Wetting in the presence of the electric field: The classical density functional theory study for a model system

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by  Vasyl Myhal

Wetting in the presence of the electric field:
The classical density functional theory study for a model system
Vasyl Myhal1 and Oleg Derzhko2,1,3
1Ivan Franko National University of L’viv, Department for Theoretical Physics, Drahomanov Street 12, 79005 L’viv, Ukraine
2Institute forCondensed Matter Physics, National Academyof Sciences ofUkraine, Svientsitskii Street 1, 79011 L’viv,Ukraine
3Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34011 Trieste, Italy
(Dated: January 25, 2017)
We discuss the effect of an external electric field on the wetting of a solid surface by liquid. To
this end, we use a model of the two-level-atom fluid for which the changes in interatomic interac-
7
1 tions due to the presence of thefield can befound using quantum-mechanicalperturbation theory.
0 Constructingthegrand potentialfunctional,weperform thestandardcalculations ofYoung’sequi-
2 librium contact angle. The switching on of the electric field |E| > 0 may increase noticeably the
contact angle θ.
n
a
PACSnumbers: 68.03.Cd;68.08.Bc
J
Keywords: classicaldensityfunctionaltheory,contact angle,wetting

4
2

I. INTRODUCTORY REMARKS
]
t
f
o Wetting of solid surfaces by liquids is important both from fundamental and practical points of view [1–4]. Liquid
s wets a solid surface, if the Young equilibrium contact angle θ (i.e., the angle between the surface of the liquid and
.
t the outline of the contact solid surface at thermodynamic equilibrium) vanishes, i.e., θ =0◦ (complete wetting). The
a
m surface is nonwetted (partial wetting) for any 0◦ < θ < 180◦ and it is completely dry for θ = 180◦. The contact
angle may vary under the change of externalparameters. If θ while increasing crossesthe value 90◦, the change from
-
d hydrophilicity (θ <90◦) to hydrophobicity (θ >90◦) occurs.
n An interesting problem in the theory of inhomogeneous fluids is to examine a dependence of θ on external param-
o eters starting from a microscopic picture within the frames of which one can follow how external influences modify
c interparticle interactions. The classicaldensity functional method [5–11] provides such a possibility since it allows to
[
calculate the properties of a nonuniform fluid on the basis of interparticle interactions.

1 It is well known that an external electric field is a simple and effective way to change wetting properties. The
v most drastic changes in the presence of the electric field occur for ionic or polar liquids (see, for example, Refs. 12–
8 15). The shape and stability of droplets in the electric field, contact angle phenomena in the electric field, as well
9
as electrocapilarity have been of interest for a long time and receive renewed attention because of electrowetting
9
(see, e.g., review papers [16, 17] and references therein). In electrowetting, one is generically dealing with droplets
6
0 of partially wetting conductive liquids (electrolytes) on planar solid substrates and the applied voltage changes the
. contact angle. However, even in the case of noble liquids the electric field can affect the macroscopic properties, and
1
in particular the wetting properties, via coupling to the transition electric dipole moment of atoms.
0
In the present paper, we intend to follow starting from the microscopic level how an external electric field affects
7
1 the Young equilibrium contact angle for a fluid of atoms. To this end, we consider a simple model of two-level-atom
: fluid in which the interatomic interactions are changed because of the presence of the field. Furthermore, within the
v
framesoftheclassicaldensityfunctionaltheoryapproachwecalculatethecontactangleθ whichdependsonthevalue
i
X of the electric field strength |E|. We show that while the value of the electric field strength increases, the contact
r angle may increase and cross 90◦. In other words, an increase of the field may lead to hydrophobicity.
a The outline ofthe paper is as follows. First, we justify the choice of a grandpotential functional whichdepends on
anexternalelectric field. Thenwe reportsome results forthe bulk properties ofthe system, as wellas for the density
profilesfor two-phasecases: liquid – vapor,substrate(solid wall)– liquid, andsubstrate – vapor. Knowing the grand
potential allows us to find the surface tensions, and then, via the Young equation, to obtain the required contact
angle θ. Our main results are shown in Figs. 1 and 2. From these plots one can see that an increase of the value of
the electric field strength |E| increases the wetting temperature T (i.e., the temperature T for which the contact
w w
angle θ vanishes), see Fig. 1, increases the contact angle θ at fixed temperature, see Fig. 2, may replace wetting by
partial wetting, see Figs. 1, 2, and may lead to a changeoverfrom hydrophilicity to hydrophobicity, see Figs. 1, 2.
