sp

,

dm . The involved correlators of dipole

moments are then calculated using the fluctuation –dissipation theorem [2]. The induced dipole

moments

indind

,

dm

have to be expressed through the fluctuating fields

spsp

,

EB , being

composed of the sum of the contributions from the surface and vacuum background. The induced

dipole moments are related with electric and magnetic polarizabilities of the paricle and the

fields

sp sp

,

EB via linear integral relations. Arising correlators are calculated with the help of the

components of the retarded photon Green function [2]. Finally, after some tedious but

straightforward algebra, the needed components of the interaction force and rate of the particle

heating are given by

Page 6

6

()()

()()

()()

zxy0

2

/

B 1

k T

γ ω

h

B 1

k T

ω

h

B 2

k T

ω

h

exp( 2)

coth Re( , )

ω

Re( , )

ω

2

coth Re( , )

ω

Re ( , )

ω

2

cothIm( , )

ω

Im( , )

ω

2

cot

kc

eemm

eemm

eemm

Fddk dk

ω

q z

RR

RR

RR

ω

>

+

γ

h

π

γω

α

⋅

γωα γω

α

⋅

γωα γω

α

⋅

γωα γω

++++

−

−−−−

++++

= −−⋅

′′′′

++

′′′′

+++

⋅

′′

+++

+

∫∫∫

kk

kk

kk

h

()()

{}

()

B 2

k T

xy0

2

/

xy0

2

/

B 1

k T

γ ω

h

h Im( , )

ω

Im( , )

ω

2

cos(2),,

sin( 2)

cothIm ( , )

ω

Im( , )

ω

2

eemm

emem

kc

kc

γω

eem

RR

d dk dk

ω

q z

%

RRR

%

R

%

d dk dk

ω

q z

%

R

%

R

%

ω<

∫∫∫

ω<

α⋅ γωα γω

γ

h

π

h

γ

π

+

−

−

α

⋅

γω

−−−−

±±±±

+

+++

′′

+

−⋅→−

−−⋅

′′

+

⋅

∫∫∫

kk

kk

h

()

()()

()()

()()

B 1

k T

ω

h

B 2

k T

ω

h

B 2

k T

coth Im( , )

ω

Im( , )

ω

2

coth Re( , )

ω

Re ( , )

ω

2

coth Re( , )

ω

Re( , )

ω

2

m

eemm

eemm

eemm

R

%

R

%

R

%

R

%

R

%

R

%

α γω

α⋅ γωα γω

α⋅ γωα γω

α⋅ γωα γω

+

−

−−−−

++++

−−−−

′′

+

′′′′

++

′′

++

′′

+

kk

kk

kk

(12)

[

α

]

[]

[]

(){}

[]

k

[]

′ ′

m

+

′ ′

e

−

−

′ ′

α

+

′ ′

α

⋅

⋅−

−→−−

−

′ ′

m

+

′ ′

e

−

−

′ ′

α

+

′ ′

α

⋅

⋅−−

−

′ ′

m

+

′ ′

e

+−=

+

−

−

−−

+

+

+

++

<

−

±±±±

<

∫∫∫

ω

−

−

−

−

−−

+

+

+

++

>

−

∞

∫

0

−

∫∫∫

ω

∫∫∫

ω

∫

k

kk

kk

kk

,(

~

R

Re)(),(

~

R

Re)()/,/(

,(

~

R

Re)(),(

~

R

Re)()/,/(

)

~

q

2 cos(

~

q

~

R

,

~

R

,)

~

q

2sin(

~

q

),(Im)(),(Im)()/,/(

),(Im

γ

)(),

ω

( Im

γ

)()/

γ

,/(

)2exp(

)/,/()()()1 (

12

12

/

0

1

0

2

2

/

0

1

0

2

2

12

12

/

0

1

0

2

2

1

1

11211

24

4

ωγωωγωγωω

ωγωαωγωαγωω

ω

d

π

γ

ω

d

π

h

γ

ωωωωω

ωγωαωγωαγωω

ω

d

π

γ

ωωωαωβ ωω

d

π

γ

c

me

mmee

ck

x

meme

ck

x

me

m

R

me

R

e

ck

x

x

TTW

TTW

zkkd

RRzkkd

TTW

RRTTW

zqqkkd

TTWxdxxF

h

h

h

(13)

Page 7

7

[

α

]

W

[]

k

[]

(){}

[

[

]