2
FIG.1: Dependenceofthecontactangleθ onthetemperatureτ/τc(0). τc(0)denotesthedimensionlesscriticaltemperatureat
E=0, i.e., τc(0)=Tc(E=0)/(E1−E0). The lower curve(with squares) corresponds to E =0, theuppercurve(with circles)
corresponds toE =0.2; E =|E|r03/|p| is the dimensionless value of theelectric field strength. For further explanations see the
main text.
FIG. 2: Dependence of the contact angle θ on E =|E|r03/|p| at different temperatures τ =T/(E1−E0): 0.6τc(0)≈0.001179
(curve1 with squares), 0.7τc(0)≈0.001376 (curve2 with circles), 0.8τc(0)≈0.001572 (curve3 with up-triangles), 0.9τc(0)≈
0.001769 (curve4 with down-triangles), 0.95τc(0)≈0.001867 (curve 5 with diamonds), and 0.9574τc(0)≈0.001881 (curve 6
with left-triangles). For furtherexplanations see themain text.
II. INTERATOMIC INTERACTIONS AND THE GRAND POTENTIAL FUNCTIONAL
Inordertofollowhowthe electricfieldaffectsthe contactanglewehavetobeginwithwritingdowntheinteraction
energy of neutral atoms with a time-independent spatially uniform electric field. This might be a puzzle since the
atoms have no permanent electric dipole moment. Therefore, we start from the first principles to show how the
electric field modifies the interatomic interactions. To be free of secondary complications, we shall consider a simple
model of a two-level-atomfluid. We assume that the energy of excitation of the atom is E −E , the atom does not
1 0
havethe electric dipole momentin the groundstate orin the excited state,and the transitionelectric dipole moment
between the ground and excited states is p. We are interested in how the electric field E modifies the long-range
interatomic interactions, while the short-range interactions are described by introducing the atom radius r = σ/2.
0
After switching on the electric field |E| > 0, one can calculate within the framework of the quantum-mechanical
perturbation theory with respect to the interaction with the field the second-order results for the energy of a single
atom, E ,orofagroupoftwoatomsat(asufficiently large)distanceR=|R |, E ,seeAppendix andRef.18.
N=1 12 N=2
3
We find
γ2
E =E − 1 (E −E )+...,
N=1 0 1 0
4
α2 γ2+γ2 3α2
E =2E − 12 + 1 2 1+ 12 −γ γ α (E −E )+...,
N=2 0 1 2 12 1 0
2 4 2
(cid:18) (cid:18) (cid:19) (cid:19)
|p|2
γ (E −E )=2|p||E|χ , α (E −E )= Φ , (1)
i 1 0 i 12 1 0 R3 12
where χ and Φ are well known functions which depend on p /|p| and E/|E| or on p /|p|, p /|p|, and R /|R |,
i 12 i 1 2 12 12
see Appendix and Ref. 18. E and E given in Eq. (1) are the only results one must know in order to find the
N=1 N=2
secondvirialcoefficientofthe fluid. The statistical-mechanicalaveragecontainsalso the averagesoverorientationsof
p (andthereforeno preferentialdirectioncreatedby the field is expected). Bearinginmindthat weareinterestedin
i
the lowesttermin|E|only,the orientationalaveragescanbedoneusingacumulantexpansion. Afterstraightforward
butcumbersomecalculations(fordetailsseeAppendix)wefindthesecondvirialcoefficientofthetwo-level-atomfluid
[18]:
∞ 3a(|E|)σ3 a(|E|)
B (T,|E|)=4v−2π dRR2 exp −1 ≈4v− ,
2 2πTR6 T
Zσ (cid:18) (cid:18) (cid:19) (cid:19)
2π 2|p|2|E|2 |p|4 v(E −E )ℵ2
a(|E|)= 1+ = 1 0 1+2ℵ2E2 . (2)
9 (E −E )2 (E −E )σ3 48
(cid:18) 1 0 (cid:19) 1 0
(cid:0) (cid:1)
Here v = πσ3/6, ℵ = |p|2/(r3(E −E )) is the dimensionless parameter which characterizes the two-level atom (in
0 1 0
what follows we set ℵ = 1 for convenience), E = |E|r3/|p| is the dimensionless value of the electric field strength.
0
For |E| = 0 one immediately recognizes in Eq. (2) the contribution of van der Waals interactions to the second
virial coefficient. For |E|>0 the interaction constant of van der Waals interactions increases in accordance with the
rescaling a(E=0)→a(|E|)=a(E=0)(1+2ℵ2E2).
Equation(2)allowsustoconstructanextrapolatedequationofstatewhichalreadycontainstheliquid-vaporphase
transition,andtofindthecorrespondingHelmholtzfreeenergyandthegrandpotential,aswellastoextendthelatter
findings to a nonuniform case, see reviews [5–11] and recent density functional theory studies of wetting [19–21]. We
will start from the following grand potential functional:
Ω[ρ(r)]=F [ρ(r)]+F [ρ(r)]+ dr ρ(r )(V(r )−µ),
sr lr 1 1 1
Z
−1+6vρ(r )−4v2ρ2(r )
F [ρ(r)]= dr ρ(r ) ln Λ3ρ(r ) + 1 1 ,
sr Z 1 1 1 (1−vρ(r1))2 !
(cid:0) (cid:1)
1 3a(|E|)σ3 1
F [ρ(r)]= dr dr ρ(r )ρ(r ) − , (3)
lr 2 1 2 1 2 2π |r −r |6
Z|r1−r2|≥σ (cid:18) 1 2 (cid:19)
which accounts for the short-range repulsion F [ρ(r)] of hard-core spheres having the diameter σ and the long-range
sr
attraction F [ρ(r)], which depends on the external electric field. Moreover, V(r) is the external potential and µ is
lr
the chemicalpotential. ForF [ρ(r)]weusethe localdensityapproximationwhichwouldyieldthe Carnahan-Starling
sr
equationofstateintheuniformlimit,seeAppendix. ForF [ρ(r)]weusethemean-fieldapproximation. Suchapprox-
lr
imations completely neglect the local correlation structure around an atom and more refined treatments are known
for both contributions,of the short-rangerepulsion[9, 10] andof the long-rangeattraction[22,23]. Nevertheless,the
adoptedtreatmentissuitableforthe purposesofthepresentstudyandmoresophisticatedapproximationsgobeyond
the scope of the present paper.
InwhatfollowswealsoneedtoknowtheexplicitformfortheexternalpotentialV(r)whichdescribestheinteraction
betweenthe solidwall(substrate)andtheatomsoffluid. We assumethatthe solidwall,say,forz <0is formedwith
uniformly distributed two-levelatoms with the density ρ , which interact with the fluid two-levelatoms via the same
s
potential as in the fluid (see, e.g., Ref. 24). The long-range (i.e., z ≥σ) contribution of the semi-infinite planar solid
wall to V (x,y,z)=V (z) is calculated by integrating the long-range interatomic interaction −3a(|E|)σ3/(2πR6) (cf.
s s
Eq. (2))
∞ ∞ 0 3a(|E|)σ3 1 ρ a(|E|)σ3 1
V (z)=ρ dx′ dy′ dz′ − =− s . (4)
s sZ−∞ Z−∞ Z−∞ 2π (x−x′)2+(y−y′)2+(z−z′)26! 4 z3
p
4
In what follows we assume η = ρ v = 1 for convenience. Moreover, we set V (z) = ∞ for 0 ≤ z < σ. Clearly, we
s s s
have assumed that all three phases are influenced by the electric field. Such a case is also experimentally realizable,
see Ref. 15, where the used experimental setup was designed in such a way that the electric field was applied to all
three interfaces.