]

′ ′

m

+

′ ′

e

⋅

−

′ ′

m

+

′ ′

e

⋅

⋅

⋅+

−→−+

−

′ ′

~

R

+

′ ′

e

⋅+

+

′ ′

α

+

′ ′

α

⋅

⋅

⋅−+

−

′ ′

m

+

′ ′

e

+=

+

−

−

−−−

+

+

+

+++

<

W

−

±±±±

<

∫∫∫

ω

−

−

−

−

−−−

+

+

+

+++

>

W

−

∞

∫

0

−

∫∫∫

ω

∫∫∫

ω

∫

kk

kk

k

kk

,(

~

R

Re)(),(

~

R

Re)()/,/(

,(

ω

~

R

Re)(),(

~

R

Re)()/,/(

)

~

q

2 cos(

~

q

,

~

R

,)

~

q

2 sin(

~

q

),( Im)(),( Im)()/,/(

),

ω

( Im

γ

)(),

ω

( Im

γ

)()/

γ

,/(

)2 exp(

)/,/()()() 1 (

12

12

/

0

1

0

2

2

/

0

1

0

2

2

12

12

/

0

1

0

2

2

1

1

11211

34

3

γωαωγωαγωωω

ωγωαωγωαγωωω

ω

d

π

γ

ω

d

π

h

γ

ωωωωω

ωγωαωγωαγωωω

ω

d

π

γ

ωωωαωβ ωω

d

π

γ

c

me

me

ck

meme

ck

mme

e

R

me

R

e

ck

TTW

TT

zkd

RRzkd

TTW

RRTT

zqqkd

TTx dxQ

&

h

h

h

(14)

where h,

B k are the Planck’s and Boltzmann’s constants; the one and doubly primed functions

denotes the corresponding real and imaginary parts,

Vkx

x

±=+=

±

ωωβγωω

, ) 1 (

1

(15)

()()

()(

( )

( )

)

1/21/2

222222222

00xy

1/21/2

222222

/,/,

( ) ( )/ , ( ) ( )/

qkcq

%

ckkkk

qkcq

%

ck

ωω

ε ω µ ω ωε ω µ ω ω

=−=−=+

=−=−

(16)

00

ee

0

q

q

0

( )

( )

( )

ω

, ( )

ω

q

q

q

q

q

q

%

q

q

%

−

+

ω

ε ω

ε ω

µ ω

µ ω

ε ω

ε ω

µ ω

µ ω

q

q

%

±

−

+

−

+

∆=∆

%

=

%%

(17)

00

mm

00

( )

( )

( )

( )

( )

ω

,( )

ω

q

q

q

q

%

2

−

+

∆=∆

%

=

%%

(18)

( )

±

222222

x

2

()

( , )

ω

2( )(1/)

e

kk k c

c

χβω=−−+

k

(19)

2

( )

±

22222

my

2

()

( , )

ω

2(1/)

kk c

c

ω

χβω

±

=−+

k

(20)

( )

±

( )

±

( )

±

( , )

ω

( , )

ω

( )

ω

( , )

ω

( )

ω

eeemm

R

R

~

R

~

R

χχ=∆+∆

kkk

(21)

)(),()(),(),(

)()()(

ωωχωωχω

emmem

∆+∆=

±±±

kkk

(22)

)(

~

∆

),()(

~

∆

),(),(

)()()(

ωωχωωχω

mmeee

+=

±±±

kkk

(23)

)(

~

∆

),()(

~

∆

),(),(

)()()(

ωωχωωχω

emmem

+=

±±±

kkk

(24)

−

=

12

12

2

coth

2

coth)/,/(

Tk

b

Tk

a

TbTaW

BB

hh

(25)

Page 8

8

In addition, it should be noted that all the integrals over the frequency ω are taken in the limits

(0, )

quadrant. The structure of the integrand functions within the brackets {

∞ , while over the projections

,

xy

k k of the planar wave vector k –in the first coordinate

}

±±±±

→

meme

RRRR

~

,

~

,

is

identical (with account of the replacement to be done) to the structure of the corresponding

functions within the figure brackets of the integrals at

ck

/

ω

>

. In definitions frequently used

by other authors, the coefficients

e

∆ ,

m

∆ ,

em

,

∆ ∆

%% coincide with the reflection factors for

electromagnetic waves with different polarization [4, 9,10].

The first integral terms in (13), (14) describe interaction of a particle with vacuum

background of temperature

2 T and do not depend on the distance to the surface z. For the first

time, they have been obtained in [15] without magnetic polarization terms. The second and third

terms (and both terms in (12)) depend on z and describe interaction with the surface. In this

case, the integrals computed over the domain of wave vectors /

kc

ω

>

ω

are related with

evanescent surface modes, while the integrals computed at /

kc

<

, are related with

propagating surface modes.

Unlike the similar formulas which have been reported in [13,14], formulas (12) –(14)

account for the contributions related with magnetic polarization of the particle,

m

α . On total, we

see that Eqs. (12) -(14) manifest complete transposition symmetry over the electric (marked by

the subscript “e”) and magnetic (marked by the subscript “m”) quantities. Quite recently,

Eqs.(13), (14) have been obtained in [27, 28], and Eq.(12) (at 0

=

V

) –in [16]. For neutral

atoms formulas (12)-(13) must be taken at

0

1=

T

. A heating of a neutral atom can be interpreted

as some kind of Lamb –shift [7,8]. This question needs further theoretical elaboration.

5. Attraction force in the nonrelativistic case

In the nonrelativistic limit 1,1

βγ

<<= formula (12) becomes simpler. Assuming the

equilibrium case

TTT

==

21

, the result is [16]

()

[]

()

()

()

0

0

2

2

2

0/

2i

2

2

0/

2

π

coth /2Im( , )

ω

( )( , )

ω

( )

ω

2

π

coth /2Im ( , )

ω

( )( , )

ω

( )

ω

q z

zBeemm

kc

q z

%

Beemm

kc

Fd k T d keRkRk

d k Td keR

%

kR

%

k

ω

>

ω

<

ωω

h

α ωα

ωω

h

α ωα

∞

∫

−

∞

∫

= −+−

−+

∫

∫

h

h

(26)

Page 9

9

where

),(

,

kR

me

ω

and ),(

~

R

,

k

me

ω

are given by (21)-(24) assuming

0

=

β

and making use the

replacement )(

~

∆

)(

,,

ωω

meme

→∆

. Using the identity ) 1

−

) /(exp(21) coth(

+=

xx

, Eq.(26) can also

be written as the sum of the zero point contribution

) , 0 (

zFz

and thermal contribution

),(

zTFz

:

()

()

()

(

+

()

()

)

2

4

2

01

2

4

1

∫

00

( ) (0, ) ( , )

F T z

21 ( , )

u

( , )

u

(i )

ω

exp2/

21( , )

u

( , )

u

(i )

ω

1 2

−

(

(

(

( , )

u

( , )

u

(

2

π

( , )

ω

Im exp 2i

/

zzz

eme

mem

eme

F zFz

u

d duu zu c

ω

c

u

u

dT duuzu c

ω

c

εεα

ω

ω

π

εεα

εεα

)

)

ε

ω

ω

∞

∫

∞

∫

∞

∫

=+=

−∆

%

−∆+

−−⋅−

+−∆

%

−∆

∆

%

)

)

+

+∆

−Π

%

h

%

%

h

(

()

(

+

)

()

2

2

4

2

00

)

1 2

−

( , )

u

( , )

u

( )

ω

21 ( , )

u

( , )

u

( )

2

π

( , )

ω

Im exp2/

21 ( , )

u

( , )

u

( )

ω

mem

eme

mem

u

u

dT duu zu c

ω

c

u

ω

εεα

εε α ω

ω

ω

εα

∞

∫

∞

∫

−

+

+

∆

%

+∆

+∆ +∆

−Π−

∆ +∆

%

h

(27)

where

()

),(

~

∆

,

1/ exp

1

),(

,

ε

ω

h

ω

u

Tk

T

me

B

−

=Π

in the first integral (27) is computed at imaginary

frequency iω with the substitution

1/2

22

2

1

(i )

ω

k c

u

=+

; ),(

,

ε

u

me

∆

and ),(

~

∆

,

ε

u

me

in the second

and third integrals (27) must be computed at real frequency ω making use the substitutions

1/2

22

2

1

k c

ω

u

=−

and

1/2

22

2

1

k c

ω

u

=−

, respectively (see (17),(18)).

In the case

0

21

==TT

and 0

m

α = formula (27) reduces to the “cold” Casimir (van –der –

Waals) force between a particle and the surface, related to the zero –point fluctuations of

electromagnetic field [29].