Thefollowingremarkabouttheelaboratedtheoryisinorderhere. Ascanbeseenfromtheconsiderationabove,the
electric field enters the theory only through the increase of the van der Waals interactions constant which is simply
multiplied by 1+2ℵ2E2. This means that the electric field may be eliminated from the theory after introducing
an appropriate energy unit. For example, after introducing the critical temperature T (|E|) as the energy unit all
c
dimensionlessquantitiesshouldbe alreadyindependentofthe field. Calculationsreportedinthe nextsectionconfirm
this observation. Of course, this feature would be not present in more advanced consideration of the electric field.
The equation for the equilibrium density ρ(r) is given by δΩ[ρ(r)]/δρ(r) = 0 [5–11]. Substituting its solution into
Eq. (3) we get the value of the grand potential of the nonuniform system under consideration Ω(T,µ,V). Here V is
the volume of the system [5–11].
III. BULK AND SURFACE PROPERTIES
Considering on the basis of Eq. (3) with V(r) = 0 the bulk properties, when ρ(r) = ρ, we find the critical
density η ≈0.13044, the critical temperature τ (E) ≈ 0.00196518ℵ2(1+2ℵ2E2), and the critical pressure π (E) ≈
c c c
0.00009202ℵ2(1+2ℵ2E2) of the fluid at hand; here we have introduced the dimensionless variables η = ρv, τ =
T/(E −E ), π =pv/(E −E ), see Ref. 18. Within the adopted approach,the critical density is independent of the
1 0 1 0
field but the critical temperature and the critical pressure increase by the factor 1+2ℵ2E2. At temperatures below
the critical temperature T the fluid can be in the form of two coexisting phases (liquid and vapor). In what follows
c
we consider just such temperatures T <T .
c
Letusexplainhowtogetthecontactangleθ. Firstwecalculatetheliquid–vaporsurfacetensionγ . Tothisend,
lv
we consider a nonuniform fluid at T <T in the form of two phases in equilibrium with the planar interface (see also
c
Ref. 18). For computation purposes, it is useful to assume that the fluid is within the cylindric vessel of the radius
R → ∞ and the height L, direct the z axis of the coordinate system along the cylinder axis, and take the origin of
the coordinate system in the middle of the height. We seek for the equilibrium density ρ(z)=ρ(x=0,y =0,z) (i.e.,
alongthe cylindric axis)andthe equationfor ρ(z)after taking the limit R→∞has no traces ofthe adoptedspecific
(cylindric) geometry. Moreover,we know the pressure p(T) and the chemical potential µ(T) of the two-phasesystem
at hand. We put V(z)=0, but seek for the solution for the equilibrium density ρ(z) which depends on the height z.
The solution for the density profile ρ(z) gives the value of the grand potential of the two-phase fluid in the cylinder
Ω(T,µ(T),πR2L). The surface tension follows from the relation γ (T)=(Ω(T,µ(T),πR2L)+p(T)πR2L)/(πR2).
lv
The interface surface tensions substrate – liquid γ or substrate – vapor γ are calculated along the same lines,
sl sv
however,onehastotakeintoaccountthepotentialofsubstrateV (r)(4),whichissituated,say,atz =0. Weinitialized
s
the system in the cylindric vessel with the liquid density (i.e., ρ(z) = ρ ) or the vapor density (i.e., ρ(z) = ρ ) if
l v
z ≫0. Then we find the equilibrium density ρ(z), the grand potential Ω(T,µ,πR2L/2), and, as a result, the values
of γ (T) or γ (T).
sl sv
Finally, the contact angle θ is defined by Young’s equation
γ −γ −γ cosθ =0. (5)
sv sl lv
Equation (5) completes the calculation of the contact angle θ(T,|E|) starting from the interparticle interactions.
Next, we turn to our findings. Density profiles are shown in Figs. 3 and 4. Some dependences of the dimensionless
surface tensions Γ = γσ2/(E −E ) and those of the contact angle θ on the dimensionless value of the electric field
1 0
strengthE arereportedinTableI.TheresultsinFigs.3,4andTableIrefertoaparticularrepresentativevalueofthe
dimensionless temperature τ ≈0.001769(this is 0.9τ (0), where τ (0) denotes the dimensionless critical temperature
c c
withoutthefield,i.e.,atE =0). Theresultsforthecontactangleθ(τ,E)obtainedonthebasisofEq.(5)arereported
in Figs. 1 and 2. Bearing in mind a plausible experimental setup when the electric field is switching on at constant
temperature, we presentall calculations atfixed T,or more precisely, in the units proportionalto T (E=0), but not
c
T (|E|).