The thermal part of the Casimir force is given by the second and third terms of Eq. (27). The

first one results from propagating modes of the surface, the second –from evanescent modes of

the surface. In total, Eq.(27) represents the most general expression for the Casimir force

between a resting particle and the surface at 1/

<<

zR

, which accounts for both electric and

magnetic coupling, thermal and retardation effects.

It is worthwhile noticing that in metallic contacts the magnetic coupling effect is not small.

Thus, in the case of ideal conductors at

0

21

==TT

, it follows from (28) that the magnetic term

equals 2 / 1 of the electric one, and consequently, its contribution to the total Casimir force

equals 30%. The corresponding force is given by

5

3

4

9

), 0 (

z

Rc

zFz

π

h

−=

in accordance with

calculation from the first principles [30].

Page 10

10

The principal non vanishing velocity expansion terms in Eq.(12) turn out to be proportional to

V and may be of two kinds. The first one behaves like (

2

)2

0

/

zV ω

in comparison with the zero

temperature force (27) (

0

ω is the characteristic absorption frequency), the second one behaves

like

2)/(

cV

. Correspondingly, the former correction dominates in the nonretarded regime of

interaction, at

1/

<

λ

z

, where

0

/ωλ

c

=

is the wave length of the absorption line. Typically, the

dynamic corrections are small. However, at

λβ

/ z

>

they can be dominating.

5. Nonrelativistic dissipative (frictional) force

In what follows we assume 1

µ = and the dielectric function of the surface is assumed to be

( )

ε ω

( ) i

ε ω

′

( )

ε ω

′′

=+

. In the nonrelativistic limit 1,1

βγ

<<= formula (13) is simplified, too.

Expanding the integrand functions in (13) up to terms linear in velocity V we get

S

x

Vac

xx

FFF

+=

(28)

(

)

() [

+ Π

]

2

121

1

5

5

0

sinh /2( ,

ω

) ( , )

ω

4

⋅

4

3

1

2

1

2

emB

B

Vac

x

emem

k TTT

k T

V

c

Fd

dd

dd

ααω

h

ωω

π

ααα

ω

α

ωω

−

∞

∫

′′ ′′

+ −Π⋅

= −⋅−

′′ ′′ ′′′′

+

++

h

h

(29)

() ()

[]

(

)()

()

() ()

5

2

0 0

21

2

1

1

1

5

2

0 0

2

/ 1 exp

2

2

2 ( ,

ω

)( , )

ω

Im Im

sinh /2Im Im

2

/1

2

2

S

x

em

ememem

Beemm

B

V

π

z

F d du

ω

cuu

c

dd

TTff

dd

k Tff

k T

V

π

d du

ω

cu

ω

ω

α

ω

α

ω

αα

ω

ω

h

α

α

ω

∞ ∞

∫∫

−

∞

∫∫

= −+−⋅

′′ ′′

′′ ′′′′′′

Π−Π⋅+++∆+∆+

⋅−

′′′′

+⋅⋅+⋅

−−⋅

⋅

h

h

h

[]

(

)

()

2i2i

21

2i

2i2i

2

1

1

Re Re

( ,

ω

) ( , )

ω

2Re

ω

()

sinh/2 Re Re

2

zz

uu

em

cc

em

zu

c

emem

zz

uu

cc

Beemm

B

dd

ef

%

ef

%

dd

TT

e

k Tef

%

ef

%

k T

ωω

ω

ωω

α

ω

α

ω

αα

ω

h

α

α

−

′′′′

++

Π −Π⋅+

′′′′

+ ∆ +∆+

′′′′

+⋅+

h

(30)

where the coefficients

,,

,

e me m

f%

f

are given by

Page 11

11

2

(2 1)( , )

u

( , )

u

ee

′′

m

′′

fu

εε

= + ∆ +∆

(31)

2

(21) ( , )

u

( , )

u

mm

′′

e

′′

fu

εε

= + ∆ +∆

(32)

2

(1 2

−

) ( , )

u

( , )

u

ee

′′

m

′′

f

%

u

εε

=∆

%

+∆

%

(33)

2

(1 2

−

) ( , )

u

( , )

u

mm

′′

e

′′

f

%

u

εε

=∆

%

+∆

%

(34)

For brevity, the corresponding arguments of the functions in the integrands (27) are omitted. In

Eq. (27) and (31)-(34) the imaginary components of the functions ),(

,

ε

u

me

∆

and ),(

~

∆

,

ε

u

me

must

be computed from (17), (18) at real frequency with the substitution

1/2

22

2

1

k c

ω

u

=−

.