c
DensityprofilesinFig.3showadiffusedboundarybetweentwophases,liquidandvapor,whichbecomessharperas
E increases(comparethe solidanddottedcurves). Thiscanbe explainedby anincreaseofthe interatomicattraction
asE >0whichresultsinanincreaseofT andthereforethefluidatthefixedtemperatureτ =0.9τ (0)turnsouttobe
c c
fartherfromthecriticalregion. SomestructurearoundthesolidsubstrateseeninFig.4isduetothehard-core-sphere
repulsion: It manifests itself for 1 ≤ z/σ ≤ 2 [25]. It is better pronounced in the case of a more dense liquid phase
(curves 1, 3, 5) and almost disappears for vapor (curves 2, 4, 6). Furthermore, from Fig. 1 one concludes that the
wetting temperature T increases after the field has been switched on. The temperature dependences of the contact
w
5
FIG. 3: Liquid – vapor density profile η(z) at the temperature τ =0.9τc(0)≈0.001769: E =0 (solid), E =0.2 (dashed), and
E =0.5 (dotted).
FIG. 4: Density profile η(z) near substrate (at z = 0) at the temperature τ = 0.9τc(0) ≈ 0.001769 without the electric field
E =0 (solid curves 1 and 2), at E =0.2 (dashed curves 3 and 4), and at E =0.5 (dotted curves 5 and 6). Curves 1, 3, and 5
correspond to thecase of liquid near substrate, curves2, 4, 6 correspond to thecase of vapor nearsubstrate.
TABLEI:DependenceofthedimensionlessinterfacesurfacetensionsΓlv(liquid–vapor),Γsl(substrate–liquid),Γsv(substrate
– vapor), and the contact angle θ (in degrees) on the dimensionless value of the electric field strength E at the temperature
τ =0.9τc(0)≈0.001769.
E Γlv Γsl Γsv θ
◦
0 0.0001689 −0.0002818 −0.0002009 61
◦
0.1 0.0002196 −0.0002817 −0.0001959 67
◦
0.2 0.0003912 −0.0002649 −0.0001799 77
◦
0.3 0.0007257 −0.0001939 −0.0001519 87
◦
0.4 0.0012646 −0.0000295 −0.0001147 94
0.5 0.0020416 0.0002607 −0.0000753 99◦
6
angle in Fig. 1 satisfy the relation 1−cosθ ∝ (Tw −T)2−αs with αs = 1, thus implying the wetting transitions of
first order [26]. From Fig. 2 one concludes that the contact angle grows with the increasing of the field strength.
For temperatures close to T < T the change of θ is rather steep. Moreover, θ may cross 90◦ indicating that the
w c
hydrophilic surface (θ < 90◦) becomes hydrophobic (θ > 90◦). For example, for τ/τ (0) = 0.7284 the contact angle
c
crosses90◦ asE variesfrom0.005to0.010. Clearly,the actualvalueofθ followsfromEq.(5)andhenceisdetermined
by the interplay of surface tensions γ , γ , and γ at a given temperature and electric field strength magnitude.
lv sl sv
IV. DISCUSSION AND CONCLUSIONS
Let us discuss the obtained results. For the case of the considered two-level-atom fluid, a nonzero electric field
increases the long-range attraction in the system. This leads to an increase of the critical temperature of the fluid
T (|E|) > T (E = 0) and therefore, after the field is switched on at constant temperature the two-phase state gets
c c
farther from the critical region. All surface tensions increase with the increase of the field, see Table I. According to
Eq.(5), γ >0influencesthevalueofcosθ butnotthe changeofitssign. As canbe seenfromTableI,γ growsand
lv sl
changes its sign as the field increases. As a result, cosθ may change its sign and the hydrophilic surface (cosθ > 0)
changetothe hydrophobicone(cosθ <0). We adoptedaverysimplemodelforthesubstrate. The externalpotential
V (r) representing the substrate may be made smaller (e.g., by a decrease of η ). Then the role of the substrate
s s
diminishes: Itbehavesasahydrophobicsurfaceeveninthe absenceofthe fieldandis lesssensitiveto the presenceof
the field. However, qualitatively the effect of the field remains the same: The contact angle grows with the increase
of the field. Finally it is worth noting, that the growth of θ slows down for relatively large fields, see Fig. 2, and our
consideration (which is valid for small E only) does not give hints for complete drying induced by the field.