Eq.(29) represents the net vacuum component of the friction force, which does not depend on

the distance to the surface. Eq.(30) represents the friction force due to the presence of the surface

and, therefore, it depends on z. The first integral term (30) corresponds to the evanescent modes

of the surface, the second –to the wave modes of the surface. At thermal equilibrium

TTT

==

21

from (28)-(30) we get

()

()

()(

)

()

()

25

52

0

25

2

52

0 0

1

2i 2i

25

2

52

0 0

()

3 sinh /2

2

1 exp ImIm

4 sinh/2

1 ImIm

4 sinh/2

xem

BB

eemm

BB

zz

uu

cc

eemm

BB

V

Fd

k Tc

π

k T

Vz

d du u

ω

uff

k Tc

π

c k T

V

d du

ω

uef

%

ef

%

k Tc

π

k T

ωω

ω

ω

h

ωαα

ωω

ω

h

αα

ω

ω

h

αα

∞

∫

∞ ∞

∫∫

∞

∫∫

′′′′

= −+−

′′ ′′

−+−+−

′′′

−−+

h

h

h

′

(35)

It is important that, contrary to the Casimir force (27) in the “cold” limit

0

21

==TT

, the friction

force proves to be zero. Eq.(35) is in agreement with the nonrelativistic results obtained by

several authors [4,7,8,17,20,22,23] , still generalizing them with account of magnetic effects (via

~

,

and the structure of the coefficients

meme

ff

,,

m

α ), the vacuum and surface wave modes of

electromagnetic field.

According to (29), (30), (35), and taking into account the relations

3

,

R

me

∝αα

, it appears

that the leading dependencies of the friction force components

Vac

x

F

and

S

x

F vs. parameters

W

RVz

ω

,,, are the following (

h

/

TkBW=ω

implies the characteristic thermal (Wien)

frequency) [27]:

i) vacuum component,

3

2

2

∝

c

R

c

V

F

wW

Vac

x

ωω

h

, (36)

ii) surface component,

Page 12

12

>>

−

<<

−

∝

1/,

1/,

3

3

2

3

2

cz

c

z

z

R

z

V

cz

z

R

z

V

F

w

W

W

S

x

ω

ω

ω

h

h

(37)

Comparing the nonretarded friction force (37) with the “cold” Casimir force

5

3

) , 0 (

z

cR

zFz

h

−∝

(the latter follows from (27)) we get

cVFF

z

S

x

//

≈

<<1 (assuming the

nonrelativistic case ). In the ultrarelativistic case

1

>>

γ

one should use Eqs.(12), (13) and the

corresponding situation must be elaborated in more details.

As far as the force

Vac

x

F

is concerned, the corresponding relativistic limit has been recently

discussed in [15]. For a dielectric particle having a narrow absorption line with frequency

0

ω ,

one may write down

)( 5 . 0)(

0

3

0

ωωδωπωα

−=

″

R

e

, and the final result is

[]

∫

Π−Π−⋅

−=

2

2

2 / 1

1020

3

0

c

4

2

0

),(),/() 1 (

γ

ωγω

ω

γ

c

ω

h

TTxx dx

R

F

Vac

x

(38)

Eq.(38) manifests a very intriguing possibility for the particle to be accelerated in the hot

vacuum background, or even under the thermal equilibrium [15]. The involved dynamics

equation

x

F dt mcd

2/3

/

−

= γβ

(m is the particle mass) must be solved simultaneously with the

equation for the particle temperature

1 T in its rest frame.

7. Nonrelativistic heating rate

The calculation of the particle heating rate results in the sum of the velocity independent heat

flux,

dtdQ /

0

, and the dynamics corrections, of which the lowest –order one turns out to be

proportional to

2

V . It follows from (14),

[]

[]

(

)

[]

4

012

3

0

4

12

3

00

1

∫

2i2i

4

12

3

00

4

( )

ω

( )

ω

( , )

ω

( ,

ω

)

22

( , )

ω

( ,

ω

) exp ImIm

2

( , )