It is also in order to make here a remark concerning the electric-field-strength scale. This scale is defined by
|E | ≡ |p|/r3 and is of the order of 1010 volts per meter. Such large values of |E | may be expected, since we deal
0 0 0
with atomic-scale electric fields. However, if the temperature is very close to (just below) the wetting temperature
T <T even small electric field strengths can produce noticeable changes in θ.
w c
According to our study, the treatment of the electric field effects on the basis of the Lennard-Jones fluid (see, e.g.,
Refs.19,27)shouldimplyachangeoftheLennard-Jonespotentialparameterstobeinagreementwiththeincreaseof
the van der Waals interactions constantby 1+2ℵ2E2. Finally, the elaboratedscheme can be also applied to examine
the wetting in the presence of excited atoms which may appear as a result of resonance irradiation [28].
A few words about a comparison with the outcomes of alternative approaches which permits to test the quality
of the obtained results are in order here. The most important test requires ab initio calculations of the effective
interatomic interactions in the presence of the electric field since this information would check the dependence of the
initial grand potential functional (3) on the field. However, such simulations are far beyond the scope of the present
study. On the other hand, the quality of the classical density functional theory results based on the simple grand
potential functional (3) for a system with the hard-core repulsion and the van der Waals attraction is known from
previous studies (e.g., Ref. 5).
To summarize, we applied a classical density functional theory to a simple two-level-atom fluid to examine the
effect of an external electric field on the wetting properties. In the considered model the electric field couples to the
transition electric dipole moment of atoms resulting in the increase of the long-range interatomic attraction in the
system. Just below the wetting temperature the electric field can increase noticeably the contact angle and lead to a
passagefromhydrophilicitytohydrophobicity. Ourcalculationsmayrefertothenoblefluidsinastrongelectricfield.
Acknowledgments
O. D. acknowledges the kind hospitality of Prof. Jozef Streˇcka (P. J. Sˇaf´arik University, Koˇsice, Slovakia) during
the CSMAG’16 conference in June of 2016. O. D. is grateful to the Organizing Committee of the 26th IUPAP
International Conference on Statistical Physics (Lyon, July 18-22, 2016) for a financial support for attending the
Conference. O. D. would also like to thank the Abdus Salam International Centre for Theoretical Physics (Trieste,
Italy) for partial support of this study through the Senior Associate Award.
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TheoreticalPhysics50,1848(1973);I.R.Yukhnovskii,R.M.Kadobyanskii,R.R.Levitskii,andO.V.Derzhko,Ukr.Fiz.
Zh.34, 300 (1989) (in Russian).
Appendix: The second virial coefficient of the two-level-atom fluid (2)
For the sake of being self-contained, in this appendix, we provide some details necessary to understand the initial
grand potential functional, see Eq. (3).