ω

( ,

ω

) Re Re

em

eemm

zz

uu

cc

eemm

Q

&

dTT

c

z

dTT duuff

cc

dTT duef

%

ef

%

c

ωω

ωωα

α

π

ω

ωωαα

π

ωωα

α

π

∞

∫

∞

∫

∞

∫

∞

∫

′′′′

= −+ ⋅ Π−Π−

′′′′

−Π −Π⋅−⋅+−

′′ ′′

−Π −Π⋅+

h

h

h

(39)

Of course, 0

0≠

Q&

only when

21

TT ≠

. The first integral term of Eq. (39) is related with the net

heat exchange between the particle and vacuum background and does not depend on z , the

Page 13

13

second is related with the evanescent wave modes of the surface and the third –with the surface

wave modes, both depending on distance

The vacuum contribution to the heat flux can be simply obtained using the Kirchhgoff law of

thermal radiation and the particle cross –section for the absorption of electromagnetic radiation

[24]. However, the surface contributions to the heat flux can be worked out only on the basis of

general theory of electromagnetic fluctuations. In the presented form, Eq.(39) has been firstly

obtained in [29]. In comparison with the results reported by several authors [6,9,12,31], Eq. (39)

incorporates both electric and magnetic effects of the particle polarization, and all contributions

to the heat flux resulting from interactions with vacuum modes, evanescent and wave modes of

the surface. On the contrary, the formulas given in [6,9,12,31] take into account only effects

related with the evanescent surface modes, while the involved integrand functions do not yet

contain important contributions resulting from the currents of magnetic polarization.

At

0

≠

V

, the process of radiative heat transfer manifests new features. Thus, the heat flux

can be observed even at thermal equilibrium

TTT

==

21

. This is an obvious consequence of the

energy conservation law: even if the dissipation integral in (7) equals zero, one still has a squared

velocity term arising from work of friction force − ⋅

F V. Another contribution results from Joule

dissipation integral. The explicit formula can be obtained making use the velocity expansion in

Eq. (14). The sign of the heat flux at thermal equilibrium may be different. In the case of small

temperature difference T

∆ between a particle and the surface, the velocity correction dominates,

to a rough estimate, at

TT /

2

∆>β

.

In the ultrarelativistic limit

1

>>

γ

, at the same conditions as in sect. 6 ()(~

0

ωωδα

−

′ ′e

), the

vacuum background contributes to the heat flux, as follows [15]

() )

1

T

,(),/(

2

020

2

2 / 1

3

43

5

0

2

Tx dx

c

R

Q

&

ωγω

γω

h

γ

∫

Π−Π=

−

(40)

According to (38) and (40), the work of fluctuating electromagnetic field over the particle leads

to its heating/cooling and slowing down (accelerating). The energy balance is given by

()

0),(),/(//

00

2

2 / 1

3

43

5

0

2

>Π−Π=+=−

∫

−

TTx dxx

c

R

dt dQVF dtdW

x

ωγω

γω

h

γ

(41)

where W is the electromagnetic field energy. This is in accordance with the general equation (7).

The same behavior, of course, must be characteristic for the process of FEI regarding the surface

modes.

8. Some numerical estimations

i) Damping of an AFM tip

Page 14

14

A development of the spectroscopy of dissipative (friction) forces in dynamic vacuum contacts

of the AFM tips with surfaces and an extraction of meaningful physical information from the

measurements are of great importance. A crucial factor in the interpretation of the experiments

is that, in the noncontact dynamic regime with compensation for the contact potential difference,

the conservative and dissipative interaction of a tip with the sample is expected to be associated

with the forces of FEI (predominantly, of van –der –Waals type at nanometer separations).

However, the theoretical calculations made by several authors have demonstrated that the

vacuum friction is smaller than the observed friction by many orders of magnitude (see [7-9] and

references there in). This enforces considering special mechanisms and conditions for the tip

damping [9].

Here we propose one possible explanation for the damping effect in line with the obtained

theoretical results. Let us consider the case of two coinciding resonance absorption peaks

characterizing the particle and surface, both locating at

WW

tt

ωωω

, 1,000

<<=

is the Wien

frequency. Assuming the nonretarded regime of the interaction, a dominating contribution to the

friction force (35) will be related with the second term. The corresponding structure of the

integrand function is simplified if use is made of the relations which hold close to