We consider N two-level atoms at sufficiently large interatomic distances |R | = |R − R |, adopt the dipole
ij i j
approximation and use a convenient spin-1/2 representation [29] to write the electron subsystem Hamiltonian as
N N N
N 1
H(R ,...,R )= (E +E )+(E −E ) sz + C sxsx+ B sx, (A1)
1 N 2 0 1 1 0 i 2 ij i j i i
Xi=1 i,j=X1(i6=j) Xi=1
where E and E are the energies of the ground and excited states,
0 1
|p ||p |
i j
C ≡4α (E −E )=4 Φ ,
ij ij 1 0 |R |3 ij
ij
Φ =sinθ sinθ cos φ −φ +cosθ cosθ
ij pi pj pi pj pi pj
−3 sinθpisinθnijcos φpi −φnij +cosθpicosθnij sinθpjsinθnijcos φp(cid:0)j −φnij +(cid:1)cosθpjcosθnij , (A2)
(cid:0) (cid:0) (cid:1) (cid:1)(cid:0) (cid:0) (cid:1) (cid:1)
8
B ≡γ (E −E )=−2|p ||E|χ ,
i i 1 0 i i
χ =sinθ sinθ cos(φ −φ )+cosθ cosθ , (A3)
i pi E pi E pi E
θ , φ , θ , φ , θ , φ are the angles that determine the orientation of the transition electric dipole moment of
pi pi nij nij E E
the i-th atom p , the unit vector n =R /|R |, and the electric field E. The first two terms in Eq. (A1) describe
i ij ij ij
a system of noninteracting two-level atoms, the third one represents the dipole-dipole interaction between them, and
the last one corresponds to the interaction with the field. To find the effective long-range interactions, one has to
calculate the eigenvalues of the Hamiltonian given in Eq. (A1). Although this calculation is straightforward within
the used spin-1/2representationfor not too largeN, in what follows we are interestedin the case ofsmall fields, and
therefore we may use the standard quantum-mechanical perturbation theory assuming the interaction with the field
to be small, i.e., γ ≪1. A correctionto the ground-stateenergy of a single atom (N =1,the third termin Eq. (A1)
i
dropsout)appearsinthe secondorderandis givenbythe formulaforE inEq.(1). ForN =2 itis reasonableto
N=1
assume in addition that α ≪1 (after such an assumption one gets the usual van der Waals interactions for E=0)
12
and the second-ordercorrectionto the ground-stateenergyof two atoms is givenby the formula for E in Eq.(1).
N=2
Let us turn to statistical mechanics. Presenting the grand partition function in the exponential form,
∞ ∞
Ξ≡ zNZ =exp V zlb ,
N l
!
N=0 l=1
X X
1
Vb =Z , Vb =Z − Z2, ..., (A4)
1 1 2 2 2 1
where z is the activity and V is the volume of the system, we obtain the cluster expansion for the grand potential
Ω
− =zb +z2b +..., (A5)
1 2
TV
which results in the virial equation of state
p b
=ρ+B ρ2+..., B =− 2, (A6)
T 2 2 b2
1
where ρ denotes the density of the system. For the required canonical partition functions one has
V E
N=1
Z = exp − ,
1 Λ3 T
(cid:28) (cid:18) (cid:19)(cid:29)
V E
Z = dR exp − N=2 , (A7)
2 2Λ6 12 T
Z|R12|≥σ (cid:28) (cid:18) (cid:19)(cid:29)
where Λ stands for the thermal de Broglie wavelength, E and E are defined in Eqs. (1), (A2), (A3), and the
N=1 N=2
angle brackets mean the average over the orientations of transition dipole moments
1 2π π
h(...)i= dΩ ... dΩ (...), dΩ = dφ dθ sinθ . (A8)
p1 pN pi 4π pi pi pi
Z Z Z Z0 Z0
Bearing in mind that we assume the field to be small, the orientational average (A8) can be done using cumulant
expansion hexpxi = exp hxi+ hx2i−hxi2 /2+... . Keeping the terms up to O(E2) only, we would need the
following averages:
(cid:0) (cid:0) (cid:1) (cid:1)
1 2 1 1
hχ2i= , hΦ2 i= , hχ χ Φ i= 1−cos2θ , hχ2Φ i= 8+6cos2θ . (A9)
i 3 12 3 1 2 12 9 n12 i 12 45 n12
(cid:0) (cid:1) (cid:0) (cid:1)
Equations (A4), (A7), (A9) give the explicit result for b [18] and the formula for b as a two-fold integral [18] which
1 2
besides the integration over R = |R | contains the integration over θ , see Eq. (A7). Introducing the variable
12 R12
y = cosθ one can do the integration over y again with the help of the cumulant expansion with the accuracy up
R12
to the terms O(E2). The obtained cluster integrals give for the second virial coefficient B in Eq. (A6) the formula
2
for B (T,|E|) in Eq. (2).
2
Next, instead of the virial equation of state (A6), (2) we introduce an extrapolated equation of state
p 1+vρ+v2ρ2−v3ρ3 a(|E|)
=ρ −ρ2 (A10)
T (1−vρ)3 T
and treating Eq. (A10) as an input after some simple standard assumptions arrive at the initial grand potential
functional given in Eq. (3).