0

ω :

s

ss

′

s

s

s

e

ε

εε

(

ε

2

ε

ε

′ ′

≈

′ ′

++

′ ′

=

+

−

≈∆′ ′

2

) 1

1

1

Im

2

2

, and

t

tt

′

t

t

t

e

R

ε

RR

εε

(

ε

ε

ε

α

′ ′

≈

′ ′

++

′ ′

=

+

−

=

′ ′

3

2

2

33

3

) 2

3

2

1

Im (42)

where

tε and

s ε are the dielectric functions of materials (the tip and the surface). Then Eq.(35)

reduces to

)()(

9

π

)()(

1

ε

) 2/( sinh

4

9

π

00

0

′ ′

2

0

5

3

2/

2/

25

3

0

0

ωεωε

ω

∆

ω

ω

ωωε

ts

W

WsWt

tt

tt

x

z

VR

tt

t

dt

z

VR

F

′ ′

−=

′ ′′ ′

−≈

∫

∆

∆+

−

hh

(43)

with

000

//

ωω

∆=∆

tt

. In the typical experimental situation [32,33] , assuming the tip cross -

section to be parabolic with the curvature radius R and the height H , Eq.(43) can be considered

as local relation in the differential volume

rd3. Then, substituting

π

4/3

33

rdR →

in (43) and

integrating over the tip volume at

1/

>>

RH

, we obtain [27]

)()(8

9

π

00

0

′ ′

2

0

0

2

0

ωεωε

ω

∆

ω

ts

B

x

Tk

z

R

z

V

F

′ ′

−=

(44)

where

0z implies the minimal tip-sample distance andR is the tip curvature radius. For

nmz nmRKT

10, 35, 300

0===

(experimental conditions [32,33]), and assuming

19

0

10

−

=

s

ω

,

01 . 0, 1 . 0/

00

=

′ ′

t

=

′ ′

s

=∆εεωω

, we get from (44) the frictional stress

1 11

10 2 . 5

−−

⋅⋅⋅

msN

, that is

close to the experimental value

1 11

10) 5 .135 . 3 (

−−

⋅⋅⋅÷

msN

[33] and agrees with the observed

Page 15

15

distance dependence

3

0

−

∝ zFx

. However, in this case the surface atomic species of the tip and

sample (in [33] both are made of gold) must have large absorption in the radio frequency range.

It is worthwhile noticing that vibrational transitions in molecular species are located in the radio

frequency domain, and the same order of magnitude has the inverse time for the damping motion

of adsorbates measured in friction experiments using the quartz crystal microbalance technique

[34]. Moreover, the order of the inverse relaxation time (

s

9

10~ in [34]) is characteristic for all

the experiments [32 -35] when using a phenomenological friction model [36].

ii) Friction forces in metallic contacts

Now we aim to demonstrate large influence of the magnetic polarization effect on friction and

heat exchange in vacuum contacts of nonmagnetic metals. Let us consider vacuum friction

between a metallic particle and the metallic surface in the case of thermal equilibrium, when

Eq.(35) is correct. First, we assume the nonretarded regime of interaction, 1/ <<

cz

W

ω

. The

dielectric function of the particle and surface materials, and the corresponding electric

(magnetic) polarizabilities are assumed to be [24]

ω

ε

πσ

4

ω

(

ε

i

0

1)

+=

(45)

2)(

1)(

ω

)(

3

+

−

=

ε

ω

ωα

R

e

(46)

cik RkR Rk

Rk

kR

R

2

mm

/2) 1 ( , )() cot(

33

2

1)(

0

3

2

3

ωσπϕωα

+=−=

+−−=

(47)

where

0

σ implies the static conductivity.

The calculated electric (including only electric polarization) and magnetic (including only

magnetic polarization) contributions to the friction force (35) are shown in Fig.2 (for a Cu –Cu

contact). We have assumed

s cmVsKT nmR

/1, 102 . 5, 300, 50

1 17

0

=⋅===

−

σ

. The accepted

value of V corresponds to the characteristic maximal velocity of the AFM tip in dynamic

regime [32,33]. As seen from Fig.2, at

nmz

10

>

the contribution due to magnetic coupling

dominates the contribution from electric coupling by 5 to 10 orders of magnitude. However, the

corresponding friction coefficient

msNVFx

/ 10/

20

⋅==

−

η

(at

KT nmz

900, 10

==

) is yet too

small to explain the observed experimental damping [32,33,35], where

msN

/ 1010

1311

⋅÷=

−−

η

.

Lines 5,6 on Fig.2 correspond to the vacuum contribution, being independent on the surface

separation.

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